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--- abstract: 'Non-commutative Henselian rings are defined and it is shown that a local ring which is complete and separated in the topology defined by its maximal ideal is Henselian provided that it is almost commutative.' author: - Masood Aryapoor title: 'Non-commutative Henselian Rings' --- We define non-commutative Henselian rings and give some examples of them. Here, all rings are assumed to be unitary. Let us start with a definition,\ A (possibly non-commutative) ring $A$ is called local if all the non-invertible elements form an (two-sided) ideal which we denote by $m$. If $A$ is a local ring, then $k=A/m$ is a skew field, called the residue field. We denote the reduction map $A\to k$ by ($a\to \bar{a}$). For a brief introduction to local rings consult  [@Lam], Chapter 7.\ Let $A[x]$ be the ring of polynomials over $A$ where the indeterminate $x$ commutes with elements of $A$. Commutative Henselian rings are defined as follows, Let $A$ be a commutative local ring with the maximal ideal $m$ and residue field $k$. $A$ is called Henselian if for every polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in A[x]$ such that $\overline{f(x)}=f_1(x)f_2(x)$ for some relatively prime monic polynomials $f_i(x)\in k[x]$ then there are unique monic polynomials $F_i(x)\in A[x]$ such that $f(x)=F_1(x)F_2(x)$ and $\overline{F_i(x)}=f_i(x)$. See  [@Ray] for a detailed discussion of commutative Henselian rings.\ The above definition makes sense as long as $k$, the residue field, is commutative. Therefore we have the following definition, Let $A$ be a (possibly non-commutative) local ring with the maximal ideal $m$ and residue field $k$. Moreover assume that $k$ is commutative. Then $A$ is called Henselian if for every polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in A[x]$ such that $\overline{f(x)}=f_1(x)f_2(x)$ for some relatively prime monic polynomials $f_i(x)\in k[x]$ then there are unique monic polynomials $F_i(x)\in A[x]$ such that $f(x)=F_1(x)F_2(x)$ and $\overline{F_i(x)}=f_i(x)$. It is well-known that every commutative local ring $A$ which is complete and separated in the $m$-adic topology is Henselian. This is not true for non-commutative local rings which are complete and separated in the topology defined by the maximal ideal. However, it holds if the local ring has an extra property which we explain in what follows.\ To each local ring one can associate an associative ring as follows, Let $A$ be a local ring with the maximal ideal $m$. Then $gr(A)=\frac{A}{m}\bigoplus \frac{m}{m^2}\bigoplus \cdots$ is defined to be the graded associated ring coming from the filtration $\cdots\subset m^{n+1}\subset m^{n} \subset \cdots \subset m \subset A$. $A$ is called almost commutative if $gr(A)$ is commutative. For basic facts regarding $gr(A)$ see \[Lang\].\ Clearly if $A$ is almost commutative, then $k$ is commutative. The main theorem is, Let $A$ be an almost commutative local ring such that $A$ is both separated, i.e. $\bigcap m^{n}=\{0\}$, and complete in the $m$-adic topology. Then $A$ is a Henselian ring. Basically, the same proof of Hensel’s lemma works. Let $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in A[x]$ such that $\overline{f(x)}=f_1(x)f_2(x)$ for some relatively prime monic polynomials $f_i(x)\in k[x]$. We will inductively construct a sequence of monic polynomials $\{ F_{1,r}(x) \}$ and $\{ F_{2,r}(x) \}$ in $A[x]$ such that $\overline{ F_{1,r}(x)}=f_1(x)$, $\overline{ F_{2,r}(x)}=f_2(x)$, $F_{1,r+1}(x)-F_{1,r}(x) \in m^{r}[x]$, $F_{2,r+1}(x)-F_{2,r}(x) \in m^{r}[x]$ and $f(x)-F_{1,r}(x)F_{2,r}(x)\in m^{r}[x]$. Clearly this proves the existence part.\ It is easy to find $F_{1,1}(x)$ and $F_{2,1}(x)$. Having defined $F_{1,r}(x)$ and $F_{2,r}(x)$, we define $F_{1,r+1}(x)$ and $F_{2,r+1}(x)$ as follows. Writing $F_{1,r+1}(x)=F_{1,r}(x)+G_1(x)$ and $F_{2,r+1}(x)=F_{2,r}(x)+G_2(x)$, finding $F_{1,r+1}$ and $F_{2,r+1}$ is equivalent to finding $G_1(x)$ and $G_2(x)$ in $m^{r}[x]$ such that $deg(G_1(x))<deg(f_1(x))$, $deg(G_2(x))<deg(f_2(x))$ and $$f(x)-F_{1,r}(x)F_{2,r}(x)-G_1(x)F_{2,r}(x)-F_{1,r}(x)G_2(x)\in m^{r+1}[x].$$ By abuse of notations this is the same as finding $G_1(x)$ and $G_2(x)$ in $m^{r}[x]$ such that $deg(G_1(x))<deg(f_1(x))$, $deg(G_2(x))<deg(f_2(x))$ and $f(x)-F_{1,r}(x)F_{2,r}(x)-G_1(x)F_{2,r}(x)-F_{1,r}(x)G_2(x)=0$ in $m^{r}/m^{r+1}[x]$. Considering $m^{r}/m^{r+1}$ as a vector space over $k=A/m$ and using the fact that $A$ is almost commutative, one can see that this is the same as finding $G_1(x)$ and $G_2(x)$ in $m^{r}[x]$ such that $deg(G_1(x))<deg(f_1(x))$, $deg(G_2(x))<deg(f_2(x) )$ and $(f(x)-F_{1,r}(x)F_{2,r}(x))-f_2(x)G_1(x)-f_1(x)G_2(x)=0$ in $m^{r}/m^{r+1}[x]$. This is possible because $f_1(x)$ and $f_2(x)$ are relatively prime.\ The uniqueness part follows from the facts that $f_1(x)$ and $f_2(x)$ are relatively prime and $A$ is separated in the $m$-adic topology. In the commutative case, one can use Hensel’s lemma to find roots of polynomials. Next we show this connection in the non-commutative case.\ Let $A[x]$ be the ring of polynomials over $A$ where the indeterminate $x$ commutes with elements of $A$. So every element of $f(x)\in A[x]$ can be written uniquely as $f(x)=a_nx^n+\cdots+a_1x+a_0$ with $a_i\in A$. One can consider $f(x)$ as a function on $A$ as follows, $f(a):=a_na^n+\cdots+a_1a+a_0$ for $a\in A$. An element $a\in A$ is called a (right) root of $f(x)=a_nx^n+\cdots+a_1x+a_0$ if $f(a)=0$. We have the following proposition, An element $a\in A$ is a root of $f(x)=a_nx^n+\cdots+a_1x+a_0\in A[x]$ if and only if $f(x)=g(x)(x-a)$ for some $g(x)\in A[x]$. To see the proof and basic facts regarding right and left roots, see  [@Lam], Chapter 5.\ Theorem 5 together with the above proposition imply that, Let $A$ be a Henselian ring. Suppose that $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in A[x]$ is a monic polynomial such that $\overline{f(x)}$ has a simple root $r\in k$. Then $f(x)$ has a unique root $a\in A$ such that $\bar{a}=r$. In the commutative case, a local ring $A$ is Henselian if and only if every finite $A$-algebra is isomorphic to a product of local rings (See \[Ray\]). In the non-commutative case we can give a similar criterion for Henselian rings in terms of some conditions on some modules over $A$.\ We begin with a few definitions. Let $A$ be a ring and $M$ a (left) $A$-module. We say that $M$ is local if it has a unique maximal submodule. $M$ is called semi-local if $M=M_1\bigoplus \cdots \bigoplus M_k$ where $M_i$’s are local. It is called indecomposable if it cannot be written as $M=M_{1}\bigoplus M_{2}$, where $M_{i}$’s are nonzero submodule of $M$. It is called strongly indecomposable if $End_A(M)$ is a local ring. One has the following theorem.\ [(Krull-Schmidt-Azumaya)]{} Suppose that the $A$-module $M$ has the following decompositions into submodules, $$M=M_1\bigoplus \cdots \bigoplus M_r\simeq N_1\bigoplus \cdots \bigoplus N_s,$$ where $M_i$’s are indecomposable and $N_i$’s are strongly indecomposable. Then $r=s$ and after a reindexing we have $M_i\simeq N_i$. For a proof see \[Lam\], chapter seven.\ From now on, we suppose that $A$ is a local ring as before. Let $M$ be an $A$-module. Set $\bar{M}=\frac{M}{mM}$ which is a $k$-module. We need a few lemmas. Let $M$ be an $A[x]$-module which is a finitely generated $A$-module. Then, $M$ is a local $A[x]$-module if and only if $\frac{M}{mM}$ is a local $k[x]$-module. By Nakayama’s lemma, every maximal submodule of $M$ contains $mM$. Let $M,N$ be finitely generated $A$-modules. Let $\alpha:M\to N$ be an $A$-module homomorphism and $\overline{\alpha}:\bar{M}\to \bar{N}$ be the induced $k$-linear map. If $ker(\overline{\alpha})\neq 0$ and $\bar{\alpha}$ is onto, then $ker(\alpha)\neq 0$. Suppose that $v_1,...,v_n$ are elements of $M$ such that $\overline{\alpha}(\overline{v_1}),...,\overline{\alpha}(\overline{v_n})$ form a basis for $\bar{N}$ over $k$. Then by Nakayama’s lemma we have that $\alpha(v_1),...,\alpha(v_n)$ generate $N$ as an $A$-module. Since $ker(\bar{\alpha})\neq 0$, $\overline{v_1},...,\overline{v_n}$ do not generate $\bar{M}$. So $v_1,...,v_n$ do not generate $M$ which follows that $ker(\alpha)\neq 0$. We also need the following lemma, Let $A$ be a local ring whose residue field $k$ is commutative. Suppose that $p,q\in A[x]$ are polynomials of degrees $r,s$ respectively and $p$ is monic. If $A[x]p+A[x]q=A[x]$, then there are polynomials $p_1,q_1\in A[x]$ such that $deg(p_1)=deg(q)$, $deg(p)=deg(q_1)$, $p_1p=q_1q$ and $q_1$ is monic. Let $\alpha:A^{s+1}\bigoplus A^{r+1}\to A^{r+s+1}$ be the following map, $$\alpha(a_0,a_1,...,a_s,b_0,b_1,...,b_r)=(\sum_{i=0}^{s}{a_ix^i})p-(\sum_{i=0}^{r}{b_ix^i})q.$$ Using lemma 12, we have that $ker(\alpha)\neq 0$. This shows that there are nonzero polynomials $p_1,q_1\in A[x]$ such that $deg(p_1)\leq deg(q)$, $deg(q_1)\leq deg(p)$, $p_1p=q_1q$. Since $\bar{p}$ and $\bar{q}$ are prime in $k[x]$ and $p$ is monic, we must have $deg(\bar{q_1})=deg(p)$, hence $deg(q_1)=deg(p)$ and $deg(p_1)=deg(q)$. Finally, it is clear that $q_1$ can be chosen to be monic. If $p,q\in A[x]$ are polynomials such that $p$ is monic and $A[x]p+A[x]q+m[x]=A[x]$ then $A[x]p+A[x]q=A[x]$. In fact we have that $M=\frac{A[x]}{A[x]p+A[x]q}$ is a finitely generated $A$-module and $mM=M$. So, by Nakayama’s lemma, $M=0$. We have the following theorem, Suppose that $A$ is a local ring whose residue field $k$ is commutative. Then the following are equivalent,\ (1) $A$ is Henselian.\ (2) For any monic polynomial $p\in A[x]$ the $A[x]$-module $M=\frac{A[x]}{A[x]p}$ is semi-local.\ First we show that (1) implies (2). If $\bar{p}$ is a power of an irreducible polynomial in $k[x]$ then $\bar{M}=\frac{M}{mM}=\frac{k[x]}{(\bar{p})}$ is a local $k[x]$-module and by lemma 11, $M$ is local. Suppose $\bar{p}=f_1f_2$ where $f_1$ and $f_2$ are relatively prime polynomials of $k[x]$. By (1) we have $p=p_1p_2=q_2q_1$ where $p_i,q_i$ are monic polynomials in $A[x]$ such that $\overline{p_i}=\overline{q_i}=f_i$. This implies that $M\simeq \frac{A[x]}{A[x]p_2}\bigoplus \frac{A[x]}{A[x]q_1}$ because $A[x]p_2+A[x]q_1=A[x]$(above remark) and it is easy to see that $A[x]p_2\cap A[x]q_1=A[x]p$. Now we can use induction on $deg(p)$.\ Conversely, let $p\in A[x]$ be a monic polynomial. Then we have $M=\frac{A[x]}{A[x]p}=M_1\bigoplus \cdots \bigoplus M_r$ where $M_i$’s are local. So we have $\bar{M}=\bar{M_1}\bigoplus \cdots \bigoplus \bar{M_r}$. On the other hand, if $\bar{p}=f_1\cdots f_s$ where $f_i$’s are powers of irreducible monic polynomials in $k[x]$, then $\bar{M}\simeq \frac{k[x]}{(f_1)}\bigoplus \cdots \bigoplus \frac{k[x]}{(f_s)}$. It is easy to see that $ \frac{k[x]}{(f_i)}$’s are strongly indecomposable as $k[x]$-modules and $\bar{M_i}$’s are local, in particular indecomposable. So by Krull-Schmidt-Azumaya theorem $r=s$ and $\bar{M_{i}}\simeq \frac{k[x]}{(f_i)}$ possibly after a reindexing $M_{i}$’s. If $v_i\in M_i$ is the image of $1\in A[x]$ then $(Av_i+Axv_i+\cdots+Ax^{n_{i}-1}v_i)+mM_i=M_i$ where $n_i$ is the degree of $f_i$. By Nakayama’s lemma $(Av_i+Axv_i+\cdots+Ax^{n_{i}-1}v_i)=M_i$. Also $p_iv_i=0$ for some monic polynomial $p_i$ of degree $n_i$ such that $\bar{p_i}=f_i$. By lemma 13, there is a monic polynomial $p'=q_1q_2\cdots q_r$ where $q_i$’s are monic polynomials and $\bar{q_i}=f_i$ and $p'\in A[x]p_i$ for each $i$. This implies that $p'\in A[x]p$. Since $deg(p)=deg(p')$ and they are monic we have $p'=p$. Finally we give some examples. Let $k$ be a field with a derivation. The ring of Volterra operators $k[[\partial^{-1}]]$ is defined as follows(See \[Lebedev\] for more on Volterra operators). It is the set of formal series $a_0+a_{1}\partial^{-1}+\cdots$ with $a_i \in k$ where $\partial^{n} a=\sum_{i=0}^{\infty}{{n\choose{i}} a^{(i)}\partial^{n-i}}$ for $n<0$. One can see that $k[[\partial^{-1}]]$ is a local ring with the maximal ideal $m=k[[\partial^{-1}]]\partial^{-1}$ which is both separated and complete in the $m$-adic topology. Moreover $gr(k[[\partial^{-1}]])$ is isomorphic to $k[x]$ the ring of polynomials over $k$, hence commutative. So $k[[\partial^{-1}]]$ is a Henselian ring. If $A$ is not almost commutative but complete and separated in the $m$-adic topology then there might not be any lifting of simple roots. Here is one example. Let $k$ be a field and $\sigma$ an automorphism of $k$. Let $A$ be the set of all series of the form $a_0+a_1\tau+a_2\tau^2+\cdots$ where $a_i \in k$. One can make $A$ into a ring using the relation $\tau a=\sigma(a)\tau$ for $a\in k$. Then $A$ is a local ring which is both separated and complete in the $m$-adic topology and $A/m=k$ is commutative. However if $\sigma$ is not the identity map then $gr(A)$ is isomorphic to the skew polynomial ring $k[x;\sigma]$, hence not commutative. Suppose $k=\mathbb{C}$ and $\sigma$ is the complex conjugation. Consider the polynomial $f(x)=x^2+1+\tau$ in $A[x]$. Then $\overline{f(x)}$ has a simple root in $k$, namely $\sqrt{-1}$. However $f(x)$ does not have any root in $A$. Since if $a=a_0+a_1\tau+a_2\tau^2+\cdots$ is a root of $f(x)$ then we have $0=a^2+1+\tau=a_0^2+1+(a_0a_1+\overline{a_0}a_1+1)\tau+\cdots$. This implies that $a_0=\sqrt{-1}$ or $a_0=-\sqrt{-1}$. Therefor $a_0a_1+\overline{a_0}a_1+1=1\neq0$, a contradiction. In the commutative case, for any local Noetherian ring $A$, there is a (unique) Henselian ring $A^{h}$, called the Henselization of $A$, and a local homomorphism $i:A\to A^{h}$ with the following universal property, given any local homomorphism $f$ from $A$ to some Henselian ring $B$ there is a unique local homomorphism $f^{h}:A^{h}\to B$ such that $f=f^{h}i$.\ One can ask the same question in the non-commutative case. If $A$ is a local ring such that $gr(A)$ is commutative, then the completion of $A$ with respect to the $m$-adic topology is Henselian provided that it is separated. It is easy to see that the intersection of all local Henselian rings $H$ in the completion $\hat{A}$, with the maximal ideal $m_H$ such that $A\subset H\subset \hat{A}$ and $m_{\hat{A}}\cap H=m_H$, denoted by $\bar{A}$, is a Henselian local ring. In the commutative case it is not hard to see that $\bar{A}$ is the Henselization. Therefore one might propose the following conjecture, The Henselization exists for any almost commutative separated local ring $A$ and $A^h\simeq \bar{A}$. [ZZZZ]{} T.Y. Lam, A first course in noncommutative rings, Graduate texts in mathematics, volume 131, Springer, New York,1991. S. Lang, Algebra, third ed., Addison-Wesley, Reading, MA, 1991. D.R. Lebedev, and Ju.I. Manin, The Gelfand-Dikii Hamiltonian operator and the coadjoint representation of the Volterra group, Funktsional. Anal. i Prilozhen. 13(1979), no.4, 40-46, 96. Michel Raynaud, [*Anneaux locaux Henseliens*]{}, Lecture Notes in Mathematics 169, Springer, Heidelberg, 1970. Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven CT 06520 USA, email: masood.aryapoor@yale.edu

--- abstract: 'We study the use of greedy feature selection methods for morphosyntactic tagging under a number of different conditions. We compare a static ordering of features to a dynamic ordering based on mutual information statistics, and we apply the techniques to standalone taggers as well as joint systems for tagging and parsing. Experiments on five languages show that feature selection can result in more compact models as well as higher accuracy under all conditions, but also that a dynamic ordering works better than a static ordering and that joint systems benefit more than standalone taggers. We also show that the same techniques can be used to select which morphosyntactic categories to predict in order to maximize syntactic accuracy in a joint system. Our final results represent a substantial improvement of the state of the art for several languages, while at the same time reducing both the number of features and the running time by up to 80% in some cases.' author: - | Bernd Bohnet$^{\spadesuit}$   Miguel Ballesteros$^{\diamondsuit\clubsuit}$   Ryan McDonald$^{\spadesuit}$  Joakim Nivre$^{\heartsuit}$\ $^{\spadesuit}$Google Inc. London, United Kingdom.\ $^{\diamondsuit}$NLP Group, Pompeu Fabra University, Barcelona, Spain\ $^{\clubsuit}$School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA\ $^{\heartsuit}$Uppsala University. Department of Linguistics and Philology. Uppsala, Sweden\ [{bohnetbd,ryanmcd}@google.com, miguel.ballesteros@upf.edu, joakim.nivre@lingfil.uu.se]{} bibliography: - 'xample.bib' - 'main3.bib' title: Static and Dynamic Feature Selection in Morphosyntactic Analyzers --- Introduction ============ Morphosyntactic tagging, whether limited to basic part-of-speech tags or using rich morphosyntactic features, is a fundamental task in natural language processing, used in a variety of applications from machine translation [@habash2006] to information extraction [@banko2007]. In addition, tagging can be the first step of a syntactic analysis, providing a shallow, non-hierarchical representation of syntactic structure. Morphosyntactic taggers tend to belong to one of two different paradigms: *standalone taggers* or *joint taggers*. Standalone taggers use narrow contextual representations, typically an $n$-gram window of fixed size. To achieve state-of-the-art results, they employ sophisticated optimization techniques in combination with rich feature representations [@brants00anlp; @toutanova00; @gimenez04; @muller2013]. Joint taggers, on the other hand, combine morphosyntactic tagging with deeper syntactic processing. The most common case is parsers that predict constituency structures jointly with part-of-speech tags [@charniak05; @petrov06] or richer word morphology . In dependency parsing, pipeline models have traditionally been the norm, but recent studies have shown that joint tagging and dependency parsing can improve accuracy of both [@lee11; @hatori11; @bohnet12emnlp; @tacl-bbjn]. Unfortunately, joint models typically increase the search space, making them more cumbersome than their pipeline equivalents. For instance, in the joint morphosyntactic transition-based parser of , the number of parser actions increases linearly by the size of the part-of-speech and/or morphological label sets. For some languages this can be quite large. For example, report morphological tag sets of size 1,000 or more. The promise of joint tagging and parsing is that by trading-off surface morphosyntactic predictions with longer distance dependency predictions, accuracy can be improved. However, it is unlikely that every decision will benefit from this trade-off. Local $n$-gram context is sufficient for many tagging decisions, and parsing decisions likely only benefit from morphological attributes that correlate with syntactic functions, like case, or those that constrain agreement, like gender or number. At the same time, while standalone morphosyntactic taggers require large feature sets in order to make accurate predictions, it may be the case that fewer features are needed in a joint model, where these predictions are made in tandem with dependency decisions of larger scope. This naturally raises the question as to whether we can advantageously optimize feature sets at the tagger and parser levels in joint parsing systems to alleviate their inherent complexity. We investigate this question in the context of the joint morphosyntactic parser of , focusing on optimizing and compressing feature sets via greedy feature selection techniques, and explicitly contrasting joint systems with standalone taggers. The main findings emerging from our investigations are: - Feature selection works for standalone taggers but is more effective in a joint system. This holds for model size as well as tagging accuracy (and parsing accuracy as a result). - Dynamic feature selection strategies that take feature redundancy into account often lead to more compact models than static selection strategies with little loss in accuracy. - Similar selection techniques can also reduce the set of morphological attributes to be predicted jointly with parsing, reducing the size of the output space at no cost in accuracy. The key to all our findings is that these techniques simultaneously compress model size and/or decrease the search space while increasing the underlying accuracy of tagging and parsing, even surpassing the state of the art in a variety of languages. With respect to the former, we observe empirical speed-ups upwards of 5x. With respect to the latter, we show that the resulting morphosyntactic taggers consistently beat state-of-the-art taggers across a number of languages. Related Work {#sec:background} ============ Since morphosyntactc tagging interacts with other tasks such as word segmentation and syntactic parsing, there has been an increasing interest in joint models that integrate tagging with these other tasks. This line of work includes joint tagging and word segmentation [@zhang08acl], joint tagging and named entity recognition [@conf/tsd/MoraV12], joint tagging and parsing [@lee11; @li11; @hatori11; @bohnet12emnlp; @tacl-bbjn], and even joint word segmentation, tagging and parsing [@hatori12]. These studies often show improved accuracy from joint inference in one or all of the tasks involved. Feature selection has been a staple of statistical NLP since its beginnings, notably selection via frequency cut-offs in part-of-speech tagging [@ratnaparkhi96]. Since then efforts have been made to tie feature selection with model optimization. For instance, used greedy forward selection with respect to model log-likelihood to select features for named entity recognition. Sparse priors, such as L1 regularization, are a common feature selection technique that trades off feature sparsity with the model’s objective [@gao2007]. extended such sparse regularization techniques to allow a model to deselect entire feature templates, potentially saving entire blocks of feature extraction computation. However, current systems still tend to employ millions of features without selection, relying primarily on model regularization to combat overfitting. Selection of morphological attributes has been carried out previously in and selection of features under similar constraints was carried out by . Feature Selection {#featselection} ================= The feature selection methods we investigate can all be viewed as greedy forward selection, shown in Figure \[forward\]. This paradigm starts from an empty set and considers features one by one. In each iteration, a model is generated from a training set and tested on a development set relative to some accuracy metric of interest. The feature under consideration is added if it increases this metric beyond some threshold and discarded otherwise. This strategy is similar to the one implemented in MaltOptimizer [@BallesterosNivre2014]. It differs from classic forward feature selection [@dellapietra97] in that it does not test all features in parallel, but instead relies on an ordering of features as input. This is primarily for efficiency, as training models in parallel for a large number of feature templates is cumbersome. The set of features, $F$, can be defined as fully instantiated input features, e.g., *suffix=ing*, or as feature templates, e.g., *prefix*, *suffix*, *form*, etc. Here we always focus on the latter. By reducing the number of feature templates, we are more likely to positively affect the runtime of feature extraction, as many computations can simply be removed. Static Feature Ordering ----------------------- Our feature selection algorithm assumes a given order of the features to be evaluated against the objective function. One simple strategy is for a human to provide a *static* ordering on features, that is fixed for traversal. This means that we are testing feature templates in a predefined order and keeping those that improve the accuracy. Those that do not are discarded and never visited again. In Figure \[forward\], this means that the Order($F$) function is fixed throughout the procedure. In our experiments, this fixed order is the same as in Table \[table:templates\]. Dynamic Feature Ordering {#mutual} ------------------------ In text categorization, static feature selection based on correlation statistics is a popular technique [@yang1997]. The typical strategy in such offline selectors is to rank each feature by its correlation to the output space, and to select the top K features. This strategy is often called *max relevance*, since it aims to optimize the features based solely to their predictive power. Unfortunately, the $n$ best features selected by these algorithms might not provide the best result [@Hanchuan2005]. Redundancy among the features is the primary reason for this, and Peng et al.  develop the *minimal redundancy maximal relevance (MRMR)* technique to address this problem. The MRMR method tries to keep the redundancy minimal among the features. The approach is based on mutual information to compute the relevance of features and the redundancy of a feature in relation to a set of already selected features. The mutual information of two discrete random variables $X$ and $Y$ is defined as follows $I(X;Y)=\sum\limits_{x\in X} \sum\limits_{y\in Y} p(x,y)log_2\frac{p(x,y)}{p(x)p(y)}$ Max relevance selects the feature set $X$ that maximizes the mutual information of feature templates $X_i \in X$ and the output classes $c \in C$. $$\max~D(X,C), ~~D(X,C)=\frac{1}{|X|}\sum\limits_{X_i \in X} I(X_i;C).$$ To account for cases when features are highly redundant, and thus would not change much the discriminative power of a classifier, the following criterion can be added to minimize mutual information between selected feature templates: $$\min~R(X), ~~R(X)=\frac{1}{|X^2|}\sum\limits_{X_i,X_j\in X} I(X_i;X_j)$$ Minimal redundancy maximal relevance (MRMR) combines both objectives: $$\max~\Phi(D,R), ~~\Phi=D(X,C)-R(X)$$ For the greedy feature selection method outlined Figure \[forward\], we can use the MRMR criteria to define the Order($F$) function. This leads to a *dynamic* feature selection technique as we update the order of features considered dynamically at each iteration, taking into account redundancy amongst already selected features. This technique can be seen in the same light as greedy document summarization [@carbonell1998], where sentences are selected for a summary if they are both relevant and minimally redundant with sentences previously selected. Morphosyntactic Tagging ======================= In this section, we describe the two systems for morphosyntactic tagging we use to compare feature selection techniques. Standalone Tagger ----------------- The first tagger is a standalone SVM tagger, whose training regime is shown in Figure \[tagger\]. The tagger iterates up to $k$ times (typically twice) over a sentence from left to right (line 2). This iteration is performed to allow the final assignment of tags to benefit from tag features on both sides of the target token. For each token of the sentence, the tagger initializes an $n$-best list and extracts features for the token in question (line 4-7). In the innermost loop (line 9-11), the algorithm computes the score for each morphosyntactic tag and inserts a pair consisting of the morphosyntactic tag and its score into the $n$-best list. The algorithm returns a two dimensional array, where the first dimension contains the tokens and the second dimension contains the sorted lists of tag-score pairs. The tagger is trained online using MIRA [@crammer06]. When evaluating this system as a standalone tagger, we select the 1-best tag. This can be viewed as a multi-pass version of standard SVM-based tagging [@marquez2004]. Joint Dependency-Based Tagger ----------------------------- The joint tagger-parser follows the design of , who augment an arc-standard transition-based dependency parser with the capability to select a part-of-speech tag and/or morphological tag for each input word from an $n$-best list of tags for that word. The tag selection is carried out when an input word is shifted onto the stack. Only the $k$ highest-scoring tokens from each $n$-best list are considered, and only tags whose score is at most $\alpha$ below the score of the best tag. In all experiments of this paper, we set $k$ to 2 and $\alpha$ to 0.25. The tagger-parser uses beam search to find the highest scoring combined tagging and dependency tree. When pruning the beam, it first extracts the 40 highest scoring distinct dependency trees and then up to 8 variants that differ only with respect to the tagging, a technique that was found by to give a good balance between tagging and parsing ambiguity in the beam. The tagger-parser is trained using the same online learning algorithm as the standalone tagger. When evaluating this system as a part-of-speech tagger, we consider only the finally selected tag sequence in the dependency tree output by the parser. Part-of-Speech Tagging Experiments {#sec:experiments} ================================== To simplify matters, we start by investigating feature selection for part-of-speech taggers, both in the context of standalone and joint systems. The main hypotheses we are testing is whether feature selection techniques are more powerful in joint morphosyntactic systems as opposed to standalone taggers. That is, the resulting models are both more compact and accurate. Additionally, we wish to empirically compare the impact of static versus dynamic feature selection techniques. Data Sets {#exsetup} --------- We experiment with corpora from five different languages: Chinese, English, German, Hungarian and Russian. For Chinese, we use the Penn Chinese Treebank 5.1 (CTB5), converted with the head-finding rules, conversion tools and with the same split as in .[^1] For English, we use the WSJ section of the Penn Treebank, converted with the head-finding rules of and the labeling rules of .[^2] For German, we use the Tiger Treebank [@brants02] in the improved dependency conversion by . For Hungarian, we use the Szeged Dependency Treebank [@farkas12]. For Russian we use the SynTagRus Treebank [@boguslavsky00; @boguslavsky02]. Feature Templates ----------------- Table \[table:templates\] presents the feature templates that we employed in our experiments (second column). The name of the functor indicates the purpose of the feature template. For instance, the functor [*form*]{} defines the word form. The argument specifies the location of the token, for instance, [*form(w+1)*]{} denotes the token to the right of the current token [*w*]{}. When more than one argument is given, the functor is applied to each defined position and the results are concatenated. Thus, [*form(w,w+1)*]{} expands to [*form(w)+form(w+1)*]{}. The functor *formlc* denotes the form with all letters converted to lowercase and *lem* denotes the lemma of a word. The functors [*suffix1, suffix2,...*]{} and [*prefix1,...*]{} denote suffixes and prefixes of length [*1, 2, ..., 5*]{}. The [*suffix1+uc, ...*]{} functors concatenates a suffix with a value that indicates uppercase or lowercase word. The functors [*pos*]{} and [*mor*]{} denote part-of-speech tags and morphological tags, respectively. The tags to the right of the current position are available as features in the second iteration of the standalone tagger as well as in the final resolution stage in the joint system. Patterns of the form $c_i$ denote the $i$th character. Finally, the functor [*number*]{} denotes a sequence of numbers, with optional periods and commas. Main Results {#fsa} ------------ In our experiments we make a division of the training corpora into 80% for training and 20% for development. Therefore, in each iteration a model is trained over 80% of the training corpus and tested on 20%.[^3] For feature selection, if the outcome of the newly trained model is better than the best result so far, then the feature is added to the feature model; otherwise, it is not. A model has to show improvement of at least 0.02 on part-of-speech tagging accuracy to count as better.[^4] Table \[table:templates\] (columns under Part-of-Speech) shows the features that the algorithms selected for each language and each system, and Table \[table:pos-results-dev\] shows the performance on the development set. We primarily report part-of-speech tagging accuracy (POS), but also report unlabeled (UAS) and labeled (LAS) attachment scores [@buchholz06] to show the effect of improved taggers on parsing quality. Additionally, Table \[table:pos-results-dev\] contains the number of features selected (\#). The first conclusion to draw is that the feature selection algorithms work for both standalone and joint systems. The number of features selected is drastically reduced. The dynamic MRMR feature selection technique for the joint system compresses the model by as much as 78%. This implies faster inference (smaller dot products and less feature extraction) and a smaller memory footprint. In general, joint systems compress more over their standalone counterpart, by about 20%. Furthermore, the dynamic technique tends to have slightly more compression. The accuracies of the joint tagger-parser are in general superior to the ones obtained by the standalone tagger, as noted by . In terms of tagging accuracy, static selection works slightly better for Chinese, German and Hungarian while dynamic MRMR works best for English and Russian (Table \[table:pos-results-dev\]). Moreover, the standalone tagger selects several feature templates that requires iterating over the sentence, such as [*pos(w+1), pos(w+2)*]{}, whereas the feature templates selected by the joint system contain significantly fewer of these features. This shows that a joint system is less reliant on context features to resolve many ambiguities that previously needed a wider context. This is almost certainly due to the global contextual information pushed to the tagger via parsing decisions. As a consequence, the preprocessing tagger can be simplified and we need to conduct only one iteration over the sentence while maintaining the same accuracy level. Interestingly, the dynamic MRMR technique tends to select less *form* features, which have the largest number of realizable values and thus model parameters. Table \[table:tagger-state-of-art\] compares the performance of our two taggers with two state-of-the-art taggers. Except for English, the joint tagger consistently outperforms the Stanford tagger and MarMot: for Chinese by 0.3, for German by 0.38, for Hungarian by 0.25 and for Russian by 0.75. Table \[table002\] compares the resulting parsing accuracies to state-of-the-art dependency parsers for English and Chinese, showing that the results are in line with or higher than the state of the art. ---------- ------- ------------------------------------------------------- ----- ----- ----- -- -- -- -- -- POS POS POS POS POS Stanford 93.75 **97.44 & 97.51 & 97.55& 98.16\ MarMot & 93.84 & 97.43 & 97.57& 97.63 & 98.18\ Standalone & 94.04 & 97.33 & 97.56 & 97.69 & 98.73\ Joint & **94.14 & 97.42 &**97.95 & **97.88 & **98.93\ ********** ---------- ------- ------------------------------------------------------- ----- ----- ----- -- -- -- -- -- : State-of-the-art comparison for tagging on the test set.[]{data-label="table:tagger-state-of-art"} \[table002\] Morphological Tagging Experiments {#morphology} ================================= The joint morphology and syntactic inference requires the selection of morphological attributes (case, number, etc.) and the selection of features to predict the morphological attributes. In past work on joint morphosyntactic parsing, all morphological attributes are predicted jointly with syntactic dependencies [@tacl-bbjn]. However, this could lead to unnecessary complexity as only a subset of the attributes are likely to influence parsing decisions, and vice versa. In this section we investigate whether feature selection methods can also be used to reduce the set of morphological attributes that are predicted as part of a joint system. For instance, consider a language that has the following attributes: case, gender, number, animacy. And let us say that language does not have gender agreement. Then likely only case and number will be useful in a joint system, and the gender and animacy attributes can be predicted independently. This could substantially improve the speed of the joint model – on top of standard feature selection – as the size of the morphosyntactic tag set will be reduced significantly. Data Sets {#data-sets} --------- We use the data sets listed in subsection \[exsetup\] for the languages that provide morphological annotation, which are German, Hungarian and Russian. Main Results: Attribute Selection --------------------------------- For the selection of morphological attributes (e.g. case, number, tense), we explore a simple method that departs slightly from those in Section \[featselection\]. In particular, we do not run greedy forward selection. Instead, we compute accuracy improvements for each attribute offline. We then independently select attributes based on these values. Our initial design was a greedy forward attribute selection, but we found experimentally that independent attribute selection worked best. We run 10-fold cross-validation experiments on the training set where 90% of training set is used for training and 10% for testing. Here we simply test for each attribute independently whether its inclusion in a joint morphosyntactic parsing system increases parsing accuracy (LAS/UAS) by a statistically significant amount. If it does, then it is included. We applied cross validation to obtain more reliable results than with the development sets as some improvements where small, e.g., gender and number in German are within the range of the standard deviation results on the development set. We use parsing accuracy as we are primarily testing whether a subset of attributes can be used in place of the full set in joint morphosyntactic parsing. Even though this method only tests an attribute’s contribution independently of other attributes, we found experimentally that this was never a problem. For instance, in German, without any morphologic attribute, we get a baseline of 89.18 LAS; when we include the attribute case, we get 89.45 LAS; and when we include number, we get 89.32 LAS. When we include both case and number, we get 89.60 LAS. Table \[attribute-selection\] shows which attributes were selected. We include an attribute when the cross-validation experiment shows an improvement of at least 0.1 with a statistical significance of 0.01 or better (indicated in the table by \*\*). Some borderline cases remain such as for Russian passive where we observed an accuracy gain of 0.2 but only a low statistical significance. [|r|r|r|l||c|]{}\ & &\ attribute & LAS & UAS & stat. sig. & Sel.\ *[none]{}& 89.2 & 91.8 & – &\ case & 89.5 & 91.9 & yes$^{***}$ &\ gender & 89.2& 91.8 & no &\ number & 89.3& 91.9 & yes$^{***}$ &\ mode & 89.2& 91.8 & no &\ person & 89.2& 91.8 & no &\ tense & 89.2& 91.8 & no &\  \ \ & &\ attribute & LAS & UAS & stat. sig. & Sel.\ *[none]{}& 84.5& 88.3 & – &\ case & 85.7& 89.0 & yes$^{****}$&\ degree & 84.6& 88.4 & yes$^{*}$ &\ number & 84.7& 88.5 & yes$^{**}$ &\ mode & 84.6& 88.4 & no &\ person P & 84.6& 88.7 & yes$^{*}$ &\ person & 85.0& 88.9 & yes$^{**}$ &\ subpos & 85.4& 88.9 & yes$^{***}$ &\ tense & 84.6& 88.4 & yes$^{*}$ &\ ** [|r|r|r|l||c|]{}\ & &\ attribute & LAS & UAS & stat. sig. & Sel.\ *[none]{}& 79.4 & 88.2 & &\ act & 80.4 & 89.1 & yes$^{****}$ &\ anim & 79.8 & 88.3 & yes$^{****}$ &\ aspect & 79.4 & 88.2 & no &\ case & 80.9 & 89.3 & yes$^{****}$ &\ degree & 79.4 & 88.2 & no &\ gender & 80.1 & 88.6 & yes$^{****}$ &\ mode & 80.0 & 88.7 & yes$^{***}$ &\ number & 82.2 & 88.6 & yes$^{****}$ &\ passive & 79.6 & 88.2 & yes$^{*}$ &\ tense & 79.8 & 88.4 & yes$^{**}$ &\ typo & 79.4 & 88.1 & no &\ * Main Results: Feature Selection ------------------------------- Having fixed the set of attributes to be predicted jointly with the parser, we can turn our attention to optimizing the feature sets for morphosyntactic tagging. To this end, we again consider greedy forward selection with the static and dynamic strategies. Table \[table:templates\] shows the selected features for the different languages where the grey boxes again mean that the feature was selected. Table \[table:morph-results-dev\] shows the performance on the development set. For German, the full template set performs best but only 0.04 better than static selection which performs nearly as well while reducing the template set by 68%. For Hungarian, all sets perform similarly while dynamic selection needs 86% less features. The top performing feature set for Russian is dynamic selection in a joint system which needs 81% less features. We observe again that dynamic selection tends to select less feature templates compared to static selection, but here both the full set of features and the set selected by static selection appear to have better accuracy on average. The feature selection methods obtain significant speed-ups for the joint system. On the development sets we observed a speedup from 0.015 to 0.003 sec/sentence for Hungarian, from 0.014 to 0.004 sec/sentence for German, and from 0.015 to 0.006 sec/sentence for Russian. This represents a reduction in running time between 50 and 80%. Table \[table003\] compares our system to other state-of-the-art morphosyntactic parsers. We can see that on average the accuracies of our attribute/feature selection models are competitive or above the state-of-the art. The key result is that state of the art accuracy can be achieved with much leaner and faster models. Conclusions {#sec:conclusion} =========== There are several methodological lessons to learn from this paper. First, feature selection is generally useful as it leads to fewer features and faster tagging while maintaining state-of-the-art results. Second, feature selection is even more effective for joint tagging-parsing, where it leads to even better results and smaller feature sets. In some cases, the number of feature templates is reduced by up to 80% with a correponding reduction in running time. Third, dynamic feature selection strategies [@Hanchuan2005] lead to more compact models than static feature selection, without significantly impacting accuracy. Finally, similar methods can be applied to morphological attribute selection leading to even leaner and faster models. Acknowledgement {#acknowledgement .unnumbered} =============== Miguel Ballesteros is supported by the European Commission under the contract numbers FP7-ICT-610411 (project MULTISENSOR) and H2020-RIA-645012 (project KRISTINA) [^1]: Training: 001–815, 1001–1136. Development: 886–931, 1148–1151. Test: 816–885, 1137–1147. [^2]: Training: 02-21. Development: 24. Test: 23. [^3]: There is also a held-out test set for evaluation, which is the standard test set provided and depicted in Section \[exsetup\]. [^4]: All the experiments were carried out on a CPU Intel Xeon 3.4 Ghz with 6 cores. Since the feature selection experiments require us to train a large number of parsing and/or tagging models, we needed to find a realistic training setup that gives us a sufficient accuracy level while maintaining a reasonable speed. After some preliminary experiments, we selected a beam size of 8 and 12 training iterations for the feature selection experiments while the final models are tested with a beam size of 40 and 25 training iterations. The size $k$ of the second beam for alternative tag sequences is kept at 8 for all experiments and the threshold $\alpha$ at 0.25.

--- abstract: 'This is an updated and expanded version of our earlier survey article [@Gut5]. Section $\S 1$ introduces the subject matter. Sections $\S 2 - \S 4$ expose the basic material following the paradigm of elliptic, hyperbolic and parabolic billiard dynamics. In section $\S 5$ we report on the recent work pertaining to the problems and conjectures exposed in the survey [@Gut5]. Besides, in section $\S 5$ we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded.' address: 'Nicolaus Copernicus University, Chopina 12/18, Torun 87-100; IM PAN, Sniadeckich 8, Warszawa 10, Poland' author: - Eugene Gutkin title: 'Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems' --- Introduction {#intro} ============ Billiard dynamics broadly understood is the geodesic flow on a Riemannian manifold with a boundary. But even this very general framework is not broad enough, e.g., for applications in physics. In these applications the manifold in question is the configuration space of a physical system. Often, it is a [*manifold with corners*]{} and [*singularities*]{}. Some physics models lead to the Finsler billiard [@GT]: The manifold in question is not Riemannian; it is Finslerian. The simplest examples of Riemannian manifolds with corners are plane polygons, and some basic physical models yield the billiard on triangles [@Kolan; @GuTr; @Glashow; @Gut3] The configuration space of the famous gas of elastic balls [@Sinai-; @Sas1] is structured combinatorially like a euclidean polyhedron of a huge number of dimensions. In fact, this configuration space is much more complicated, because the polyhedron is not flat. The mathematical investigation of this system produced the celebrated Boltzmann Ergodic Hypothesis. After Sinai’s seminal papers [@Sinai-; @Sinai], a modified version of the original conjecture became known as the Boltzmann-Sinai Hypothesis. However, the bulk of our exposition is restricted to the billiard in a bounded planar domain with piecewise smooth boundary. The reason is threefold. First of all, this setting allows us to avoid lengthy preliminaries and cumbersome formalism: It immediately leads to qualitative mathematical questions. (This was also the opinion of G. D. Birkhoff [@Birk].) Second, there are basic physical models that correspond to planar billiards [@Gut3]. Third, and most important, there are fundamental problems on the plane billiard that are still open. The problems are indeed fundamental: They concern the main features of these dynamical systems. In the body of the paper we introduce several open problems of billiard dynamics. Our choice of the questions is motivated partly by the personal taste and partly by the simplicity of formulation. We review the preliminaries, discuss the motivation, and outline possible angles of attack. We also point out partial results and other evidence toward the answer. Formally, the exposition is self-contained, but the reader may want to consult the literature [@KH; @HK; @KozTres; @Tabach; @Chernov4; @Gut2; @Gut3]. For obvious reasons, we will call the planar domain in question the [*billiard table*]{}. Its geometric shape determines the qualitative character of the motion. Historically, three classes of shapes have mostly attracted attention. First, it is the class of smooth and strictly convex billiard tables. For several reasons, the corresponding billiard dynamics is called elliptic. Second, it is the piecewise concave and piecewise smooth billiard tables. The corresponding dynamics is hyperbolic.[^1] Billiard tables of the third class are the polygons. The corresponding dynamics is parabolic. The three types of the billiard are exposed in § \[elliptic\] – § \[hyperbolic\] – § \[parabolic\] repectively. In the rest of the introduction we describe the basic notation and the terminology. Let $Y\subset{{\mathbb R}^2}$ be a compact, connected billiard table. See Figure \[fig1\]. Its boundary ${\partial}Y$ is a finite union of $C^1$ curves. It may have several connected components. The [*billiard flow*]{} on $Y$ is modelled on the motion of a material point: The “particle" or the “billiard ball". At each time instant, the state of the system is determined by the position of the ball, $y\in Y$, and its velocity, a unit vector $v\in{{\mathbb R}^2}$. (It suffices to consider the motion with the unit speed.) The ball rolls along the ray emanating from $y$, in the direction $v$. At the instant the ball reaches $\partial Y$, its direction changes. Let $x\in\partial Y$ be the point in question, and let $v'$ be the new direction. The transformation, $v\mapsto v'$, is the [*orthogonal reflection*]{} about the tangent line to $\partial Y$ at $x$. The vector $v'$ is directed inward, and the ball keeps rolling. These rules define: 1) The [*phase space*]{} $\Psi$ of the billiard flow, as the quotient of $Y\times S^1$ by the identification $(x,v)=(x,v')$ above; 2) The billiard flow $T^t:\Psi\to\Psi$. If $Y$ is simply connected, and $\partial Y$ is $C^1$, then $\Psi$ is homeomorphic to the three-dimensional sphere.[^2] In any way, $\dim\Psi=3$, and the reader may think of $\Psi$ as the set of pairs $(y,v)$, such that $v$ is directed inward. A few remarks are in order. The rules defining the billiard flow stem from the assumptions that the billiard motion is frictionless, and that the boundary of the billiard table is perfectly elastic. The orthogonal reflection rule $v\mapsto v'$ insures that billiard orbits are the local minimizers of the distance functional. (This property extends to the Finsler billiard [@GT].) The reflection rule is not defined at the corners of the boundary. The standard convention is to “stop the ball" when it reaches a corner. Thus, if ${\partial}Y$ is not $C^1$, then there are billiard orbits that are not defined for all times. Their union has zero volume with respect to the [*Liouville measure*]{} defined below. Set $X={\partial}Y$, and endow it with the positive orientation. Choosing a reference point on each connected component, and using the arc length parameter, we identify $X$ with the disjoint union of $k\ge 1$ circles. In this paper, with the exception of § \[parabolic\], $k=1$. The set $\Phi\subset\Psi$ given by the condition $y\in\partial Y$ is a cross-section for the billiard flow. The Poincaré mapping ${\varphi}:\Phi\to\Phi$ is the [*billiard map*]{} and $\Phi$ is its phase space. The terminology is due to G. D. Birkhoff who championed the “billiard ball problem" [@Birk]. Let $x$ be the arc length parameter on $X$. For $(x,v)\in\Phi$ let $\theta$ be the angle between $v$ and the positive tangent to $\partial Y$ at $x$. Then $0\le\theta\le\pi$, where $0$ and $\pi$ correspond to the forward and the backward tangential directions respectively. This coordinate system fails at the corners of ${\partial}Y$. If ${\partial}Y$ is $C^1$, then $\Phi=X\times[0,\pi]$. We will use the notation ${\varphi}(x,\theta)=(x_1,\theta_1)$. Let $p,q$ be the euclidean coordinates in ${{\mathbb R}^2}$, and let $0\le \alpha < 2\pi$ be the angle coordinate on the unit circle. The Liouville measure on $\Psi$ has the density $d\nu=dpdqd\alpha$. It is invariant under the billiard flow. The [*induced Liouville measure*]{} $\mu$ on $\Phi$ is invariant under the billiard map, and has the density $d\mu=\sin\theta dxd\theta$. Both measures are finite. Straightforward computations yield $$\label{volumes-eq} \nu(\Psi)=2\pi\mbox{Area}(Y),\ \mu(\Phi)=2\,\mbox{Length}(\partial Y).$$ Smooth, strictly convex billiard: elliptic dynamics {#elliptic} =================================================== The first deep investigation of this framework is due to G. D. Birkhoff [@Birk]. For this reason, it is often called the [*Birkhoff billiard*]{}. The billiard map is an [*area preserving twist map*]{} [@KH]. An [*invariant circle*]{} is a ${\varphi}$-invariant curve $\Gamma\subset\Phi$ which is a noncontractible topological circle. Recall that $\Phi$ is a topological annulus. Both components of $\partial\Phi$ are the trivial invariant circles. From the geometric optics viewpoint, $\Phi$ is the space of light rays (i.e., directed lines), and $Y$ is a room whose walls are the perfect mirrors. Then $\Gamma\subset\Phi$ is a one-parameter family of light rays in $Y$, and its [*envelope*]{} $F(\Gamma)$ is the set of focusing points of light rays in this family. Note that $F(\Gamma)$ is not a subset of $Y$, in general. For instance, if $Y$ is an ellipse, then there are invariant curves $\Gamma$ such that $F(\Gamma)$ are confocal hyperbolas. Let $\Gamma$ be an invariant circle, and let $\gamma=F(\Gamma)$. Then $\gamma\subset\mbox{Int}(Y)$ [@GutKat]. These curves are the [*caustics*]{} of the billiard table. When ${\partial}Y$ is an ellipse, the caustics are the confocal ellipses. Their union is the region $Y\setminus[ff']$, where $f,f'$ are the foci of $\in Y$. If $Y$ is not a disc, the invariant circles fill out a region, $C(\Phi)\subset\Phi$, whose complement looks like a pair of “eyes". See Figure \[fig3\]. \[integrable-def\] A billiard table $Y$ is [*integrable*]{} if the set of invariant circles has nonempty interior. The most famous open question about caustics is known as the [*Birkhoff conjecture*]{}. It first appeared in print in a paper by Poritsky [@Porit], several years after Birkhoff’s death.[^3] [**Problem 1 (Birkhoff conjecture)**]{}. Ellipses are the only integrable billiard tables.\ A disc is a degenerate ellipse, with $f=f'$. The preceding analysis applies, and the invariant circles fill out all of the phase space. M. Bialy proved the converse: If all of $\Phi(Y)$ is foliated by invariant circles, then $Y$ is a disc [@Bialy]. See [@Bialy1] for an extension of this theorem to the surfaces of arbitrary constant curvature. We refer the reader to section §  \[update\] for elaborations and updates on the Birkhoff conjecture. Let $Y$ be any oval. If $X=\partial Y$ is sufficiently smooth, and its curvature is strictly positive, then the invariant circles fill out a set of positive measure. This was proved by V. Lazutkin under the assumption that $\partial Y$ was of class $C^{333}$ [@Laz]. Lazutkin’s proof crucially uses a famous theorem of J. Moser [@Moser1].[^4] The number $333$ is chosen in order to satisfy the assumptions in [@Moser1]. The required smoothness was eventually lowered to $C^6$  [@Douad]. By a theorem of J. Mather [@Mather1], the positive curvature condition is necessary for the existence of caustics. An invariant noncontractible topological annulus, $\Omega\subset\Phi$, whose interior contains no invariant circles, is a [*Birkhoff instability region*]{}. This is a special case of an important concept for area preserving twist maps [@KH]. Assume the Birkhoff conjecture, and let $Y$ be a non-elliptical billiard table. Then $\Phi$ contains Birkhoff instability regions. The dynamics in an instability region has positive [*topological entropy*]{} [@Angen]. Hence, the Birkhoff conjecture implies that any non-elliptical billiard has positive topological entropy. By the [*(metric) entropy*]{} of a billiard we will mean the entropy of the Liouville measure. The only examples of convex billiard tables with positive entropy are the Bunimovich stadium [@Buni+] and its generalizations. These billiard tables are not strictly convex, and their boundary is only $C^1$. The corresponding billiard dynamics is hyperbolic. See § \[hyperbolic\]. This leads to our next open question.\ [**Problem 2**]{}. a) Construct a strictly convex $C^1$-smooth billiard table with positive entropy. b) Construct a convex $C^2$-smooth billiard table with positive entropy.\ Using an ingenuous variational argument, Birkhoff proved the existence of certain periodic billiard orbits [@Birk]. His approach extends to area preserving twist maps, and thus yields a more general result on periodic orbits of these dynamical systems [@KH]. In the billiard framework the relevant considerations are especially transparent. A periodic orbit of period $q$ corresponds to an (oriented) closed polygon with $q$ sides, inscribed in $Y$, and satisfying the obvious condition on the angles it makes with $\partial Y$. Birkhoff called these the [*harmonic polygons*]{}. Vice versa, any oriented harmonic $q$-gon $P$ determines a periodic orbit of period $q$. Let $1\le p <q$ be the number of times the pencil tracing $P$ goes around $\partial Y$. The ratio $0 < p/q < 1$ is the [*rotation number*]{} of a periodic orbit. Fix a pair $1\le p <q$, with $p$ and $q$ relatively prime. Let $X(p,q)$ be the set of all inscribed $q$-gons that go $p$ times around $\partial Y$. The space $X(p,q)$ is a manifold with corners. For $P\in X(p,q)$ let $f(P)$ be the circumference of $P$. Then harmonic polygons are the critical points of the function $f:X(p,q)\to{{\mathbb R}}$. Birkhoff proved that $f$ has at least two distinct critical points. One of them delivers the maximum, and the other a minimax to the circumference. The corresponding periodic billiard orbits are the [*Birkhoff periodic orbits*]{} with the rotation number $p/q$. By way of example, we take the rotation number $1/2$. Then the maximal Birkhoff orbit yields the [*diameter of $Y$*]{}. The minimax orbit corresponds to the [*width of $Y$*]{}. When the diameter and the width of $Y$ are equal, the boundary $\partial Y$ is a [*curve of constant width*]{}; then we have a one-parameter family of periodic orbits with the rotation number $1/2$. They fill out the “equator" of $\Phi$. There are other examples of ovals with one-parameter families of periodic orbits having the same length and the same rotation number. See [@Innami] and [@Gut+; @Gut10] for different approaches. One of the basic characteristics of a dynamical system is the [*growth rate of the number of periodic points*]{}. In order to talk about it, we need a [*counting function*]{}. The standard counting function $f_Y(n)$ for the billiard map is the number of periodic points of the period at most $n$. (See § \[hyperbolic\] and § \[parabolic\] for other examples.) The set of periodic points is partitioned into periodic orbits, and let $F_Y(n)$ be the number of periodic orbits of period at most $n$. Birkhoff’s theorem bounds $F_Y(n)$ from below by the number of relatively prime pairs $1\le p < q \le n$. This implies a universal cubic lower bound $f_Y(n)\ge cn^3$. See, e. g., [@Hardy]. Since an oval may have infinitely many periodic points of the same period, there is no universal upper bound on $f_Y(n)$. The size of a measurable set is naturally estimated by its measure. Let ${\mathcal P}\subset\Phi$ (resp. ${\mathcal P}_n\subset\Phi$) be the set of periodic points (resp. periodic points of period $n$). For example, if $Y$ is a table of constant width, then ${\mathcal P}_2\subset\Phi$ is the equator. Although it is infinite, $\mu({\mathcal P}_2)=0$. Since ${\mathcal P}=\cup_{n=2}^{\infty}{\mathcal P}_n$, a disjoint union, $\mu({\mathcal P})=\sum_{n=2}^{\infty}\mu({\mathcal P}_n)$. Thus, $\mu({\mathcal P})=0$ iff $\mu({\mathcal P}_n)=0$ for all $n=2,3,4,\dots$. The famous [*Weyl formula*]{} gives the leading term and the error estimate for the spectral asymptotics of the Laplace operator (with either Dirichlet or Neumann boundary conditions) in a bounded domain of the euclidean space (of any number of dimensions). The (also famous) [*Weyl conjecture*]{} predicts the second term of the asymptotic series [@Weyl]. A theorem of V. Ivrii [@Ivrii] establishes the Weyl conjecture for a euclidean domain under the assumption that the set of periodic billiard orbits has measure zero.[^5] Ivrii conjectured that the assumption $\mu({\mathcal P})=0$ was superfluous: It should hold for any euclidean domain with a smooth boundary. Members of the Sinai’s dynamics seminar in Moscow promised to him in 1980 to prove the desideratum in a few days ... The question is still open. Problem 3 below states the conjecture for plane domains.\ [**Problem 3 (Ivrii conjecture)**]{}. Let $Y$ be a piecewise smooth billiard table. i) Prove that $\mu({\mathcal P})=0$. ii) Prove that $\mu({\mathcal P}_n)=0$ for all $n$.\ Although Problem 3 concerns arbitrary billiard tables, it is especially challenging for the Birkhoff billiard, hence we have put the problem into this section. It is convenient to designate by, say, $I_n$ the claim $\mu({\mathcal P}_n)=0$. Thus, Ivrii conjecture amounts to proving $I_n$ for all $n\ge 2$. Claim $I_2$ is obvious. Proposition $I_3$ is a theorem of M. Rychlik [@Rych]. His proof depends on a formal identity, verified using Maple. L. Stojanov simplified the proof, and eliminated the computer verification [@Stoj2]. Ya. Vorobets gave an independent proof [@Vor1]. His argument applies to higher dimensional billiards as well. M. Wojtkowski [@Wojt3] obtained Rychlik’s theorem as an application of the [*mirror equation*]{} of the geometric optics and the [*isoperimetric inequality*]{}. See [@GutKat] for other applications. Ivrii’s conjecture is known to hold in many special cases, e.g., for hyperbolic and parabolic billiard tables. See § \[hyperbolic\] and § \[parabolic\]. It holds for billiard tables with real analytic boundary [@SaVa]. For the generic billiard table the sets ${\mathcal P}_n$ are finite for all $n$ [@PetStoj]. The billiard map for a Birkhoff billiard table is an area preserving twist map. However, there are smooth area preserving twist maps such that $\mu({\mathcal P})>0$. Thus, Ivrii’s conjecture is really about the billiard map! Recently Glutsyuk and Kudryashov announced a proof of Proposition $I_4$ [@GluKudr]. See Section  \[elliptic\_sub\] for further comments. Hyperbolic billiard dynamics {#hyperbolic} ============================ It is customary to say that a billiard table is hyperbolic if the associated dynamics is hyperbolic. The dynamics in question may be the billiard flow or the billiard map or the induced map on a subset of the phase space. For concreteness, we will call a billiard table hyperbolic if the corresponding billiard map is hyperbolic. The modern approach to hyperbolic dynamics crucially uses the Oseledets multiplicative ergodic theorem [@Osel]. See [@KH; @HK] for a general introduction into the hyperbolic dynamics and [@KS; @Buni; @Tabach] (resp. [@Chernov4]) for introductory (resp. thorough) expositions of the hyperbolic billiard. The first hyperbolic billiard tables were made from concave arcs. As a motivation, let us consider the following construction. Let $P$ be a convex polygon. Replace some of the sides of $P$ by circular arcs whose centers are sufficiently far from $P$. The result is a “curvilinear polygon", $Y$, approximating $P$. Choosing appropriate center points, we insure that the “curved sides" of $Y$ are convex inward. It is not important that they be circular, as long as they are smooth and convex inward. This class of billiard tables arose in the work of Ya. Sinai on the [*Boltzmann-Sinai gas*]{} [@Sinai-].[^6] In the [*Boltzmann gas*]{} the identical round molecules are confined by a box. Sinai has replaced the box by periodic boundary conditions. Thus, the molecules of the Boltzmann-Sinai gas move on a flat torus. In the “real world", the confining box is three-dimensional and the number of moving molecules is enormous. In the Sinai “mathematical caricature", there are only two molecules on a two-torus. The system reduces to the geodesic flow on a flat torus with a round hole. Represent the flat torus by the $2\times 2$ square, so that the hole is the central disc of radius $1/2$. By the four-fold symmetry, the problem reduces to the billiard on the unit square with the deleted quarter-disc of radius $1/2$, centered at a vertex. See Figure \[fig4\]. This domain is known as [*the Sinai billiard*]{}.[^7] Let now $P$ be a (not necessarily convex) $n$-gon. Let $Y$ be the region obtained by replacing $1\le m < n$ (resp. all) of the sides of $P$ by circular arcs, satisfying the conditions above. Then $Y$ is a [*semi-dispersive (resp. dispersive) billiard table*]{}. The circular arcs (resp. the segments) of ${\partial}Y$ are its [*dispersive (resp. neutral) components*]{}. This terminology extends to the billiard tables whose dispersive boundary components are smooth, convex inward curves. These are the (semi)dispersive billiard tables. The table in Figure 4 has one dispersive and four neutral boundary components. In [@Sinai] Sinai proved the hyperbolicity of dispersive billiard tables. After the discovery by L. Bunimovich that the [*stadium*]{} and similar billiard tables are hyperbolic [@Buni+], mathematicians started searching for geometric criteria of hyperbolicity. The notion of an [*invariant cone field*]{} [@Wojt+; @KB] proved to be very useful. Denote by $V_z$ the tangent plane to the phase space at $z\in\Phi$. The differential ${\varphi}_*$ is a linear map from $V_z$ to $V_{{\varphi}(z)}$. By our convention, a subset of a vector space is a [*cone*]{} if it is invariant under multiplications by all scalars. \[cone-def\] A family ${\mathcal C}=\{C_z\subset V_z:z\in\Phi\}$ is an [*invariant cone field*]{} if the following conditions are satisfied. - 1\. The closed cone $C_z$ is defined for almost all $z\in\Phi$, and the map $z\mapsto C_z$ is measurable. - 2\. The cone $C_z$ is has nonempty interior.\ - 3\. We have ${\varphi}_*(C_z)\subset C_{{\varphi}(z)}$.\ - 4\. There exists $n=n(z)$ such that ${\varphi}_*^n(C_z)\subset \mbox{int}(C_{{\varphi}^n(z)})$.\ The hyperbolicity is equivalent to the existence of an invariant cone field [@Wojt+]. Wojtkowski constructed invariant cone fields for several classes of billiard tables [@Wojt1]. In addition to the dispersive tables and the generalized stadia, he found invariant cone fields for a wide class of locally strictly convex tables. Wojtkowski’s approach was further extended by Bunimovich, V. Donnay, and R. Markarian [@Chernov4]. Using these ideas, B. Gutkin, U. Smilanski and the author constructed hyperbolic billiard tables on surfaces of arbitrary constant curvature  [@GuSm].\ [**Problem 4**]{}. Is every semi-dispersive billiard table hyperbolic?\ Let $Y$ be a semi-dispersive $n$-gon with only one neutral component. Let $Y'$ be the reflection of $Y$ about this side, and set $Z=Y\cup Y'$. Since $Z$ is a dispersive billiard table, it is hyperbolic. By the reflection symmetry, the table $Y$ is also hyperbolic. In special cases, the reflection trick yields the hyperbolicity of semi-dispersive $n$-gons with $m<n-1$ dispersive components. For instance, let $P$ be a triangle with an angle $\pi/n$. Let $Y$ be the semi-dispersive triangle, whose only dispersive component is located opposite the $\pi/n$ angle. Reflecting $Y$ successively $2n$ times, we obtain a dispersive billiard table, $Z$. Thus, $Z$ is hyperbolic. By the symmetry, the table $Y$ is hyperbolic as well. A suitable generalization of the reflection trick will work if $P$ is a [*rational polygon*]{}. See § \[parabolic\]. The special case $m=1$ of Problem 4 is closely related to Problem 9 of § \[parabolic\]. Dispersive billiard tables are ergodic [@BSC2]. There are examples of hyperbolic, but nonergodic billiard tables [@Wojt1]. The consensus is that a typical hyperbolic billiard is ergodic. For instance, the stadium and its relatives are ergodic [@Szasz]. There are no examples of strictly convex hyperbolic billiard tables. See Problem 2. For the rest of this section, we consider only dispersive billiard tables. Referring the reader to [@BSC1; @BSC2; @Chernov1; @Sas1; @Chernov2; @LaiSun] for a discussion of their chaotic properties and to open questions about, e.g., the [*decay of correlations*]{}, we concentrate on the statistics of periodic orbits in hyperbolic billiards. The set of periodic points of any period is finite; let $f_Y(n)$ be the number of periodic points, whose period is less than or equal to $n$. The asymptotics of $f_Y(n)$, as $n\to\infty$, is an important dynamical characteristic. By theorems of Stojanov and Chernov [@Stoj; @BSC1], there are $0<h_-<h_+<\infty$ such that $$\label{lower-upper-eq} 0<h_-\le\liminf_{n\rightarrow\infty}\frac{\log f_Y(n)}{n}\le \limsup_{n\rightarrow\infty}\frac{\log f_Y(n)}{n}\le h_+<\infty.$$ The following two problems were contributed by N. Chernov.\ [**Problem 5**]{}. Does the limit $$\label{asympt-eq} h=\lim_{n\rightarrow\infty}\frac{\log f_Y(n)}{n}$$ exist?\ [**Problem 6**]{}. If the limit in equation (\[asympt-eq\]) exists, is $0<h<\infty$ the [*topological entropy*]{} of the billiard map?\ Problems 5 and 6 fit into the general relationship between the distribution of periodic points and the topological entropy [@Kat2]. However, the singularities, which constitute the paramount feature of billiard dynamics, preclude the applicability of smooth ergodic theory. Other techniques have to be developed [@Chernov3; @GuHa95; @GuHa]. Polygonal billiard: parabolic dynamics {#parabolic} ====================================== The polygon $P$ that serves as a billiard table is not required to be convex or simply connected. It may also have [*barriers*]{}, i. e., obstacles without interior. It is [*rational*]{} if the angles between its sides are of the form $\pi m/n$. Let $N=N(P)$ be the least common denominator of these rational numbers. A classical construction associates with $P$ a closed surface $S=S(P)$ tiled by $2N$ copies of $P$. The surface $S$ has a finite number of cone points; the cone angles are integer multiples of $2\pi$. Suppose that $P$ is a [*simple polygon*]{},[^8] and let $m_i\pi/n_i,1\le i \le p,$ be its angles. The genus of $S(P)$ satisfies [@Gut2] $$\label{genus-eq} g(S(P)) = 1+\frac{N}{2}\sum_{i=1}^p\frac{m_i-1}{n_i}.$$ Equation (\[genus-eq\]) implies that $S(P)$ is a torus if and only if $P$ tiles the plane under reflections. The billiard in $P$ is essentially equivalent to the geodesic flow on $S(P)$. This observation was first exploited by A. Katok and A. Zemlyakov [@KZ], and $S(P)$ is often called the “Katok-Zemlyakov surface". However, the construction has been in the literature (at least) since the early 20-th century [@Stackel; @Fox]. We refer to the surveys [@Gut2; @Gut3; @Smillie; @MasTab; @Tabach] for extensive background material. Surfaces $S(P)$ are examples of [*translation surfaces*]{}, which are of independent interest [@GuJ]. From the viewpoint of classical analysis, a translation surface is a closed Riemann surface with a holomorphic linear differential. Using holomorphic quadratic (as opposed to linear) differentials, we arrive at the notion of [*half-translation surfaces*]{} [@Guj; @GuJ]. Billiard orbits on a polygon become geodesics on the corresponding translation (or the half-translation) surface. Since billiard orbits change directions at every reflection, the notion of the direction of an orbit is not well defined. Geodesics on a translation surface, on the contrary, do not change their directions. This yields a technical advantage of translation surfaces over polygons [@KZ]. The crucial advantage comes, however, from the natural action of the group ${\text{SL}(2,{{\mathbb R}})}$ on translation surfaces [@KMS; @Mas1; @Mas2; @Veech2; @Smillie]. See section  \[update\] for elaborations. The geodesic flow of any translation surface, $S$, decomposes into the one-parameter family of [*directional flows*]{} $b_{\theta}^t,\ 0\le \theta < 2\pi$. The flow $b_{\theta}^t$ is identified with the [*linear flow on $S$ in direction $\theta$*]{}. The Lebesgue measure on $S$ is preserved by every $b_{\theta}^t$. Thus, not only is the billiard flow of a rational polygon not ergodic, it decomposes as a one-parameter family of [*directional billiard flows*]{}. Let $S$ be an arbitrary translation surface. A theorem of Kerckhoff, Masur, and Smillie [@KMS] says that the flows $b_{\theta}^t$ are uniquely ergodic for Lebesgue almost all $\theta$. In particular, the directional billiard flow of a rational polygon is ergodic for almost every direction. The set ${\mathcal N}(S)\subset[0,2\pi)$ of non-uniquely ergodic directions has positive Hausdorff dimension for the typical translation surface [@MaSmi]; for particular classes of rational polygons and translation surfaces the sets ${\mathcal N}(S)$ are countably infinite [@Gut1; @Veech2; @Yitwah]. We point out that a typical translation surface does not correspond to any polygon, which illustrates the limitations of this relationship for the study of polygonal billiard. See section  \[update\] for elaborations on the polygonal billiard and translation surfaces.\ Much less is known about the billiard in irrational (i. e., arbitrary) polygons. Denote by ${{\mathcal T}}(n)$ the moduli space of simple euclidean $n$-gons. Since the billiard dynamics is invariant under scaling, in ${{\mathcal T}}(n)$ we identify polygons that coincide up to scaling. The space ${{\mathcal T}}(n)$ is a finite union of components which correspond to particular combinatorial data. We will refer to them as the [*combinatorial type components*]{}. Each component is homeomorphic to a relatively compact set of the maximal dimension in a Euclidean space. Let $\lambda$ be the probability measure on ${{\mathcal T}}(n)$, such that its restrictions to the combinatorial type components are the corresponding Lebesgue measures. For instance, the space ${{\mathcal T}}(3)\subset{{\mathbb R}}^2$ is given by ${{\mathcal T}}(3)=\{(\alpha,\beta): 0< \alpha \le\beta <\pi/2\}$. Thus, ${{\mathcal T}}(3)$ itself is a plane triangle. By a theorem in [@KMS], the set ${{\mathcal E}}(n)\subset{{\mathcal T}}(n)$ of ergodic $n$-gons is residual in the sense of Baire category [@Oxtoby].\ [**Problem 7**]{}. Is $\lambda({{\mathcal E}}(n))>0$ ?\ The case of $n=3$ is especially interesting, since the mechanical system of three elastic point masses moving on a circle (see Figure \[fig5\]) leads to the billiard in an acute triangle [@Glashow; @Casati2]. Let $m_1,m_2,m_3$ be the masses. Then the angles of the corresponding triangle $\Delta(m_1,m_2,m_3)$ satisfy $$\label{angles-eq} \tan\alpha_i=m_i\sqrt{\frac{m_1+m_2+m_3}{m_1m_2m_3}}.$$ We point out that the rationality of the triangle corresponding to a mechanical system of point masses does not have any obvious physical meaning. In the limit, when $m_3\rightarrow\infty$, we obtain the physical system of two elastic particles on an interval. The limit of $\Delta(m_1,m_2,m_3)$ is the right triangle whose angles satisfy $\tan\alpha_1=\sqrt{m_1/m_2}, \tan\alpha_2=\sqrt{m_2/m_1}$. Let $P$ be an irrational polygon. Let ${\alpha}_1,\dots,{\alpha}_k$ be its angles. If the numbers ${\alpha}_i/\pi$ simultaneously admit a certain super-exponentially fast rational approximation, then $P$ is ergodic [@Vorob]. This remarkable theorem yields explicit examples of ergodic polygons. However, it does not help with the above problem. There is some numerical evidence that irrational polygons are ergodic and have other stochastic properties [@Casati1; @Casati2]. So far, there are no theorems confirming or precluding this.\ [**Problem 8**]{}. Give an example of an irrational but nonergodic polygon.\ Let $P$ be an arbitrary $n$-gon, and let $a_1,\dots,a_n$ be its sides. For $1\le i \le n$ let $\Phi_i\subset\Phi$ be the set of elements whose base points belong to the side $a_i$. Then $\Phi=\cup_{i=1}^n\Phi_i$, a disjoint decomposition. By equation (\[volumes-eq\]), $\mu(\Phi_i)=2\,\mbox{Length}(a_i)$. The next question/conjecture concerns the structure of invariant sets in the phase space of a nonergodic polygon. (Compare with Problem 4 in § \[hyperbolic\].) By [@KMS], the conjecture holds for rational polygons.\ [**Problem/Conjecture 9**]{}. Let $P$ be an irrational $n$-gon, and let $M\subset\Phi$ be an invariant set of positive measure. If $\Phi_i\subset M$ for some $1\le i \le n$, then $M=\Phi$.\ The subject of periodic billiard orbits in polygons requires only elementary euclidean geometry, and has immediate applications to physics. For instance, let $\Delta(m_1,m_2,m_3)$ be the acute triangle corresponding to the system of three elastic point masses equation (\[angles-eq\]). Periodic billiard orbits in $\Delta(m_1,m_2,m_3)$ correspond to the periodic motions of this mechanical system. [^9] Ironically, periodic orbits in polygons turned out to be especially elusive.\ [**Problem 10**]{}. Does every polygon have a periodic orbit?\ Every rational polygon has periodic orbits, and much is known about them. Certain classes of irrational polygons have periodic orbits [@Kolan; @GuTr]. Every acute triangle has a classical periodic orbit - the Fagnano orbit [@Gut4]. It corresponds to the inscribed triangle of minimal perimeter. It is not known if every acute triangle has other periodic orbits; it is also not known if every obtuse triangle has a periodic orbit [@GaStVor; @Hunger]. See §  \[update\] for updates and elaborations. A periodic orbit with an even number of segments is contained in a [*parallel band*]{} of periodic orbits of the same length. The boundary components of a band are concatenations of singular orbits, the so-called [*generalized diagonals*]{} [@Kat]. These are the billiard orbits with endpoints at the corners. Periodic orbits with an odd number of segments (e. g., the Fagnano orbit) are isolated. They seem to be rare; a rational polygon has at most a finite number of them. Denote by $f_P(\ell)$ the number of periodic bands of length at most $\ell$. This counting function for periodic billiard orbits in polygons grows subexponentially [@Kat; @GuHa95; @GuHa]. Conjecturally, there should be a universal polynomial upper bound on $f_P(\cdot)$. See section  \[update\] for elaborations.\ [**Problem 11**]{}. Find efficient upper and lower bounds on $f_P$ for irrational polygons.\ From now until the end of this section we consider only rational polygons. By results of Masur [@Mas1; @Mas2] and Boshernitzan [@Bosh1; @Bosh2], there exist numbers $0 < c_*(P) \le c^*(P) <\infty$ such that $c_*(P)\ell^2 \le f_P(\ell) \le c^*(P)\ell^2$ for sufficiently large $\ell$. We will refer to these inequalities as the [*quadratic bounds on periodic billiard orbits*]{}. The numbers $c_*(P),c^*(P)$ are the [*quadratic constants*]{}.\ [**Problem 12**]{}. Find efficient estimates for quadratic constants.\ In all known examples $f_P(\ell)/\ell^2$ has a limit, i. e., $c_*(P)=c^*(P)=c(P)$. In this case, we say that the [*polygon has quadratic asymptotics*]{}. The preceding definitions and questions have obvious counterparts for translation surfaces, where periodic billiard orbits are replaced by closed geodesics. There is a special class of polygons and surfaces: Those satisfying the [*lattice property*]{}, or, simply, the [*lattice polygons and translation surfaces*]{} [@Veech1; @Guj; @GuJ]. They have quadratic asymptotics, and there are general expressions for their quadratic constants [@Gut1; @Veech1; @Veech3; @Guj; @GuJ]. Besides, the quadratic constants for several lattice polygons are explicitly known [@Veech1; @Veech3]. These formulas contain rather subtle arithmetic. The (normalized) quadratic constant of any lattice polygon is an algebraic number [@GuJ]. There is a lot of information about lattice polygons and lattice translation surfaces. Lattice polygons seem to be very rare. All acute lattice triangles have been determined [@KenSmi; @Pu]. For obtuse triangles the question is still open. There is a ${\text{SL}(2,{{\mathbb R}})}$-invariant Lebesgue-class finite measure on the [*strata*]{} of the [*moduli space*]{} ${{\mathcal M}}_g$ of translation surface of fixed genus [@EskMas]. The strata correspond to the partitions $2g-2=p_1+\cdots+p_t:p_i\in{{\mathbb N}}$. Hence, we can speak of a [*generic translation surface*]{} in ${{\mathcal M}}_g(p_1,\dots,p_t)$. The quadratic constant of the generic translation surface $S\in{{\mathcal M}}_g(p_1,\dots,p_t)$ depends only on the stratum [@EskMas; @EskOk; @EskMasZor]. These results crucially use the relationship between the [*Teichmüller flow*]{} [@ImaTani] on ${{\mathcal M}}_g$ and linear flows on translation surfaces. See the surveys [@MasTab; @DeM; @Avi] for this material. The results for generic translation surfaces have no consequences for rational polygons, since the generic translation surface does not correspond to a polygon. However, an extension of the Teichmüller flow approach establishes the quadratic asymptotics for a special, but nontrivial class of rational polygons [@EskMasSchm].\ [**Problem 13**]{}. Does every rational polygon have quadratic asymptotics?\ The following section contains updates, elaborations, and extensions of the preceding material. Comments, updates, and extensions {#update} ================================= At the time of writing this text, the problems discussed in the survey [@Gut5] remain open. However, the works that have since appeared contain substantial relevant material. The main purpose of this section is to comment on this material. At the same time, we take the opportunity to add a few extensions and ramifications that for some reasons did not appear in the survey [@Gut5]. Accordingly, we have updated and expanded the bibliography. A few recent books contain discussions of the billiard ball problem. The book [@Chernov4] is a thorough exposition of the hyperbolic billiard dynamics. See § \[hyperbolic\]. The treatise [@Berger1] contains several discussions of connections between the billiard ball problem and geometry. The material in [@Berger1] is relevant for all three types of billiard tables discussed above. The latter can be also said about [@Tabach1]. However, the two books are fundamentally different. The book [@Berger1] is a comprehensive treatise on geometry, where the billiard ball problem is but one of a multitude of illustrative examples and applications, while [@Tabach1] is addressed primarily to young American students, and discusses a few instances pertaining to the geometric aspect of the billiard. The book [@Gruber] exposes several applications of convex geometry to the billiard ball problem. See the material in § \[elliptic\]. The Birkhoff conjecture and related material {#birkhoff_sub} -------------------------------------------- The traditional formulation of the Birkhoff integrability conjecture is in terms of the billiard map. See Definition  \[integrable-def\] and Problem $1$. The billiard flow on a smooth, convex table $Y$ is a Hamiltonian system with $2$ degrees of freedom. The Hamiltonian version of Problem $1$ is as follows: [**Problem 1a**]{}. Let $Y$ be a smooth, convex billiard table. If the billiard flow on $Y$ is an integrable Hamiltonian system, then $Y$ is an ellipse.\ Although Problem 1 and Problem 1a are obviously related, a resolution of either one of them would not directly imply a resolution of the other. The results of S. Bolotin [@Bolotin1] provide evidence supporting a positive resolution of Problem 1a. Let $\gamma\subset{{\mathbb R}^2}$ be a closed convex curve. Denote by $|\gamma|$ its perimeter. The [*string construction*]{} associates with any $\ell>|\gamma|$ a closed convex curve $G(\gamma,\ell)$ containing $\gamma$ in its interior. The curve $G(\gamma,\ell)$ is obtained by the following “physical process”. We take a ring of length $\ell$ made from a soft, non-stretchable material and wrap it around $\gamma$. We pull the ring tight with a pencil; then, holding it tight, we rotate the pencil all the way around $\gamma$. The moving pencil will then trace the curve $G(\gamma,\ell)$. A gardener could use this process to design fences around his flower beds. For this reason, the procedure is sometimes called the [*gardener construction*]{} [@Berger1]. Let $Y=Y(\gamma,\ell)\subset{{\mathbb R}^2}$ be the billiard table whose boundary is $G(\gamma,\ell)$. Then $\gamma$ is a caustic for the billiard on $Y$ [@Laz; @GutKat; @Tabach1]. If $\gamma_1$ is an ellipse and $\ell_1>|\gamma_1|$, then $\gamma_2=G(\gamma_1,\ell_1)$ is a confocal ellipse. Let now $\ell_2>|\gamma_2|$. Then $\gamma_3=G(\gamma_2,\ell_2)$ is a confocal ellipse containing $\gamma_1$. There is a unique $\ell_3$ such that $\gamma_3=G(\gamma_1,\ell_3)$. This transitivity property of gardener’s construction is a consequence of the integrability of billiard on an ellipse. The Birkhoff conjecture suggests the following problem. [**Problem 1b**]{}. Ellipses are the only closed convex curves satisfying the above transitivity.\ This entirely geometric variant of the Birkhoff conjecture is due to R. Melrose. A positive solution of Problem 1b is the subject of the PhD thesis of Melrose’s student E. Amiran [@Amiran1]. However, the work [@Amiran1] contains a serious gap, and the question remains open. Let now $\gamma$ be any closed convex curve. For $\ell>|\gamma|$ let $Y(\ell)=Y(\gamma,\ell)$ be the corresponding family of convex billiard tables, and let $0<\rho_{\gamma}(\ell)<1/2$ be the rotation number of the caustic $\gamma\subset Y(\ell)$. The [*rotation function*]{} $\rho(\ell)=\rho_{\gamma}(\ell)$ is continuous and monotonically increasing, but not strictly, in general. Let $r\in(0,1/2)$ be such that $\rho^{-1}(r)=[a(r),b(r)]$ is a nontrivial interval. Then i) $r$ is rational; ii) for $\ell\in[a(r),b(r)]$ the billiard map of $Y(\ell)$ restricted to the caustic $\gamma$ is not a rotation. The converse also holds; $[a(r),b(r)]$ are the [*phase locking intervals*]{}. The above situation is a special case of the dynamical phenomenon called [*phase locking*]{}. It is characteristic for one-parameter deformations in elliptic dynamics. See [@GutKni] for a study and a detailed discussion of this phase locking when $\gamma$ is a triangle. Let now $\gamma$ be a convex polygon. Then the $C^1$ curve $G(\gamma,\ell)$ is a concatenation of arcs of ellipses with foci at the corners of $\gamma$. At the points of transition between these elliptic arcs, typically, only one of the two foci changes, causing a jump in the curvature. Thus, a typical billiard table, say $Y$, obtained by this construction, is strictly convex, piecewise analytic, but not $C^2$. The boundary ${\partial}Y$ contains a finite number of points where the curvature jumps. A. Hubacher studied billiard tables of this class [@Hubacher]. She proved that there is an open neighborhood $\Omega\subset Y$ of ${\partial}Y$ such that any caustic in $Y$ belongs to the complement of $\Omega$ [@Hubacher]. Recall that $\gamma$ is a caustic of $Y(\gamma,\ell)$ for any $\ell>|\gamma|$. There is an analogy between Hubacher’s theorem and a result in [@GutKat] which says that billiard caustics stay away from the table’s boundary if it contains points of very small curvature. This result is a quantitative version of Mather’s theorem [@Mather1] that insures nonexistence of caustics if ${\partial}Y$ has points of zero curvature. Hubacher’s theorem replaces them with jump points of the curvature. As opposed to [@GutKat], the work [@Hubacher] does not estimate the size of the [*region $\Omega\subset Y$ free of caustics*]{}. It is plausible that for the typical table $Y({\gamma},\ell)$ the free of caustics region $\Omega=\Omega({\gamma},\ell)$ is the annulus between ${\partial}Y(\gamma,\ell)$ and $\gamma$. There are polygons ${\gamma}$ such that for special values of the string length $\ell$ the boundary ${\partial}Y(\gamma,\ell)$ is a $C^2$ curve. Let ${\gamma}$ be the regular hexagon with the unit side length. Then ${\partial}Y(\gamma,14)$ is a $C^2$ curve. The work of H. Fetter [@Fet2012] studies the billiard on $Y(\gamma,14)$. Fetter suggests that the billiard on $Y(\gamma,14)$ is integrable, and thus $Y(\gamma,14)$ is a counterexample to the Birkhoff conjecture. However, the evidence of integrability of $Y(\gamma,14)$ presented in [@Fet2012] is mostly numerical. The present author believes that the further investigation of the billiard on $Y(\gamma,14)$ will confirm the Birkhoff conjecture. The subject of [@KalSor] is a billiard version of the famous question of Marc Kac: [*“Can one hear the shape of a drum?*]{}” [@Kac]. As an application of their results, the authors prove a [*conditional version*]{} of the Birkhoff conjecture. The Ivrii conjecture and related material {#ivrii_sub} ----------------------------------------- Investigations of Problem 3i) (i.e., the Ivrii conjecture) and Problem 3ii) (i.e., the claims $I_n$ for $n>2$), as well as related questions, remain active. The work [@Baryshnikov1] develops a functional theoretic approach to study billiard caustics. As a byproduct, [@Baryshnikov1] contains yet another proof of Rychlik’s theorem stating that the set of $3$-periodic billiard orbits has measure zero. The preprint [@Gut9] announced a solution of the Ivrii conjecture. Unfortunately, the work [@Gut9] contains a mistake; thus the conjecture remains open. The paper [@GluKudr] announced a proof of the claim $I_4$: The set of $4$-periodic billiard orbits has measure zero. The work [@GluKudr] states several propositions implying the claim, and explains the strategy of their proofs. The approach of [@GluKudr] is based on a study of certain foliations, on one hand, and a very detailed analysis of singularities of certain mappings, on the other hand. Complete proofs should appear shortly. There is a curious connection between the Ivrii conjecture and the subject of [*invisibility*]{} [@PlakRosh]. It is natural to investigate the counterparts of the Ivrii conjecture for the billiard on (simply connected) surfaces of constant curvature. The billiard on ${{\mathbb R}^2}$ corresponds to the zero curvature, $\kappa=0$. Multiplying a constant curvature by a positive factor does not qualitatively change the geometry; thus, it suffices to consider the two cases $\kappa=\pm 1$. The surfaces in question are the hyperbolic plane, $\kappa=-1$, and the round unit sphere, $\kappa=1$. Let us denote them by ${{\mathbb H}}^2$ and ${{\mathbb S}}^2$ respectively. On ${{\mathbb S}}^2$ the immediate analog of the Ivrii conjecture fails. The paper [@GT] contains an example of a (not strictly) smooth, convex billiard table in ${{\mathbb S}}^2$ with an open set of periodic orbits. This observation shows the subtlety of the Ivrii conjecture for ${{\mathbb R}^2}$. The work [@BlKiNaZh] contains a detailed study of $3$-periodic orbits for billiard tables in ${{\mathbb H}}^2$ and ${{\mathbb S}}^2$. It shows, in particular, that the set of $3$-periodic billiard orbits on a Birkhoff billiard table in ${{\mathbb H}}^2$ has measure zero. This is the counterpart of Rychlik’s theorem for the hyperbolic plane. Extensions of the material in § \[elliptic\] {#elliptic_sub} -------------------------------------------- Nontrivial billiard properties can be roughly divided into three categories: i) Those that hold for all tables; ii) Those that hold for the typical table; iii) Those that hold for special billiard tables. Studies in category iii) can be described as follows: Let P be a property satisfied by a very particular billiard table, e. g., the round disc. Are there non-round tables that have property P? If the answer is “yes”, then describe the billiard tables having property P. The following example illustrates the situation. Let $0<\alpha\le\pi/2$ be an angle. Let $Y\subset{{\mathbb R}^2}$ be a Birkhoff billiard table. We say that the table $Y$ has property $P_{\alpha}$ if every chord in $Y$ that makes angle $\alpha$ with $\partial Y$ at one end also makes angle $\alpha$ with $\partial Y$ at the other end. The round table has property $P_{\alpha}$ for any $\alpha$. A billiard table with property $P_{\alpha}$ has a very special caustic $\Gamma_{\alpha}$; we will say that $\Gamma_{\alpha}$ is a [*constant angle caustic*]{}. Let $\rho({\theta}),0\le{\theta}\le 2\pi,$ be the radius of curvature for $\partial Y$. Tables with the caustic $\Gamma_{\pi/2}$ are well known to geometers: Their boundaries are the [*curves of constant width*]{} [@Berger1; @Gruber]. A curve $\partial Y$ has constant width if an only if its radius of curvature satisfies the identity $$\rho({\theta})+\rho({\theta}+\pi)={\mbox{const}}.$$ In particular, there are non-round infinitely smooth, and even real analytic billiard tables in this class. For $0<\alpha<\pi/2$ let ${{\mathcal P}}_{\alpha}$ be the class of non-round tables with the property $P_{\alpha}$. The author has investigated the class ${{\mathcal P}}_{\alpha}$ about 20 years ago and reported the results at the 1993 Pennsylvania State University Workshop on Dynamics [@Gut+]. The main results in [@Gut+] are as follows: The class ${{\mathcal P}}_{\alpha}$ is nonempty if and only if $\alpha$ satisfies $$\label{alfa-eq} \tan(n\alpha)=n\tan\alpha$$ for some $n>1$. The set $A_n\subset(0,\pi/2)$ of solutions of equation   is finite and nonempty for $n\ge 4$. For every $\alpha\in A_n$ there is an analytic family of (nonround) distinct, convex, real analytic tables $Y_{\alpha}(s)\in{{\mathcal P}}_{\alpha}:0<s<1$. As $s\to 0$, the tables $Y_{\alpha}(s)$ converge to the unit disc. The limit $Y_{\alpha}(1)$ of $Y_{\alpha}(s)$, as $s\to 1$, also exists, but has corners. It turns out that planar regions satisfying property $P_{\alpha}$ for some angle $\alpha$ are of interest in the mathematical fluid mechanics. Besides the concept of Archimedean floating,[^10] there is a concept of capillary [*floating in neutral equilibrium at a particular contact angle*]{}. This concept goes back to Thomas Young [@Young] and was further developed and investigated by R. Finn [@Finn1]. If $Y\in{{\mathcal P}}_{\alpha}$, then the infinite cylinder $C=Y\times{{\mathbb R}}$ floats in neutral equilibrium at the contact angle $\pi-\alpha$ [*at every orientation*]{}. The work [@Gut7] is a detailed exposition of the results in [@Gut+] aimed, in particular, at the mathematical fluid mechanics readership. See [@Tabach3; @Cyr] for related investigations and [@Gut12] for additional comments. Comments and updates for the material in § \[parabolic\] {#parabolic_sub} -------------------------------------------------------- Although Problem 10 remains open, recent publications [@Schw06; @Schw09; @HooSchw] provided substantial evidence toward the positive answer, i.e., that every polygon does have a periodic orbit. These papers investigate periodic billiard orbits in obtuse triangles. Let $\Delta(\alpha,\beta,{\gamma})$ be the triangle with the angles $\alpha\le\beta\le{\gamma}$. In [@Schw06; @Schw09] R. Schwartz proves that if $\gamma$ is less than or equal $100$ degrees, then $\Delta(\alpha,\beta,{\gamma})$ has a periodic orbit. In [@HooSchw] Hooper and Schwartz prove that if the angles $\alpha$ and $\beta$ are sufficiently close, then $\Delta(\alpha,\beta,{\gamma})$ has a periodic orbit. If $\alpha=\beta$, then $\Delta$ is an isosceles triangle. It is well known and elementary that isosceles triangles have periodic orbits. The main theorem in [@HooSchw] says that any triangle which is sufficiently close to an isosceles one, has periodic orbit. Note that in the Hooper-Schwartz theorem the angles $\alpha$ and $\beta$ can be arbitrarily small, thus $\gamma$ can be arbitrarily close to 180 degrees. Let $\Delta$ be a triangle, and let $a,b,c$ be the sides of $\Delta$. The works [@Schw06; @Schw09; @HooSchw] build on the approach of [@GaStVor], where periodic billiard orbits were coded by words on the alphabet $\{a,b,c\}$. The paper [@GaStVor] studied relationships between periodic orbits and the associated words. A periodic billiard orbit is [*stable*]{} if it persists under all sufficiently small deformations of $\Delta$. By [@GaStVor], the stability of an orbit is equivalent to a combinatorial property of the associated word. For the sake of brevity, I will simply say that the words on the alphabet $\{a,b,c\}$ satisfying this property are [*stable*]{}. Let ${{\mathcal T}}\subset{{\mathbb R}^2}$ be the moduli space of triangles. Slightly simplifying the situation, we assume that ${{\mathcal T}}=\{(x,y)\in{{\mathbb R}^2}:0\le x,y\le 1\}$. The subsets of obtuse (resp. isosceles) triangles are given by $x+y<1$ (resp. $x=y$). Let $W_k$ be the set of stable words of length $k$, and let $W=\cup_kW_k$. For $w\in W$ let ${{\mathcal T}}_w\subset{{\mathcal T}}$ be the open set of triangles having a periodic orbit with the code $w$. These are the [*tiles*]{} in the terminology of [@HooSchw]. The approach of Hooper and Schwartz is to exhibit a [*sufficient set*]{} $W_{suff}\subset W$ so that the tiles $\{{{\mathcal T}}_w:w\in W_{suff}\}$ cover the targeted part of ${{\mathcal T}}$. It goes without saying that this idea cannot be implemented without substantial computer power. The computer program “MacBilliards” created by Hooper and Schwartz does the job. Besides providing us with ample evidence towards the conjecture that every polygon has periodic billiard orbits, the works [@Schw06; @Schw09; @HooSchw] have established several facts that show just how intricate the matter is. In some cases, every finite set is insufficient; then [@HooSchw] sufficient infinite sets. Unfortunately, the survey [@Gut5] has omitted the subject of [*complexity of billiard orbits*]{} in polygons, which is very close to Problem 11. We will briefly discuss it below. Let $P$ be a polygon, and let ${{\mathcal A}}=\{a,b,c,\ldots\}$ be the set of its sides. Following a finite billiard orbit $\gamma$ and recording the sides that it encounters, we obtain a word, $w(\gamma)$, on the alphabet ${{\mathcal A}}$. We say that $w(\gamma)$ is the [*code*]{} of $\gamma$; the number of letters in $w(\gamma)$ is the [*combinatorial length*]{} of $\gamma$. Let $W_n(P)$ be the set of codes of all billiard orbits with combinatorial length $n$. The function $F(n)=|W_n(P)|$ is the [*full complexity*]{} of the billiard on $P$. Imposing various restrictions on the billiard orbits, we obtain conditional or [*partial complexities*]{} $F_*(n)$. For instance, the function $f_P(n)$ in Problem 11 is the [*periodic complexity*]{} for the billiard on $P$. Thus, Problem 11 is a special case of the following open question.\ [**Problem 14**]{}. Find nontrivial, explicit bounds on partial billiard complexities for the [*general polygon*]{}.[^11]\ By [@Kat; @GuHa95; @GuHa], the full complexity of the billiard in any polygon is subexponential. This means that for $n$ sufficiently large $F(n)<e^{an}$ for any $a>0$. Note that this result provides no subexponential bound on $F(n)$. Very few nontrivial bounds on billiard complexities are known. Let $f_P(n)$ be the number of codes for periodic orbits in $P$ of length at most $n$. For any $k\in{{\mathbb N}}$ Hooper constructed open sets ${{\mathcal X}}_k$ in the moduli space of polygons such that for $P\in{{\mathcal X}}_k$ the function $f_P(n)$ grows faster than $n\log^kn$ [@Hoop]. Let $g_P(n)$ be the number of generalized diagonals in $P$ of length at most $n$. Let ${{\mathcal T}}_3$ be the moduli space of triangles, endowed with the Lebesgue measure. Scheglov [@Sche2012] showed that for almost every $P\in{{\mathcal T}}_3$ and any ${\varepsilon}>0$ the inequality $g_p(n)<{\mbox{const}}\exp(n^{\sqrt{3}-1+{\varepsilon}})$ holds. Note that Scheglov’s result yields explicit subexponential upper bounds on the full and the periodic complexities for almost every triangle. To make Problem 14 more concrete, we state below a widely accepted conjecture. [**Conjecture 1**]{}. There is $d\ge 3$ such that the full billiard complexity for any polygon has a cubic lower bound and a degree $d$ upper bound.\ The claim is established only for rational polygons, with $d=3$. This is a consequence of the results of Masur [@Mas1; @Mas2] about saddle connections on translation surfaces. We will now briefly discuss recent results on partial complexities that provide support for it [@GutRam]. Let $P$ be an arbitrary polygon. Let $0\le\theta< 2\pi$ be a direction, and let $z\in P$ be a point. Coding those billiard orbits that start off in direction $\theta$ (resp. from the point $z$) and counting the number of words of length less than or equal to $n$, we obtain the [*directional complexity*]{} $F_{\theta}(n)$ (resp. [*position complexity*]{} $F_z(n)$) for the billiard on $P$. It was known that the directional complexity grows polynomially, i.e., there is $d>0$ depending only on $P$ such that $F_{\theta}(n)=O(n^d)$ for all directions $\theta$ [@GuTr]. By [@GutRam], for any $P$ and any ${\varepsilon}>0$ for almost all directions $\theta$, we have $F_{\theta}(n)=O(n^{1+{\varepsilon}})$. Another result in [@GutRam] says that for any $P$ and any ${\varepsilon}>0$, for almost all points $z\in P$ we have $F_z(n)=O(n^{2+{\varepsilon}})$.[^12] The proofs are based on the relationships between the average complexity and individual complexities. The concept of a [*piecewise convex polygon exchange*]{} introduced in [@GT1] yields a new approach to the billiard complexities. This approach works for the polygonal billiard on surfaces of arbitrary constant curvature. Ramifications and extensions of the polygonal billiard {#noncomp_sub} ------------------------------------------------------ In this section we briefly report on several generalizations of the subject in section § \[parabolic\] that did not get discussed in [@Gut5]. The main direction in these developments is to replace the usual polygons by noncompact or infinite polygons. The work in this direction started already at the end of the last century [@DegDelLen98; @DegDelLen2000], and flourished after the turn of the century. The noncompact polygons in [@DegDelLen98; @DegDelLen2000] are semi-infinite stairways. A stairway in these works is a noncompact polygon, say $P$, with infinitely many vertical and horizontal sides, and of finite area. In particular, $P$ is a [*rational noncompact polygon*]{}, and we have the obvious family of directional billiard flows $b_{{\theta}}^t,0\le{\theta}\le\pi/2,$ on $P$. In addition to studying the ergodicity of these flows, the papers [@DegDelLen98; @DegDelLen2000] investigate [*escaping orbits*]{} in $P$, a new phenomenon caused by the noncompactness of $P$. There are many kinds of noncompact polygons, both rational and irrational, bounded or unbounded, with finite or infinite area. More generally, there are [*noncompact polygonal surfaces*]{} [@Gut09], and in particular, [*noncompact translation surfaces*]{} [@Hoop08; @Gut09]. In this [*brave new world*]{}, there are infinite (as opposed to semi-infinite) stairways [@HooHubWei], infinite coverings of compact polygonal surfaces [@Gut09; @FrUl11; @CoGu], and even fractals, e.g., the Koch snowflake [@LapNie]. Besides being of interest on its own, the billiard on noncompact polygons comes up in physics and engineering [@GuKac; @Gut85; @Gut87; @DanGu; @BaKhMaPl]. All of the questions concerning the dynamics and geometry for compact polygons, translation surfaces, etc, have obvious counterparts in the noncompact world.[^13] The answers to these questions are sometimes unexpected [@FrUl11; @CoGu; @Trevino], not at all analogous to the answers in the compact world. Besides, there are problems in the noncompact world that do not arise in the classical setting. One of them is the conservativity of billiard dynamics. In the classical setting the conservativity is ensured by the Poincaré recurrence theorem. We conclude this necessarily incomplete survey of noncompact polygonal billiard with a brief discussion of the billiard for a classical family of noncompact, doubly periodic polygons. Let $0<a, b <1$, and let $R(a,b)$ be the $a\times b$ rectangle. Denote by $R_{(0,0)}$ the upright rectangle $R(a,b)$ centered at $(1/2,1/2)$. For $(m,n)\in{{\mathbb Z}^2}$ set $R_{(m,n)}=R_{(0,0)}+(m,n)$. The region ${\tilde{P}}(a,b)={{\mathbb R}^2}\setminus\left(\cup_{(m,n)\in{{\mathbb Z}^2}}R_{(m,n)}\right)$ is a noncompact rational polygon of infinite area. We will refer to it as the [*rectangular Lorenz gas*]{}. The randomized version of ${\tilde{P}}(a,b)$ is the famous [*wind-tree model*]{} of statistical physics [@Eh]. For $0\le{\theta}<\pi/2$ denote by $b_{{\theta}}^t$ the directional billiard flows on ${\tilde{P}}(a,b)$. The work [@CoGu] works out the ergodic decomposition of $b_{{\theta}}^t$ on ${\tilde{P}}(a,b)$ for particular directions ${\theta}$ and sufficiently small $a$ and $b$, provided that $a/b$ be irrational. The directions in question are ${\theta}=\arctan(q/p)$ corresponding to $(p,q)\in{{\mathbb N}^2}$ such that the flow line of $b_{{\theta}}^t$ emanating from a corner of $R_{(0,0)}$ reaches the homologous corner of $R_{(p,q)}$ bypassing the obstacles. Although these directions form a finite set, it is asymptotically dense as $a,b$ go to zero. The dissipative component of $b_{{\theta}}^t$ is spanned by the straight lines[^14] avoiding the obstacles. The conservative part of $b_{{\theta}}^t$ decomposes as a direct sum of $2pq$ identical ergodic flows. The decomposition is as follows. There is a certain subgroup $H_{(p,q)}\subset{{\mathbb Z}^2}$ of index $2pq$. Let $S_1,\dots,S_{2pq}\subset{{\mathbb Z}^2}$ be its cosets. The ergodic component of $b_{{\theta}}^t$ corresponding to the coset $S_i,1\le i\le 2pq,$ is spanned by the billiard orbits that encounter the obstacles $R_{(m,n)}$, where $(m,n)\in S_i$. Thus, the ergodic decomposition of the flow in the $(p,q)$ direction is induced by a natural partition of the set of obstacles in the configuration space. For instance, the two ergodic components of the flow in the direction $\pi/4$ correspond to the billiard orbits on ${\tilde{P}}(a,b)$ that encounter the obstacles $R_{(m,n)}$ with even and odd $m+n$ respectively. These results follow from the ergodicity of certain ${{\mathbb Z}^2}$-valued cocycles over irrational rotations [@CoGu] established by the classical methods of ergodic theory for infinite invariant measures [@Ar; @Co; @Schm]. The explicit ergodic decomposition of the (conservative part of) the directional flows yields nontrivial consequences for the recurrence and the spatial distribution of typical billiard orbits [@CoGu]. The recurrence in the direction $\pi/4$ has been discussed in the physics literature [@HW80]. Judging by the results in [@CoGu; @FrUl11] etc, the noncompact polygonal billiard may yield further surprises. Security for billiard tables and related questions {#security_sub} -------------------------------------------------- This subject arose quite recently. It has to do with the geometry of billiard orbits as curves on the configuration space. Let ${\Omega}\subset{{\mathbb R}^2}$ be any billiard table. For any pair $x,y\in{\Omega}$ (in particular for pairs $x,x$) let ${\Gamma}(x,y)$ be the set of billiard orbits in ${\Omega}$ that connect $x$ and $y$. We say that the [*pair $x,y$ is secure*]{} if there exists a finite set $\{z_1,\ldots,z_n\}\subset{\Omega}\setminus\{x,y\}$ such that every ${\gamma}\in{\Gamma}(x,y)$ passes through some $z_i$. We say that $z_1,\ldots,z_n$ are [*blocking points*]{} for $x,y$. We call ${\Omega}$ secure if every pair of points is secure. Thus, to show that ${\Omega}$ is [*insecure*]{} means to find a pair $x,y$ that cannot be blocked with a finite set of blocking points. The questions that arise are: i) What tables ${\Omega}$ are secure?; ii) If ${\Omega}$ is insecure, how insecure is it? For instance, is it true that every pair $x,y\in{\Omega}$ is insecure, that almost all pairs $x,y\in{\Omega}$ are insecure, etc. The subject, in disguise of [*problems about bodyguards*]{} came up in the Mathematical Olympiad literature. The recent interest in security got triggered by [@HS]. The authors, who were then students at Cambridge University, studied the security of polygons. The main claim in [@HS] is that every rational polygon is secure. Regular $n$-gons provide a counterexample to the claim: A regular $n$-gon is secure if and only if $n=3,4,6$ [@Gut6]. See [@Gut7] for related results. Although the statement is elementary, the proof is not. The claim follows from a study of security in translation surfaces, and is based on [@Gut1; @Veech1; @Veech3; @Guj; @GuJ; @GuHuSc]. However, the general study of security for polygons has just begun. [**Problem 15**]{}. To characterize secure polygons. In particular, establish a criterion of security for rational polygons.\ Triangles with the angles of $30,60,90$ and $45,45,90$ degrees, as well as the equilateral triangle and the rectangles are the only polygons whose translation surfaces are flat tori, and hence their billiard flows are integrable [@Gut1]. Following [@Gut1], we will call them [*integrable polygons*]{}. A polygon $P$ is [*almost integrable*]{} if it is tiled under reflections by one of the integrable polygons [@GutKat1]. By [@Gut6], every almost integrable polygon is secure.\ [**Conjecture 2**]{}. A polygon is secure if and only if it is almost integrable.\ Almost integrable polygons are certainly rational. Conjecture 2 would imply, in particular, that every irrational polygon is insecure. At present, the problem of insecurity for general polygons seems hopeless. However, the security framework makes sense for arbitrary billiard tables, and more generally, for arbitrary riemannian manifolds (with boundary, corners, and singularities, in general). Security of riemannian manifolds is related to the growth of the number of connecting geodesics, and hence to their [*topological entropy*]{} [@BuGut; @Gut8]. Not much is known about the security in non-polygonal billiard tables ${\Omega}$. Let ${\Omega}$ be a Birkhoff billiard table. Approximating ${\partial}{\Omega}$ locally by the arcs of its osculating circles, it is intuitively clear that pairs of sufficiently close points $x,y\in{\partial}{\Omega}$ are insecure; the work [@Tabach2] confirms it. Much more is known about the security of compact, smooth Riemannian manifolds. Flat manifolds are secure [@Gut6; @BuGut]. For a flat torus, the proof is elementary [@Gut6]. For general flat manifolds, this follows from the Bieberbach theorem [@BuGut]. [**Conjecture 3**]{}. A compact, smooth riemannian manifolds is secure if and only if it is flat.\ Conjecture 3 has been established for various classes of compact Riemannian manifolds [@BuGut; @GutSch; @BaGu]. In particular, it holds for all compact surfaces of genus greater than zero [@BaGu]. Thus, among surfaces, it remains to prove it for arbitrary smooth Riemannian metrics on the two-sphere. 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Young, [*An essay on the cohesion of fluids*]{}, Philos. Trans. Roy. Soc. Lond. [**95**]{} (1805), 65 - 87. [^1]: There are also convex billiard tables that yield hyperbolic dynamics. See section  \[hyperbolic\]. [^2]: We are not aware of any uses of this observation in the billiard literature. [^3]: In the introduction to [@Porit], the author says that many years ago, when he was a doctoral student of Birkhoff, his advisor communicated the conjecture to him. [^4]: This is a seminal paper on the [*KAM theory*]{}. See also [@Moser2]. The name KAM stands for Kolmogorov, Arnold and Moser. See [@deLla] for an exposition. [^5]: A more general formula for the spectral asymptotics of the Laplacean, due to Safarov and Vassiliev, contains a term accounting for periodic orbits [@SaVa]. If periodic points yield a set of measure zero, this term vanishes. [^6]: See the Appendix by D. Szasz in [@Sas1]. [^7]: Unfortunately, there is a fair amount of confusing terminology in the literature. Mathematicians often use the expressions like “Sinai’s billiard" or “the Sinai table" or “a dispersive billiard" interchangeably. Physicists tend to mean by “the Sinai billiard" a special billiard table, although not necessarily that of Figure 4. [^8]: A polygon $P$ is simple if ${\partial}P$ is connected. [^9]: Another connection between billiards and physics arises in the study of mechanical linkages [@Sossinsky]. [^10]: In fact, this concept goes back to Aristotle. [^11]: The designation [*general polygon*]{} may be replaced by [*irrational polygon*]{}. [^12]: This establishes, in particular, the unpublished results of Boshernitzan [@Bosh2]. [^13]: Except for the billiard on a fractal table where these counterparts are not obvious [@LapNie]. [^14]: With the slopes $\pm q/p$.

--- abstract: 'We consider construction of Lagrangians which are candidates for $p$-adic sector of an adelic open scalar string. Such Lagrangians have their origin in Lagrangian for a single $p$-adic string and contain the Riemann zeta function with the d’Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach which takes into account all $p$-adic Lagrangians. The very attractive feature of this new Lagrangian is that it is an analytic function of the d’Alembertian. Investigation of the field theory with Riemann zeta function is interesting in itself as well.' author: - | Branko Dragovich[^1]\ \ *[Institute of Physics]{}\ *[Pregrevica 118, P.O. Box 57, 11001 Belgrade, Serbia]{}** date:   title: '**On $p$-Adic Sector of Adelic String**' --- Introduction ============ The first notion of a $p$-adic string was introduced by I. V. Volovich in 1987 [@volovich1]. After that, various versions of $p$-adic strings were developed. The most interest have attracted strings whose only world sheet is $p$-adic and all other properties are described by real or complex numbers. Such $p$-adic strings are connected with ordinary ones by product of their scattering amplitudes and notion of an adelic string has been considered. Adelic string enables to treat ordinary and $p$-adic strings simultaneously and on an equal footing. Adelic strings can be regarded as more fundamental than ordinary and $p$-adic ones (for a review of the early days developments, see e.g. [@freund1; @volovich2]). Some $p$-adic structures have been also observed in many other parts of modern mathematical physics (for a recent review we refer to [@dragovich]). One of the greatest achievements in $p$-adic string theory is an effective field description of open scalar $p$-adic string tachyon [@freund2; @frampton1]. The corresponding Lagrangian is nonlocal, nonlinear, simple and exact. It describes four-point scattering amplitudes as well as all higher ones at the tree-level. In the last decade the Lagrangian approach to $p$-adic string theory has been significantly advanced and many aspects of $p$-adic string dynamics have been investigated, compared with dynamics of ordinary strings and applied to nonlocal cosmology (see, e.g. [@sen; @zwiebach; @vladimirov; @arefeva; @barnaby] and references therein). Adelic approach to the string scattering amplitudes connects $p$-adic and ordinary counterparts, eliminates unwanted prime number parameter $p$ contained in $p$-adic amplitudes and cures the problem of $p$-adic causality violation. Adelic quantum mechanics [@dragovich2] was also successfully formulated, and it was found a connection between adelic vacuum state of the harmonic oscillator and the Riemann zeta function. There is also successful application of adelic analysis to Feynman path integral [@dragovich-b], quantum cosmology [@dragovich-c], summation of divergent series [@dragovich-d], and dynamical systems [@dragovich-e]. The present paper is a result of investigation towards construction of an effective field theory Lagrangian for $p$-adic sector of adelic open scalar string. At the beginning, we give a brief review of Lagrangian for $p$-adic string and also of our previous work on this subject. Then, we present a new Lagrangian, which also contains Riemann zeta function, but in such way that Lagrangian is now an analytic function of the d’Alembertian $\Box$. Note that $p$-adic sector of the four point adelic string amplitude contains the Riemann zeta function. $p$-Adic and Adelic Strings =========================== Let us recall the crossing symmetric Veneziano amplitude for scattering of two ordinary open strings: $$A_\infty (a, b) = g_\infty^2\, \int_{\mathbb{R}} |x|_{\infty}^{a-1} \, |1-x|_{\infty}^{b-1}\, d_\infty x = g_\infty^2 \frac{\zeta (1-a)}{\zeta (a)}\, \frac{\zeta (1-b)}{\zeta (b)}\, \frac{\zeta (1-c)}{\zeta (c)}\,, \label{1.1}$$ where $a = - \alpha (s) = - \frac{s}{2} - 1,\, b = - \alpha (t),\, c = - \alpha (u)$ with the condition $a + b + c = 1$, i.e. $s + t + u = - 8$. In (\[1.1\]), $\, |\cdot|_\infty$ denotes the ordinary absolute value, $\mathbb{R}$ is the field of real numbers, kinematic variables $a, b, c \in \mathbb{C}$, and $\zeta$ is the Riemann zeta function. The corresponding Veneziano amplitude for scattering of $p$-adic strings was introduced as $p$-adic analog of the integral in (\[1.1\]), i.e. $$A_p (a, b) = g_p^2\, \int_{\mathbb{Q}_p} |x|_p^{a-1} \, |1-x|_p^{b-1}\, d_p x \,, \label{1.2}$$ where $\mathbb{Q}_p$ is the field of $p$-adic numbers, $|\cdot |_p$ is $p$-adic absolute value and $d_p x$ is the additive Haar measure on $\mathbb{Q}_p$. In (\[1.2\]), kinematic variables $a, b, c$ maintain their complex values with condition $a + b + c = 1$. After integration in (\[1.2\]) one obtains $$A_p (a, b) = g_p^2\, \frac{1- p^{a-1}}{1- p^{-a}}\, \frac{1- p^{b-1}}{1- p^{-b}}\,\frac{1- p^{c-1}}{1- p^{-c}}\,, \label{1.3}$$ where $p$ is any prime number. Recall the definition of the Riemann zeta function $$\zeta (s) = \sum_{n= 1}^{+\infty} \frac{1}{n^{s}} = \prod_p \frac{1}{ 1 - p^{- s}}\,, \quad s = \sigma + i \tau \,, \quad \sigma >1\,, \label{1.4}$$ which has analytic continuation to the entire complex $s$ plane, excluding the point $s=1$, where it has a simple pole with residue 1. According to (\[1.4\]) one can take product of $p$-adic string amplitudes $$\prod_p A_p (a, b) = \frac{\zeta (a)}{\zeta (1-a)}\, \frac{\zeta (b)}{\zeta (1-b)}\, \frac{\zeta (c)}{\zeta (1-c)} \, \prod_p g_p^2\,, \label{1.5}$$ what gives a nice simple formula $$A_\infty (a, b) \, \prod_p A_p (a, b) = g_\infty^2 \, \prod_p g_p^2\,. \label{1.6}$$ To have infinite product of amplitudes (\[1.6\]) finite it must be finite product of coupling constants, i.e. $g_\infty^2 \, \prod_p g_p^2 = const.$ From (\[1.6\]) it follows that the ordinary Veneziano amplitude, which is rather complex, can be expressed as product of all inverse $p$-adic counterparts, which are much more simpler. Moreover, expression (\[1.6\]) gives rise to consider it as the amplitude for an adelic string, which is composed of the ordinary and $p$-adic ones. Lagrangian for a $p$-Adic Open String ------------------------------------- The exact tree-level Lagrangian of the effective scalar field $\varphi$, which describes the open $p$-adic string tachyon, is [@freund2; @frampton1] $${\cal L}_p = \frac{m^D}{g_p^2}\, \frac{p^2}{p-1} \Big[ -\frac{1}{2}\, \varphi \, p^{-\frac{\Box}{2 m^2}} \, \varphi + \frac{1}{p+1}\, \varphi^{p+1} \Big]\,, \label{2.1}$$ where $p$ is a prime, $\Box = - \partial_t^2 + \nabla^2$ is the $D$-dimensional d’Alembertian. An infinite number of spacetime derivatives follows from the expansion $$p^{-\frac{\Box}{2 m^2}} = \exp{\Big( - \frac{1}{2 m^2} \log{p}\, \Box \Big)} = \sum_{k = 0}^{+\infty} \, \Big(-\frac{\log p}{2 m^2} \Big)^k \, \frac{1}{k !}\, \Box^k \,.$$ The equation of motion for (\[2.1\]) is $$p^{-\frac{\Box}{2 m^2}}\, \varphi = \varphi^p \,, \label{2.4}$$ and its properties have been studied by many authors (see, [@vladimirov] and references therein). Lagrangians for $p$-Adic Sector =============================== Now we want to consider construction of Lagrangians which are candidates to describe entire $p$-adic sector of an adelic open scalar string. In particular, an appropriate such Lagrangian should describe scattering amplitude (\[1.5\]), which contains the Riemann zeta function. Consequently, this Lagrangian has to contain the Riemann zeta function with the d’Alembertian in its argument. Thus we have to look for possible constructions of a Lagrangian which contains the Riemann zeta function and has its origin in $p$-adic Lagrangian (\[2.1\]). We have found and considered two approaches: additive and multiplicative. Additive approach ----------------- Prime number $p$ in (\[2.1\]) can be replaced by any natural number $n \geq 2$ and consequences also make sense. Now we want to introduce a Lagrangian which incorporates all the above Lagrangians (\[2.1\]), with $p$ replaced by $n \in \mathbb{N}$. To this end, we take the sum of all Lagrangians ${\cal L}_n$ in the form $$L = \sum_{n = 1}^{+\infty} C_n\, {\cal L}_n = \sum_{n= 1}^{+\infty} C_n \frac{ m^D}{g_n^2}\frac{n^2}{n -1} \Big[ -\frac{1}{2}\, \phi \, n^{-\frac{\Box}{2 m^2}} \, \phi + \frac{1}{n + 1} \, \phi^{n+1} \Big]\,, \label{3.1}$$ whose explicit realization depends on particular choice of coefficients $C_n$ and coupling constants $g_n$. To avoid a divergence in $1/(n-1)$ when $n = 1$ one has to take that ${C_n}/{g_n^2}$ is proportional to $n -1$. Here we consider some cases when coefficients $C_n$ are proportional to $n-1$, while coupling constants $g_n$ do not depend on $n$, i.e. $ g_n = g$. In fact, according to (\[1.6\]), in this case $ g_n^2 = g^2 = 1$. Another possibility is that $C_n$ is not proportional to $n-1$, but $g_n^2 = \frac{n^2}{n^2 - 1}$ and then $\prod_p g_p^2 = \zeta (2) = \frac{\pi^2}{6}$, what is consistent with (\[1.6\]). To differ this new field from a particular $p$-adic one, we use notation $\phi$ instead of $\varphi$. We have considered three cases for coefficients $C_n$ in (\[3.1\]): (i) $C_n = \frac{n-1}{n^{2+h}}$, where $h$ is a real parameter; (ii) $C_n = \frac{n^2 -1}{n^2}$; and (iii) $C_n = \mu (n)\, \frac{n-1}{n^2}$, where $\mu (n)$ is the Möbius function. Case (i) was considered in [@dragovich3; @dragovich4]. Obtained Lagrangian is $$L_{h} = \frac{m^D}{g^2} \Big[ \,- \frac{1}{2}\, \phi \, \zeta\Big({\frac{\Box}{2 \, m^2} + h }\Big) \, \phi + {\cal{AC}} \sum_{n= 1}^{+\infty} \frac{n^{- h}}{n + 1} \, \phi^{n+1} \Big]\,, \label{3.2}$$ where $\mathcal{AC}$ denotes analytic continuation. Case (ii) was investigated in [@dragovich5] and the corresponding Lagrangian is $$L = \frac{m^D}{g^2} \Big[ \, - \frac{1}{2}\, \phi \, \Big\{ \zeta\Big({\frac{\Box}{2\, m^2} - 1}\Big)\, + \, \zeta\Big({\frac{\Box}{2\, m^2} }\Big) \Big\} \, \phi \, + \, \frac{\phi^2}{1 - \phi} \, \Big]\,. \label{3.3}$$ Case with the Möbius function $\mu (n)$ is presented in [@dragovich6] and the corresponding Lagrangian is $$L = \frac{m^D}{g^2} \Big[ - \frac{1}{2}\, \phi \, \frac{1}{ \zeta\Big({\frac{\Box}{2 m^2}}\Big)} \,\phi + \int_0^\phi {\cal M}(\phi) \, d\phi\Big] , \label{3.6}$$ where ${\cal M}(\phi) = \sum_{n= 1}^{+\infty} {\mu (n)} \, \phi^{n} = \phi - \phi^2 - \phi^3 - \phi^5 + \phi^6 - \phi^7 + \phi^{10} - \phi^{11} - \dots $. Multiplicative approach ----------------------- In the multiplicative approach the Riemann zeta function emerges through its product form (\[1.4\]). Our starting point is again $p$-adic Lagrangian (\[2.1\]). It is useful to rewrite (\[2.1\]), first in the form, $${\cal L}_p = \frac{m^D}{g_p^2}\, \frac{p^2}{p^2-1} \Big\{ -\frac{1}{2}\, \varphi \, \Big[ p^{-\frac{\Box}{2 m^2}+1} + p^{-\frac{\Box}{2 m^2}} \Big]\, \varphi + \, \varphi^{p+1} \Big\}\, \label{3.2.1}$$ and then, by addition and substraction of $\varphi^2$, as $${\cal L}_p = \frac{m^D}{g_p^2}\, \frac{p^2}{p^2-1} \Big\{ \frac{1}{2}\, \varphi \, \Big[ \Big(1 - p^{-\frac{\Box}{2 m^2}+1} \Big) + \Big( 1 - p^{-\frac{\Box}{2 m^2}}\Big) \Big]\, \varphi - \varphi^2 \Big( 1 - \varphi^{p-1} \Big) \Big\}\,. \label{3.2.2}$$ Taking products $$\prod_p g_p^2 = C \,, \, \prod_p \frac{1}{1 - p^{-2}}\,, \, \prod_p (1 - p^{-\frac{\Box}{2 m^2}+1}) \,, \, \prod_p (1 - p^{-\frac{\Box}{2 m^2}}) \,, \, \prod_p ( 1 - \varphi^{p-1}) \label{3.2.3}$$ in (\[3.2.2\]) at the relevant places one obtains Lagrangian $${\mathcal L} = \frac{m^D}{C}\, \zeta (2)\, \Big\{ \frac{1}{2} \, \phi \Big[ \zeta^{-1} \Big( \frac{\Box}{2 m^2} - 1 \Big) + \zeta^{-1} \Big( \frac{\Box}{2 m^2} \Big)\Big] \, \phi - \phi^2 \prod_p \Big( 1 - \phi^{p-1} \Big) \Big\} \,, \label{3.2.4}$$ where $\zeta^{-1} (s) = 1/\zeta (s)$. It is worth noting that from Lagrangian (\[3.2.4\]) one can easily reproduce its $p$-adic ingredient (\[3.2.1\]). Lagrangian (\[3.2.4\]) was introduced and considered in [@dragovich7]. In particular, it was shown that very similar Lagrangian can be obtained from the additive approach with the Möbius function and that these two Lagrangians describe the same field theory in the week field approximation. A new Lagrangian with Riemann zeta function ------------------------------------------- Here we present a new Lagrangian constructed by additive approach taking $C_n = (-1)^{n-1}\, \frac{n^2 -1}{n^2}$ in (\[3.1\]). This choice of coefficients $C_n$ is similar to the above case (ii) and distinction is in the sign $(-1)^{n-1}$. The starting $p$-adic Lagrangian is in the form (\[3.2.1\]) and it gives $$L = \sum_{n= 1}^{+\infty} C_n \frac{ m^D}{g_n^2}\frac{n^2}{n^2 -1} \Big[ -\frac{1}{2}\, \phi \, n^{-\frac{\Box}{2 m^2} +1} \, \phi -\frac{1}{2}\, \phi \, n^{-\frac{\Box}{2 m^2}} \, \phi + \, \phi^{n+1} \Big]\,. \label{3.3.1}$$ Recall that $$\sum_{n= 1}^{+\infty} (-1)^{n-1} \frac{1}{n^{s}} = (1 - 2^{1-s}) \, \zeta (s), \quad s = \sigma + i \tau \,, \quad \sigma > 0\,, \label{3.3.2}$$ which has analytic continuation to the entire complex $s$ plane without singularities. At point $s = 1$, one has $\lim_{s\to 1} (1 - 2^{1-s}) \, \zeta (s)\, = \, \sum_{n= 1}^{+\infty} (-1)^{n-1} \frac{1}{n} \, = \, \log 2$. Applying (\[3.3.2\]) to (\[3.3.1\]) and using analytic continuation one obtains $$L = - {m^D} \Big[ \, \frac{1}{2}\, \phi \, \Big\{ \, \Big(1 - 2^{2 - \frac{\Box}{2 m^2}}\Big)\, \zeta\Big({\frac{\Box}{2\, m^2} - 1}\Big)\, + \, \Big(1 - 2^{1 - \frac{\Box}{2 m^2}}\Big)\, \zeta\Big({\frac{\Box}{2\, m^2} }\Big) \Big\} \, \phi \, - \, \frac{\phi^2}{1 + \phi} \, \Big]\,, \label{3.3.3}$$ where it was taken $g_n^2 = g^2 = 1$. The corresponding equation of motion is $$\Big[ \, \Big(1 - 2^{2 - \frac{\Box}{2 m^2}}\Big)\, \zeta\Big({\frac{\Box}{2\, m^2} - 1}\Big)\, + \, \Big(1 - 2^{1 - \frac{\Box}{2 m^2}}\Big)\, \zeta\Big({\frac{\Box}{2\, m^2} }\Big) \Big] \, \phi \, = \, \frac{\phi^2 + 2 \phi}{(1 + \phi)^2} \,, \label{3.3.4}$$ which in the week field approximation gives equation $$\Big(1 - 2^{2 - \frac{M^2}{2 m^2}}\Big)\, \zeta\Big({\frac{M^2}{2\, m^2} - 1}\Big)\, + \, \Big(1 - 2^{1 - \frac{M^2}{2 m^2}}\Big)\, \zeta\Big(\frac{M^2}{2\, m^2} \Big) - 2 = 0 \, \label{3.3.5}$$ for the spectrum of masses $M^2$ as function of string mass $m^2$. Equation (\[3.3.4\]) has three $\phi=const.$ solutions, which are $\phi = 0,\, 1, \, -\frac{5}{3}$. The potential can be obtained by equality $V (\phi) = - L (\Box = 0)$, i.e. $$V (\phi) = m^D \, \frac{3 \,\phi - 5}{8 \,(1 + \phi)}\, \phi^2 \label{3.3.6}$$ which has two local minima at $\phi = 1$ and $\phi = - \frac{5}{3}$, and it has one local maximum $V(0) = 0$. These values of $\phi$ coincide with constant solutions of equation of motion (\[3.3.4\]). Potential (\[3.3.6\]) is singular at $\phi = -1$. Note that sign $(-1)^{p-1}$ in front of ${\cal L}_p$ in (\[3.3.1\]) is positive when $p$ is an odd prime and it has as a result that $V (\phi) \to + \infty$ when $\phi \to \pm \infty$. Concluding remarks ================== The main result of this paper is construction of the Lagrangian (\[3.3.3\]). Unlike previously constructed Lagrangians, this one has no singularity with respect to the d’Alembertian $\Box$ and it enables to apply easier pseudodifferential approach. This analyticity of the Lagrangian should be also useful in its application to nonlocal cosmology, which uses linearization procedure (see, e.g. [@koshelev] and references therein). It is worth mentioning that an interesting approach towards foundation of a field theory and cosmology based on the Riemann zeta function was proposed in [@volovich3]. 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--- abstract: | Let $\Lambda$ be a finite measure on the unit interval. A $\Lambda$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($\Lambda$-coalescent) in analogy to the duality known for the classical Fleming-Viot process and Kingman’s coalescent, where $\Lambda$ is the Dirac measure in $0$. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($\Xi$-coalescent) with mutation, the $\Xi$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $\Xi$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a population is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $\Xi$-Fleming-Viot processes naturally arise as limiting models. author: - | Matthias Birkner[^1] , Jochen Blath[^2] , Martin Möhle[^3] ,\ Matthias Steinrücken$^\dag$ and Johanna Tams$^\dag$ date: '27th of October, 2008' title: 'A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks' --- AMS subject classification. [*Primary:*]{} 60K35; 60G09; 92D10 [*Secondary:*]{} 60C05; 92D15 Keywords: coalescent, duality, Fleming-Viot process, measure-valued process, modified\ lookdown construction Running head: Lookdown construction for Fleming-Viot processes Introduction and main results {#sec:introduction} ============================= Motivation ---------- One of the fundamental aims of mathematical population genetics is the construction of population models in order to describe and to analyse certain phenomena which are of interest for biological applications. Usually these models are constructed such that they describe the evolution of the population under consideration forwards in time. A classical and widely used model of this kind is the Wright-Fisher diffusion, which can be used for large populations to approximate the evolution of the fraction of individuals carrying a particular allele. On the other hand it is often quite helpful to look from the present back into the past and to trace back the ancestry of a sample of $n$ individuals, genes or particles. In many situations, the Kingman coalescent [@K82a; @K82b] turns out to be an appropriate tool to approximate the ancestry of a sample taken from a large population. It is well known that the Wright-Fisher diffusion is dual to the block counting process of the Kingman coalescent [@D86; @M01]. More general, the Fleming-Viot process [@FV79], a measure-valued extension of the Wright-Fisher diffusion, is dual to the Kingman coalescent. Such and similar duality results are quite common in particular in the physics literature on interacting particle systems [@L85] and in the more theoretical literature on mathematical population genetics [@AH07; @AS05; @DK96; @DK99; @EK95; @H00; @M99; @M01]. Donnelly and Kurtz [@DK96] established a so-called lookdown construction and used this construction to show that the Fleming-Viot process is dual to the Kingman coalescent. This construction and corresponding duality results have been extended [@DK99; @BLG03; @BLG05; @BLG06] to the $\Lambda$-Fleming-Viot process, which is the measure-valued dual of a coalescent process allowing for multiple collisions of ancestral lineages. For more information on coalescent processes with multiple collisions, so-called $\Lambda$-coalescents, we refer to Pitman [@P99] and Sagitov [@S99]. There exists a broader class of coalescent processes [@MS01; @S00; @S03] in which many multiple collisions can occur with positive probability simultaneously at the same time. These processes can be characterized by a measure $\Xi$ on an infinite simplex and are hence called $\Xi$-coalescents. It is natural to further extend the above constructions and results to this full class of coalescent processes and, in particular, to provide constructions of the dual processes, called $\Xi$-Fleming-Viot processes. Although such extensions have been briefly indicated in [@DK99] and [@BLG03], these extensions have not been carried out in detail yet. $\Xi$-coalescents have also recently been applied to study population genetic problems, see [@TV08; @SW08]. The motivation to present this paper is hence manifold. We explicitly construct the $\Xi$-Fleming-Viot process and provide a representation via empirical measures of an exchangeable particle system in the spirit of Donnelly and Kurtz [@DK96; @DK99]. We furthermore establish corresponding convergence results and pathwise duality to the $\Xi$-coalescent. We also provide an alternative, more classical functional-analytic construction of the $\Xi$-Fleming-Viot process based on the Hille-Yosida theorem and present representations for the generator of the $\Xi$-Fleming-Viot process. Our approaches include neutral mutations. The results give insights into the pathwise structure of the $\Xi$-Fleming-Viot process and its dual $\Xi$-coalescent. Examples and situations are presented in which certain $\Xi$-Fleming-Viot processes and their dual $\Xi$-coalescents occur naturally. Moran models with (occasionally) large families {#GenMoranMod} ----------------------------------------------- Consider a population of fixed size $N\in{\mathbb N}:=\{1,2,\ldots\}$ and assume that each individual is of a certain type, where the space $E$ of possible types is assumed to be compact and Polish. Furthermore assume that for each vector ${\bf k}=(k_1,k_2,\ldots)$ of integers satisfying $k_1\ge k_2\ge\cdots\ge 0$ and $\sum_{i=1}^\infty k_i\le N$ a non-negative real quantity $r_N({\bf k})\ge 0$ is given. The population is assumed to evolve in continuous time as follows. Given a vector ${\bf k}=(k_1,\ldots,k_m,0,0,\ldots)$, where $k_1\ge \cdots\ge k_m\ge 1$ and $k_1+\cdots+k_m\le N$, with rate $r_N({\bf k})$ we choose randomly $m$ groups of sizes $k_1,\ldots,k_m$ from the present population. Inside each of these $m$ groups we furthermore choose randomly a ‘parent’ which forces all individuals in its group to change their type to the type of that parent. We say that a ${\bf k}$-reproduction event occurs with rate $r_N({\bf k})$. The classical Moran model corresponds to $r_N(2,0,0,\ldots)=N$. Except for the fact that these models are formulated in continuous time, they essentially coincide with the class of neutral exchangeable population models with non-overlapping generations introduced by Cannings [@C74; @C75]. Starting with the seminal work of Kingman [@K82a; @K82b], the genealogy of samples taken from such populations is well understood, in particular for the situation when the total population size $N$ tends to infinity. Genealogies and exchangeable coalescents {#ExCoal} ---------------------------------------- For neutral population models of large, but fixed population size and finite-variance reproduction mechanism, Kingman [@K82a] showed that the genealogy of a finite sample of size $n$ can be approximately described by the so called $n$-coalescent $(\Pi^{\delta_{\bf 0},(n)}_t)_{t\ge 0}$. The $n$-coalescent is a time-homogeneous Markov process taking values in $\mathcal{P}_n$, the set of partitions of $\{1,\ldots,n\}$. If $i$ and $j$ are in the same block of the partition $\Pi^{\delta_{\bf 0},(n)}_t$, then they have a common ancestor at time $t$ ago. $\Pi^{\delta_{\bf 0},(n)}_0$ is the partition of $\{1,\ldots,n\}$ into singleton blocks. The transitions are then given as follows: If there are $b$ blocks at present, then each pair of blocks merges with rate 1, thus the overall rate of seeing a merging event is ${b \choose 2}$. Note that only binary mergers are allowed and that at some random time, all individuals will have a (most recent) common ancestor. Kingman [@K82a] also showed that there exists a $\mathcal{P}_{\ensuremath{\mathbb{N}}}$-valued Markov process $(\Pi^{\delta_{\bf 0}}_t)_{t\ge 0}$, where $\mathcal{P}_{\ensuremath{\mathbb{N}}}$ denotes the set of partitions of ${\ensuremath{\mathbb{N}}}$. This process, the so-called Kingman coalescent, is characterised by the fact that for each $n$ the restriction of $(\Pi^{\delta_{\bf 0}}_t)_{t\ge 0}$ to the first $n$ natural numbers is the $n$-coalescent. The process can be constructed by an application of the standard Kolmogoroff extension theorem, since the restriction of every $n$-coalescent to $\{1,\ldots,m\}$, where $1\le m\le n$, is an $m$-coalescent. Whereas the Kingman coalescent allows only for binary mergers, the idea of a time-homogeneous $\mathcal{P}_{\ensuremath{\mathbb{N}}}$-valued Markov process that evolves by the coalescence of blocks was extended by Pitman [@P99] and Sagitov [@S99] to coalescents where multiple blocks are allowed to merge at the same time, so-called $\Lambda$-coalescents, which arise as the limiting genealogy of populations where the variance of the offspring distribution diverges as the population size tends to infinity. Möhle and Sagitov [@MS01] and Schweinsberg [@S00] introduced the even larger class of coalescents with simultaneous multiple collisions, also called exchangeable coalescents or $\Xi$-coalescents, which describe the genealogies of populations allowing for large family sizes. Schweinsberg [@S00] showed that any exchangeable coalescent $(\Pi^\Xi_t)_{t\ge 0}$ is characterised by a finite measure $\Xi$ on the infinite simplex $$\Delta\ :=\ \{{\ensuremath{\boldsymbol\zeta}}=(\zeta_1,\zeta_2,\ldots):\zeta_1\ge\zeta_2\ge\cdots\ge 0, \mbox{$\sum_{i=1}^\infty$}\zeta_i\le 1\}.$$ Throughout the paper, for ${\ensuremath{\boldsymbol\zeta}}\in\Delta$, the notation $|{\ensuremath{\boldsymbol\zeta}}|:=\sum_{i=1}^\infty\zeta_i$ and $({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}}):=\sum_{i=1}^\infty \zeta_i^2$ will be used for convenience. Note that Möhle and Sagitov [@MS01] provide an alternative (though somewhat less intuitive) characterisation of the $\Xi$-coalescent based on a sequence of finite symmetric measures $(F_r)_{r\in{\ensuremath{\mathbb{N}}}}$. Coalescent processes with multiple collisions ($\Lambda$-coalescents) occur if the measure $\Xi$ is concentrated on the subset of all points ${\ensuremath{\boldsymbol\zeta}}\in\Delta$ satisfying $\zeta_i=0$ for all $i\ge 2$. The Kingman-coalescent corresponds to the case $\Xi = \delta_{\bf 0}$. It is convenient to decompose the measure $\Xi$ into a ‘Kingman part’ and a ‘simultaneous multiple collision part’, that is, $\Xi=a\delta_{\mathbf 0}+\Xi_0$ with $a:=\Xi(\{\mathbf 0\})\in[0,\infty)$ and $\Xi_0(\{\mathbf 0\})=0$. The transition rates of the $\Xi$-coalescent $\Pi^\Xi$ are given as follows. Suppose there are currently $b$ blocks. Exactly $\sum_{i=1}^r k_i$ blocks collide into $r$ new blocks, each containing $k_1,\dots,k_r\ge 2$ original blocks, and $s$ single blocks remain unchanged, such that the condition $\sum_{i=1}^r k_i+s=b$ holds. The order of $k_1,\dots,k_r$ does not matter. The rate at which the above collision happens is then given as (Schweinsberg [@S00 Theorem 2]) $$\label{rates} \lambda_{b;k_1,\dots,k_r;s} \ =\ a{\ensuremath{\mathbbm{1}}}_{\{r=1,k_1=2\}} + \int_\Delta\sum_{l=0}^s {s\choose l}(1-|{\ensuremath{\boldsymbol\zeta}}|)^{s-l} \sum_{i_1\neq\cdots\neq i_{r+l}} \zeta_{i_1}^{k_1}\cdots\zeta_{i_r}^{k_r}\zeta_{i_{r+1}}\cdots\zeta_{i_{r+l}} \frac{\Xi_0(d{{\ensuremath{\boldsymbol\zeta}}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})}.$$ An intuitive explanation of (\[rates\]) is given below in terms of Schweinsberg’s [@S00] Poisson process construction of the $\Xi$-coalescent. If $\Xi(\Delta)\ne 0$, then without loss of generality it can be assumed that $\Xi$ is a probability measure, as remarked after Eq. (12) of [@S00]. Otherwise simply divide each rate by the total mass $\Xi(\Delta)$ of $\Xi$. Poisson process construction of the $\Xi$-coalescent {#Erhard} ---------------------------------------------------- It is convenient to give an explicit construction of the $\Xi$-coalescent in terms of Poisson processes. Indeed, Schweinsberg [@S00 Section 3] shows that the $\Xi$-coalescent can be constructed from a family of Poisson processes $\{\mathfrak{N}^K_{i,j}\}_{i,j\in{\ensuremath{\mathbb{N}}},i<j}$ and a Poisson point process $\mathfrak{M}^{\Xi_{0}}$ on ${\ensuremath{\mathbb{R}}}_+\times\Delta\times [0,1]^{{\ensuremath{\mathbb{N}}}}$. The processes $\mathfrak{N}^K_{ij}$ have rate $a=\Xi(\{\mathbf 0\})$ each and govern the binary mergers of the coalescent. The process $\mathfrak{M}^{\Xi_{0}}$ has intensity measure $$\label{intensity_measure} dt\otimes\frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})} \otimes({\ensuremath{\mathbbm{1}}}_{[0,1]}(t)dt)^{\otimes{\ensuremath{\mathbb{N}}}}.$$ These processes can be used to construct the $\Xi$-coalescent as follows: Assume that before the time $t_m$ the process $\Pi$ is in a state $\{B_1,B_2,\ldots\}$. If $t_m$ is a point of increase of one of the processes $\mathfrak{N}^K_{i,j}$ (and there are at least $i\vee j$ blocks), then we merge the corresponding blocks $B_i$ and $B_j$ into a single block (and renumber). This mechanism corresponds to the Kingman-component of the coalescent. The non-Kingman collisions are governed by the points $$(t_m,{{\ensuremath{\boldsymbol\zeta}}}_m,{\bf u}_m) \ =\ (t_m,(\zeta_{m1},\zeta_{m2},\ldots),(u_{m1},u_{m2},\ldots))$$ of the Poisson process $\mathfrak{M}^{\Xi_0}$. The random vector ${{\ensuremath{\boldsymbol\zeta}}}_m$ denotes the respective asymptotic family sizes in the multiple merger event at time $t_m$ and the ${\bf u}_m$ are “uniform coins”, determining the blocks participating in the respective merger groups; see or [@S00 Section 3] for details. $\Xi$-Fleming-Viot processes ---------------------------- An in many senses dual approach to population genetics is to view a population of finite size as a vector of types $(Y^N_1,\ldots,Y^N_N)$ with values in $E^N$ or as an empirical measure of that vector $\frac{1}{N}\sum_{i=1}^N\delta_{Y^N_i}$ and look at the evolution under mutation and resampling forwards in time. When $N$ tends to infinity one obtains the Fleming-Viot process [@FV79]. This process has been extended to incorporate other important biological phenomena and has found wide applications, see [@EK93] for a survey. Donnelly and Kurtz [@DK96] embedded an $E^\infty$-valued particle system into the classical Fleming-Viot process, via a clever lookdown construction, and showed that it is dual to the Kingman-coalescent. This construction and the duality has been extended to the so-called $\Lambda$-Fleming-Viot process, dual to the $\Lambda$-coalescents, and investigated by several authors, see, e.g., [@DK99; @BBC05; @BLG03; @BLG05; @BLG06], or [@BB07] for an overview. Let $f\in C_b(E^p)$, $\mu \in {\ensuremath{\mathcal{M}}}_1(E)$ and $G_f(\mu) := \langle f , \mu^{\otimes p} \rangle$. The generator of the $\Lambda$-Fleming-Viot process without mutation has the form (see [@BBC05 Equation (1.11)]) $$\label{eq_lambda_fleming_viot} L^{\Lambda}G_f(\mu) = \sum_{J\subset\{1,\ldots,p\},|J|\ge 2} \lambda_{p;|J|;p-|J|} \int \big(f({\bf x}^J) - f({\bf x})\big)\,\mu^{\otimes p}(d{\bf x}),$$ where $$({\bf x}^J)_i\ =\ \begin{cases} x_{\min(J)} & \mbox{if $i \in J$,}\\ x_i & \mbox{otherwise.} \end{cases}$$ Note that (\[eq\_lambda\_fleming\_viot\]) includes the generator of the classical Fleming-Viot process (without mutation) if the summation is restricted to sets $J$ satisfying $|J|=2$. Our aim in this paper is to present the modified lookdown construction for a measure-valued process that we will call the $\Xi$-Fleming-Viot process with mutation, or the $(\Xi,B)$-Fleming-Viot process. The symbol $B$ stands here for an operator describing the mutation process. We will establish its duality to the $\Xi$-coalescent with mutation. The modified lookdown construction will also enable us to establish some path properties of the $(\Xi,B)$-Fleming-Viot process. A modified lookdown construction of the $(\Xi,B)$-Fleming-Viot process ---------------------------------------------------------------------- Consider a population described by a vector $Y^N(t)=(Y^N_1(t),\ldots,Y^N_N(t))$ with values in $E^N$, where $Y^N_i(t)$ is the type of individual $i$ at time $t$. The evolution of this population (forwards in time) has two components, namely reproduction and mutation. During its lifetime, each particle undergoes mutation according to the bounded linear mutation operator $$\label{mutop} Bf(x)\ =\ r\int_E(f(y)-f(x))\,q(x,dy),$$ where $f$ is a bounded function on $E$, $q(x,dy)$ is a Feller transition function on $E\times\mathcal B(E)$, and $r\ge 0$ is the global mutation rate. The resampling of the population is governed by the Poisson point process $\mathfrak{M}^{\Xi_{0}}$, which was introduced as a driving process for the $\Xi$-coalescent. In particular, the resampling events allow for the simultaneous occurrence of one or more large families. The resampling procedure is described in detail in Section \[exchangeable\]. An important fact is that this resampling is made such that it retains exchangeability of the population vector. In Section \[exchangeable\], we introduce another particle system $X^N=(X^N_1,\ldots,X^N_N)$ again with values in $E^N$. Each particle mutates according to the same generator as before. For the resampling event, we will use the same driving Poisson point process $\mathfrak{M}^{\Xi_{0}}$, but we will use the modified lookdown construction of Donnelly and Kurtz introduced in [@DK99], suitably adapted to our scenario. This $(\Xi,B)$-lookdown process will be introduced in Section \[orderedmodel\]. It is crucial that the resampling events retain exchangeability of the population vector and that the process $\{X^N(t)\}$ has the same empirical measure $\sum_{i=1}^N \delta_{X^N_i(t)}$ as the process $\{Y^N(t)\}$. The construction of the resampling events allows us to pass to the limit as $N$ tends to infinity and obtain an $E^\infty$-valued particle system $X=(X_1,X_2,\ldots)$. Since this particle system is also exchangeable, this procedure enables us to access the almost sure limit of the empirical measure as $N$ tends to infinity by the De Finetti Theorem (which is not possible for the $Y^N$). Results {#suse_results} ------- Let $\mathcal{D}(B)$ denote the domain of the mutation generator $B$ and let $f_1,f_2,\ldots\in\mathcal{D}(B)$ be functions that separate points of ${\ensuremath{\mathcal{M}}}_1(E)$ in the sense that $\int f_k\,d\mu=\int f_k\,d\nu$ for all $k\in{\ensuremath{\mathbb{N}}}$ implies that $\mu=\nu$. Such sequences exist, see, e.g. Section 1 (Lemma 1.1 in particular) of [@DK96]. We use the metric $d$ on ${\ensuremath{\mathcal{M}}}_1(E)$ defined via $$\label{metric_on_measures} d(\mu,\nu)\ :=\ \sum_k\frac{1}{2^k} \Big|\int f_k\,d\mu-\int f_k\,d\nu\Big|, \qquad\mu,\nu\in{\ensuremath{\mathcal{M}}}_1(E)$$ and equip the topology of locally uniform convergence on $D_{{\ensuremath{\mathcal{M}}}_1(E)}([0,\infty))$ with the metric $$\label{metric_on_paths_of_measures} d_p(\mu,\nu)\ :=\ \int_0^\infty e^{-t}d(\mu(t),\nu(t))\,dt.$$ \[main\] The $\mathcal M_1(E)$-valued process $(Z_t)_{t\ge 0}$, defined in terms of the [*ordered*]{} particle system $X=(X^1,X^2,\dots)$ by $$Z_t\ :=\ \lim_{n\to\infty}Z^n_t \ =\ \lim_{n\to\infty}\frac 1n \sum_{i=1}^n \delta_{X_i(t)},\quad t\ge 0,$$ is called the [*$\Xi$-Fleming-Viot process with mutation operator $B$*]{} or simply the [*$(\Xi,B)$-Fleming-Viot process*]{}. Moreover, the empirical processes $(Z_t^n)_{t\ge 0}$ converge almost surely on the path space $D_{\mathcal M_1(E)}([0,\infty))$ to the càdlàg process $(Z_t)_{t\ge 0}$. Since the empirical measures of $X^N$ and $Y^N$ are identical, we arrive at the following corollary. \[maincor\] Define, for each $n$, $$\tilde Z^n_t:= \frac 1n \sum_{i=1}^n \delta_{Y_i(t)}, \quad t\ge 0,$$ the empirical process of the $n$-th unordered particle system, and assume that $\tilde Z_0^n \to Z_0$ weakly as $n\to\infty$. Then, $(\tilde Z_t^n)_{t\ge 0}$ converges weakly on the path space $D_{\mathcal M_1(E)}([0,\infty))$ to the $(\Xi,B)$-Fleming-Viot process $(Z_t)_{t\ge 0}$. The Markov process $(Z_t)_{t\ge 0}$ is characterized by its generator as follows. \[mainprop\] The $(\Xi,B)$-Fleming-Viot process $(Z_t)_{t\ge 0}$ is a strong Markov process. Its generator, denoted by $L$, acts on test functions of the form $$\label{eq:testfunction} G_f(\mu)\ :=\ \int_{E^n} f(x_1,\dots,x_n)\,\mu^{\otimes n}(dx_1,\dots,dx_n), \quad \mu \in {\ensuremath{\mathcal{M}}}_1(E),$$ where $f:E^n\to{\ensuremath{\mathbb{R}}}$ is bounded and measurable, via $$\label{eq_genXiFV} L G_f(\mu)\ :=\ L^{a\delta_{\bf 0}}G_f(\mu) + L^{\Xi_{0}}G_f(\mu) + L^B G_f(\mu),$$ where $$\begin{aligned} L^{a\delta_{\bf 0}} G_f(\mu) & := a\sum_{1\le i < j \le n} \int_{E^n} \Big( f(x_1,\!.., x_i,\!.., x_i,\!.., x_n)- f(x_1,\!.., x_i,\!.., x_j,\!.., x_n) \Big)\mu^{\otimes n}(d{\bf x}),\label{eq_genXiFV_kingman}\\ L^{\Xi_{0}} G_f(\mu) & := \int_\Delta \int_{E^{\ensuremath{\mathbb{N}}}} \left[ G_f\big( (1-|{\ensuremath{\boldsymbol\zeta}}|)\mu + {\textstyle\sum_{i=1}^\infty \zeta_i \delta_{x_i}} \big) - G_f(\mu) \right] \mu^{\otimes {\ensuremath{\mathbb{N}}}}(d\mathbf{x}) \frac{\Xi_{0}(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})}, \label{eq_genXiFV_xi}\\ L^{B} G_f(\mu) & := r\sum_{i=1}^n \int_{E^n} B_i(f(x_1, \dots, x_n)) \mu^{\otimes n}(d{\bf x}),\label{eq_genXiFV_mutation} \end{aligned}$$ and $B_i f$ is the mutation operator $B$, defined in (\[mutop\]), acting on the $i$-th coordinate of $f$. 1\) In [@DK99], Donnelly & Kurtz established a construction and pathwise duality for the $\Lambda$-Fleming-Viot process. In some sense, their paper works under the general assumption “allow simultaneous and/or multiple births and deaths, but we assume that all the births that happen simultaneously come from the same parent” (p. 166), even though they very briefly in Section 2.5 mention a possible extension to scenarios with simultaneous multiple births to multiple parents. In essence, the present paper converts these ideas into theorems. 2\) Note that in a similar direction, Bertoin & Le Gall remark briefly on p. 277 of [@BLG03] how their construction of the $\Lambda$-Fleming-Viot process via flows of bridges can be extended to the simultaneous multiple merger context (but leave details to the interested reader). We are not following this approach, as it is hard to combine with a general type space and general mutation process. 3\) The $\Xi$-Fleming-Viot process has recently been independently constructed by Taylor and Véber (personal communication, 2008) via Bertoin and Le Gall’s flow of bridges (see [@BLG03]) and Kurtz and Rodriguez’ Poisson representation of measure-valued branching processes (see [@KR08]). In this context we refer to Taylor and Véber [@TV08] for a larger study of structured populations, in which $\Xi$-coalescents appear under certain limiting scenarios. 4\) Note that the modified lookdown construction of the $\Lambda$-Fleming-Viot process contains all information available about the genealogy of the process and therefore also provides a pathwise embedding of the [*$\Lambda$-coalescent measure tree*]{} considered by Greven, Pfaffelhuber and Winter [@GPW07]. A similar statement holds for the $\Xi$-coalescent. The rest of the paper is organised as follows: In Section \[exchangeable\] we use the Poisson point process $\mathfrak{M}^{\Xi_0}$ to introduce the finite unordered $(\Xi,B)$-Moran model $Y^N$ and the finite ordered $(\Xi,B)$-lookdown model $X^N$. It is shown that the ordered model is constructed in such a way that we can let $N$ tend to infinity and obtain a well defined limit. We will also show that the reordering preserves the exchangeability property, which will be crucial for the proof in Section \[tightness\]. In this section, we will introduce the empirical measures of the process $Y^N$ and $X^N$, show that they are identical and converge to a limiting process having nice path properties, which is the statement of Theorem \[main\]. Section \[markov\_semigroup\] will be concerned with the generator of the $\Xi_{0}$-Fleming-Viot process. We will give two alternative representations and show that it generates a strongly continuous Feller semigroup. Furthermore, we will show that the process constructed in Section \[tightness\] solves the martingale problem for this generator. One representation of the generator will then be used in Section \[dualities\] to establish a functional duality between the $\Xi$-coalescent and the $\Xi$-Fleming-Viot process on the genealogical level. Due to the Poissonian construction, this duality can also be extended to a “pathwise” duality. We will also give a function-valued dual, which incorporates mutation. In Section \[examples\], we look at two examples: The first example is concerned with a population model with recurrent bottlenecks. Here, a particular $\Xi$-coalescent, which is a subordination of Kingman’s coalescent, arises as a natural limit of the genealogical process. The second example discusses the Poisson-Dirichlet-coalescent and obtains explicit expressions for some quantities of interest. Exchangeable $E^\infty$-valued particle systems {#exchangeable} =============================================== The canonical $(\Xi,B)$-Moran model {#canon_moran_modell} ----------------------------------- We can use the Poisson process from Section \[Erhard\] governing the $\Xi$-coalescent to describe a corresponding forward population model in a canonical way, simply reversing the construction of the coalescent by interpreting the merging events as birth events. Consider the points $$(t_m,{{\ensuremath{\boldsymbol\zeta}}}_m,{\bf u}_m)\ =\ (t_m,(\zeta_{m1},\zeta_{m2},\ldots),(u_{m1},u_{m2},\ldots))$$ of $\mathfrak{M}^{\Xi_0}$ defined by (\[intensity\_measure\]). The $t_m$ denote the times of reproduction events. Define $$\label{eq_definition_g} g({\ensuremath{\boldsymbol\zeta}},u)\ :=\ \begin{cases} \min\{j\,|\,\zeta_1+\cdots+\zeta_j\ge u\} & \text{if }u \le\sum_{i\in{\ensuremath{\mathbb{N}}}}\zeta_i,\\ \infty & \text{else}. \end{cases}$$ At time $t_m$, the $N$ particles are grouped according to the values $g({{\ensuremath{\boldsymbol\zeta}}}_m, u_{ml})$, $l=1,\dots,N$ as follows: For each $k \in {\ensuremath{\mathbb{N}}}$, all particles $l \in \{1,\dots,N\}$ with $g({{\ensuremath{\boldsymbol\zeta}}}_m, u_{ml})=k$ form a family. Among each non-trivial family we uniformly pick a ‘parent’ and change the others’ types accordingly. Note that although the jump times $(t_m)$ may be dense in ${\ensuremath{\mathbb{R}}}_+$, the condition $$\int_\Delta \sum_i\zeta_i^2\,\frac{\Xi_0(d{{\ensuremath{\boldsymbol\zeta}}})}{({{\ensuremath{\boldsymbol\zeta}}},{{\ensuremath{\boldsymbol\zeta}}})} \ =\ \Xi(\Delta)\ <\ \infty$$ guarantees that in a finite population, in each finite time interval only finitely many non-trivial reproduction events occur. As above, each particle follows an independent mutation process, according to , in between reproductive events. We describe the population corresponding to the $N$-particle $(\Xi,B)$-Moran model at time $t\ge 0$ by a random vector $$Y^N(t)\ :=\ (Y^N_1(t),\ldots,Y^N_N(t))$$ taking values in $E^N$. Note that this model is completely symmetric, thus, for each $t$, the population vector $Y^N(t)$ is exchangeable if $Y^N(0)$ is exchangeable. The ordered model and exchangeability {#orderedmodel} ------------------------------------- We now define an ordered population model with the same family size distribution, extending the ideas of Donnelly and Kurtz [@DK99] in an obvious way. This time each particle will be attached a “level” from $\{1,2,\dots\}$ in such a way that we obtain a nested coupling of approximating $(\Xi,B)$-Moran models as $N$ tends to infinity. It will be crucial to show that this ordered model retains initial exchangeability, so that the limit as $N\to\infty$ of the empirical measures of the particle systems, at each fixed time, exists by De Finetti’s theorem. We will refer to this model as the the $(\Xi,B)$-lookdown-model. If the population size is $N$, it will be described at time $t$ by the $E^N$-valued random vector $$\label{loddef} X^N(t)\ :=\ (X^N_1(t),\ldots,X^N_N(t)).$$ The dynamics works as in the $(\Xi,B)$-Moran model above, including the distribution of family sizes and the mutation processes for each particle. In each reproduction step, for each family, a “parental” particle will be chosen, that then superimposes its type upon its family. This time, however, the parental particle will not be chosen uniformly among the members of each family (as in the $(\Xi,B)$-Moran model). Instead, the parental particle will always be the particle with the lowest level among the members of a family (hence each family member “looks down” to their relative with the lowest level). The attachment of types to levels is then rearranged as follows (see Figure \[fig\_reproduction\] for an illustration): \[items:atoc\] - All parental particles of all families (including the trivial ones) will retain their type and level. - All levels of members of families will assume the type of their respective parental particle. - All levels which are still vacant will assume the pre-reproduction types of non-parental particles retaining their initial order. Once all $N$ levels are filled, the remaining types will be lost. In this way, the dynamics of a particle, at level $l$, say, will only depend on the dynamics of the particles with [*lower*]{} levels. This consistency property allows to construct all approximating particle systems, as well as their limit as $N \to \infty$, [*on the same probability space.*]{} Exchangeability of the modified $(\Xi,B)$-lookdown model is crucial in order to pass to the De Finetti limit of the associated empirical particle systems. For each $N$, we will show that if $X(0)$ is exchangeable, then $X$ is exchangeable at fixed times and at stopping times. The proof will rely on an explicit construction of uniform random permutations $\Theta(t)$ which maps $X^N$ to $Y^N$. \[exchangeable\_fixed\_times\] If the initial distribution of the population vector $(X^N_1(0),\dots,X^N_N(0))$ in the $(\Xi,B)$-lookdown-model is exchangeable, then $(X^N_1(t),\dots,X^N_N(t))$ is exchangeable for each $t\ge 0$. For the rest of this section, we omit the superscript $N$ for the population models in an attempt not to get *lost in notation*. The proof of Theorem \[exchangeable\_fixed\_times\] follows that of Theorem 3.2 in [@DK99]. We will construct a coupling via a permutation-valued process $\Theta(t)$ such that $$\label{models_equal1} (Y_{1}(t),\dots,Y_N(t))=(X_{\Theta_{1}(t)}(t),\dots,X_{\Theta_N(t)}(t))$$ and $\Theta(t)$ is uniformly distributed on all permutations of $\{1,\dots,N\}$ for each $t$ and independent of the empirical process up to time $t$ and the “demographic information” in the model (see for a precise definition). It suffices to construct the skeleton chain $(\theta_m)_{m \in {\ensuremath{\mathbb{N}}}_0}$ of $\Theta$. As a guide through the following notation, we have found it useful to occasionally remember that $\Theta(t)$ (and its skeleton chain) is built to the following aim: ** $\Theta$ maps a position of an individual in the vector $Y$ ($(\Xi,B)$-Moran-model) to the level of the corresponding individual in the ordered vector $X$ ($(\Xi,B)$-lookdown-model). #### Notation and ingredients For $N>0$ let $S_{N}$ denote the collection of all permutations of $\{1,\dots,N\}$, let $P_{N}=\mathcal{P} (\{1,\dots,N\})$, the set of all subsets of $\{1,\dots,N\}$, and let $P_{N,k}\subset P_{N}$ be the subcollection of subsets with cardinality $k$. For a set $M$, $M(i)$ will denote the $i$th largest element in $M$. At time $m$ (for the skeleton chain) let $c_m$ the total number of children. Let $a_m$ be the number of families and $c_m^i$ the number of children born to family $i$, hence $$\label{sum_km} \sum_{i=1}^{a_m} c^i_m = c_m.$$ Note that we allow $c_m^i = 0$ for some, but not all $i$. These are the trivial families where only the parental particle is below level $N$ and all potential children are above. Furthermore, we need to keep track of these “one-member families” in order to match the rates of our model to those of the $\Xi$-coalescent later on. Let $\theta_{0}$ be uniformly distributed over $S_N$. For each $m \in {\ensuremath{\mathbb{N}}}$, pick (independently, and independent of $\theta_0$) - $\Phi_{m}$ a random set, uniformly chosen from $P_{N,c_m+a_m}$, - $\big(\phi_{m}^{1},\dots,\phi_{m}^{a_{m}}\big)$ a random (ordered) partition of $\Phi_m$, such that each $\phi_m^i$ has size $c_m^i+1$, - $\sigma_m^i$, $i=1,\dots,a_m$ random permutations, each $\sigma_{m}^i$ uniformly distributed over $S_{c_m^i + 1}$, independently of $\Phi_m$ and the $\phi_m^i$. Denote - $\mu_{m}^{i}:=\min\phi_{m}^{i}, \; i \in \{1,\dots,a_{m}\}$, - write $\Delta_m$ for the set of the highest $c_m$ integers from $\{1,\ldots,N\}\setminus\bigcup_{i=1}^{a_m}\mu_m^i$. Proceeding inductively we assume that $\theta_{m-1}$ has already been defined. We then construct $\theta_m$ as follows: Let - $\nu_{m}^{i}:=\theta_{m-1}^{-1}(\mu_{m}^{i})$, - $\psi_m := \theta_{m-1}^{-1}(\Delta_m), \quad \mbox{and}$ - a random ordered “partition” $\big(\psi_{m}^{1},\dots,\psi_{m}^{a_{m}}\big)$ of $\psi_m$ such that $|\psi_m^i|=c^i_m$, chosen independently of everything else. In view of our intended application of $\theta_m$ to transfer from the Moran model to the lookdown model, we will later on interpret these quantities as follows: In the $m$-th event, $\mu_m^i$ will be the level of the parental particle of family $i$ in the lookdown-model, and $\nu_m^i$ will be the corresponding index in the (unordered) Moran model. $\Delta_m$ will specify the levels in the lookdown-model at which individuals die. We do not just pick the highest $c_m$ levels, because we wish to retain parental particles. $\psi_m$ will be the corresponding indices in the Moran model. $\big(\phi_{m}^{1},\dots,\phi_{m}^{a_{m}}\big)$ describes the family decomposition (including the respective parents) in this event in the lookdown model, and $\psi_m^i$ are the indices of the children in the $i$-th family in the Moran model. Thus, $\theta_m$ will map $\phi_m^i$ to $\psi_m^i \cup \{\nu_m^i\}$ (in a particular order). Finally, define $\theta_m$ as follows: Put $\Psi_m := \{\nu_m^1,\dots,\nu_m^{a_m}\} \cup \psi_m$. On $\Psi_m$, $$\label{construction_theta_m_1} \theta_{m}(\nu_{m}^{i}) := \phi_{m}^{i}(\sigma_m^i(1)), \; i=1,\dots,a_m,$$ and $$\label{construction_theta_m_2} \theta_{m}(\psi_m^i(j)) := \phi_{m}^{i}(\sigma_m^i(j+1)) \quad \forall j \in \{1,\ldots,c_m^i\}$$ for each $i\in\{1,\ldots,a_m\}$ with $c_m^i \neq 0$. On $\{1,\dots,N\}\setminus\Psi_m$ let $\theta_m$ be the mapping onto $\{1,\dots,N\}\setminus\Phi_m$ with the same order as $\theta_{m-1}$ restricted to $\{1,\dots,N\}\setminus\Psi_m$, that is, whenever $\theta_{m-1}(i)<\theta_{m-1}(j)$ for some $i,j\in\{1,\dots,N\} \setminus\Psi_m$, then $\theta_m(i)<\theta_m(j)$ should also hold. \[ex\_permutation\] We consider a realisation of the $m$-th event of a population of size $N=8$, as illustrated in Figure \[fig\_reproduction\]. There are $a_m=2$ families (depicted by “triangle” and “star”, respectively). The first family $\phi_m^1=\{3,6,8\}$ has size $c_m^1+1=3$, the second, $\phi_{m}^2=\{2,5\}$, has size $c_m^2+1=2$. Hence, the set of levels involved in this birth event is $\Phi_m=\{2,3,5,6,8\}$, and $\mu_{m}^1=3$, $\mu_m^2=2$ are the levels of the parental particles. Since there is no parental particle among the highest three levels, the particles at levels $\Delta_m=\{6,7,8\}$ “die”. Now let us assume that $\theta_{m-1}$ is as given in Figure \[theta\_m-1\]. Thus, $\nu_m^1=4$, $\nu_m^2=1$, $\psi_m=\{3,5,7\}$. The set of indices $\psi_m$ of individuals in the Moran model who will get replaced by offspring in this event is partitioned according to the family sizes, for example let $\psi_m^1=\{3,7\}$ and $\psi_m^2=\{5\}$. We construct $\theta_m$ as follows: Let $\sigma_m^1={ 1\: 2\: 3 \choose 3\: 1\: 2}$ and $\sigma_m^2={ 1\: 2 \choose 2\: 1}$. For the restriction of $\theta_m$ to $\Psi_m=\{1,3,4,5,7\}$, we read from that $\theta_m(4)=\phi_m^1(3)=8$, $\theta_m(1)=\phi_m^2(2)=5$ and from that $\theta_m(3)=\theta_m(\psi_m^1(1))=\phi_m^1(\sigma_m^1(1+1))=\phi_m^1(1)=3$, $\theta_m(7)=\theta_m(\psi_m^1(2))=\phi_m^1(\sigma_m^1(2+1))=\phi_m^1(2)=6$ and $\theta_m(5)=\theta_m(\psi_m^2(1))=\phi_m^2(\sigma_m^2(1+1))=\phi_m^2(1)=2$. This leads to the partial permutation which is given in Figure \[add\_families\]. Restricted to the complementary set $\{2,6,8\}$, $\theta_m$ is a mapping onto $\{1,4,7\}$ with the same order as $\theta_{m-1}$ restricted to $\{2,6,8\}$. The resulting permutation $\theta_m$ is given in Figure \[theta\_m\].$\blacksquare$ For notational convenience, let $$\chi_m := (\nu_m^1,\psi_m^1,\ldots,\nu_m^{a_m},\psi_m^{a_m}),$$ which summarises the combinatorial information generated in the $m$-th step (namely, the family structure we would observe in the Moran model). \[distributed\] For each $m$, $\chi_1,\ldots,\chi_m,\theta_m$ are independent. Furthermore $\theta_m$ is uniformly distributed over $S_N$ and $$\Upsilon_m := \bigcup_{i=1}^{a_m} \{\nu_m^i\} \cup \psi_m^i$$ is uniformly distributed over $P_{N,c_m+a_m}$, and each $\chi_m$ is, given $\Upsilon_m$, uniformly distributed on all ordered partitions of $\Upsilon_m$ with family sizes consistent with the $c_m^i$. We prove the statement by induction. Denoting $\mathcal{F}_m = \sigma(\theta_k, \chi_k: 0 \le k \le m)$, we have $$\label{dependence_through_theta} {\ensuremath{\mathbb{E}}}[f(\theta_m,\chi_m)\mid \mathcal{F}_{m-1}]={\ensuremath{\mathbb{E}}}[f(\theta_m,\chi_m)\mid \theta_{m-1}],$$ since $\theta_m$ and $\chi_m$ are only based on $\theta_{m-1}$ and additional independent random structure. This implies, for any choice of $f\colon S_n \to {\ensuremath{\mathbb{R}}}$ and $h_k \colon \cup_{n=1}^N \big(\{1,\dots,N\} \times \mathcal{P}(\{1,\dots,N\})\big)^n \to {\ensuremath{\mathbb{R}}}$, $$\begin{aligned} {\ensuremath{\mathbb{E}}}\left[f(\theta_m)\prod_{k=1}^{m}h_k({\chi_k})\right] & = & {\ensuremath{\mathbb{E}}}\left[{\ensuremath{\mathbb{E}}}[f(\theta_m)h_m({\chi_m})\mid\mathcal{F}_{m-1}]\prod_{k=1}^{m-1}h_k({\chi_k})\right]\\ & = & {\ensuremath{\mathbb{E}}}\left[{\ensuremath{\mathbb{E}}}[f(\theta_m)h_m({\chi_m})\mid\theta_{m-1}]\prod_{k=1}^{m-1}h_k({\chi_k})\right]\\ & = & {\ensuremath{\mathbb{E}}}[f(\theta_m)h_m({\chi_m})]\prod_{k=1}^{m-1}{\ensuremath{\mathbb{E}}}[h_k({\chi_k})],\end{aligned}$$ where we used in the second and the induction hypothesis in the third equality. It remains to show that $\theta_m$ and $\chi_m$ are independent and have the correct distributions. $\theta_{m-1}$ is uniformly distributed by the induction hypothesis and independent of the distributions of the parental-levels $\mu_m^i$ and the “death-levels” $\Delta_m$ by construction. It is immediate from the construction that $\Phi_m$ and $\Upsilon_m$ are uniformly distributed over $P_{N,c_m+a_m}$ and the family structure $\chi_m$ is uniformly distributed among all admissible configurations. Furthermore, conditioning on $\chi_m$ and $\Phi_m$, $\theta_m$ is uniformly distributed over all permutations that map $\Upsilon_m$ onto $\Phi_m$. This follows from the fact that $\Phi_m$ is uniform on $P_{N,c_m+a_m}$ and that this set is uniformly divided into the families $\phi_m^i$. Since uniform and independent permutations $\sigma_m^i$ are used for the construction of $\theta_m$ and the non-participating levels remain uniformly distributed, $\theta_m$ is uniform under these conditions. Finally, conditioning on $\chi_m$ does not alter the fact that $\Phi_m$ is uniformly distributed over $P_{N,c_m+a_m}$. This implies that given $\chi_m$, $\theta_m$ is also uniformly distributed over $S_N$. Since $$\mathcal{L}(\theta_m\vert\chi_m) = \text{unif}(S_N) = \mathcal{L}(\theta_m),$$ $\theta_m$ and $\chi_m$ are independent of each other. Suppose a realization $X$ of the $N$-particle $(\Xi,B)$-lookdown-model is given and let $\{t_m\}$ denote the times at which the birth events occur. The families involved in the $m$-th birth event are denoted by $\phi^i_m$. Note that by definition of the lookdown-dynamics, the “ingredients” $\Phi_m, c_m, a_m, c_m^i, \mu_m^i, \Delta_m$ introduced earlier can be obtained from this, and that their joint distributions is as discussed above. Moreover, let the initial permutation $\theta_0$ be independent of $X$ and uniformly distributed on $S_N$. Let $\sigma^i_m$ be independent of all other random variables and uniformly distributed on $S_{c^i_m+1}$, $1\le i \le a_m$, $m\in{\ensuremath{\mathbb{N}}}$. Define $\theta_m$ as above, and $$\Theta(t) := \theta_m \quad\text{for}\:t_m \le t < t_{m+1}.$$ Observe that, by Lemma \[distributed\], $$\label{define_y_through_theta} (Y_1(t),\ldots,Y_N(t)) := (X_{\Theta_1(t)}(t),\ldots,X_{\Theta_N(t)}(t))$$ is a version of the $(\Xi,B)$-Moran-model. Note that “one-member families” are in this construction simply treated as non-participating individuals in the $(\Xi,B)$-Moran-model. $Y(t)$ depends only on $Y(0)$, $\{\chi_m\}_{t_m \le t}$ and the the evolution of the type processes between birth and death events, so $\Theta(t)$, and hence $\Theta(t)^{-1}$ is independent of $$\label{eq_xi_filtration} \mathcal{G}_t := \sigma\big((Y_1(s),\ldots,Y_N(s)):s\le t\big) \vee \sigma(\chi_m : m \in {\ensuremath{\mathbb{N}}})$$ due to Lemma \[distributed\]. Therefore, we see from $$(X_1(t),\ldots,X_N(t)) = (Y_{\Theta^{-1}_1(t)}(t),\ldots,Y_{\Theta^{-1}_N(t)}(t))$$ that $(X_1(t),\ldots,X_N(t))$ is exchangeable. Starting from the same exchangeable initial condition, the laws of the empirical processes of the $(\Xi,B)$-Moran-model and the $(\Xi,B)$-lookdown-model coincide. The exchangeability property does not only hold for fixed times, but also for stopping times. \[exchangeable\_stopping\_times\] Suppose that the initial population vectors $Y^N(0)$ in the $(\Xi,B)$-Moran-model and $X^N(0)$ in the $(\Xi,B)$-lookdown-model have the same exchangeable distribution, and let $\tau$ be a stopping time with respect to $(\mathcal{G}_t)_{t\ge 0}$ given by . Then, $(X^N_1(\tau),\dots,X^N_N(\tau))$ is exchangeable. We show that $\Theta(\tau)$ is independent of the $\sigma$-algebra $\mathcal{G}_\tau$ (the $\tau$-past) and uniformly distributed over $S_N$. First, assume that $\tau$ takes only countable many values $t_k$, $k\in{\ensuremath{\mathbb{N}}}$. Let $A\in\mathcal{G}_\tau$ and $h:S_N\to\mathbb{R}_+$, then $$\label{discrete_stopping_times} \begin{split} \mathbb{E}\Big( h\big(\Theta(\tau)\big){\ensuremath{\mathbbm{1}}}_A\Big) & = \mathbb{E} \Big( \sum_{k=1}^\infty h\big(\Theta(t_k)\big) {\ensuremath{\mathbbm{1}}}_{A\cap\{ \tau = t_k \}} \Big)\\ & = \sum_{k=1}^\infty \Big(\mathbb{E} h\big(\Theta(t_k)\big)\Big) \Big(\mathbb{E}{\ensuremath{\mathbbm{1}}}_{A\cap\{\tau = t_k\}}\Big)\\ & = \int h(\Theta)\,\mathfrak{U}(d\Theta) \sum_{k=1}^\infty\mathbb{E}{\ensuremath{\mathbbm{1}}}_{A\cap\{\tau = t_k\}}\\ & = \int h(\Theta)\,\mathfrak{U}(d\Theta)\,\mathbb{E}{\ensuremath{\mathbbm{1}}}_A, \end{split}$$ where $\mathfrak{U}$ denotes the uniform distribution on $S_N$. To see that the second equality holds, observe that, for fixed $t_k$, $\Theta(t_k)$ is independent of $\mathcal{G}_{t_k}$ in the proof of Theorem \[exchangeable\_fixed\_times\]. By approximating an arbitrary stopping time from above by a sequence of discrete stopping times, we see that holds in the general case as well. Now, exchangeability of $(X^N_1(\tau),\dots,X^N_N(\tau))$ follows as in the proof of Theorem \[exchangeable\_fixed\_times\]. One can also define a variant of the $(\Xi,B)$-lookdown model which is more in the spirit of the ‘classical’ lookdown construction from [@DK96], where, instead of a)–c) on page , at a jump time each particle simply copies the type of that member of the family it belongs to with the lowest level (and no types get shifted upwards). This variant, which is (up to a renaming of levels by the points of a Poisson process on ${\ensuremath{\mathbb{R}}}$) also the one suggested by adapting [@KR08] to the ‘simultaneous multiple merger’-scenario, has been considered by Taylor & Véber (2008, personal communication). The same results as above hold for this variant, with only minor modifications of the proofs. Note that the flavour of the lookdown process described above is easily adaptable to a set-up with time-varying total population size, which is not obvious for the other variant. The limiting population {#limitpop} ----------------------- We now construct the limiting $E^\infty$-valued particle system $X = (X_1,X_2,\ldots)$ by formulating a stochastic differential equation for each level $l$. These exist for each level and are well defined, since the equation for level $l$ needs only information about lower levels. The generator of a pure jump process can be written in the form $$Bf(x) = r\int_0^1 \big( f(m(x,u)) - f(x)\big)\,du,$$ where $r$ is the global mutation rate and $m\colon E\times[0,1] \to E$ transforms a uniformly distributed random variable on $[0,1]$ into the jump distribution $q(x,dy)$ of the process. The random times and uniform “coins” for the mutation process at each level $l$ are given by a Poisson point process $\mathfrak{N}^{\text{Mut}}_l$ on ${\ensuremath{\mathbb{R}}}_+\times[0,1]$ with intensity measure $r dt \otimes du$. As in Section \[canon\_moran\_modell\], denote by $$(t_m,{{\ensuremath{\boldsymbol\zeta}}}_m, {\bf u}_m) = (t_m, (\zeta_{m1}, \zeta_{m2}, \ldots), (u_{m1}, u_{m2}, \ldots))$$ the points of the Poisson point process $\mathfrak{M}^{\Xi_{0}}$ and recall the definition of the “colour” function $g$. Based on this, define $$\label{eq_llj} L_J^l(t)\ := \sum_{m:t_m\le t} \prod_{j\in J}{\ensuremath{\mathbbm{1}}}_{\{g({{\ensuremath{\boldsymbol\zeta}}}_m, u_{mj})< \infty\}} \prod_{j\in\{1,\ldots,l\}\setminus J}{\ensuremath{\mathbbm{1}}}_{\{g({{\ensuremath{\boldsymbol\zeta}}}_m,u_{mj})=\infty\}},$$ for $J \subset \{1, \dots ,l\}$ with $|J|\ge 2$. $L^l_J(t)$ counts how many times, among the levels in $\{1,\dots,l\}$, exactly those in $J$ were involved in a birth event up to time $t$. Moreover, let $$\label{eq_lljk} L_{J, k}^l(t)\ := \sum_{m:t_m\le t} \prod_{j\in J}{\ensuremath{\mathbbm{1}}}_{\{g({{\ensuremath{\boldsymbol\zeta}}}_m, u_{mj})=k\}} \prod_{j\in\{1,\ldots,l\}\setminus J}{\ensuremath{\mathbbm{1}}}_{\{g({{\ensuremath{\boldsymbol\zeta}}}_m,u_{mj})\neq k\}}.$$ $L^l_{J,k}(t)$ counts how many times, among the levels in $\{1,\dots,l\}$, exactly those in $J$ were involved in a birth event up to time $t$ and additionally assumed “colour” $k$. To specify the new levels of the individuals not participating in a certain birth event, we construct a function $J_m$ as follows: Denote by $\mu_m^k := \min\{l \in {\ensuremath{\mathbb{N}}}\,\vert\, g({{\ensuremath{\boldsymbol\zeta}}}_m,u_{ml}) = k\}$ the level of the parental particle of family number $k$ and by $M_m := \{\mu_m^k\}_{k \in {\ensuremath{\mathbb{N}}}}$ the set of all levels of parental particles involved in the $m$-th birth event. Furthermore $U_m := \{ l \in {\ensuremath{\mathbb{N}}}\,\vert\, g({{\ensuremath{\boldsymbol\zeta}}}_m,u_{ml}) = \infty\}$ denotes the set of the levels not participating in the birth event $m$. Define the mapping $$\label{eq_function_jm} J_m\,:\,U_m\to{\ensuremath{\mathbb{N}}}\setminus M_m$$ that maps the $i$-th smallest element of the set $U_m$ to the $i$-th smallest element of the set ${\ensuremath{\mathbb{N}}}\setminus M_m$ for all $i$. Assuming for the moment that $E$ is an Abelian group, the (infinite) vector describing the types in the $(\Xi,B)$-lookdown-model is defined as the (unique) strong solution of the following system of stochastic differential equations. The lowest individual on level 1 just evolves according to mutation, i.e., $$\label{eq_stochastic_differential_equation_lvel_one} X_1(t)\ :=\ \int_{[0,t]\times[0,1]} (m(X_1(s-),u) - X_1(s-))\,d\mathfrak{N}^{\text{Mut}}_1(s,u).$$ The individuals above level one can look down during birth events. Thus, for $l\ge 2$, define $$\label{eq_stochastic_differential_equation} \begin{split} X_l(t) := & X_l(0) + \int_{[0,t]\times[0,1]} \big(m(X_l(s-),u) - X_l(s-)\big)\,d\mathfrak{N}^{\text{Mut}}_l(s,u)\\ & + \sum_{1 \leq i <l} \int_0^t (X_i(s-) - X_l(s-))\,d\mathfrak{N}^K_{il}(s)\\ & + \sum_{1 \leq i < j < l} \int_0^t (X_{l-1}(s-) - X_l(s-))\,d\mathfrak{N}^K_{ij}(s)\\ & + \sum_{k \in {\ensuremath{\mathbb{N}}}} \sum_{K \subset \{1,\ldots,l\},l \in K} \int_0^t (X_{\min(K)}(s-) - X_l(s-))\,dL^l_{K,k}(s)\\ & + \sum_{K \subset \{1,\ldots,l\},l \notin K} \int_0^t (X_{J_m(l)}(s-) - X_l(s-))\,dL^l_K(s).\\ \end{split}$$ The second and third lines describe the “Kingman events”, where only pairs of individuals are involved. The first part copies the type from level $i$ when $l$ looks down to this level, because it is involved in a birth event and the parental particle is at level $i$. The second part handles the event that the parental particle places a child on a level below $l$. In this case, $l$ has to copy the type from the level $l-1$, since the new individual is inserted at some level below $l$ and pushes all particles above that level one level up. The fourth and fifth lines describe the change of types for a birth event with large families in a similar way. If the particle at level $l$ is involved in the family $k$, it copies the type from the parental particle which resides at the lowest level of the family. If level $l$ is not involved in any family, then $J_m(l)$ ($\le l$) gives the level from where the type is copied (which comes from shifting particles not involved in the lookdown event upwards). Since the equation for $X_l$ involves only $X_1,\dots,X_l$ and finitely many Poisson processes, it is immediate that there exists a unique strong solution of –. In the case where $E$ has no group structure, one may still construct suitable jump-hold processes $X_i$, using the driving Poisson processes in an obvious extension of –. These stochastic differential equations determine an infinitely large population vector $$X(t)\ :=\ (X_1(t),X_2(t),\ldots)$$ in a consistent way, and for each $N\in{\ensuremath{\mathbb{N}}}$, the dynamics of $(X_1,\ldots,X_N)$ is identical to that defined in Section \[orderedmodel\]. In particular, we see from Theorem \[exchangeable\_fixed\_times\] that, for each $t\ge 0$, $X(t)$ is exchangeable and the empirical distribution $$\label{eq_emp_dist_fixed_time_exists} Z(t) \ :=\ \lim_{l\to\infty} Z^l(t) \ :=\ \lim_{l\to\infty}\frac{1}{l} \sum_{i=1}^l \delta_{X_i(t)}$$ exists almost surely. Let $F$ be the set of bounded measurable functions $\varphi:[0,\infty)\times [0,1]^{\ensuremath{\mathbb{N}}}\times [0,1]^\infty\to{\ensuremath{\mathbb{R}}}$ such that $\varphi(t,{\ensuremath{\boldsymbol\zeta}},\bf{u})$ does not depend on $\bf{u}$, and put $$\label{eq_def_Ht} \mathcal{H}_t\ :=\ \sigma\bigg( \big(Z(s):s\le t\big), \Big({\textstyle\int\varphi\,d\mathfrak{M}^{\Xi_0}}:\varphi\in F\Big) \bigg).$$ Let $\tau$ be a stopping time with respect to $(\mathcal{H}_t)_{t\ge 0}$. Then $$X(\tau)\ =\ (X_1(\tau),X_2(\tau),\ldots)$$ is exchangeable. We claim that for $t\ge 0$, $A\in\mathcal{H}_t$ with ${\ensuremath{\mathbb{P}}}\{A\}>0$ and $n\in{\ensuremath{\mathbb{N}}}$, $$\label{eq_exch_conditioned} (X_1(t),\dots,X_n(t))\mbox{ is exchangeable under ${\ensuremath{\mathbb{P}}}\{\cdot|A\}$}.$$ Observe that, taking $A=\{\tau=t_k\}$, immediately implies the result for discrete stopping times $\tau$, from which the general case can be deduced by approximation as in the proof of Theorem \[exchangeable\_stopping\_times\]. Obviously, is equivalent to $$\label{eq_exch_conditioned2} {\ensuremath{\mathbb{P}}}\big\{A \cap \{ (X_1(t),\dots,X_n(t)) \in C \} \big\} = {\ensuremath{\mathbb{P}}}\big\{A \cap \{ (X_{\sigma(1)}(t),\dots,X_{\sigma(n)}(t)) \in C \} \big\} \quad \forall\, C \subset E^n, \sigma \in S_n.$$ As the collection of sets $A$ from $\mathcal{H}_t$ satisfying is a Dynkin system, it suffices to verify for events of the form $$\label{eq_A_cap_stable} A\ =\ \{Z(s_1)\in B_1,\ldots,Z(s_k)\in B_k\}\cap H',$$ where $H' \in \sigma\big( \int \varphi \, d\mathfrak{M}^{\Xi_{0}} : \varphi \in F \big)$, $k\in{\ensuremath{\mathbb{N}}}$, $s_1<\cdots<s_k\le t$, $B_i\in\mathcal{B}(s_i)$ for $i\in\{1,\ldots,k\}$, and $\mathcal{B}(s_i)$ is a $\cap$-stable generator of $\mathcal{B}_{\mathcal{M}_1(E)}$ with the property that ${\ensuremath{\mathbb{P}}}\{Z(s_i) \in \partial B'\}=0$ for all $B'\in \mathcal{B}(s_i)$. For $A$ as given in , $\varepsilon > 0$ and $n \in{\ensuremath{\mathbb{N}}}$, $\sigma\in S_n$, $C\subset E^n$ appearing in , by there exists $l$ ($l \gg n$) such that $$A_l\ :=\ \{Z^l(s_1)\in B_1,\ldots,Z^l(s_k)\in B_k\}\cap H'$$ satisfies ${\ensuremath{\mathbb{P}}}\{(A\setminus A_l)\cup (A_l\setminus A)\}\le\varepsilon$. By the arguments given in the proof of Theorem \[exchangeable\_stopping\_times\], holds with $A$ replaced by $A_l$. Finally, take $\varepsilon\to 0$ to conclude. Pathwise convergence: Proof of Theorem \[main\] {#tightness} =============================================== Recall the empirical processes $Z^l$, and their limit $Z$, from . Obviously, for each $l\in{\ensuremath{\mathbb{N}}}$, the process $(Z^l(t))_{t\ge 0}$ has càdlàg paths. To verify the corresponding property for $Z$, we introduce the following auxiliary (Lévy) process $U$, derived from Poisson point process $\mathfrak{M}^{\Xi_{0}}$ which governs the large family birth events of the population $X$: If $\big\{(t_m,{{\ensuremath{\boldsymbol\zeta}}}_m,{\bf u}_m)\big\}$ are the points of the process $\mathfrak{M}^{\Xi_0}$, we define $$U(t)\ :=\ \sum_{t_m\le t}v_m^2,$$ where $v_m:=\sum_{i=1}^\infty\zeta_{mi}$. The jumps of $U:=(U(t))_{t\ge 0}$ are the squared total fractions of the population which are replaced in large birth events. The generator of $U$ is given by $$Df(u)\ =\ \int_0^1(f(u+v^2)-f(u))\,\nu(dv),$$ where the measure $\nu$ on $[0,1]$, defined via $$\nu(A)\ :=\ \int_\Delta {\ensuremath{\mathbbm{1}}}_{\{\sum_{i=1}^\infty\zeta_i\in A\}} \frac{\Xi(d{\ensuremath{\boldsymbol\zeta}})}{({{\ensuremath{\boldsymbol\zeta}}},{{\ensuremath{\boldsymbol\zeta}}})},$$ governs the jumps. We need the following version of Lemma A.2 from [@DK99]. \[A2\] a) Let $e_1,e_2,\dots$ be exchangeable and suppose there exists a constant $K$ such that $|e_i|\le K$ almost surely. Define $$M_k\ :=\ \frac{1}{k}\sum_{i=1}^k e_i$$ and let $M_\infty$ be the almost sure limit of $(M_k)_{k\in{\ensuremath{\mathbb{N}}}}$, whose existence is guaranteed by the de Finetti Theorem. Let $\varepsilon>0$. Then there exists $\eta_1>0$ depending only on $K$ and $\varepsilon$, such that, for $l<n \in {\ensuremath{\mathbb{N}}}\cup \{\infty\}$, $${\ensuremath{\mathbb{P}}}\{\left|M_n-M_l\right|\ge \varepsilon\} \le 2e^{-\eta_1(K,\varepsilon) l}.$$ b\) Let $(e_i(t))_{t\in[0,1]}$ be centered martingales such that $\max_{i \in{\ensuremath{\mathbb{N}}}} \sup_{t \in [0,1]} |e_i(t)| \le K$ almost surely and $(e_1(1),e_2(1),\ldots)$ is exchangeable. Put $$M_k(t)\ :=\ \frac{1}{k}\sum_{i=1}^k e_i(t).$$ Let $\varepsilon>0$. Then there exists $\eta_2>0$ depending only on $K$ and $\varepsilon$, such that, for $l\in{\ensuremath{\mathbb{N}}}$ $${\ensuremath{\mathbb{P}}}\{\sup_{t\in[0,1]} |M_k(t)| \ge \varepsilon\} \le 2e^{-\eta_2(K,\varepsilon) l}.$$ The proof of part a) is a straightforward extension of that of Lemma A.2 from [@DK99], which employs the fact that an infinite exchangeable sequence is conditionally i.i.d. together with standard arguments based on the moment generating function. For part b) observe that by Doob’s submartingale inequality, $$\label{doob_inequality} {\ensuremath{\mathbb{P}}}\Big\{\sup_{0\le t<1} |M_{k}(t)|\ge\varepsilon\Big\} \ \le\ \inf_{\lambda>0} \frac{1}{e^{\varepsilon\lambda}} {\ensuremath{\mathbb{E}}}e^{\lambda |M_{k}(1)|} \ \le\ \inf_{\lambda>0} \frac{1}{e^{\varepsilon\lambda}} {\ensuremath{\mathbb{E}}}\exp\Big(\frac{\lambda}{k}\sum_{i=1}^k |e_i(1)|\Big). $$ Now proceed as in part a). The following lemma provides the technical core of the argument and replaces Lemma 3.4 and Lemma 3.5 in [@DK99]. The proof given below follows closely the arguments of Donnelly and Kurtz [@DK99]. \[lemma\_for\_bound\] In the setting of Theorem \[main\], for all $c,T,\epsilon > 0$ and $f \in \mathcal{D}(B)$ (the domain of the mutation generator) there exists a sequence $\delta_l$ such that $\sum_{l=1}^\infty\delta_l < \infty$ and $${\ensuremath{\mathbb{P}}}\Big\{ \sup_{0\le t\le T}\big\vert\langle f,Z(t) \rangle - \langle f,Z^l(t) \rangle\big\vert \ge 11\epsilon, U(T)\le c \Big\} \le \delta_l.$$ By Lemma \[A2\] and the exchangeability properties of $X$, we have $$\label{eq:fstopempir} {\ensuremath{\mathbb{P}}}\{|\langle f,Z(\alpha)\rangle - \langle f,Z^l(\alpha)\rangle| \ge\epsilon\}\ \le\ 2e^{-\eta l},$$ if $\alpha$ is a stopping time with respect to $\widetilde{\mathcal{H}}:=(\widetilde{\mathcal{H}}_t)_{t\ge 0} := \big(\sigma(U(s):s\ge0) \vee \sigma(Z(s):0\le s\le t)\big)_{t\ge 0}$ (observe that $\widetilde{\mathcal{H}}_t \subset \mathcal{H}_t$, where $\mathcal{H}_t$ is defined in ). Now fix $l$ and $\epsilon$. Define the $\widetilde{\mathcal{H}}$-stopping times $$\alpha_1\ :=\ \inf \left\{t: U(t)>\frac{1}{l^4}\right\}\wedge\frac{1}{l^4}$$ and $$\alpha_{o+1}\ :=\ \inf\left\{t: U(t)>U(\alpha_o)+\frac{1}{l^4} \right\}\wedge\left(\alpha_o+\frac{1}{l^4}\right), \quad o=1,2,\dots,$$ which yield a decomposition of the interval $[0,T]$. Note that on the event $\big\{ U(T) \le c\big\}$ there exist at most $$\label{definition_o_l} o_l:=2(c+T)l^4$$ such $\alpha_o$, i.e., we have $${\ensuremath{\mathbb{P}}}\big\{\alpha_{o_l} < T, U(\alpha_{o_l}) < c, U(T) \le c\big\}=0.$$ We define a second kind of $\widetilde{\mathcal{H}}$-stopping times depending on $\alpha_k$ via $$\label{definition_alpha_o} \tilde{\alpha}_o := \inf\{t>\alpha_o :\vert \langle f,Z(t)\rangle - \langle f,Z(\alpha_o)\rangle \vert \ge 6\epsilon\}.$$ We see from (\[eq:fstopempir\]) that $$H_o:=\vert\langle f,Z(\alpha_o)\rangle - \langle f,Z^l(\alpha_o)\rangle\vert \vee \vert\langle f,Z(\tilde{\alpha}_o)\rangle - \langle f,Z^l(\tilde{\alpha}_o)\rangle\vert$$ satisfies $$\label{bounded_h_supremum} {\ensuremath{\mathbb{P}}}\Big\{\sup_{o\le o_l}H_o\ge\varepsilon, U(T) \le c \Big\}\le \sum_{o=1}^{o_l} {\ensuremath{\mathbb{P}}}\left\{H_o\ge\varepsilon, U(T) \le c \right\}\le 8(c+T)l^4e^{-\eta l}.$$ It remains to estimate the variation of $Z^l$ and $Z$ in between the stopping times $\alpha_o$. For $u\in [\alpha_o,\alpha_{o+1})$ let $\beta_{jo}(u)$ denote the smallest index of a descendant of $X_j(\alpha_o)$, let the stopping time $\gamma_{jo}$ be the time when the smallest descendant of $X_j(\alpha_o)$ is shifted above the level $l$. Put $$\tilde{X}_j(u)= \begin{cases} X_{\beta_{jo}(u)}(u) & \mbox{if} \; u < \gamma_{jo},\\ X_{\beta_{jo}(\gamma_{jo}-)}(\gamma_{jo}-) & \mbox{if} \; u \ge \gamma_{jo}. \end{cases}$$ Observe that $$\begin{split} \langle f,Z^l(u)\rangle &- \langle f,Z^l(\alpha_o)\rangle= \langle f,Z^l(u)\rangle-\frac{1}{l} \sum_{j=1}^l f(\tilde{X}_j(u))+\frac{1}{l} \sum_{j=1}^l \Big(f(\tilde{X}_j(u)) - f(\tilde{X}_j(\alpha_o)) \Big). \end{split}$$ It will be useful to treat the two parts of the sum separately. Define $$K_1:=\max_{o\le o_l}\sup_{u\in[\alpha_o,\alpha_{o+1})}\bigg|\langle f, Z^l(u)\rangle - \frac 1 l \sum_{j=1}^l f(\tilde{X}_j(u))\bigg|$$ and $$K_2:=\max_{o\le o_l}\sup_{u\in[\alpha_o, \alpha_{o+1})}\bigg|\frac 1 l \sum_{j=1}^l \big(f(\tilde{X}_j(u))-f(\tilde{X}_j(\alpha_o))\big)\bigg|.$$ Note that the law of $K_2$ depends only on the mutation mechanism, since $\tilde{X}_j(u)$ follows the line of the individual $\tilde{X}_j(\alpha_o) = X_j(\alpha_o)$ and thus only evolves independently according to a mutation process with generator $B$. Begin with $K_1$ and note that, for $u\in[\alpha_o,\alpha_{o+1})$, $$\label{part_of_k_bounded} \langle f,Z^l(u)\rangle-\frac{1}{l}\sum_{j=1}^l f(\tilde{X}_j(u)) = \frac{1}{l}\Big(\sum_{j=1}^l f(X_j(u))-\sum_{j=1}^l f(\tilde{X}_j(u))\Big) \le \frac{2\|f\|}{l} N^l[\alpha_o,\alpha_{o+1}),$$ where $N^l[\alpha_o,\alpha_{o+1})$ is the total number of births occurring in the time interval $[\alpha_o,\alpha_{o+1})$ with index less than or equal to $l$. To see this note that at time $\alpha_o$ the two sums in the second expression cancel. A birth event in the interval $[\alpha_o,\alpha_{o+1})$ means that one type is removed from the second sum and another one is added, thus the expression can be altered by up to $2||f||/l$. There are two mechanisms which can increase $N^l[\alpha_o,\alpha_{o+1})$. It can either increase during a large birth event given by a “jump” of $\mathfrak{M}^{\Xi_{0}}$ or during a small birth event which is given by one of the “Kingman-related” Poisson-Processes $\mathfrak{N}^K_{ij}$. We first consider large birth events. Let $(v_i)$ be the jumps of $U$ in the interval $[\alpha_o, \alpha_{o+1})$, and condition on this configuration for the rest of this paragraph. At the time of the $m$-th jump, a Binomial($l, v_m$)-distributed number of levels $\le l$ participates in this event, hence $k_m$, the total number of children below level $l$ in the $m$-th birth event, satisfies $$k_m \le (b_m-1)_+,$$ where $b_m$ is Binomial($l, v_m$)-distributed. Note that we can subtract 1 from the binomial random variable, since at least one of the levels participating in the birth event must be a mother. This subtraction will be crucial later on. By elementary calculations with Binomial distributions, involving fourth moments, similar to [@DK99 p. 186], we can estimate $$\label{eq:sprich} {\ensuremath{\mathbb{P}}}\Big\{ \sum_m k_m > \epsilon l \Big\} \le {\ensuremath{\mathbb{P}}}\Big\{ \sum_m (b_m-1)_+ > \epsilon l \Big\} \le \frac{C_1}{l^6}$$ for some $0 < C_1 < \infty$. As we mentioned before, $N^l[\alpha_o,\alpha_{o+1})$ and thus $K_1$ can also be increased by the Kingman part of the birth process, but only if the parental particle and its offspring are placed below level $l$. The number of times this happens in the interval $[\alpha_o,\alpha_{o+1})$ is stochastically dominated by a Poisson distributed random variable $R$ with parameter ${l \choose 2}l^{-4}$ since the length of the interval is bounded by $l^{-4}$. So, the probability that $\frac{2\|f\|}{l}N^l[\alpha_o,\alpha_{o+1})$ exceeds $2\epsilon$ due to this mechanism is bounded by the probability that $R$ exceeds $\frac{l\epsilon}{\|f\|}$. By elementary estimates on the tails of Poisson random variables, we have $$\label{eq:Poisstail} {\ensuremath{\mathbb{P}}}\Big\{R > \frac{l\epsilon}{\|f\|}\Big\} \le e^{-\eta_1 l},$$ for some $\kappa>0$ and $l$ large enough. Combining (\[eq:sprich\]) and (\[eq:Poisstail\]), we obtain $$\label{k1_final_bound} \begin{aligned} {\ensuremath{\mathbb{P}}}\big\{ K_1 > 2\epsilon, U(T) \le c \big\} & = {\ensuremath{\mathbb{P}}}\Big\{ \max_{o\le o_l}\sup_{u\in[\alpha_o,\alpha_{o+1})}\big\vert\langle f, Z^l(u)\rangle - \frac 1 l \sum_{j=1}^l f(\tilde{X}_j(u)) \big\vert > 2\epsilon, U(T) \le c \Big\}\\ & \le o_l \Big( \frac{C_1}{l^{6}} + e^{-\eta_1 l} \Big), \end{aligned}$$ for $l$ large enough. This controls the increments of $\langle f,Z^l\rangle$ in the intervals $[\alpha_o,\alpha_{o+1})$. We now consider $K_2$. Observe that $$\begin{aligned} \frac 1 l \sum_{j=1}^l (f(\tilde{X}_j(u))-f(\tilde{X}_j(\alpha_o))) & = \frac{1}{l} \sum_{j=1}^l \bigg(f(\tilde{X}_j(u))-f(\tilde{X}_j(\alpha_o)) - \int^u_{\alpha_o} Bf(\tilde{X}_j(s))ds \bigg) \notag \\ & \quad + \frac{1}{l} \sum_{j=1}^l \int^u_{\alpha_o} Bf(\tilde{X}_j(s))ds,\end{aligned}$$ and that, for $u \ge \alpha_o$ and each $o$, $$M_{lo}(u \wedge \alpha_{o+1}) := \frac{1}{l} \sum_{j=1}^l \bigg(f(\tilde{X}_j(u\wedge \alpha_{o+1}))-f(\tilde{X}_j(\alpha_o)) - \int^{u\wedge \alpha_{o+1}}_{\alpha_o} Bf(\tilde{X}_j(s))ds \bigg)$$ is a martingale. For $l$ so large that $l^{-4}\|Bf\| \le \varepsilon$, we have $$\begin{split}\label{k2_first_bound} {\ensuremath{\mathbb{P}}}\big\{K_2 \ge 2\varepsilon, U(T) \le c \big\} & \le \sum^{o_l-1}_{o=0} {\ensuremath{\mathbb{P}}}\Big\{\sup_{\alpha_o \le u < \alpha_{o+1}} \vert M_{lo}(u) + \frac{1}{l} \sum_{j=1}^l \int^u_{\alpha_o} Bf(\tilde{X}_j(s))ds \vert \ge 2\varepsilon, U(T) \le c\Big\}\\ & \le \sum^{o_l-1}_{o=0} {\ensuremath{\mathbb{P}}}\Big\{\sup_{\alpha_o \le u < \alpha_{o+1}} \vert M_{lo}(u) \vert + l^{-4}\|Bf\| \ge 2\varepsilon, U(T) \le c\Big\}\\ & \le \sum^{o_l-1}_{o=0} {\ensuremath{\mathbb{P}}}\Big\{\sup_{\alpha_o \le u < \alpha_{o+1}} \vert M_{lo}(u) \vert \ge \varepsilon, U(T) \le c \Big\}. \end{split}$$ We now need to bound each summand. Using the notation $$M_{lo}(u) = \frac{1}{l}\sum_{j=1}^l e_j(u),$$ where $$e_j(u) := f(\tilde{X}_j(\alpha_{o+1} \wedge u))-f(\tilde{X}_j(\alpha_o)) - \int^{\alpha_{o+1}\wedge u}_{\alpha_o} Bf(\tilde{X}_j(s))ds, \quad u \in [0,1],$$ each $(e_i(u))_u$ is a martingale with ${\ensuremath{\mathbb{E}}}e_j(u)=0$ and $|e_j(u)|\le 2\|f\|+\|Bf\|/l^4=:K$ almost surely. Moreover, the $e_j(u)$ are exchangeable. We obtain from Lemma \[A2\] $$\label{condlargedev} {\ensuremath{\mathbb{P}}}\Big\{\sup_{\alpha_o \le u < \alpha_{o+1}} \vert M_{lo}(u) \vert \ge \varepsilon\Big\} \le 2 e^{-\eta_2 l},$$ for some $\eta_2>0$. Combining this result with , we arrive at $$\label{k2_final_bound} {\ensuremath{\mathbb{P}}}\big\{ K_2 \ge 2\varepsilon, U(T) \le c \big\} \le o_l C_2 e^{-\eta_2 l}.$$ Now observe that if $\max_{o\le o_l} H_o <\epsilon$, $K_1<2\epsilon$ and $K_2<2\epsilon$, then $\tilde{\alpha}_o\ge \alpha_{o+1}$. This can easily be seen by contradiction. Indeed, if we assume that $\tilde{\alpha}_o < \alpha_{o+1}$, this would imply $$\label{restating_definition} \vert \langle f,Z(\alpha_o)\rangle - \langle f,Z(\tilde{\alpha}_o)\rangle \vert \ge 6\epsilon,$$ according to . But on the other hand we know that $$\vert\langle f,Z(\alpha_o)\rangle - \langle f,Z^l(\alpha_o)\rangle\vert < \epsilon \text{ and }\vert\langle f,Z(\tilde{\alpha}_o)\rangle - \langle f,Z^l(\tilde{\alpha}_o)\rangle\vert < \epsilon \quad \forall o$$ due to our bound on $H_o$. Since the distance between $\langle f,Z\rangle$ and $\langle f,Z^l\rangle$ was at most $\epsilon$ at the beginning of the interval and $\langle f,Z^l\rangle$ can only have moved by at most $4\epsilon$ on the event $\{ K_1 \le 2\epsilon \}\cap\{ K_2 \le 2\epsilon \} \cap \{ \max_{o\le o_l}H_o \le \epsilon\}$, $$\vert \langle f,Z(\alpha_o)\rangle - \langle f,Z^l(\tilde{\alpha}_o)\rangle \vert < 5\epsilon$$ must hold if $\tilde{\alpha}_o \le \alpha_{o+1}$. But equation states that, $\langle f,Z(\tilde{\alpha}_o)\rangle$ is more than $6\epsilon$ away from its starting point, so this contradicts that it can only be $\epsilon$ away from $\langle f,Z^l(\tilde{\alpha}_o)\rangle$ which is ensured by our condition on $H_o$. Thus $\tilde{\alpha}_o$ has to be greater than $\alpha_{o+1}$ which in turn implies that $$\label{eq_sup_for_z} \sup_{\alpha_o \le u < \alpha_{o+1}}\Big\{\big\vert \langle f,Z(u)\rangle - \langle f,Z(\alpha_o)\rangle \big\vert\Big\} \le 6\epsilon$$ holds on the event $\{ K_1 \le 2\epsilon \}\cap\{ K_2 \le 2\epsilon \} \cap \{ \max_{o\le o_l}H_o \le \epsilon\}$. Putting observations and , the bound and the bound together, we finally obtain $${\ensuremath{\mathbb{P}}}\Big\{ \sup_{0\le t\le T}\big\vert\langle f,Z(t) \rangle - \langle f,Z^l(t) \rangle\big\vert \ge 11\epsilon, U(T)\le c \Big\} \le \delta_l$$ with $$\delta_l := 8(c+T)l^4e^{-\eta l} + o_l C_1 l^{-6} + o_l e^{-\eta_1 l} + o_l C_2 e^{-\eta_2 l}$$ which is the statement of the lemma since due to equation $o_l\sim l^4$ holds and therefore the $\delta_l$ are summable. Almost sure convergence of $Z^l$ to $Z$ with respect to the metric follows directly from Lemma \[lemma\_for\_bound\] and the Borel-Cantelli Lemma, completing the proof of Theorem \[main\]. The Hille-Yosida approach ========================= In this section we provide two alternative representations of the $\Xi_0$-Fleming-Viot generator, leading to the distributional duality to the $\Xi$-coalescent discussed in Section \[dualities\], and we show that they generate a Markov semigroup on ${\ensuremath{\mathcal{M}}}_1(E)$, hence leading to a classical construction of the $\Xi_0$-Fleming-Viot process as a Markov process. Two representations of the $\Xi_0$-Fleming-Viot generator --------------------------------------------------------- Recall that if the type space $E$ is a compact Polish space (which is assumed in this paper), then the set ${\ensuremath{\mathcal{M}}}_1(E)$ of all probability measures on $E$, equipped with the weak topology, is again a Polish space. We briefly recall the notation from Section \[sec:introduction\]. For $f:E^n\to{\ensuremath{\mathbb{R}}}$ bounded and measurable consider the test function $$\label{eq:testfunction1} G_f(\mu)\ :=\ \int_{E^n} f(x_1,\dots,x_n)\,\mu^{\otimes n}(dx_1,\dots,dx_n), \quad\mu\in{\ensuremath{\mathcal{M}}}_1(E).$$ The linear operator $L^{\Xi_0}$ was defined via $$\label{eq:genXiFV1} L^{\Xi_{0}} G_f(\mu) = \int_\Delta \int_{E^{\ensuremath{\mathbb{N}}}} \left[ G_f\big( (1-|{\ensuremath{\boldsymbol\zeta}}|)\mu + {\textstyle\sum_{i=1}^\infty \zeta_i \delta_{x_i}} \big) - G_f(\mu) \right] \mu^{\otimes {\ensuremath{\mathbb{N}}}}(d\mathbf{x}) \frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})}.$$ This operator is the $\Xi_0$-Fleming-Viot generator from Proposition \[mainprop\]. The following representation will be useful to establish the duality with the $\Xi_0$-coalescent. Note that if $\Xi$ is concentrated on $\{{\ensuremath{\boldsymbol\zeta}}\in\Delta\,:\,\zeta_i=0\mbox{ for all } i\ge 2\}$, i.e., if the corresponding coalescent is a $\Lambda$-coalescent, then this result has already been obtained by Bertoin and Le Gall [@BLG03 Eqs. (16) and (17)]. For convenience, we will denote the transition rates by $$\label{eq:xi.rate} \lambda(k_1,\dots,k_p)\ =\ \lambda_{b;k_1,\dots,k_r;s},$$ where $k_1\ge\dots\ge k_r\ge 2$, $p-r=s$ and $k_{r+1}=\ldots=k_p=1$. Furthermore, define for $p,n_1,\dots,n_p\in{\ensuremath{\mathbb{N}}}$ such that $n_1+\dots+n_p>p$ ($\Leftrightarrow$ not all $n_i=1$) $$\label{eq:xi.rate.alt} \lambda(n_1,\dots,n_p)\ :=\ \lambda(k_1,\dots,k_p),$$ where $k_1\ge\dots\ge k_p$ is the re-arrangement of $n_1,\dots,n_p$ in decreasing order. \[lemma\_generator\_alt\] The operator $L^{\Xi_0}$ has the alternative representation $$\label{eq:genXiFV.alt} L^{\Xi_{0}} G_f(\mu) \ =\ \sum_{\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n \atop \text{not all singletons}} \hspace{-1em} \lambda(|A_1|,\dots,|A_p|)\int_{E^n} \left(f\big(\mathbf{x}[\pi]\big)-f(\mathbf{x})\right) \mu^{\otimes n}(dx_1,\dots,dx_n),$$ where $\mathbf{x}[\{A_1,\dots,A_p\}]\in E^n$ has entries $$(\mathbf{x}[\{A_1,\dots,A_p\}])_i\ :=\ x_{\min A_j} \quad\text{if\ \ $i\in A_j$, $i=1,\dots,n$.}$$ Note that basically boils down to , if $|A_i|=1$ for all but one $A_i$. First note that for fixed ${\ensuremath{\boldsymbol\zeta}}$ and $\mathbf{x}$, $$\label{eq:funnysum1} \begin{aligned} G_f\big( &(1-|{\ensuremath{\boldsymbol\zeta}}|)\mu + {\textstyle\sum_{i=1}^\infty \zeta_i \delta_{x_i}}\big) \\ & = \sum_{\phi : \{1,\dots,n\}\to{\ensuremath{\mathbb{Z}}}_+} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{a(\phi)} \prod_{j \le n\, : \phi(j)>0} \zeta_{\phi(j)} \int_{E^{a(\phi)}} f\big(\eta(\phi, \mathbf{x}, \mathbf{y})\big) \mu^{\otimes a(\phi)}(dy_1,\dots,dy_{a(\phi)}), \end{aligned}$$ where $a(\phi):=\#\{1\le j\le n:\phi(j)=0\}$ and $\eta(\phi,\mathbf{x},\mathbf{y})\in E^n$ is given by $$\eta(\phi, \mathbf{x}, \mathbf{y})_j \ =\ \left\{\begin{array}{cl} x_{\phi(j)} & \mbox{if}\; \phi(j) > 0, \\[1ex] y_k & \mbox{if}\; \phi(j) = 0, \; \mbox{where}\; k = \#\{ 1 \le j' \le j : \phi(j') = 0\}. \end{array}\right.$$ Identity (\[eq:funnysum1\]) can be understood as follows: Expanding the $n$-fold product of $(1-|{\ensuremath{\boldsymbol\zeta}}|)\mu + \sum_{i=1}^\infty \zeta_i \delta_{x_i}$, we put $\phi(j)=0$ if in the $j$-th factor, we use $(1-|{\ensuremath{\boldsymbol\zeta}}|)\mu$, and we put $\phi(j)=i$ if we use $\zeta_i \delta_{x_i}$ in the $j$-factor. Each $\phi:\{1,\dots,n\}\to{\ensuremath{\mathbb{Z}}}_+$ is uniquely described by a partition $\pi=\{A_1,\dots,A_p\} \in \mathcal{P}_n$ with labels $\ell_1,\dots,\ell_p \in {\ensuremath{\mathbb{Z}}}_+$ by defining $j \sim_\phi j'$ if and only if $\phi(j)=\phi(j')>0$ and putting $\ell_i := \phi(A_i)$, $i=1,\dots,p$. Note that for a given partition $\{A_1,\dots,A_p\}$, any vector $(\ell_1,\dots,\ell_p) \in {\ensuremath{\mathbb{Z}}}_+^p$ of labels with the properties $$\ell_i=0 \; \Rightarrow \; |A_i|=1 \quad \mbox{and} \quad i \neq j, \ell_i, \ell_j \neq 0 \; \Rightarrow \; \ell_i \neq \ell_j$$ is admissible. Thus we have $$\begin{aligned} \notag \lefteqn{\int_{E^{\ensuremath{\mathbb{N}}}} G_f\big( (1-|{\ensuremath{\boldsymbol\zeta}}|)\mu + {\textstyle\sum_{i=1}^\infty \zeta_i \delta_{x_i}}\big) \mu^{\otimes {\ensuremath{\mathbb{N}}}}(d\mathbf{x})} \\ \label{eq:altform1} & = & \sum_{\pi=\{A_1,\dots,A_p\} \in \mathcal{P}_n} \sum_{(\ell_1,\dots,\ell_p) \atop \text{admissible}} \hspace{-1em} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{\#\{1\le i\le p : \ell_i=0\}} \prod_{i=1 \atop \ell_i>0}^p \zeta_{\ell_i}^{|A_i|} \int_{E^n} f(\mathbf{x}[\pi])\,\mu^{\otimes n}(d\mathbf{x}). \end{aligned}$$ Note that, for a given partition with $p$ blocks, the integration appearing in the last line runs effectively only over $E^p$. For further simplification assume that the blocks $A_1,\ldots,A_p$ of $\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n$ are enumerated according to decreasing block size, and write $s(\pi)$ for the number of singleton blocks of the partition $\pi=\{A_1,\dots,A_p\}$. Then, for a given $\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n$, the last sum in (\[eq:altform1\]) can be written as $$\sum_{l=0}^{s(\pi)} {{s(\pi)}\choose l} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l} \sum_{i_1,\dots,i_{p-s(\pi)+l}\in{\ensuremath{\mathbb{N}}}\atop\text{all distinct}} \hspace{-2em} \zeta_{i_1}^{|A_1|}\cdots\zeta_{i_{p-s(\pi)+l}}^{|A_{p-s(\pi)+l}|} \int_{E^n} f\big(\mathbf{x}[\pi]\big)\mu^{\otimes n}(d\mathbf{x}).$$ Furthermore, for any ${\ensuremath{\boldsymbol\zeta}}\in\Delta$ and $n\in{\ensuremath{\mathbb{N}}}$, $$\begin{aligned} 1 & = & \Big( \big(1-|{\ensuremath{\boldsymbol\zeta}}|\big) + {\textstyle\sum_{i=1}^\infty} \zeta_i \Big)^n\\ & = & \sum_{\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n} \sum_{l=0}^{s(\pi)} {s(\pi)\choose l} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l} \hspace{-2em} \sum_{i_1,\dots,i_{p-s(\pi)+l}\in{\ensuremath{\mathbb{N}}}\atop\text{all disticnt}} \hspace{-2em} \zeta_{i_1}^{|A_1|}\cdots\zeta_{i_{p-s(\pi)+l}}^{|A_{p-s(\pi)+l}|}. \end{aligned}$$ This allows us to re-express the inner integral in (\[eq:genXiFV1\]) as $$\begin{aligned} \sum_{\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n} & \sum_{l=0}^{s(\pi)} {s(\pi) \choose l} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l} \hspace{-2em} \sum_{i_1,\dots,i_{p-s(\pi)+l} \in {\ensuremath{\mathbb{N}}}\atop \text{all distinct}} \hspace{-2em} \zeta_{i_1}^{|A_1|}\cdots\zeta_{i_{p-s(\pi)+l}}^{|A_{p-s(\pi)+l}|} \int_{E^n} [f\big(\mathbf{x}[\pi]\big) -f(\mathbf{x})] \, \mu^{\otimes n}(d\mathbf{x}) \\[2ex] = & \sum_{\pi=\{ A_1,\dots,A_p\} \in \mathcal{P}_n \atop \text{not all singletons}} \hspace{0em} \sum_{l=0}^{s(\pi)} {s(\pi) \choose l} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l} \hspace{-2em} \sum_{i_1,\dots,i_{p-s(\pi)+l} \in {\ensuremath{\mathbb{N}}}\atop \text{all distinct}} \hspace{-2em} \zeta_{i_1}^{|A_1|}\cdots\zeta_{i_{p-s(\pi)+l}}^{|A_{p-s(\pi)+l}|} \\ & \hspace{10em} {} \times \int_{E^n} [f\big(\mathbf{x}[\pi]\big) -f(\mathbf{x})] \, \mu^{\otimes n}(d\mathbf{x}), \end{aligned}$$ because $\mathbf{x}[\{\{1\},\dots,\{n\}\}] = \mathbf{x}$. Integrating this equation over $\Delta$ with respect to the measure $({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})^{-1}\Xi_0$ yields (\[eq:genXiFV.alt\]). Note that (see also [@S03 p. 844]) $$\begin{aligned} & & \hspace{-15mm} \sum_{\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n\atop\text{not all singletons}} \sum_{l=0}^{s(\pi)} {s(\pi)\choose l} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l} \hspace{-2em} \sum_{i_1,\dots,i_{p-s(\pi)+l}\in{\ensuremath{\mathbb{N}}}\atop\text{all distinct}} \hspace{-2em} \zeta_{i_1}^{|A_1|}\cdots\zeta_{i_{p-s(\pi)+l}}^{|A_{p-s(\pi)+l}|}\\ & \le & \sum_{\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n\atop\text{not all singletons}} \Big(\sum_{i_1=1}^\infty \zeta_{i_1}^2\Big) \sum_{l=0}^{s(\pi)} {s(\pi)\choose l} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l} \hspace{-2em} \sum_{i_{p-s(\pi)+1},\dots,i_{p-s(\pi)+l}\in{\ensuremath{\mathbb{N}}}} \hspace{-2em} \zeta_{i_{p-s(\pi)+1}}\cdots\zeta_{i_{p-s(\pi)+l}}\\ & = & \sum_{\pi=\{A_1,\dots,A_p\}\in\mathcal{P}_n\atop\text{not all singletons}} ({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})\sum_{l=0}^{s(\pi)} {{s(\pi)}\choose l}(1-|{\ensuremath{\boldsymbol\zeta}}|)^{s(\pi)-l}|{\ensuremath{\boldsymbol\zeta}}|^l \ = \ (|\mathcal{P}_n|-1)\,({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}}) \end{aligned}$$ to verify that there is no singularity near ${\ensuremath{\boldsymbol\zeta}}=\mathbf{0}$. Construction of the Markov semigroup and proof of Proposition \[mainprop\] {#markov_semigroup} -------------------------------------------------------------------------- The following proposition ensures that there exists a Markov process attached to the $\Xi_{0}$-Fleming-Viot generator. \[prop:semigroup\] The closure of $\{(G_f, L^{\Xi_{0}}G_f):n\in{\ensuremath{\mathbb{N}}}, f:E^n\to{\ensuremath{\mathbb{R}}}\; \mbox{bounded and measurable}\}$ generates a Markov semigroup on ${\ensuremath{\mathcal{M}}}_1(E)$. We write $G$ instead of $G_f$ for convenience. By the Hille-Yosida theorem (see, for example, [@EK86 p. 165, Theorem 2.2]) it is sufficient to verify that 1. the domain $D$ is dense in $C({\mathcal M}_1(E))$, 2. the operator $L^{\Xi_0}$ satisfies the positive maximum principle, i.e., $L^{\Xi_0}G(\mu)\le 0$ for all $G\in D$, $\mu\in{\mathcal M}_1(E)$ with $\sup_{\nu\in{\mathcal M}_1(E)}G(\nu)=G(\mu)\ge 0$, and that 3. the range of $\lambda-L^{\Xi_{0}}$ is dense in $C({\mathcal M}_1(E))$ for some $\lambda>0$. In order to verify (i) and (iii) we mimic the proof of Proposition 3.5 in Chapter 1 of [@EK86] and construct a suitable sequence $D_1,D_2,\ldots$ of finite-dimensional subspaces of $C({\mathcal M}_1(E))$ such that $D:=\bigcup_{k\in{\ensuremath{\mathbb{N}}}}D_k$ is dense in $C({\mathcal M}_1(E))$ and $L^{\Xi_{0}}:D_k\to D_k$ for all $k\in{\ensuremath{\mathbb{N}}}$ as follows. For $n\in{\ensuremath{\mathbb{N}}}$ and $f:E^n\to{\ensuremath{\mathbb{R}}}$ bounded and measurable let $D_f$ denote the set of all linear combinations of elements from the set $$\{G:G(\mu)={\textstyle\int f(\mathbf{x}[\pi])\, \mu^{\otimes n}(d\mathbf{x})},\pi\in{\mathcal P}_n\}.$$ Since $|{\mathcal P}_n|<\infty$, it is easily seen that $D_f$ is a finite-dimensional subspace of $C({\ensuremath{\mathcal{M}}}_1(E))$. From (\[eq:genXiFV.alt\]) it follows that $L^{\Xi_0}:D_f\to D_f$. For each $n\in{\ensuremath{\mathbb{N}}}$ let $\{g_{nm}:m\in{\ensuremath{\mathbb{N}}}\}\subset C(E^n)$ be dense, and let $\{f_k:k\in{\ensuremath{\mathbb{N}}}\}$ be an enumeration of $\{g_{nm}:n,m\in{\ensuremath{\mathbb{N}}}\}$. Then, $D_k:=D_{f_k}$, $k\in{\ensuremath{\mathbb{N}}}$, has the desired properties. Note that $D:=\bigcup_{k\in{\ensuremath{\mathbb{N}}}}D_k$ is dense in $C({\mathcal M}_1(E))$ (Stone-Weierstrass), i.e. condition (i) holds. We have $(\lambda-L^{\Xi_{0}})(D_k)=D_k$ for all $\lambda$ not belonging to the set of eigenvalues of $L^{\Xi_{0}}|_{D_k}$, i.e., for all but at most finitely many $\lambda>0$. Thus, $(\lambda-L^{\Xi_{0}})(D)=(\lambda-L^{\Xi_{0}})(\bigcup_{k\in{\ensuremath{\mathbb{N}}}}D_k) =\bigcup_{k\in{\ensuremath{\mathbb{N}}}}D_k=D$ is dense in $C({\mathcal M}_1(E))$ for all but at most countably many $\lambda>0$. In particular, condition (iii) is satisfied. Condition (ii) follows from the fact that the expression inside the integrals in (\[eq\_genXiFV\_xi\]) satisfies $$G((1-|{\ensuremath{\boldsymbol\zeta}}|)\mu+{\textstyle\sum_{i=1}^\infty\zeta_i\delta_{x_i}}) - G(\mu)\ \le \sup_{\nu\in{\mathcal M}_1(E)}G(\nu)-G(\mu) \ =\ G(\mu)-G(\mu)\ =\ 0$$ for all ${\mathbf x}=(x_1,x_2,\ldots)\in E^{\ensuremath{\mathbb{N}}}$, ${{\ensuremath{\boldsymbol\zeta}}}\in\Delta$, $G\in D$ and $\mu\in{\mathcal M}_1(E)$ with $\sup_{\nu\in{\mathcal M}_1(E)}G(\nu)=G(\mu)$. Thus, the Hille-Yosida theorem ensures that the closure $\overline{L^{\Xi_0}}$ of $L^{\Xi_0}$ on $C({\mathcal M}_1(E))$ is single-valued and generates a strongly continuous, positive, contraction semigroup $\{T_t\}_{t\ge 0}$ on ${\mathcal M}_1(E)$. Note that from (iii) it follows that $D$ is a core for $\overline{L^{\Xi_{0}}}$ ([@EK86 p. 166]). The operator $L^{\Xi_{0}}$ maps constant functions to the zero function, i.e., $L^{\Xi_{0}}$ is conservative. Thus, $\{T_t\}_{t\ge 0}$ is a Feller semigroup and corresponds to a Markov process with sample paths in $D_{{\mathcal M}_1(E)}([0,\infty))$. \[remark\_generator\] i) If the finite measure $\Xi$ on $\Delta$ allows for some mass $a:=\Xi(\{\mathbf 0\})$ at zero, then $L^{\Xi_{0}}$ has to be replaced by $L^\Xi := L^{\Xi_{0}} + L^{a\delta_{\bf 0}}$, where $L^{\Xi_{0}}$ is defined as before and $L^{a\delta_{\bf 0}}$ is the generator of the classical Fleming-Viot process [@FV79] given by . The existence of a Markov process $Z=(Z_t)_{t\ge 0}$ with generator $L^\Xi$ can be deduced as in the proof of Proposition \[prop:semigroup\] via the Hille-Yosida theorem.\ ii) The construction of the Markov process attached to the ‘full’ generator $L$, including the Kingman component and the mutation component , works via the standard Trotter approach.\ iii) Note that $\int (L^\Xi)G\,d\delta_{\delta_x}=0$, $x\in E$, where $\delta_{\nu}\in{\mathcal M}_1({\mathcal M}_1(E))$ denotes the unit mass at $\nu\in{\mathcal M}_1(E)$. Thus, see [@EK86 p. 239, Proposition 9.2], the states $\delta_x$, $x\in E$, are absorbing for the $\Xi$-Fleming-Viot process. We now turn to the proof of Proposition \[mainprop\]. Indeed, we verify the following [*Claim:*]{} The distribution of the measure valued Markov process with generator $L$, as defined in Remark \[remark\_generator\] ii), coincides with the distribution of the $(\Xi,B)$-Fleming-Viot process, as defined in Theorem \[main\]. It suffices to verify the following lemma. \[thm\_martingale\_problem\] The $(\Xi,B)$-Fleming-Viot process defined in Theorem \[main\] solves the martingale problem for the generator $L$ given in . To prepare this, let us concentrate on the case when there is no mutation and no Kingman-component ($L=L^{\Xi_0}$). Fix $l$ and suppose we are at the $m$-th birth event. As in the previous section, let $\{\phi_m^1,\ldots,\phi_m^{a_m}\}$ denote the assignments of the levels to one of the $a_m$ families. So $\phi_m^i \subset \{1,\ldots,l\}$ and $\phi_m^i\cap\phi_m^i\neq\emptyset$ for all $i,j$. Furthermore, we again denote by $\Phi_m := \bigcup_{i=1}^{a_m}\phi_m^i$ all individuals participating in the birth event. Note, that this can be a strict subset of $\{0,\ldots,l\}$, and $\{\phi_m^1,\ldots,\phi_m^{a_m}\}$ holds all information about what is going on at the birth event. The function $g({\ensuremath{\boldsymbol\zeta}},u)$ is defined as in . We introduce a Poisson process counting the number of times a specific birth event $\{\phi_m^1,\ldots,\phi_m^{a_m}\}$ happens. With $(t_m,{\ensuremath{\boldsymbol\zeta}}_m,{\mathbf u}_m)$ denoting the points of the Poisson point process $\mathfrak{M}^{\Xi_{0}}$ we define $$L_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}}(t) \ :=\ \sum_{t_m\le t} \sum_{b_1,\ldots,b_{a_m}\in{\ensuremath{\mathbb{N}}}\atop \text{all\,distinct}} \prod_{i=1}^{a_m} \prod_{j\in\phi^i_m} {\ensuremath{\mathbbm{1}}}_{\{g({\ensuremath{\boldsymbol\zeta}}_m,u_{mj})=b_i\}} \prod_{j \in \{1,\ldots,l\}\setminus\Phi_m} {\ensuremath{\mathbbm{1}}}_{\{g({\ensuremath{\boldsymbol\zeta}}_m,u_{mj})=\infty\}}.$$ To describe the effect of the birth event $\{\phi_m^1,\ldots,\phi_m^{a_m}\}$ on the population vector $x \in E^l$ we introduce the function $\mathfrak{T}$ defined by $$\big(\mathfrak{T}_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}} ({\mathbf x})\big)_i := \begin{cases} x_{\min(\phi_m^j)} &\text{if}\, k \in \phi_m^j,\\ x_{J_m(i)} &\text{else} \end{cases}$$ for all $k \in \{1,\ldots,l\}$, where $J_m$ is the function defined in that holds the information on where the non-participating particles should look down to. With this notation we can use equation and the dependence between the $L_{J,k}^l$ and $L_{J}^l$ to show that $$\label{eq_level_stochastic_differential_equation} X^l(t) := X^l(0) + \sum_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}, \atop \dot{\bigcup}\phi_m^i \subset \{1,\ldots,l\}} \int_0^t \Big( \mathfrak{T}_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}}\big(X^l(s-)\big) - X^l(s-) \Big)\, dL_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}}(s)$$ describes the evolution of the first $l$ levels $X^l \in E^l$, if we assume no mutation and no Kingman part. Note that for simplicity we use the notation $X^l = (X_1, \ldots, X_l)$. Since the $L_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}}(t)$ are Poisson processes derived from the Poisson point process $\mathfrak{M}^{\Xi_{0}}$ it is straightforward to verify that their rates are given by $$\label{eq_rates_of_poisson} r\big(\{\phi_m^1,\ldots,\phi_m^{a_m}\}\big) \ :=\ \sum_{i_1,\ldots,i_{a_m} \atop \text{all\ distinct}} \int_\Delta \zeta_{i_1}^{k_m^1+1}\cdots\zeta_{i_r}^{k_m^r+1} \zeta_{i_{r+1}}\cdots\zeta_{i_{a_m}}(1-|{\ensuremath{\boldsymbol\zeta}}|)^{(l-|\Phi|)} \frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})},$$ where $k_m^i+1=|\phi_m^i|$ as before and the sets are ordered, such that $k_m^1\ge\cdots\ge k_m^r\ge 1$ and $k_m^{r+1}=\cdots=k_m^{a_m}=0$ hold. Assume that at least $k_m^1\ge 1$ holds, because otherwise $\mathfrak{T}$ is the identity. Note that under this assumption the integral in is finite (c.f.[@S00] or [@S03]). We now turn to the actual proof of the lemma. We will prove the result for the generator $L^{\Xi_0}$. The full result can then be obtained in analogy to the proof of Theorem 2.4 in [@DK96]. Indeed, we have to show that for each function $G_f\in\mathcal{D}(L^{\Xi_0})$ of the form $$G_f(\mu)\ =\ \langle f,\mu^{\otimes l}\rangle,$$ for $\mu\in\mathcal{M}_1(E)$ and $f\colon E^l \to {\ensuremath{\mathbb{R}}}$ bounded and measurable, $$\label{martingale} G_f(Z(t)) - G_f(Z(0)) - \int_0^t (L^{\Xi_0}G_f)(Z(s))\,ds$$ is a martingale with respect to the natural filtration of the Poisson point process $\mathfrak{M}^{\Xi_{0}}$ given by $$\{\mathcal{J}_t\}_{t\ge 0} := \Big\{ \sigma\big( \mathfrak{M}^{\Xi_{0}}\Big\vert_{[0,t]\times\Delta\times[0,1]^{\ensuremath{\mathbb{N}}}}\big) \Big\}_{t\ge 0}.$$ Note that $$\label{f_equals_b} \mathbb{E}\Big[ f\big( X_1(s),\ldots,X_l(s)\big)\Big\vert \mathcal{J}_t \Big] = \mathbb{E}\Big[ \big\langle f,Z(s)^{\otimes l}\big\rangle \Big\vert \mathcal{J}_t\Big]$$ holds for all $s,t\ge 0$, which will be crucial in the following steps. We start by observing that, for $0\le w\le t$, the representation leads to $$\begin{gathered} \label{eq_stochastic_martingale} 0 = \mathbb{E} \Big[ f\big(X^l(t)\big) - f\big(X^l(w)\big)\\ - \sum_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}, \atop \dot{\bigcup}\phi_m^i \subset \{1,\ldots,l\}}\int_w^t \Big( f\Big(\mathfrak{T}_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}}\big(X^l(s)\big)\Big) - f\big(X^l(s)\big) \Big)r\big(\{\phi_m^1,\ldots,\phi_m^{a_m}\}\big)\, ds \Big\vert \mathcal{J}_w \Big],\end{gathered}$$ since this is a martingale. Using the definition of the rates and the fact that due to the exchangeability of $X^l$, the action of $\mathfrak{T}_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}}$ and the $[\pi]$ operation under the expectation is the same, we can now rewrite the last term (without the substraction of $f(X^l(s))$ from the integrand) as $$\begin{aligned} & & \hspace{-8mm} \mathbb{E} \bigg[ \int_w^t \sum_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}, \atop \dot{\bigcup}\phi_m^i \subset \{1,\ldots,l\}} r\big(\{\phi_m^1,\ldots,\phi_m^{a_m}\}\big) f\Big(\mathfrak{T}_{\{\phi_m^1,\ldots,\phi_m^{a_m}\}} \big(X^l(s)\big)\Big)\, ds \bigg| \mathcal{J}_w \bigg]\nonumber\\ & = & \mathbb{E} \bigg[ \int_w^t \sum_{\pi=\{A_1,\dots,A_p\} \in \mathcal{P}_n} \sum_{(r_1,\dots,r_p) \atop \text{admissible}} \int_\Delta (1-|{\ensuremath{\boldsymbol\zeta}}|)^{\#\{r_i=0\}} \prod_{i=1 \atop r_i>0}^p \zeta_{r_i}^{|A_i|} \frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})} f\Big( \big(X^l(s)\big)[\pi]\Big)\,ds \bigg| \mathcal{J}_w \bigg]\nonumber\\ & = & \mathbb{E} \bigg[ \int_w^t \int_\Delta \sum_{\pi=\{A_1,\dots,A_p\} \in \mathcal{P}_n} \sum_{(r_1,\dots,r_p) \atop \text{admissible}} (1-|{\ensuremath{\boldsymbol\zeta}}|)^{\#\{r_i=0\}} \prod_{i=1 \atop r_i>0}^p \zeta_{r_i}^{|A_i|} \langle f\circ [\pi],Z(s)^{\otimes l} \rangle \frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})} \,ds \bigg| \mathcal{J}_w \bigg]\nonumber\\ & = & \mathbb{E} \bigg[ \int_w^t \int_\Delta \int_{E^{\ensuremath{\mathbb{N}}}} G_f\big( (1-|{\ensuremath{\boldsymbol\zeta}}|)Z(s) + {\textstyle\sum_{i=1}^\infty \zeta_i \delta_{x_i}}\big) Z(s)^{\otimes {\ensuremath{\mathbb{N}}}}(d\mathbf{x})\frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})} \,ds\bigg| \mathcal{J}_w \bigg], \label{eq_really_weird_stuff}\end{aligned}$$ since the sum about the configurations $\{\phi_m^1,\ldots,\phi_m^{a_m}\}$ and the distinct indices $i_1,\ldots,i_{a_m}$ can be rewritten as the sum about the partitions $\pi$ and the admissible vectors $(r_1,\ldots,r_p)$. The last equality holds due to equation . Combining equation with equation we see that $$\begin{aligned} 0 & = & \mathbb{E}\bigg[ f\big(X^l(t)\big) - f\big(X^l(w)\big)\nonumber\\ & & - \int_w^t \int_\Delta \int_{E^{\ensuremath{\mathbb{N}}}} \big(G_f\big((1-|{\ensuremath{\boldsymbol\zeta}}|)Z(s) + {\textstyle\sum_{i=1}^\infty \zeta_i \delta_{x_i}}\big) - G_f(Z(s))\big) Z(s)^{\otimes {\ensuremath{\mathbb{N}}}}(d\mathbf{x})\frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})} \,ds \bigg|\mathcal{J}_w \bigg]\nonumber\\ & = & \mathbb{E}\bigg[ \langle f,Z(t)^{\otimes l}\rangle - \langle f,Z(w)^{\otimes l}\rangle - \int_w^t (L^{\Xi_{0}}G_f)(Z(s))\,ds\bigg|\mathcal{J}_w \bigg]\nonumber\\ & = & \mathbb{E}\bigg[ G_f\big(Z(t)\big) - G_f\big(Z(w)\big) - \int_w^t(L^{\Xi_0}G_f)(Z(s))\,ds\bigg|\mathcal{J}_w \bigg]\end{aligned}$$ holds, where we use in the second equality. Thus, (\[martingale\]) is a martingale. Dualities ========= Distributional duality versus pathwise duality {#sec_distributional_duality} ---------------------------------------------- We first establish a [*distributional duality*]{} in the classical sense of [@L85]. Indeed, (\[eq:genXiFV.alt\]) and results about the classical Fleming-Viot process bring forth the following duality between a $\Xi$-coalescent $\Pi=(\Pi_t)_{t\ge 0}$ and a $\Xi$-Fleming-Viot process $Z=(Z_t)_{t\ge 0}$. (Duality) \[lemma:duality\] For $n\in{\ensuremath{\mathbb{N}}}$, $f:E^n\to{\ensuremath{\mathbb{R}}}$ bounded and measurable, $\mu\in{\mathcal M}_1(E)$, $\pi\in{\mathcal P}_n$ and $t\ge 0$, $$\label{analyticduality} \mathbb{E}^\mu \Big[ \int_{E^n} f\big(\mathbf{x}[\pi]\big) Z_t^{\otimes n}(d\mathbf{x}) \Big] = \mathbb{E}^\pi \Big[ \int_{E^n} f\big(\mathbf{x}[\Pi_t^{(n)}]\big) \mu^{\otimes n}(d\mathbf{x}) \Big],$$ where $\Pi_t^{(n)}$ is the restriction of $\Pi_t$ to ${\mathcal P}_n$. To obtain a [*pathwise duality*]{}, we use the driving Poisson processes of the modified lookdown construction to construct realisation-wise a $\Xi$-coalescent embedded in the $\Xi$-Fleming-Viot process. More explicitly, recall the Poisson processes $L^l_J$ and $L^l_{J,k}$ from equation and in Section \[limitpop\] and the Poisson process $\mathfrak{N}^K_{ij}$ defined in Section \[ExCoal\]. For each $t\ge 0$ and $l\in{\ensuremath{\mathbb{N}}}$, let $N_t^l(s), 0\le s\le t$, be the level at time $s$ of the ancestor of the individual at level $l$ at time $t$. In terms of the $L^l_J$ and $L^l_{J,k}$, the process $N^l_t(\cdot)$ solves, for $0\le s\le t$, $$\begin{aligned} \label{genealllogy} N_t^l(s)\ =\ l &- \sum_{1 \le i < j < l} \int_{s-}^t {\ensuremath{\mathbbm{1}}}_{\{N_t^l(u+) > j\}} \, d\mathfrak{N}^K_{ij}(u) \notag \\ & - \sum_{1 \le i < j < l} \int_{s-}^t (j-i) {\ensuremath{\mathbbm{1}}}_{\{N_t^l(u+) = j\}} \, d\mathfrak{N}^K_{ij}(u) \notag \\ & - \sum_{K \subset \{1,\ldots,l\}} \int_{s-}^t (N_t^l(u+) - J_m(N_t^l(u+))){\ensuremath{\mathbbm{1}}}_{\{N_t^l(u+) \notin K\}} \, dL^l_K(u)\notag\\ & - \sum_{k \in {\ensuremath{\mathbb{N}}}} \sum_{K \subset \{1,\ldots,l\}} \int_{s-}^t (N_t^l(u+)-\min(K)) {\ensuremath{\mathbbm{1}}}_{\{N_t^l(u+) \in K\}}\,dL^l_{K,k}(u),\end{aligned}$$ where $J_m(\cdot)=J_{m(u)}(\cdot)$ is defined by and $m(u)$ is the index of the jump at time $u$. Fix $0\le T$ and, for $t\le T$, define a partition $\Pi_t^T$ of ${\ensuremath{\mathbb{N}}}$ such that $k$ and $l$ are in the same block of $\Pi_t^T$ if and only if $N_T^l(T-t) = N_T^k(T-t)$. Thus, $k$ and $l$ are in the same block if and only if the two levels $k$ and $l$ at time $T$ have the same ancestor at time $T-t$. Then ([@DK99], Section 5), $$\label{eq:pathwisedual} \mbox{the process $(\Pi_t^T)_{0\le t\le T}$ is a $\Xi$-coalescent run for time $T$}.$$ Note that by employing a natural generalisation of the lookdown construction using driving Poisson processes on ${\ensuremath{\mathbb{R}}}$ and e.g.using $T=0$ above, one can use the same construction to find an $\Xi$-coalescent with time set ${\ensuremath{\mathbb{R}}}_+$. We would like to emphasise that the lookdown construction provides a realisation-wise coupling of the type distribution process $(Z_t)_{t\ge 0}$ and the coalescent describing the genealogy of a sample, thus extending , which is merely a statement about one-dimensional distributions. The function-valued dual of the $(\Xi,B)$-Fleming-Viot process -------------------------------------------------------------- The duality between the $ \Xi$-Fleming-Viot process and the $\Xi$-coalescent established in Section \[sec\_distributional\_duality\] worked only on the genealogical level, the mutation was not taken into account. However, it is possible to define a function-valued dual to the $(\Xi,B)$-Fleming-Viot process such that not only the genealogical structure, but also the mutation is part of the duality. This kind of duality is well known for the classical Fleming-Viot process, see, e.g., Etheridge [@E00 Chapter 1.12]. First note, that due to Lemma \[lemma\_generator\_alt\] we can rewrite the generator of the $(\Xi,B)$-Fleming-Viot process given by equation to obtain $$\begin{aligned} LG_f(\mu) &:= a\sum_{1\le i < j \le n} \int_{E^n} \Big(f(x_1,\!.., x_i,\!.., x_i,\!.., x_n)- f(x_1,\!.., x_i,\!.., x_j,\!.., x_n) \Big)\mu^{\otimes n}(d{\bf x}) \nonumber\\ &\quad + \sum_{\pi = \{ A_1,\dots,A_p\} \in \mathcal{P}_n \atop \text{ not all singletons}} \lambda(|A_1|,\dots,|A_p|) \int_{E^n} \left( f\big(\mathbf{x}[\pi]\big) -f(\mathbf{x})\right) \mu^{\otimes n}(d{\bf x}), \nonumber\\ &\quad + r \sum_{i=1}^n \int_{E^n} B_i(f(x_1, \dots, x_n)) \mu^{\otimes n}(d{\bf x}). \label{eq_real_alt_generator}\end{aligned}$$ We can now reinterpret the function $G_f(\mu)$ acting on measures as a function $G_\mu(f)$ acting on the functions $C_b(E^n)$. This reinterpretation transfers the operator $L$ acting on $C\big({\ensuremath{\mathcal{M}}}_1(E)\big)$ to an operator $L^*$ acting on $C_b\big(C_b(E^n)\big)$. Let $\mathcal{C} := \bigcup_{n=1}^\infty C_b(E^n)$. A $\mathcal{C}$-valued Markov process $(\rho_t)_{t\ge 0}$ solving the martingale problem for $L^*$ can then be constructed as follows: - If $\rho_t({\bf x})\in C_b(E^n)$ and $n\ge 2$, then the process $(\rho_t)_{t\ge 0}$ jumps to $\rho_t\big({\bf x[\pi]}\big)$ with rate $\lambda(|A_1|,\dots,|A_p|) + a{\ensuremath{\mathbbm{1}}}_{\{\exists!|A_i| =2;\forall j\neq i: |A_j|=1\}}$, for all $\pi = \{ A_1,\dots,A_p\} \in \mathcal{P}_n$, where $|A_j| \ge 1$ for at least one $j$. - If $\rho_t\in C_b(E)$, that is it is a function of a single variable, then no further jumps occur. - Between jumps the process evolves deterministically according to the “heat flow” generated by the mutation operator , independently for each coordinate. Note that this process is not literally a coalescent, but has coalescent-like features. The duality relation between $\rho_t$ and $Z_t$ immediately follows from and can be written in integrated form as $${\ensuremath{\mathbb{E}}}_{Z_0} \langle \rho_0, Z_t^{\otimes n} \rangle \ =\ {\ensuremath{\mathbb{E}}}_{\rho_0} \langle \rho_t , Z_0^{\otimes n} \rangle.$$ It can be used for example to show uniqueness of the martingale problem for $L$ via the existence of $(\rho_t)_{t\ge 0}$ or to calculate the moments of the $(\Xi,B)$-Fleming-Viot process. The dual of the block counting process -------------------------------------- In this section, we specialise to the case where the type space $E$ consists of two types only, say $E=\{0,1\}$. Define the real-valued process $Y=(Y_t)_{t\ge 0}$ via $Y_t:=Z_t(\{\text{1}\})$, $t\ge 0$. Define $g:{\mathcal M}_1(E)\to [0,1]$ via $g(\mu):=\mu(\{1\})$. The generator $A$ of $Y$ is then given by $Af(x)=(L^\Xi(f\circ g))(\mu)$, $f\in C^2([0,1])$, where $\mu$ depends on $x\in[0,1]$ and can be chosen arbitrary, as long as $g(\mu)=x$. Thus, $$\label{eq:genA} Af(x)\ =\ a\frac{x(1-x)}{2}f''(x) + \int_\Delta\int_{\{0,1\}^{\ensuremath{\mathbb{N}}}} \big( f((1-|{\ensuremath{\boldsymbol\zeta}}|)x+{\textstyle\sum_{i=1}^\infty} \zeta_i y_i) - f(x) \big)(\mathcal{B}(1,x))^{\otimes{\ensuremath{\mathbb{N}}}}(d{\mathbf y}) \frac{\Xi_{0}(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})},$$ $x\in [0,1]$, $f\in C^2([0,1])$, where $\mathcal{B}(1,x)$ denotes the Bernoulli distribution with parameter $x$. For $x\in[0,1]$ let $V_1(x),V_2(x),\ldots$ be a sequence of independent and identically $\mathcal{B}(1,x)$-distributed random variables. Then, $$Af(x)\ =\ a\frac{x(1-x)}{2}f''(x) + \int_\Delta\int_{[0,1]} \big( f((1-|{\ensuremath{\boldsymbol\zeta}}|)x+y) - f(x) \big)Q({\ensuremath{\boldsymbol\zeta}},x,dy) \frac{\Xi_0(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})},$$ where $Q({\ensuremath{\boldsymbol\zeta}},x,.)$ denotes the distribution of $\sum_{i=1}^\infty \zeta_iV_i(x)$. Hence the process can be considered as a Wright-Fisher diffusion with jumps. The situation where $\Xi$ is concentrated on $[0,1]\times\{0\}^{\ensuremath{\mathbb{N}}}$, i.e., when the underlying $\Xi$-coalescent is a $\Lambda$-coalescent, has been studied in [@BLG05]. Note that $A f\equiv 0$ for $f(x)=x$, so $Y$ is a martingale. Furthermore, the boundary points $0$ and $1$ are obviously absorbing. In analogy to Lemma \[lemma:duality\] it follows that $Y$ is dual to the block counting process $D=(D_t)_{t\ge 0}$ of the $\Xi$-coalescent with respect to the duality function $H:[0,1]\times{\ensuremath{\mathbb{N}}}\to{\ensuremath{\mathbb{R}}}$, $H(x,n):=x^n$ (see, e.g., Liggett [@L85]), i.e., $$\mathbb{E}^y[Y_t^n]\ =\ \mathbb{E}^n[y^{D_t}],\quad n\in{\ensuremath{\mathbb{N}}},y\in [0,1],t\ge 0.$$ Thus, the moments of the ‘forward’ variable $Y_t$ can be computed via the generating function of the ‘backward’ variable $D_t$ and vice versa. Such and closely related moment duality relations are well known from the literature [@AH07; @AS05; @M99]. The duality can be used to relate the accessibility of the boundaries of $Y$ and the existence of an entrance law for $D$ with $D_{0+}=\infty$. Note that by the Markov property and the structure of the jump rates, we always have $$\label{eq:Dthits1} {\ensuremath{\mathbb{P}}}^\infty(D_t=1\mbox{ eventually})\ \in\ \{0,1\}$$ and either ${\ensuremath{\mathbb{P}}}^\infty(\bigcap_{t\ge 0}\{D_t=\infty\})=1$ (if the probability in (\[eq:Dthits1\]) equals $0$) or $\lim_{t\to\infty}{\ensuremath{\mathbb{P}}}^\infty(D_t=1)=1$ (if the probability in (\[eq:Dthits1\]) equals $1$). \[prop:hitbdry\] $\lim_{t\to\infty}{\ensuremath{\mathbb{P}}}^\infty(D_t=1)=1$ if and only if $Y$, the dual of its block counting process, hits the boundary $\{0,1\}$ in finite time almost surely, starting from any $y\in(0,1)$. Fix $y\in(0,1)$, $T>0$. Construct $(Z_t)$ starting from $y\delta_1+(1-y)\delta_0$ and no mutations, $Bf\equiv 0$, (and hence $Y$ starting from $y$) by using the lookdown construction from Section \[limitpop\]: Let $X_1(0),X_2(0),\dots$ be independent $\mathcal{B}(1,y)$-distributed random variables which are independent of the driving Poisson processes, and let $X_n(t)$, $t>0$, $n\in{\ensuremath{\mathbb{N}}}$, be the solution of (\[eq\_stochastic\_differential\_equation\]). Let $$D'_t\ :=\ |\{N^n_T(T-t):n\in{\ensuremath{\mathbb{N}}}\}|,$$ where $N^n_T(s)$ solves (\[genealllogy\]). By (\[eq:pathwisedual\]), the law of $(D'_t)_{0 \le t \le T}$ is that of the block counting process of the (standard-)$\Xi$-coalescent run for time $T$. Then by construction (as there is no mutation), $$X_n(T) = X_{N^n_T(0)}(0),$$ implying $$\{D'_T=1\} \subset \{Y_T\in\{0,1\}\} \quad\mbox{and}\quad \{D'_T=\infty \}\subset \{0<Y_T<1\}\mbox{ almost surely},$$ which easily yields the claim. This is related to the so-called ‘coming down from infinity’-property of the standard $\Xi$-coalescent (i.e., the property that starting from $D_0=\infty$, $D_t<\infty$ almost surely for all $t>0$). Recall ([@S00], p. 39f) that a $\Xi$-coalescent may have infinitely many classes for a positive amount of time and then suddenly jumps to finitely many classes. This can occur if $\Xi$ has positive mass on $\Delta_f:=\{{\mathbf u}=(u_1,u_2,\ldots)\in\Delta:u_1+\cdots+u_n=1 \mbox{ for some $n\in{\ensuremath{\mathbb{N}}}$}\}$. On the other hand [@S00 Lemma 31], if $\Xi(\Delta_f)=0$, then the $\Xi$-coalescent either comes down from infinity immediately or always has infinitely many classes. Combining this with Proposition \[prop:hitbdry\] we obtain Assume that $\Xi(\Delta_f)=0$. Then the $\Xi$-coalescent comes down from infinity if and only if the dual of its block counting process hits the boundary $\{0,1\}$ in finite time almost surely. In general, there seems to be no ‘simple’ criterion to check whether a $\Xi$-coalescent comes down from infinity (see the discussion in Section 5.5 of [@S00]). On the other side, there seems to be also no ‘handy’ criterion for accessibility of the boundary of a process with jumps (and with values in $[0,1]$), but at least Proposition \[prop:hitbdry\] allows to transfer any progress from one side to the other and vice versa. We conclude this section with a simple toy example for which most quantities of interest, in particular the generator $A$, can be computed explicitly. Fix $l\in{\ensuremath{\mathbb{N}}}$. If the measure $\Xi$ is concentrated on $\Delta_l:=\{{\ensuremath{\boldsymbol\zeta}}\in\Delta:\zeta_1+\cdots+\zeta_l=1\}$, then (\[eq:genA\]) reduces to $$Af(x)\ =\ \int_\Delta \sum_{y_1,\ldots,y_l\in\{0,1\}} x^{y_1+\dots+y_l}(1-x)^{l-(y_1+\dots+y_l)} \big(f({\textstyle\sum_{i=1}^l \zeta_i y_i})-f(x)\big) \frac{\Xi(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})}.$$ For example, assume that the measure $\Xi$ assigns its total mass $\Xi(\Delta):=1/l$ to the single point $(1/l,\ldots,1/l,0,0,\ldots) \in\Delta_l$. Then, $$Af(x)\ =\ \sum_{k=0}^l {l\choose k}x^k(1-x)^{l-k}f(k/l) - f(x) \ =\ \int (f(y/l)-f(x))\,{\cal B}(l,x)(dy),$$ where ${\cal B}(l,x)$ denotes the binomial distribution with parameters $l$ and $x$. Note that the corresponding $\Xi$-coalescent never undergoes more than $l$ multiple collisions at one time. The rates (\[eq:xi.rate\]) are $$\lambda(k_1,\ldots,k_p) \ =\ \int_\Delta \sum_{i_1,\dots,i_p\in{\ensuremath{\mathbb{N}}}\atop \text{all distinct}} \zeta_{i_1}^{k_1}\cdots \zeta_{i_p}^{k_p}\, \frac{\Xi(d{\ensuremath{\boldsymbol\zeta}})}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})} \ =\ \frac{(l)_p}{l^n},$$ where $(l)_p:=l(l-1)\cdots(l-p+1)$ and $n:=k_1+\cdots+k_p$. The block counting process $D$ has rates $$g_{np}\ =\ \frac{n!}{p!} \hspace{-2mm} \sum_{{{m_1,\ldots,m_p\in{\ensuremath{\mathbb{N}}}}\atop{m_1+\cdots+m_p=n}}} \hspace{-1em} \frac{\lambda(m_1,\ldots,m_p)}{m_1!\cdots m_p!} \ =\ S(n,p)\frac{(l)_p}{l^n},\quad 1\le p<n,$$ where the $S(n,p)$ denote the Stirling numbers of the second kind. The total rates are $g_n=\sum_{p=1}^{n-1}g_{np}=1-(l)_n/l^n$, $n\in{\mathbb N}$. Note that the corresponding $\Xi$-coalescent stays infinite for a positive amount of time (‘Case 2’ on top of [@S00 p. 39] with $\Xi_2\equiv 0$). The dual of its block counting process hits the boundary in finite time.$\blacksquare$ Examples ======== The first of the two examples in this section presents a model, where the population size varies substantially due to recurrent bottlenecks. It is shown, that the $\Xi$-coalescent appears naturally as the limiting genealogy of this model. In the second example we present the Poisson-Dirichlet-coalescent by choosing a particular measure for $\Xi$ which has a density with respect to the Poisson-Dirichlet distribution. We provide explicit expressions for several quantities of interest. An example involving recurrent bottlenecks ------------------------------------------ Consider a population, say with non-overlapping generations, in which the population size has undergone occasional abrupt changes in the past. Specifically, we assume that ‘typically’, each generation contains $N$ individuals, but at several instances in the past, it has been substantially smaller for a certain amount of time, and then the population has quickly re-grown to its typical size $N$. This is related to the models considered by Jagers & Sagitov in [@JS04], but we assume occasional much more radical changes in population size than [@JS04]. Let us assume that the demographic history is described by three sequences of positive real numbers $(s_{i})_{i\in {\ensuremath{\mathbb{N}}}}$, $(l_{i,N})_{i\in {\ensuremath{\mathbb{N}}}}$ and $(b_{i,N})_{i\in {\ensuremath{\mathbb{N}}}}$, where $0 < b_{i,N} \le 1$ holds for all $i$, and the population size $t$ generations before the present is given by $G(t)$, where $$G(t)\ =\ \begin{cases} b_{m,N}N & \text{if}\ N\big(\sum_{i=1}^{m-1}(s_i+l_{i,N})+s_m\big)<t\le N\sum_{i=1}^m (s_i+l_{i,N}),\quad m\in{\ensuremath{\mathbb{N}}},\\ N & \text{otherwise}. \end{cases}$$ Thus, back in time the population stays at size $N$ for some time $s_i N$. Then the size is reduced to $b_{i,N} N$ for the time $l_{i,N} N$. Thereafter it is again given by $N$, until the next bottleneck occurs after time $s_{i+1} N$. Note that for simplicity, we have assumed ‘instantaneous’ re-growth after each bottleneck. Furthermore, we assume that the reproduction behaviour is given by the standard Wright-Fisher dynamics, so each individual chooses its parent uniformly at random from the previous generation, independently of the other individuals. This is the case in every generation, also during the bottleneck and at the transitions between the bottleneck and the typical size. We now want to keep track of the genealogy of a sample of $n$ individuals from the present generation, and describe its dynamics in the limit $N\to\infty$. Denote by $\Pi^{(N,n)}(t)$ the ancestral partition of the sample $t$ generations before the present. \[lemma:bottleneck\] Fix $(s_i)_{i\in{\ensuremath{\mathbb{N}}}}$ and assume that $b_{i,N}\to 0$ and that $l_{i,N}\to 0$ as $N\to\infty$. Furthermore assume that $b_{i,N}N\to\infty$ and that $l_{i,N}/b_{i,N}\to\gamma_i>0$. Then $$\Pi^{(N,n)}(Nt)\ \to\ \Pi^{\delta_{\bf 0},(n)}(R_t)$$ weakly as $N\to\infty$ on $D_{\mathcal{P}_n}([0,\infty))$, where $R_t:=t+\sum_{i:s_1+\cdots+s_i\le t}\gamma_i$. Note that we assume $l_{i,N} \to 0$ as $N \to \infty$, so the duration of the bottleneck is negligible on the timescale of the ‘normal’ genealogy. We also assume $b_{i,N}\to 0$ but $Nb_{i,N}\to\infty$, i.e., in the pre-limiting scenario, the population size during a bottleneck should be tiny compared to the normal size, but still large in absolute numbers. The ratio $l_{i,N}/b_{i,N}$ is sometimes called the [*severity*]{} of the ($i$-th) bottleneck in the population genetic literature. Given sequences $(s_i), (b_{i, N})$ and $(l_{i,N})$, classical convergence results for samples of size $n$ can be applied for the time-intervals between bottlenecks and “inside” the bottlenecks. Since $b_{i,N} N \to \infty$, the probability that any of the ancestral lines of the sample converge exactly at the transition to a bottleneck is $O((b_{i,N} N)^{-1})=o(1)$, so that naïve “glueing” is feasible. Note that bottleneck events with $\gamma_i=0$ become invisible in the limit, whereas in a bottleneck with $\gamma_i=+\infty$ the genealogy necessarily comes down to only one lineage (and thus, all genetic variability is erased). Since we fixed the $s_i$ and $\gamma_i$, the limiting process described in Lemma \[lemma:bottleneck\] is not a homogeneous Markov process and thus does not fit literally into the class of exchangeable coalescent processes considered in this paper. Assume that the waiting intervals $s_i$ are exponentially distributed, say with parameter $\beta$, and that the $\gamma_i$ are independently drawn from a certain law $\mathcal{L}_\gamma$. Thus, in the pre-limiting $N$-particle model forwards in time, in each generation there is a chance of $\sim\beta/N$ that a ‘bottleneck event’ with a randomly chosen severity begins. In this situation, the genealogy of an $n$-sample from the population at present is (approximately) described by $$\label{eq:bottleneckcoal} \Pi^{\delta_{\bf 0},(n)}(S_t),\quad t\ge 0,$$ where $(S_t)_{t\ge 0}$ is a subordinator (in fact, a compound Poisson process with Lévy measure $\beta \mathcal{L}_\gamma$ and drift $1$). Let $N_\gamma$ be the number of lineages at time $\gamma>0$ in the standard Kingman coalescent starting with $N_0=\infty$, and let $D_j$ be the law of the re-ordering of a ($j$-dimensional) Dirichlet$(1,\dots,1)$ random vector according to decreasing size, padded with infinitely many zeros. The process defined in (\[eq:bottleneckcoal\]) is the $\Xi$-coalescent restricted to $\{1,\ldots,n\}$, where $$\Xi(d{{\ensuremath{\boldsymbol\zeta}}})\ =\ \delta_{\bf 0}(d{{\ensuremath{\boldsymbol\zeta}}}) + ({{\ensuremath{\boldsymbol\zeta}}},{{\ensuremath{\boldsymbol\zeta}}})\int_{(0,\infty)} \sum_{j=1}^\infty {\ensuremath{\mathbb{P}}}(N_\sigma=j) D_j(d{\ensuremath{\boldsymbol\zeta}}) \,\beta\mathcal{L}_\gamma(d\sigma).$$ Recall that the number of families of the classical Fleming-Viot process without mutation after $\sigma$ time units is $N_\sigma$. Given $N_\sigma=j$, the distribution of the family sizes is a uniform partition of $[0,1]$, hence Dirichlet$(1,\dots,1)$. Size-ordering thus leads to the above formula for $\Xi$. The Poisson-Dirichlet case -------------------------- The Poisson-Dirichlet distribution ${\rm PD}_\theta$ with parameter $\theta>0$ is a distribution concentrated on the subset $\Delta^*$ of points ${\ensuremath{\boldsymbol\zeta}}\in\Delta$ satisfying $|{\ensuremath{\boldsymbol\zeta}}|=1$. It can, for example, be obtained via size-ordering of the normalized jumps of a Gamma-subordinator at time $\theta$. For more information on this distribution we refer to [@K75] or [@ABT99]. Sagitov [@S03] considered the Poisson-Dirichlet coalescent $\Pi=(\Pi_t)_{t\ge 0}$ with parameter $\theta>0$, where (by definition) the measure $\Xi$ has density ${\ensuremath{\boldsymbol\zeta}}\mapsto({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})$ with respect to ${\rm PD}_\theta$. As the measure ${\rm PD}_\theta$ is concentrated on $\Delta^*$, the rates (\[eq:xi.rate\]) reduce to $$\lambda(k_1,\ldots,k_j)\ =\ \int_{\Delta^*} \sum_{i_1,\dots,i_j\in{\ensuremath{\mathbb{N}}}\atop\text{all distinct}} \zeta_{i_1}^{k_1}\cdots\zeta_{i_j}^{k_j}\,{\rm PD}_\theta(d{\ensuremath{\boldsymbol\zeta}}).$$ From the calculations of Kingman [@K93] it follows that the Poisson-Dirichlet coalescent has rates $$\lambda(k_1,\ldots,k_j)\ =\ \frac{\theta^j}{[\theta]_k}\prod_{i=1}^j (k_i-1)!,$$ $k_1,\ldots,k_j\in\mathbb{N}$ with $k:=k_1+\dots+k_j>j$, where $[\theta]_k:=\theta(\theta+1)\dots(\theta+k-1)$. Möhle and Sagitov [@MS01] characterised exchangeable coalescents via a sequence $(F_j)_{j\in\mathbb{N}}$ of symmetric finite measures. For each $j\in\mathbb{N}$, the measure $F_j$ lives on the simplex $\Delta_j:= \{(\zeta_1,\ldots,\zeta_j)\in [0,1]^j:\zeta_1+\dots+\zeta_j\le 1\}$ and is uniquely determined via its moments $$\lambda(k_1,\ldots,k_j)\ =\ \int_{\Delta_j} \zeta_1^{k_1-2}\cdots\zeta_j^{k_j-2}F_j(d\zeta_1,\ldots,d\zeta_j), \quad k_1,\ldots,k_j\ge 2.$$ For the Poisson-Dirichlet coalescent, an application of Liouville’s integration formula shows that the measure $F_j$ has density $f_j(\zeta_1,\ldots,\zeta_j):=\theta^j\zeta_1\cdots\zeta_j (1-\sum_{i=1}^j\zeta_i)^{\theta-1}$ with respect to the Lebesgue measure on $\Delta_j$. As $\Xi$ is concentrated on $\Delta^*$, it follows that $$\label{finite} \int_\Delta\frac{|{\ensuremath{\boldsymbol\zeta}}|}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})}\,\Xi(d{\ensuremath{\boldsymbol\zeta}}) \ =\ \int_\Delta \frac{1}{({\ensuremath{\boldsymbol\zeta}},{\ensuremath{\boldsymbol\zeta}})}\,\Xi(d{\ensuremath{\boldsymbol\zeta}}) \ =\ \int_{\Delta^*}\Pi_\theta(d{\ensuremath{\boldsymbol\zeta}})\ =\ 1\ <\ \infty.$$ By [@S00 Proposition 29], the Poisson-Dirichlet coalescent is a jump-hold Markov process with bounded transition rates and step function paths. By [@S00 Proposition 30], for arbitrary but fixed $t>0$, $\Pi_t$ does not have proper frequencies. The block counting process $D:=(D_t)_{t\ge 0}$, where $D_t:=|\Pi_t|$ denotes the number of blocks of $\Pi_t$, is a decreasing process with rates $$g_{nk} \ =\ \frac{n!}{k!}\sum_{{n_1,\ldots,n_k\in\mathbb{N}}\atop{n_1+\dots+n_k=n}} \frac{\lambda(n_1,\ldots,n_k)}{n_1!\cdots n_k!} \ =\ \frac{\theta^k}{[\theta]_n}\frac{n!}{k!} \sum_{{n_1,\ldots,n_k\in\mathbb{N}}\atop{n_1+\dots+n_k=n}} \frac{1}{n_1\cdots n_k} \ =\ \frac{\theta^k}{[\theta]_n}s(n,k),$$ $k,n\in\mathbb{N}$ with $k<n$, where the $s(n,k)$ are the absolute Stirling numbers of the first kind. The total rates are $$g_n\ :=\ \sum_{k=1}^{n-1}g_{nk}\ =\ 1-\frac{\theta^n}{[\theta]_n}, \quad n\in\mathbb{N}.$$ Note that $g_{nk}=\mathbb{P}\{K_n=k\}$, $k<n$, where $K_n$ is a random variable taking values in $\{1,\ldots,n\}$ with distribution $$\mathbb{P}\{K_n=k\} \ =\ \frac{\theta^k}{[\theta]_n}s(n,k),\qquad k\in\{1,\ldots,n\}.$$ We have $$\gamma_n\ :=\ \sum_{k=1}^{n-1}(n-k)g_{nk} \ =\ \sum_{k=1}^{n-1}(n-k){\mathbb P}\{K_n=k\} \ =\ n-{\mathbb E}K_n\ \le\ n.$$ In particular, $\sum_{n=2}^\infty\gamma_n^{-1}\ge\sum_{n=2}^\infty 1/n=\infty$. Together with (\[finite\]) and $\Xi(\Delta_f)=0$, where $\Delta_f:=\{{\ensuremath{\boldsymbol\zeta}}\in\Delta\,|\,\zeta_1+\dots+\zeta_n=1\mbox{ for some }n\}$, it follows from [@S00 Proposition 33] that the Poisson-Dirichlet coalescent stays infinite. 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[^1]: Weierstra[ß]{}-Institut für Angewandte Analysis und Stochastik, Mohrenstra[ß]{}e 39, 10117 Berlin, Germany, e-mail: [birkner@wias-berlin.de]{}, URL: [http://www.wias-berlin.de/]{}$\sim$[birkner]{} [^2]: Institut für Mathematik, Technische Universität Berlin, Stra[ß]{}e des 17. Juni 136, 10623 Berlin, Germany, e-mail: [blath@math.tu-berlin.de]{}, [steinrue@math.tu-berlin.de]{}, [johannatams@gmx.de]{}, URL: [http://www.math.tu-berlin.de/]{}$\sim$[blath]{}, [http://www.math.tu-berlin.de/]{}$\sim$[steinrue]{} [^3]: Mathematisches Institut, Universität Düsseldorf, Universitätsstra[ß]{}e 1, 40225 Düsseldorf, Germany, e-mail: [moehle@math.uni-duesseldorf.de]{}, URL: [http://www.math.uni-duesseldorf.de/Personen/indiv/Moehle]{} (corresponding author)

--- abstract: 'Hashing method maps similar data to binary hashcodes with smaller hamming distance, and it has received a broad attention due to its low storage cost and fast retrieval speed. However, the existing limitations make the present algorithms difficult to deal with large-scale datasets: (1) discrete constraints are involved in the learning of the hash function; (2) pairwise or triplet similarity is adopted to generate efficient hashcodes, resulting both time and space complexity are greater than $O(n^2)$. To address these issues, we propose a novel discrete supervised hash learning framework which can be scalable to large-scale datasets. First, the discrete learning procedure is decomposed into a binary classifier learning scheme and binary codes learning scheme, which makes the learning procedure more efficient. Second, we adopt the [*Asymmetric Low-rank Matrix Factorization*]{} and propose the [*Fast Clustering-based Batch Coordinate Descent*]{} method, such that the time and space complexity is reduced to $O(n)$. The proposed framework also provides a flexible paradigm to incorporate with arbitrary hash function, including deep neural networks and kernel methods. Experiments on large-scale datasets demonstrate that the proposed method is superior or comparable with state-of-the-art hashing algorithms.' author: - bibliography: - 'references.bib' title: Scalable Discrete Supervised Hash Learning with Asymmetric Matrix Factorization --- Introduction ============ During the past few years, hashing has become a popular tool in solving large-scale vision and machine learning problems [@li2011hashing; @liu2012supervised; @li2013sign]. Hashing techniques encode various types of high-dimensional data, including documents, images and videos, into compact hashcodes by certain hash functions, so that similar data are mapped to hashcodes with smaller Hamming distance. With the compact binary codes, we are able to compress data into very small storage space, and conduct efficient nearest neighbor search on large-scale datasets. The hashing techniques are composed of [*data-independent*]{} methods and [*data-dependent*]{} methods. Locality-Sensitive Hashing (LSH)  [@datar2004locality; @gionis1999similarity] and MinHash [@broder1998min] are the most popular [*data-independent*]{} methods. These methods have theoretical guarantees that similar data have higher probability to be mapped into the same hashcode, but they need long codes to achieve high precision. In contrast to [*data-independent*]{} hashing methods, [*data-dependent*]{} learning-to-hash methods aim at learning hash functions with training data. A number of methods are proposed in the literature, and we summarize them into two categories: [*unsupervised methods*]{}, including Spectral Hashing(SH) [@weiss2009spectral], Iterative Quantization(ITQ) [@gong2013iterative], Anchor Graph Hashing(AGH) [@liu2011hashing], Isotropic Hashing(IsoH) [@kong2012isotropic], Discrete Graph Hashing(DGH) [@liu2014discrete]; and [*supervised methods*]{}, such as Binary Reconstructive Embeddings(BRE)  [@kulis2009learning], Minimal Loss Hashing [@norouzi2011minimal], Supervised Hashing with Kernels(KSH) [@liu2012supervised], FastHash(FastH) [@lin2014fast], Supervised Discrete Hashing(SDH) [@Shen_2015_CVPR]. Experiments convey that hash functions learned by supervised hashing methods are superior to unsupervised ones. Recent works [@Shen_2015_CVPR; @lin2014fast] demonstrate that more training data can improve the performances of the learned hash functions. However, existing hashing algorithms rarely discuss training on large-scale datasets. Most algorithms use pairwise or triplet similarity to learn hash functions, so that there are intuitive guarantees that similar data can learn similar hashcodes. But there are $O(n^2)$ data pairs or $O(n^3)$ data triplets where $n$ is the number of training data, which makes both the training time and space complexity at least $O(n^2)$. These methods cannot train on millions of data, like ImageNet dataset [@deng2009imagenet]. Recent works like SDH [@Shen_2015_CVPR] reduces the training time to $O(n)$, but it lies in the assumption that the learned binary codes are good for linear classification, thus there are no guarantees that similar hashcodes correspond to data with similar semantic information. Moreover, the discrete constraints imposed on the binary codes lead to mix-integer optimization problems, which are generally NP-hard. Many algorithms choose to remove the discrete constraints and solve a relaxed problem, but they are less effective due to the high quantization error. Recent studies focus on learning the binary codes without relaxations. DGH [@liu2014discrete] and SDH [@Shen_2015_CVPR] design an optimization function in which binary constraints are explicitly imposed and handled, and the learning procedure consists of some tractable subproblems. But DGH is an unsupervised method, and SDH does not consider semantic similarity information between the training data. We consider discrete methods that can leverage the similarity information between training samples should be better for hashing. In this paper, we propose a novel discrete learning framework to learn better hash functions. A joint optimization method is proposed, in which the binary constraints are preserved during the optimization, and the hash function is obtained by training several binary classifiers. To leverage pairwise similarity information between the training data, the similarity matrix is used in the optimization function. By making use of [*Asymmetric Low-rank Similarity Matrix Factorization*]{}, we reduce the computing time and storage of similarity matrix from $O(n^2)$ to $O(n)$, so our method can deal with millions of training data. To solve the most challenging binary code learning problem, we propose a novel [*Fast Clustering-based Batch Coordinate Descent (Fast C-BCD)*]{} algorithm to convert the binary code learning problem to a clustering problem, and generate binary codes bit by bit. We name the proposed framework as [*screte upervised ashing (DISH) Framework*]{}. Recent works [@xia2014supervised; @lai2015simultaneous; @guo2016hash; @zhu2016deep] show that hashing methods with deep learning can learn better hash functions. This framework is also able to learn hash functions with deep neural networks to capture better semantic information of the training data. Our main contributions are summarized as follows: 1. We propose a novel discrete supervised hash learning framework which is decomposed into a binary classifier learning scheme and binary codes learning scheme. Discrete method makes the learning procedure more efficiently. 2. We propose the [*Fast Clustering-based Batch Coordinate Descent*]{} algorithm to train binary codes directly, and introduce the [*Asymmetric Low-Rank Similarity Matrix Factorization*]{} scheme to decompose the similarity matrix into two low-rank matrices, so that the time and space complexity is reduced to $O(n)$. 3. The proposed DISH framework succeeds in learning on millions of data and experimental results show its superiority over the state-of-the-art hashing methods on either the retrieval performance or the training time. The rest of the paper is organized as follows. Section \[sec:related\] presents the related work of recent learn-to-hash methods. Section \[sec:framework\] introduces the [*Discrete Supervised Hashing (DISH) Framework*]{}, and we discuss how to combine the framework with deep neural network in Section \[sec:deephashing\] and kernel-based methods in Section \[sec:kernelhash\]. Experiments are shown in Section \[sec:exp\], and the conclusions are summarized in Section \[sec:conclusion\]. Related Work {#sec:related} ============ Discrete Hashing Methods ------------------------ The goal of hash learning is to learn certain hash functions with given training data, and the hashcodes are generated by the learned hash function. Recently, many researches focus on discrete learning methods to learn hashcodes directly. Two Step Hashing (TSH) [@lin2013general] proposes a general two-step approach to learn hashcodes, in which the binary codes are learned by similarities within data, and hash function can be learned by a classifier. FastH [@lin2014fast] is an extension of the TSH algorithm, which improves TSH by using Boosting trees as the classifier. However, these methods learn hashcodes and hash functions separately, thus the learned binary codes may lack the relationship with the distribution of data. What’s more, pair-wise similarity matrix is involved in these methods, making them not scalable. We succeed in learning hashcodes with pairwise similarity as well as achieving $O(n)$ complexity. Some other works tried to jointly learn discrete hashcodes as well as hash functions. Discrete Graph Hashing(DGH) [@liu2014discrete] designs an optimization function in which binary constraints are explicitly imposed and handled, and the learning procedure consists of two tractable subproblems. Anchor graphs are also used in this algorithm, reducing the storage of pairwise similarity matrix to $O(n)$. But it is an unsupervised algorithm and does not make use of semantic information. Supervised Discrete Hashing(SDH) [@Shen_2015_CVPR] proposes a method in which binary codes and hash functions are learned jointly and iteratively. But it cannot tackle the case where the data have no semantic labels, and it lacks theoretical and intuitive guarantees for explaining advantages of this algorithm. Moreover, it only discusses the learning of kernel-based hash functions. Our proposed framework can deal with arbitrary hash functions. Deep Hashing with Convolutional Networks ---------------------------------------- Recently, deep convolutional neural network (CNN) have received great success in image classification [@krizhevsky2012imagenet; @simonyan2014very; @he2015deep], object detection [@ren2015faster] and so on. Recent works [@wan2014deep; @schroff2015facenet] convey that features extracted from the last hidden layers represent more semantic information, and outperform most hand-crafted features in many vision applications. [@xia2014supervised; @lai2015simultaneous] show that simultaneously learning hash functions as well as the network can generate the codes with much better semantic information. CNNH [@xia2014supervised] decomposes the hash learning process into two stages. First, the approximate hashcodes are learned with pairwise similarity matrix, then the learned hashcodes are used as the supervised information to learn the deep neural network. [@zhuang2016fast] use triplet loss function to generate approximate hashcodes. But the learned approximate hashcodes in these methods have no relevance with the data, making the learned nets not effective. Some other works use one-stage method to learn binary codes and image features simultaneously. [@lai2015simultaneous] uses triplet loss to learn hash function, and DHN [@zhu2016deep] proposes a pairwise loss function to train the network. CNNBH [@guo2016hash] is similar as SDH [@Shen_2015_CVPR], and they both assume that the learned binary codes are good for classification. For ease of back-propagation, these methods remove the discrete constraints and add some quantization penalty to decrease the quantization error. Although the penalty is introduced, the quantization error still affects the efficiency of the learned hash functions. By incorporating our DISH framework to deep neural network, we can tackle two problems mentioned above: (1) we use the discrete method to reduce the quantization error; (2) we bridge the input data and semantic information by jointly learning hashcodes and deep neural networks. The Discrete Supervised Hashing Framework {#sec:framework} =========================================  Suppose we are given $n$ data vectors $\mathbf{X}=[\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_n]^\mathrm{T}$. The goal is to learn hash function $\mathbf{H}(\mathbf{X})=[\mathbf{h}(\mathbf{x}_1),\mathbf{h}(\mathbf{x}_2),...,\mathbf{h}(\mathbf{x}_n)]^\mathrm{T} \in \{-1,1\}^{n \times r}$, where $\mathbf{h}(\mathbf{x}_i) = [h_{i1},h_{i2},...,h_{ir}]^\mathrm{T} \in \{-1,1\}^r$ is the hash function of data vector $\mathbf{x}_i$, and $r$ is the hashcode length. Denote $h_{ik}=\mathrm{sgn}(f_k(\mathbf{x}_i))$, where $\mathrm{sgn}(x)$ is $+1$ if $x>=0$ and $-1$ otherwise. Define $F(\mathbf{x})=[f_1(\mathbf{x}), f_2(\mathbf{x}),...,f_r(\mathbf{x})]^\mathrm{T} \in \mathbb{R}^{r}$ and $F(\mathbf{X})=[F(\mathbf{x}_1),F(\mathbf{x}_2),...,F(\mathbf{x}_n)]^\mathrm{T} \in \mathbb{R}^{n \times r}$, we have $\mathbf{H}(\mathbf{X})=\mathrm{sgn} (F(\mathbf{X}))$, where $\mathrm{sgn}(\cdot)$ is an element-wise sign function. Let $\mathbf{S} = \{s_{ij}\}_{n \times n}$ be pairwise similarity matrix, in which $s_{ij}=1$ if $\mathbf{x}_i$ and $\mathbf{x}_j$ are most similar and $s_{ij}=-1$ otherwise. Then the objective to learn hash function $\mathbf{H}(\mathbf{X})$ can be formulated as $$\begin{split} \min_{F} \mathcal{Q} &= \sum_{i,j=1}^n [r s_{ij}-\mathbf{h}(\mathbf{x}_i)^{\mathrm{T}} \mathbf{h}(\mathbf{x}_j)]^2 \\ &= \Arrowvert r\mathbf{S} - \mathrm{sgn}(F(\mathbf{X}))\mathrm{sgn}(F(\mathbf{X}))^\mathrm{T} \Arrowvert_F^2 \label{obj} \end{split}$$ where $\| . \|_F$ is Frobenius norm of a matrix. It should be noticed that $\mathbf{h}(\mathbf{x}_i)^{\mathrm{T}} \mathbf{h}(\mathbf{x}_j)=r$ if $\mathbf{h}(\mathbf{x}_i)$ and $\mathbf{h}(\mathbf{x}_j)$ are identical and $\mathbf{h}(\mathbf{x}_i)^{\mathrm{T}} \mathbf{h}(\mathbf{x}_j) = -r$ if Hamming distance of $\mathbf{h}(\mathbf{x}_i)$ and $\mathbf{h}(\mathbf{x}_j)$ is the largest. Optimizing Eq. (\[obj\]) means that the Hamming distance between hashcodes of similar data pairs should be small, and large otherwise. Eq. (\[obj\]) is hard to optimize because the sign function is involved. Inspired by [@liu2014discrete], we remove the sign function and add a quantization loss to hold the binary constraints as much as possible $$\begin{split} \min_{\mathbf{H}, F} \mathcal{Q} &= \Arrowvert r\mathbf{S} - \mathbf{H} \mathbf{H}^\mathrm{T} \Arrowvert_F^2 + n \nu \mathcal{L}(\mathbf{H}, F(\mathbf{X})) \\ &= \Arrowvert r\mathbf{S} - \mathbf{H} \mathbf{H}^\mathrm{T} \Arrowvert_F^2 + n \nu \sum_{i=1}^n \sum_{k=1}^r l(h_{ik}, f_k(\mathbf{x}_i)) \end{split} \label{dis_obj_ori}$$ where $n$ is the number of training samples, $\nu$ is the penalty parameter, and $\mathcal{L}(\mathbf{H}, F(\mathbf{X}))$ denotes the quantization loss. If $\mathbf{H}$ and $\mathrm{sgn}(F(\mathbf{X}))$ are the same, the objective should be zero. Another difficulty in solving Eq. (\[obj\]) is the existence of pairwise similarity matrix $\mathbf{S}$, which involves at least $O(n^2)$ memory usage and at least $O(n^2)$ time consumption in matrix multiplication for $n$ training samples. Square time and space complexity makes it impossible to learn with large-scale training samples. In what follows, we propose [*Asymmetric Low-Rank Similarity Matrix Factorization*]{}, where we introduce the product of two low-rank matrices, $\mathbf{P} \in \mathbb{R}^{n \times l}, \mathbf{R} \in \mathbb{R}^{n \times l}(l \ll n)$, to approximate the similarity matrix $\mathbf{S} \in \mathbb{R}^{n \times n}$: $$\mathbf{S} \approx \mathbf{P}\mathbf{R}^\mathrm{T} \label{lowrank_mat}$$ thus Eq. (\[dis\_obj\_ori\]) can be rewritten as $$\min_{\mathbf{H}, F} \mathcal{Q} = \Arrowvert r\mathbf{P}\mathbf{R}^\mathrm{T}- \mathbf{H} \mathbf{H}^\mathrm{T} \Arrowvert_F^2 + n \nu \sum_{k=1}^r \sum_{i=1}^n l(h_{ik}, f_k(\mathbf{x}_i)) \label{dis_obj}$$ If $\mathbf{H}$ is fixed, we can directly regard $\sum_{i=1}^n l (h_{ik}, f_k(\mathbf{x}_i))$ as a binary classification problem. For example, kernel SVM corresponds to a kernel-based hash function. A binary classifier with high classification accuracy corresponds to a good hash function. We propose a discrete learning procedure to optimize Eq. (\[dis\_obj\]), which is discussed below in detail, and is summarized in Figure \[fig:framework\]. The choice of a good similarity matrix factorization and a good binary classifier is also discussed. Discrete Learning Procedure --------------------------- Eq. (\[dis\_obj\]) is still a nonlinear mixed-integer program involving discrete variables $\mathbf{H}$ and hash function $F$. Similar with [@liu2014discrete], we decompose Eq. (\[dis\_obj\]) into two sub-problems: $\mathbf{F}$-Subproblem $$%% \setlength{\abovedisplayskip}{3pt} %% \setlength{\belowdisplayskip}{3pt} \min_{F} \mathcal{Q}_F = \sum_{k=1}^r \sum_{i=1}^n l (h_{ik}, f_k(\mathbf{x}_i)) \label{wsub}$$ and $\mathbf{H}$-Subproblem $$\min_{\mathbf{H}} \mathcal{Q}_H = \Arrowvert r\mathbf{P}\mathbf{R}^\mathrm{T} - \mathbf{H} \mathbf{H}^\mathrm{T} \Arrowvert_F^2 + n \nu \sum_{k=1}^r \sum_{i=1}^n l (h_{ik}, f_k(\mathbf{x}_i)) \label{hsub}$$ The subproblems (\[wsub\]) and (\[hsub\]) are solved alternatively. In what follows, we regard Eq. (\[wsub\]) as $r$ independent binary classification problems, and introduce a novel clustering-based algorithm to optimize (\[hsub\]) bit by bit. ### $\mathbf{F}$-Subproblem It is clear that $\sum_{i=1}^n l (h_{ik}, f_k(\mathbf{x}_i))$ can be regarded as a binary classification problem, in which $h_{ik}$ is the label of $\mathbf{x}_i$ and $f_k(\cdot)$ is the function to learn. Each learned binary classifier involves minimizing $\sum_{i=1}^n l (h_{ik}, f_k(\mathbf{x}_i))$ for any$ k=1,2,3,...,r$. Denote $f^*_k(\cdot)$ as the learned classification function, then $F^*(\cdot)=[f^*_1(\cdot),f^*_2(\cdot),...,f^*_r(\cdot)]^\mathrm{T}$ is the optimum of $\mathbf{F}$-Subproblem. ### $\mathbf{H}$-Subproblem We propose an efficient [*Fast Clustering based Batch Coordinate Descent (Fast C-BCD)*]{} algorithm to optimize $\mathbf{H}$, in which $\mathbf{H}$ is learned column by column. Let $\mathbf{b}=[b_1,...,b_n]^\mathrm{T} \in \{1,-1\}^{n}$ be the $k$th column of $\mathbf{H}$, and $\mathbf{H}'$ is the matrix of $\mathbf{H}$ excluding $\mathbf{b}$. Set $\mathbf{H}'$ fixed, and let $\mathbf{Q}=r\mathbf{P}\mathbf{R}^\mathrm{T} - \mathbf{H}{'} \mathbf{H}{'} ^{\mathrm{T}}$ the Eq. (\[hsub\]) can be rewritten as $$\begin{split} \mathcal{Q}_H(\mathbf{b}) =& \Arrowvert r\mathbf{P}\mathbf{R}^\mathrm{T} - \mathbf{H}{'} \mathbf{H}{'} ^{\mathrm{T}} - \mathbf{b}\mathbf{b}^{\mathrm{T}} \Arrowvert_F^2 \\ &+ n \nu \sum_{i=1}^n l (b_{i}, f_k(\mathbf{x}_i)) + \mathrm{const} \\ =&\Arrowvert \mathbf{Q} \Arrowvert_F^2 + \Arrowvert \mathbf{b}\mathbf{b}^\mathrm{T} \Arrowvert^2 \\ &- 2\mathbf{b}^\mathrm{T} \mathbf{Q} \mathbf{b} + n \nu \sum_{i=1}^n l (b_{i}, f_k(\mathbf{x}_i)) + \mathrm{const} \\ =& - 2\mathbf{b}^\mathrm{T} \mathbf{Q} \mathbf{b} + n \nu \sum_{i=1}^n l (b_{i}, f_k(\mathbf{x}_i)) + \mathrm{const} \end{split}$$ It should be noticed that $l (b_{i}, f_k(\mathbf{x}_i))$ is a unitary binary function, thus we can rewrite it as a binary linear function: $l (b_{i}, f_k(\mathbf{x}_i))=\frac{1}{2}[l (1,f_k(\mathbf{x}_i))-l (-1,f_k(\mathbf{x}_i))] b_i + \frac{1}{2}[l (1,f_k(\mathbf{x}_i))+l (-1,f_k(\mathbf{x}_i))]$. Let $\mathbf{q}=\frac{n \nu}{2}[l (1,f_k(\mathbf{x}_1))-l (-1,f_k(\mathbf{x}_1)),...,l (1,f_k(\mathbf{x}_n))-l (-1,f_k(\mathbf{x}_n))]^\mathrm{T}$, then we have $n \nu \sum_{i=1}^n l (b_{i}, f_k(\mathbf{x}_i))=\mathbf{q}^\mathrm{T} \mathbf{b} + \mathrm{const}$. Discarding the constant term, we arrive at the following Binary Quadratic Programming (BQP) problem: $$\min_{\mathbf{b}} g(\mathbf{b}) = - 2\mathbf{b}^\mathrm{T} \mathbf{Q} \mathbf{b} + \mathbf{q}^\mathrm{T} \mathbf{b}, \mathbf{b} \in \{-1,1\}^n \label{bcd}$$ Optimization of (\[bcd\]) is still intractable. Inspired by [@yang2013new; @kang2016column], we transform the problem above to an efficient clustering problem.  [@yang2013new] studied the following constrained BQP problem: $$\max_{\mathbf{c}} \mathbf{c}^\mathrm{T}\mathbf{Q}_0\mathbf{c} \quad \mathrm{s.t.} \mathbf{c}^\mathrm{T}\mathbf{1}=k, \mathbf{c} \in \{0,1\}^n \label{equ:bqp01}$$ where the diagonal elements of $\mathbf{Q}_0$ are zero. We should transform Eq. (\[bcd\]) to the same form as Eq. (\[equ:bqp01\]). First of all, we add the bit-balanced constraint $\mathbf{b}^\mathrm{T} \mathbf{1}=(n\mod 2)$ to Eq. (\[bcd\]), which is widely used in learning-based hashing: $$\begin{split} &\max_{\mathbf{b}} g'(\mathbf{b}) = 2\mathbf{b}^\mathrm{T} \mathbf{Q} \mathbf{b} - \mathbf{q}^\mathrm{T} \mathbf{b} \\ &\mathrm{s.t.} \quad \mathbf{b} \in \{-1,1\}^n, \mathbf{b}^\mathrm{T} \mathbf{1}=(n\mod 2) \end{split} \label{equ:bqpminus1}$$ Second, we set the diagonal elements of $\mathbf{Q}$ to zero. Third, we transform the domain from $\{-1,1\}^n$ to $\{0,1\}^n$ by executing the transformation of $\mathbf{b}=2\mathbf{c}-1,\mathbf{c} \in \{0,1\}^n$. Finally, we rewrite the BQP problem to the form without the linear term as well as removing the constant form, and we have the following BQP problem that is equivalent to Eq. (\[equ:bqpminus1\]): $$\begin{split} \max_{\tilde{\mathbf{c}}} \quad& \tilde{\mathbf{c}}^\mathrm{T} \mathbf{Q}_0 \tilde{\mathbf{c}} \\ \mathrm{s.t.} \quad& \tilde{\mathbf{c}} = [\mathbf{c},1]^\mathrm{T} \in \{0,1\}^{n+1}, \mathbf{c}^\mathrm{T} \mathbf{1} = \lfloor \frac{n+1}{2} \rfloor \\ &\mathbf{Q}_0 = \begin{pmatrix} 8[\mathbf{Q}-\mathrm{diag}(\mathbf{Q})] & \mathbf{q}_0 \\ \mathbf{q}_0 & 0 \end{pmatrix} \in \mathbb{R}^{n+1} \\ &\mathbf{q}_0 = -4[\mathbf{Q}-\mathrm{diag}(\mathbf{Q})]^\mathrm{T}\mathbf{1} - \mathbf{q} \\ \end{split} \label{equ:bqpnolinear}$$ where $\mathrm{diag}(\mathbf{Q})$ is the diagonal matrix of $\mathbf{Q}$. As illustrated in [@yang2013new], Eq. (\[equ:bqpnolinear\]) can be regarded as a specific clustering problem. Given a set of vectors $\mathcal{V} = \{\mathbf{v}_i, i=1,...,n,n+1\}$, we want to find a subset $\mathcal{V}_1$ of size $K=\lfloor \frac{n+1}{2} \rfloor+1$ such that $\mathbf{v}_{n+1} \in \mathcal{V}_1$ and the sum of square of the distances between the vectors in $\mathcal{V}_1$ and the clustering center is minimized. The objective can be formulated as $$\min_{|\mathcal{V}_1|=K, \mathbf{v}_{n+1} \in \mathcal{V}_1} \sum_{\mathbf{u} \in \mathcal{V}_1} \Arrowvert \mathbf{u} - \frac{\sum_{\mathbf{v} \in \mathcal{V}_1} \mathbf{v}}{K} \Arrowvert^2 \quad \mathrm{s.t.} \mathcal{V}_1 \in \mathcal{V} \label{equ:cluster}$$ It is clear that there exists a certain $\lambda$ such that $\mathbf{Q}_0+\lambda\mathbf{I}$ is positive semidefinite, so we have a sufficently large $\lambda$ such that $\mathbf{Q}_0+\lambda\mathbf{I}=\mathbf{V}^\mathrm{T}\mathbf{V},\mathbf{V} \in \mathbb{R}^{(n+1) \times (n+1)}$. Then we rewrite the Theorem 2.4.1 in [@yang2013new] to obtain the following theorem: If there exists $\lambda$ such that $\mathbf{Q}_0+\lambda\mathbf{I}=\mathbf{V}^\mathrm{T}\mathbf{V}, \mathbf{V} \in \mathbb{R}^{n \times n}$, and $\mathbf{v}_i$ is the $i$th column of $\mathbf{V}$, then Eq. (\[equ:bqpnolinear\]) and Eq. (\[equ:cluster\]) are equivalent. The global optimum of Eq. (\[equ:bqpnolinear\]), denote $\mathbf{c}^*=[c^*_1,...,c^*_n]^\mathrm{T} \in \{0,1\}^n$, and Eq. (\[equ:cluster\]), denote $\mathcal{V}^*_1$, have the relationship such that $c^*_i=1 \Leftrightarrow \mathbf{v}_i \in \mathcal{V}^*_1, \forall i=1,...,n$. \[the:cluster\]  [@yang2013new] proposes an iterative method to approximately solve the clustering problem. However, the Cholesky decomposition is used for getting the vectors to cluster, which involves $O(n^3)$ computational complexity. So we have to discover another efficient clustering-based algorithm. **Input:** hashcode $\mathbf{H}$, hyper-parameter $\lambda$. **Output:** optimal solution of $\mathbf{H}$-Subproblem (Eq. (\[hsub\])). Let $\textbf{b}_0$ be the $k$th column of $\mathbf{H}$. Denote $\textbf{b}_0$ as initialization of Eq. (\[equ:bqpminus1\]). Set $\mathbf{c}_0 \gets \frac{1}{2}(\mathbf{b}_0+1)$ as the initialization of Eq. (\[equ:bqpnolinear\]). Set $\mathbf{v}_i \in \mathcal{V}_1$ for any $[\mathbf{c}_0]_i=1$; Compute $\mathrm{sim}(\mathbf{v}_i,\mathbf{m})$ for all $i=1,2,...,n$, according to Eq. (\[equ:dist\]); Select $\mathbf{v}_i$ as the subset $\mathcal{V}_1$, such that they are in the first $\lfloor \frac{n+1}{2} \rfloor$ maximum value of $\mathrm{sim}(\mathbf{v}_i,\mathbf{m})$; Set $\mathbf{c}=[c_1,...,c_n]^\mathrm{T}$ such that $c_i=1$ if $\mathbf{v}_i \in \mathcal{V}_1$, and $c_i=0$ if $\mathbf{v}_i \notin \mathcal{V}_1$; **break**; $\mathbf{c}_0 \gets \mathbf{c}$; $\mathbf{b}=2\mathbf{c}-1$; Replace the $k$th column of $\mathbf{H}$ with $\mathbf{b}$. Return the updated $\mathbf{H}$. \[hsub\_algo\] We denote $[\mathbf{a}]_i$ as the $i$th column of a vector $\mathbf{a}$, and $[\mathbf{A}]_{ij}$ denotes a element of $\mathbf{A}$ at the $i$th row and the $j$th column. It can be noticed that $\| \mathbf{v}_i \|^2=\lambda$ and $\mathbf{v}_i^\mathrm{T}\mathbf{v}_j=[\mathbf{Q}_0+\lambda\mathbf{I}]_{ij}$ for arbitrary $i,j=1,2,...,n+1$, thus the square of distance of $\mathbf{v}_i \in \mathcal{V}_1$ and the clustering center $\mathbf{m}=\frac{\sum_{\mathbf{v} \in \mathcal{V}_1} \mathbf{v}}{K}$ is $$\begin{split} \Arrowvert \mathbf{v}_i-\mathbf{m} \Arrowvert^2 &= \mathbf{v}_i^2 + \mathbf{m}^2 - \frac{2 \sum_{\mathbf{v} \in \mathcal{V}_1} \mathbf{v}^\mathrm{T} \mathbf{v}_i}{K} \\ &= -\frac{2}{K} \sum_{j, \mathbf{v}_j \in \mathcal{V}_1} [\mathbf{Q}_0+\lambda\mathbf{I}]_{ij} + \mathrm{const} \end{split}$$ Denote the similarity between $\mathbf{v}_i$ and $\mathbf{m}$ as $ \mathrm{sim}(\mathbf{v}_i, \mathbf{m})=\sum_{j, \mathbf{v}_j \in \mathcal{V}_1} [\mathbf{Q}_0+\lambda\mathbf{I}]_{ij}$. Applying the [*Asymmetric Low-Rank Matrix Factorization*]{} such that $\mathbf{Q}=r\mathbf{P}\mathbf{R}^\mathrm{T} - \mathbf{H}{'} \mathbf{H}{'} ^{\mathrm{T}}$, we can greatly simplify the similarity computation: $$\begin{split} \mathrm{sim}(\mathbf{v}_i, \mathbf{m}) &= \begin{cases} 8[\mathbf{d}_0]_i + [\mathbf{q}_0]_i - 8[\mathbf{Q}]_{ii} + \lambda & \mathbf{v}_i \in \mathcal{V}_1, i \le n \\ 8[\mathbf{d}_0]_i + [\mathbf{q}_0]_i & \mathbf{v}_i \notin \mathcal{V}_1 \\ \end{cases} \\ \mathbf{d}_0&=r\mathbf{P}(\sum_{j, \mathbf{v}_j \in \mathcal{V}_1} \mathbf{R}_{j,*})^\mathrm{T}-\mathbf{H}{'} (\sum_{j, \mathbf{v}_j \in \mathcal{V}_1} \mathbf{H}{'}_{j,*})^\mathrm{T} \end{split} \label{equ:dist}$$ where $\mathbf{R}_{j,*}$ and $\mathbf{H}{'}_{j,*}$ is the $j$th row of matrix $\mathbf{R}$ and $\mathbf{H}{'}$, respectively. Thus we can compute the distance without executing the Cholesky decomposition. For a specific $\lambda$, we modify the method proposed in  [@yang2013new] and get an efficient algorithm to solve Eq. (\[equ:bqpminus1\]), which is summarized in Algorithm \[hsub\_algo\]. Asymmetric Low-Rank Similarity Matrix Factorization --------------------------------------------------- As discussed before, we should find two low-rank matrices $\mathbf{P} \in \mathbb{R}^{n \times l}, \mathbf{R} \in \mathbb{R}^{n \times l}(l \ll n)$ to approximate the similarity matrix $\mathbf{S} \in \mathbb{R}^{n \times n}$. In supervised hashing algorithms, $\mathbf{S}$ is mostly defined by semantic label. $s_{ij}=1$ if $\mathbf{x}_i$ and $\mathbf{x}_j$ share the same semantic labels, and $s_{ij}=-1$ otherwise. Suppose there are $l$ semantic labels. Denote $\mathbf{y}_i = [y_{i1},y_{i2},...,y_{il}]^\mathrm{T} \in \{0,1\}^l$ as the label vector of data $\mathbf{x}_i$, in which $y_{ik}=1$ if the label of $\mathbf{x}_i$ is $k$ and $y_{ik}=0$ otherwise. Define $\mathbf{Y}=[\mathbf{y}_1,\mathbf{y}_2,...,\mathbf{y}_n]^\mathrm{T}$ and $$\mathbf{P} = [2\mathbf{Y}; \mathbf{1}_{n \time 1}] \in \mathbb{R}^{n \times (l+1)}, \mathbf{R} = [\mathbf{Y}; -\mathbf{1}_{n \time 1}] \in \mathbb{R}^{n \times (l+1)} \label{factor}$$ then we have the following matrix factorization: $$\mathbf{P} \mathbf{R}^\mathrm{T} = 2 \mathbf{Y} \mathbf{Y}^\mathrm{T} - \mathbf{1}_{n \times n} = \mathbf{S} \label{low_rank}$$ It should be noticed that $\mathbf{Y}$ can be stored by sparse matrix, so just $O(n)$ space can contain all the information of $\mathbf{S}$. We can also apply Eq. (\[low\_rank\]) to existing pairwise supervised hashing methods including KSH [@liu2012supervised], TSH [@lin2013general], etc. Moreover, the [*Asymmetric Low-Rank Similarity Matrix Factorization*]{} scheme can be applied to any similarity matrix. For example, a variety of methods have discussed the low-rank approximation of Gaussian RBF kernel matrix [@liu2011hashing; @zhang2008improved]. Choosing Hash Functions and Initialization {#k_hash_func} ------------------------------------------ For most binary classifiers, they contain certain function that predicts 1 or -1 for given data. Learning a good hash function corresponds to training a good binary classifier. There are two kinds of widely used hash functions: kernel based and deep neural network based, which is illustrated in Figure \[fig:framework\]. Since Eq. (\[dis\_obj\]) is a mixed-integer non-convex problem, a good initial point is very important. We can choose the existing efficient and scalable hashing algorithm to get the initialization value of $\mathbf{H}$ and hash function $F$. Different hash functions have different initialization strategies. The strategy of choosing hash function and initialization is discussed in detail at Section \[sec:deephashing\] and \[sec:kernelhash\]. Analysis -------- **Input:** Training data $\{ \mathbf{x}_i, y_i \}_{i=1}^n$; code length $r$; a certain hash function $F$ and a binary classification loss function $\mathcal{L} (h_{ik}, \mathbf{w}_k^\mathrm{T} k(\mathbf{x}_i))$; max iteration number $t$. **Output:** Hash function $\mathbf{h}=\mathrm{sgn}(F(\mathbf{x}))$ 1. Initialize hash function $F(\cdot)$ with a certain efficient and scalabel hashing algorithm. Set $\mathbf{H}=\mathrm{sgn} (F(\mathbf{X}))$. 2. Loop until converge or reach maximum iterations: - **H-Subproblem:** optimize Eq. (\[hsub\]) by [*Clustering-based Batch Coordinate Descent(C-BCD)*]{} algorithm proposed in Algorithm \[hsub\_algo\]. - **F-Subproblem:** optimize Eq. (\[wsub\]) by solving $r$ binary classification problems. \[dksh\_algo\] The proposed [*Discrete Supervised Hashing(DISH) Framework*]{} is summarized in Algorithm \[dksh\_algo\]. Denote $p$ as average count of nonzeros in each row of $\mathbf{P}$ and $\mathbf{R}$. Solving $\mathbf{H}$-Subproblem involves executing Fast C-BCD algorithm $r$ times, which will cost at most $O(2(p+r+1)T_{iter}nr)$ time, where $T_{iter}$ is the maximum number of iteration in Fast C-BCD algorithm. The space complexity is just $O(2pn+rn)$ for solving $\mathbf{H}$-Subproblem. For $\mathbf{F}$-Subproblem, $r$ binary classification problems should be solved, and many efficient classification algorithm with $O(n)$ time and space complexity can be used. Therefore, our algorithm is expected to be scalable. Discrete Learning with Deep Hashing {#sec:deephashing} =================================== By applying the deep neural networks to our DISH framework, we are able to train neural nets with the discrete constraints preserved. Denote $\Phi (\mathbf{x}_i) \in \mathbb{R}^m$ as the activation of the last hidden layer for a given image $\mathbf{x}_i$, and $\mathbf{W} = [\mathbf{w}_1,...,\mathbf{w}_r] \in \mathbb{R}^{m \times r}$ as the weight of the last fully-collected layer, where $r$ is the hash codes length. Then the activation of the output hashing layer is $F(\mathbf{x}_i)=\mathbf{W}^\mathrm{T} \Phi (\mathbf{x}_i)$ and the hash function is defined as $$\mathbf{h}_i=\mathrm{sgn}(\mathbf{W}^\mathrm{T} \Phi (\mathbf{x}_i))$$ And we use the squared hinge-loss for binary classification loss function: $$\mathcal{Q}_F=\sum_{i=1}^n \sum_{k=1}^r [\max (1-h_{ik} \mathbf{w}^\mathrm{T}_k \Phi (\mathbf{x}_i))]^2$$ There are two approaches to learn better network and avoid overfitting. First of all, if the class label is provided, we use the similar approach as CNNH+ [@xia2014supervised], in which the softmax loss layer can be added above the last hidden layer: $$\sum_{i=1}^n[\mathrm{softmax}(y_i, \mathbf{W}^\mathrm{T}_1 \Phi(\mathbf{x}_i)) + \mu \sum_{k=1}^r [\max (1-h_{ik} \mathbf{w}^\mathrm{T}_k \Phi (\mathbf{x}_i))]^2]$$ where $\mu$ is a hyper-parameter, $y_i$ is the class label of $\mathbf{x}_i$, $\mathbf{W}_1$ is the parameter of fully-collected layer connecting to the softmax loss layer, and $\mathrm{softmax}(\cdot, \cdot)$ is the softmax loss. Second, recent work show that [*DisturbLabel*]{} [@xie2016disturblabel], which randomly replaces part of labels to incorrect value during iteration, can prevent the network training from overfitting. Inspired by this, we can randomly flip some bits of the binary codes with probability $\alpha$ during the training of the network. We name this process as [*DisturbBinaryCodes*]{}, which we expect to improve the quality of binary codes. For labeled data, we adopt the same procedure of CNNBH [@guo2016hash] as initialization. CNNBH binarizes the activation of a fully-connected layer at threshold 0 to generate hashcodes, which is easy to train and achieves good performance. Discrete Learning with Kernel-Based Hashing {#sec:kernelhash} =========================================== Kernel methods can embed input data to more separable space, which is widely used in machine learning algorithms such SVM, Gaussian Process, etc. As in Supervised Hashing with Kernels(KSH) [@liu2012supervised] and Supervised Discrete Hashing(SDH) [@Shen_2015_CVPR], we define hash function as follows: $$h_{ik}=f_k(\mathbf{x}_i)=\mathrm{sgn}(\mathbf{w}_k^\mathrm{T} k(\mathbf{x}_i)) \label{equ:kernelf}$$ where $k(\mathbf{x})$ is the feature vector in the kernel space. $k(\mathbf{x})$ is defined as $k(\mathbf{x}) = [\phi(\mathbf{x}, \mathbf{x}_{(1)}) - \frac{1}{n} \sum_{i=1}^n \phi(\mathbf{x}_i, \mathbf{x}_{(1)}), ..., \phi(\mathbf{x}, \mathbf{x}_{(m)}) - \frac{1}{n} \sum_{i=1}^n \phi(\mathbf{x}_i, \mathbf{x}_{(m)})]^\mathrm{T}$, where $\phi(\mathbf{x}_i, \mathbf{x}_{(j)})$ is the kernel function between $\mathbf{x}_i$ and $\mathbf{x}_{(j)}$, and $\mathbf{x}_{(j)}, j = 1,2,...,m$ are anchors. The subtraction of $\frac{1}{n} \sum_{i=1}^n \phi(\mathbf{x}_i, \mathbf{x}_{(j)})$ is to centerize the feature vectors in kernel space, so that each bit of hashcode can be more balanced. Denote $\mathbf{W}=[\mathbf{w}_1,...,\mathbf{w}_r]$ and $K(\mathbf{X})=[k(\mathbf{x}_1),...,k(\mathbf{x}_n)]^\mathrm{T}$, the hashcodes can be formulated as $\mathbf{H} = F(\mathbf{X}) = \mathrm{sgn}(K(\mathbf{X})\mathbf{W})$, and Eq. (\[dis\_obj\]) can be rewritten as $$\min_{\mathbf{H}, \mathbf{W}} \mathcal{Q} = \Arrowvert r\mathbf{P}\mathbf{R}^\mathrm{T}- \mathbf{H} \mathbf{H}^\mathrm{T} \Arrowvert_F^2 + n \nu \sum_{k=1}^r \sum_{i=1}^n l (h_{ik}, \mathbf{w}_k^\mathrm{T} k(\mathbf{x}_i)) \label{ksh_dis_obj}$$ and we can derive three types of binary classifiers: Linear Regression, SVM and Logistic Regression, each of which corresponds to a kind of loss function $\sum_{i=1}^n l (h_{ik}, \mathbf{w}_k^\mathrm{T} k(\mathbf{x}_i))$. For initialization, we proposed a relaxed method similar to  [@liu2012supervised], in which we remove the $\mathrm{sgn}$ function of Eq. (\[obj\]), and optimize each column of $\mathbf{W} = [\mathbf{w}_1,\mathbf{w}_2,...,\mathbf{w}_r]$ successively. Denote $\mathbf{H}_k$ as the first $k$ column of $\mathbf{H}$. The optimization of $\mathbf{w}_k$ is a constrained quadratic problem: $$\setlength{\abovedisplayskip}{3pt} \setlength{\belowdisplayskip}{3pt} \begin{split} \max_{\mathbf{w}_k} (K(\mathbf{X}) \mathbf{w}_k)^{\mathrm{T}} (\mathbf{P}\mathbf{R}^\mathrm{T} - \mathbf{H}_k \mathbf{H}_k ^{\mathrm{T}})(K(\mathbf{X}) \mathbf{w}_k) \\ \mathrm{s.t.} (K(\mathbf{X}) \mathbf{w}_k)^{\mathrm{T}} (K(\mathbf{X}) \mathbf{w}_k) = n \end{split} \label{spectral_obj}$$ where $n$ is the number of training samples. Eq. (\[spectral\_obj\]) is a generalized eigenvalue problem $$K(\mathbf{X})^\mathrm{T}(\mathbf{P}\mathbf{R}^\mathrm{T} - \mathbf{H}_k \mathbf{H}_k ^{\mathrm{T}})K(\mathbf{X}) \mathbf{w}_k = \lambda K(\mathbf{X})^\mathrm{T}K(\mathbf{X}) \mathbf{w}_k \label{eigen_obj}$$ After optimizing $\mathbf{w}_k$, denoted as $\mathbf{w}_k^0$, the $k$th column of $\mathbf{H}$ can be generated as $\mathrm{sgn}(K(\mathbf{X}) \mathbf{w}_k^0)$. The time and space complexity of the initialization procedure is just $O(n)$ because the asymmetric low-rank matrix factorization is involved. Experiments {#sec:exp} =========== In this section, we run large-scale image retrieval experiments on three benchmarks: CIFAR-10[^1], ImageNet[^2] [@deng2009imagenet] and Nuswide[^3] [@chua2009nus]. CIFAR-10 consists of 60,000 $32 \times 32$ color images from 10 object categories. ImageNet dataset is obtained from ILSVRC2012 dataset, which contains more than 1.2 million training images of 1,000 categories in total, together with 50,000 validation images. Nuswide dataset contains about 270K images collected from Flickr, and about 220K images are available from the Internet now. It associates with 81 ground truth concept labels, and each image contains multiple semantic labels. Following [@liu2011hashing], we only use the images associated with the 21 most frequent concept tags, where the total number of images is about 190K, and number of images associated with each tag is at least 5,000. The experimental protocols is similar with [@Shen_2015_CVPR; @xia2014supervised]. In CIFAR-10 dataset, we randomly select 1,000 images (100 images per class) as query set, and the rest 59,000 images as retrieval database. In Nuswide dataset, we randomly select 2,100 images (100 images per class) as the query set. And in ImageNet dataset, the provided training set are used for retrieval database, and 50,000 validation images for the query set. For CIFAR-10 and ImageNet dataset, similar images share the same semantic label. For Nuswide dataset, similar images share at least one semantic label. For the proposed DISH framework, we name **DISH-D** and **DISH-K** as deep hashing method and kernel-based hashing method, respectively. We compare them with some recent state-of-the-art algorithms including four supervised methods: SDH [@Shen_2015_CVPR], KSH [@liu2012supervised], FastH [@lin2014fast], CCA-ITQ [@gong2013iterative], three unsupervised methods: PCA-ITQ [@gong2013iterative], AGH [@liu2011hashing], DGH [@liu2014discrete], and five deep hashing methods: CNNH+ [@xia2014supervised], SFHC [@lai2015simultaneous], CNNBH [@guo2016hash], DHN [@zhu2016deep], FTDE [@zhuang2016fast]. Most codes and suggested parameters of these methods are available from the corresponding authors. Similar with [@liu2012supervised; @xia2014supervised], for each dataset, we report the compared results in terms of [*mean average precision*]{}(MAP), precision of Hamming distance within 2, precision of top returned candidates. For Nuswide, we calculate the MAP value within the top 5000 returned neighbors, and we report the MAP of all retrieved samples on CIFAR-10 and ImageNet dataset. Groundtruths are defined by whether two candidates are similar. Scalability and sensitivity of parameters of the proposed framework will also be discussed in the corresponding subsection. The training is done on a server with Intel(R) Xeon(R) E5-2678 v3@2.50GHz CPU, 64GB RAM and a Geforce GTX TITAN X with 12GB memory. Experiments with Kernel-based Hashing {#subsec:kernel} ------------------------------------- ------------ ---------- ----------- ----------- ----------- ----------- --------- -- -- -- -- -- \# Time/s Method Training 16 bits 32 bits 64 bits 96 bits 64 bits PCA-ITQ 59000 0.163 0.169 0.175 0.178 9.6 AGH 59000 0.150 0.148 0.141 0.137 4.1 DGH 59000 0.168 0.173 0.178 0.180 28.3 KSH 5000 0.356 0.390 0.409 0.415 756 SDH 5000 0.341 0.374 0.397 0.404 8.6 **DISH-K** 5000 **0.380** **0.407** **0.420** **0.424** 22.2 CCA-ITQ 59000 0.301 0.323 0.328 0.334 25.4 KSH 59000 0.405 0.439 0.468 0.474 16415 FastH 59000 0.394 0.427 0.451 0.457 1045 SDH 59000 0.414 0.442 0.470 0.474 76.0 **DISH-K** 59000 **0.446** **0.474** **0.486** **0.491** 83.1 ------------ ---------- ----------- ----------- ----------- ----------- --------- -- -- -- -- -- : Comparative results of kernel-based hashing methods in MAP and training time(seconds) on CIFAR-10.The results are the average of 5 trials. We use kernel-based hash function in FastH for fair comparison.[]{data-label="tab:cifar10"} ------------ ----------- ----------- ----------- ----------- --------- Time/s Method 16 bits 32 bits 64 bits 96 bits 64 bits DGH 0.413 0.421 0.428 0.429 97.5 CCA-ITQ 0.508 0.515 0.522 0.524 146.1 KSH 0.508 0.523 0.530 **0.536** 39543 SDH **0.516** **0.526** 0.531 0.531 489.1 **DISH-K** 0.512 0.524 **0.532** 0.533 216.7 ------------ ----------- ----------- ----------- ----------- --------- : Results of various kernel-based hashing methods on Nuswide dataset. 500 dimension bag-of-words features are extracted for evaluation.[]{data-label="tab:nuswide"} ![Precision at hamming distance within 2 value and precision-recall curve of different kernel-based methods on CIFAR-10 dataset.[]{data-label="fig:cifar10"}](cifar10.pdf) ![Precision at hamming distance within 2 value and precision-recall curve of different kernel-based methods on Nuswide dataset.[]{data-label="fig:nuswide"}](nuswide.pdf) Most existing hashing methods use the hand-crafted image features for evaluation, thus we first of all use kernel-based hash function as well as widely used hand-crafted image features to test the effectiveness of our DISH framework. For CIFAR-10 dataset, we extract 512 dimensional GIST descriptors. And for Nuswide dataset, the provided 500-dimensional bag-of-word features are used, and all features are normalized by $l_2$ norm. We use Gaussian RBF kernel $\phi(\mathbf{x}, \mathbf{y}) = \exp (-\| \mathbf{x} - \mathbf{y} \|/2\sigma^2)$ and $m=1000$ anchors. Anchors are randomly drawn from the training set and $\sigma$ is tuned for an approximate value. For FastH, we also use the kernel-based function for fair comparison. For KSH, Eq. (\[low\_rank\]) is applied to achieve faster training. For the proposed method, we set the number of iterations $t=5$ and $\nu=10^{-4}$. We solve the linear regression problem in $\mathbf{F}$-subproblem, thus the parameter $\mathbf{W}$ can be updated as follows: $$\mathbf{W}^* = (K(\mathbf{X})^\mathrm{T} K(\mathbf{X}) + \lambda \mathbf{I})^{-1} K(\mathbf{X})^\mathrm{T} \mathbf{H}$$ where we set $\lambda=0$. Retrieval results of different methods are shown in Table \[tab:cifar10\], \[tab:nuswide\] and Figure \[fig:cifar10\], \[fig:nuswide\]. Our DISH method achieves best performance on CIFAR-10 dataset at both MAP and precision of Hamming distance within 2 value.[^4] And it is not surprisingly that the proposed method achieves higher precision value in longer codes, showing that discrete learning can achieve higher hash lookup success rate. Although our DISH framework is just comparable with SDH and KSH on Nuswide dataset at MAP value, the training speed of DISH is faster than SDH and KSH, showing that we can generate effective codes efficiently. The last column in Table \[tab:cifar10\] and \[tab:nuswide\] show the training time. We can see that the training time of the proposed method is much faster than that of other methods involving pairwise similarity. Less than 2 minutes are consumed to train 59,000 images in our method, and it only cost 4 minutes for training nearly 200,000 data, showing that the proposed method can be easily applied to large-scale dataset. Experiments with Deep Hashing {#subsec:deep} ----------------------------- layer No. 1,2 3,4 5,6 ----------- --------- ---- ---------- ---- ---------- ---- Type Conv MP Conv MP Conv AP Size 3\*3-96 3\*3-192 3\*3-192 : The structure of 6-layer convolutional(6conv) net. Conv means convolution layer, MP means max-pooling layer, AP means average-pooling layer. The size of pooling is 3\*3 with stride 2. ReLU activation used above each convolution layer.[]{data-label="tab:6conv"} [c|c|cccc]{} & &\ Method & Net & 12 bits & 24 bits & 32 bits & 48 bits\ \ CNNH+ & 6conv & 0.633 & 0.625 & 0.649 & 0.654\ SFHC [@lai2015simultaneous] & NIN & 0.552 & 0.566 & 0.558 & 0.581\ CNNBH [@guo2016hash] & 6conv & 0.633 & 0.647 & 0.648 & 0.647\ **DISH-D(Ours)** & 6conv & **0.667** & **0.686** & **0.690** & **0.695**\ \ DHN [@zhu2016deep] & AlexNet & 0.555 & 0.594 & 0.603 & 0.621\ **DISH-D(Ours)** & AlexNet & **0.758** & **0.784** & **0.799** & **0.791**\ \ SFHC [@zhuang2016fast] & VGG-16 & N/A & 0.677 & 0.688 & 0.699\ FTDE [@zhuang2016fast] & VGG-16 & N/A & 0.760 & 0.768 & 0.769\ **DISH-D(Ours)** & VGG-16 & **0.841** & **0.854** & **0.859** & **0.857**\ [c|c|cccc]{} & &\ Method & Net & 12 bits & 24 bits & 32 bits & 48 bits\ \ DHN [@zhu2016deep] & AlexNet & 0.708 & 0.735 & 0.748 & 0.758\ **DISH-D(Ours)** & AlexNet & **0.787** & **0.810** & **0.810** & **0.813**\ \ SFHC [@zhuang2016fast] & VGG-16 & N/A & 0.718 & 0.720 & 0.723\ FTDE [@zhuang2016fast] & VGG-16 & N/A & 0.750 & 0.756 & 0.760\ **DISH-D(Ours)** & VGG-16 & **0.833** & **0.850** & **0.850** & **0.856**\ ![Top-100 precision of different deep hashing methods on CIFAR-10 and Nuswide dataset.[]{data-label="fig:deephash"}](deephash.pdf) ### Network Structure and Experimental Setup Below the output hashing layer, the network contains multiple convolutional, pooling and fully-connected layers. Different models can lead to significantly different retrieval performance. For fair comparison, we use three types of model for evaluation: a 6-layer convolutional network(6conv) which is shown in Table \[tab:6conv\]; the pre-trained AlexNet [@krizhevsky2012imagenet]; and the pre-trained VGG-16 net [@simonyan2014very]. We evaluate the deep hashing methods on CIFAR-10 and Nuswide dataset. For CIFAR-10, following [@xia2014supervised], we random sample 5,000 images (500 images per class) as training set. For Nuswide, we use the database images as training images. We resize images to $256 \times 256$ to train AlexNet and VGG-16 net. We implement the proposed model based on the **Caffe** [@jia2014caffe] framework. We set the number of iterations $t=3$ and $\nu=10^{-4}$. For 6conv model, the initial learning rate is set to $0.01$. For AlexNet and VGG-16 model, the weight before the last hidden layer is copied from pre-trained model, the initial learning rate is $0.001$ and the learning rate of last fully-connected layer is 10 times that of lower layers. The base learning rate drops 50% after each iteration. We use stochastic gradient descent (SGD) with momentum $0.9$ and the weight decay is set to $0.0005$. The parameter of [*DisturbBinaryCodes*]{} is set to $\alpha=0.3$ for CIFAR-10 dataset to avoid over-fitting. It is selected with cross-validation. ### Comparison with State-of-the-art Table \[tab:cifar10\_deep\], \[tab:nuswide\_deep\] and Figure \[fig:deephash\] shows the retrieval performance on existing deep hashing methods. Note that the results with citation are copied from the corresponding papers. On a variety of network structures, our DISH framework achieves much better MAP and precision of Hamming distance within 2 value by a margin of **4%-10%**. Compared with DHN and CNNBH, it can be seen clearly that combining discrete learning with deep hashing can greatly improve the performance compared with relaxed-based deep hashing methods. Although the network structure of the proposed framework is similar with CNNH+ and FTDE, our DISH framework is much better than the latter. It is likely that the codes generated by discrete learning procedure in DISH framework can embed more information of the distribution of data. ------------------------ -------- ---------- --------- Method Net CIFAR-10 Nuswide SFHC [@zhuang2016fast] VGG-16 174 365 FTDE [@zhuang2016fast] VGG-16 15 32 **DISH-D(Ours)** VGG-16 **4** **9** ------------------------ -------- ---------- --------- : Training time of various deep hashing methods. VGG-16 net is used for evaluation. Our DISH framework performs much faster than others.[]{data-label="tab:deep_time"} Table \[tab:deep\_time\] summarize the training time of some state-of-the-art methods. VGG-16 net is used for evaluation. As expected, the training speed is much faster than triplet-based deep hashing methods. It takes less than 1 day to generate efficient binary codes by VGG-16 net, thus we can also train binary codes efficiently with deep neural nets. Scalability: Training Hashcodes on ImageNet ------------------------------------------- --------------------------- ----------- ----------- ----------- ----------- ---------- Time/s Method 32 bits 64 bits 128 bits 256 bits 128 bits DGH 0.044 0.077 0.108 0.135 121.5 CCA-ITQ 0.096 0.183 0.271 0.340 421.0 KSH 0.131 0.219 0.282 0.319 9686 SDH 0.133 0.219 0.284 0.325 5831 **DISH-K** **0.176** **0.250** **0.313** **0.357** 257.3 $l_2$ distance FastH-Full [@lin2014fast] N/A **DISH-K-Full** 2050 --------------------------- ----------- ----------- ----------- ----------- ---------- : Comparative results of various methods on ImageNet dataset. For the first five rows, 100,000 samples are used for training. FastH-Full and DISH-K-Full used all 1.2 million training samples at 128 bits.[]{data-label="tab:imagenet"} We evaluate with the ImageNet dataset for testing the scalability of our proposed framework. We use the provided training set as the retrieval database and 50,000 validation set as the query set. For kernel-based methods, 4096-dimensional features are extracted from the last hidden layer of VGG-19 net [@simonyan2014very]. We subtract the mean of training image features, followed by normalizing the feature representation with $l_2$-norm. We found that it may not converge during training because the number of dissimilar pairs is much larger than the similar ones. To tackle the problem, we set the value of similar pairs as 9 in $\mathbf{S}$ to balance the dissimilar pairs. Thus the similarity matrix can be represented as $\mathbf{S}=10\mathbf{Y}\mathbf{Y}^\mathrm{T}-\mathbf{1}$, where $\mathbf{Y}$ is defined the same as Eq. (\[low\_rank\]). First, we sample 100,000 images (100 samples per category) as the training set. The results are shown in Table \[tab:imagenet\] and Figure \[fig:imagenet\]. Retrieval results based on $l_2$ distance of 4096-dimensional features are also reported. Similar with results on CIFAR-10, the proposed DISH achieves the best performance, especially on MAP($\sim0.03$). DGH algorithm gets higher precision of Hamming distance within 2 in longer codes, but has much poorer MAP. Overall, the performances of DISH and DGH show the power of discrete learning methods. To test the scalability of proposed framework, we train hash function with the full 1.2 million ImageNet dataset. It takes less than 1 hour for training, showing the framework can easily applied to large-scale dataset. DISH-K-Full in Table \[tab:imagenet\] and Figure \[fig:imagenet\] shows the result. The MAP value improves over **33%** if the whole training data are used, and it is shown clearly that DISH method is able to deal with millions of data. It is interesting that the proposed method outperforms the method based on $l_2$ distance, showing that DISH can not only reduce the dimensionality of data, but also embed more useful information. ![Results of precision at Hamming distance within 2 and Top-k precision of different methods on ImageNet dataset.[]{data-label="fig:imagenet"}](imagenet.pdf) ![Comparative results of different $\nu$ and different number of iterations $t$. “w/ Init” means we perform initialization before discrete optimization and “w/o Init” otherwise. **DISH-K** is used for evaluation.[]{data-label="fig:nu"}](paras.pdf) \[fig:iteration\] $\alpha$ 0 0.1 0.2 0.3 0.4 0.5 ----------- ------- ------- ------- ------- ------- ------- MAP 0.687 0.681 0.699 0.699 0.706 0.326 Precision 0.688 0.676 0.678 0.673 0.647 0.001 : Results of various [*DisturbBinaryCodes*]{} ratio $\alpha$. 6conv net is used for evaluation.[]{data-label="tab:disturb"} Sensitivity to Parameters {#subsec:discussion} ------------------------- In this subsection, the efficiency on different settings of the DISH framework (DISH-K and DISH-D) is evaluated. We test our method on CIFAR-10 dataset, and the experimental settings are the same as Section \[subsec:deep\] and \[subsec:kernel\]. 5,000 samples (500 samples per class) from the retrieval database are used for training set. The code length is 64. ### Influence of $\nu$ Figure \[fig:nu\](a) shows the performance on different values of $\nu$. It is shown that the algorithm is not sensitive to $\nu$ over a wide range, so we can choose this parameter freely. ### Influence of initialization and discrete learning procedure Figure \[fig:iteration\](b) shows the performance on the existence of initialization and different number of iterations $t$. “w/ Init” means we perform initialization before discrete optimization, and “w/o Init” otherwise. $t=0$ means we do not perform discrete optimization. It is clear that both initialization and discrete learning procedure makes the retrieval performance much better, especially for precision of Hamming distance within 2. The performance will not be improved after just a few iterations, showing our method can converge fast. ### Influence of [*DisturbBinaryCodes*]{} Table \[tab:disturb\] shows the performance on different disturb rate $\alpha$, $\alpha=0$ means no [*DistrubBinaryCodes*]{} is involved. It is seen clearly that the performance can be greatly improved if a proper $\alpha$ is set, showing that [*DistrubBinaryCodes*]{} can be performed as a regularizer on the loss-layer and avoid over-fitting. Conclusion {#sec:conclusion} ========== In this paper, we propose a novel discrete supervised hashing framework for supervised hashing problem. To deal with the discrete constraints in the binary codes, a discrete learning procedure is proposed to learn binary codes directly. We decompose the learning procedure into two sub-problems: one is to learn binary codes directly, the other is solving several binary classification problems to learn hash functions. To leverage the pairwise similarity and reduce the training time simultaneously, a novel [*Asymmetric Low-rank Similarity Matrix Factorization*]{} approach is introduced, and we propose the [*Fast Clustering-based Batch Coordinate Descent*]{} method to learn binary codes efficiently. The DISH framework can be easily adopted to arbitrary binary classifier, including deep neural networks and kernel-based methods. Experimental results on large-scale datasets demonstrate the efficiency and scalability of our DISH framework. Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by the National Basic Research Program (973 Program) of China (No. 2013CB329403), and the National Natural Science Foundation of China (Nos. 61332007, 91420201 and 61620106010). [^1]: http://www.cs.toronto.edu/\~kriz/cifar.html [^2]: http://www.image-net.org/ [^3]: http://lms.comp.nus.edu.sg/research/NUS-WIDE.htm [^4]: The reported experiments use $\mathbf{H}=\mathrm{sgn}(F(\mathbf{X}))$ to generate hashcodes of the database set. Some recent works [@kang2016column; @zhang2014supervised] directly learn codes by similarity information for database set, and use out-of-sample technique for query set. Our DISH framework can directly adopt this protocol, in which the learned $\mathbf{H}$ can be used as database and the codes for query set can be generated by hash function $F$. The MAP value of DISH framework for CIFAR-10 is **0.724** at 64 bits, compared with 0.637 in COSDISH [@kang2016column] and 0.618 in LFH [@zhang2014supervised].

--- abstract: 'Global Value Numbering (GVN) is an important static analysis to detect equivalent expressions in a program. We present an iterative data-flow analysis GVN algorithm in SSA for the purpose of detecting total redundancies. The central challenge is defining a *join* operation to detect equivalences at a join point in polynomial time such that later occurrences of redundant expressions could be detected. For this purpose, we introduce the novel concept of *value $\phi$-function*. We claim the algorithm is precise and takes only polynomial time.' author: - 'Rekha R. Pai' bibliography: - '/home/rekha/my\_Home/research/reports/my\_bib.bib' title: | Global Value Numbering:\ A Precise and Efficient Algorithm --- Introduction ============ Global Value Numbering is an important static analysis to detect equivalent expressions in a program. Equivalences are detected by assigning *value numbers* to expressions. Two expressions are assigned the same value number if they could be detected as equivalent. The seminal work on GVN by Kildall [@Kildall1973] detects all *Herbrand equivalences* [@Ruething1999] in non-SSA form of programs using the powerful concept of *structuring* but takes exponential time. Efforts were made to improve on efficiency in detecting equivalences. However the algorithms are either as precise as Kildall’s [@Saleena2014] or efficient [@Ruething1999; @Alpern1988; @Gulwani2007] but not both. The strive for combining precision with efficiency has motivated our work in this area. We propose an iterative data-flow analysis GVN algorithm to detect redundancies in SSA form of programs that is precise as Kildall’s and efficient (i.e.take only polynomial time). As in a data-flow analysis problem, the central challenge is to define a *join* operation to detect all equivalences at a join point in polynomial time such that any later occurrences of redundant expressions could be detected. We introduce the novel concept of *value $\phi$-function* for this purpose. Terminology =========== #### Program Representation Input to our algorithm is the Control Flow Graph (CFG) representation of a program in SSA. The graph has empty *entry* and *exit* blocks. Other blocks contain assignment statements of the form $x = e$, where $e$ is an expression which is either a constant, a variable, or of the form $x \oplus y$ such that $x$ and $y$ are constants or variables and $\oplus$ is a generic binary operator. An expression can also be of the form $\phi_{k}(x, y)$, called *$\phi$-functions*, where $x$ and $y$ are variables and $k$ is the block in which it appears. We assume a block can have at most two predecessors and a block with exactly two predecessors is called *join* block. The input and output points of a block are called *in* and *out* points, respectively, of the block. The *in* point of a join block is called *join point*. We may omit the subscript $k$ in $\phi_{k}$ when the join block is clear from the context. In the CFGs we draw, $\phi$-functions appear in join blocks. But for clarity in explaining some of our concepts we assume $\phi$-functions are transformed to *copy* statements and appended to appropriate predecessors of the join block. #### Equivalence Two expressions $e_{1}$ and $e_{2}$ are *equivalent*, denoted $e_{1} \equiv e_{2}$, if they will have the same value whenever they are executed. Two expressions in a path are said to be *equivalent in the path* if they are equivalent in that path. We detect only Herbrand equivalences [@Ruething1999] which is equivalence among expressions with same operators and corresponding operands being equivalent. Basic Concept ============= Our main goal is to detect equivalences with a view to detecting redundancies in a program in polynomial time. We introduce the concept of *value $\phi$-function* for the purpose which is explained in this section followed by our method to detect redundancies. Value $\phi$-function --------------------- Consider the simple code segment in Fig.1(a). Here irrespective of the path taken $x_{1}+y_{1}$ is equivalent to $a_{1}+b_{1}$. In terms of the variables being assigned to, we can say $z_{1}$ is equivalent to same variable $c_{1}$. [0.3]{} ![Concept of value $\phi$-function[]{data-label="fig:vpf"}](dc-figure0.eps "fig:"){width="17mm" height="15mm"} [0.4]{} ![Concept of value $\phi$-function[]{data-label="fig:vpf"}](dc-figure1.eps "fig:"){width="55mm" height="20mm"} Now consider the code segment in Fig.1(b). Depending on the path taken expression $x_{3}+y_{3}$ is equivalent to either $x_{1}+y_{1}$ or $x_{2}+y_{2}$. In terms of the variables being assigned to, we can say $w_{3}$ is equivalent to merge of different variables – $p_{1}$ and $q_{2}$. Inspired by the notion of $\phi$-function, we can say $w_{3}$ is equivalent to $\phi(p_{1}, q_{2})$. This notion of $\phi$-function is an extended notion of $\phi$-function as seen in the literature. In the literature, a $\phi$-function has different subscripted versions of the same non-SSA variable, say $\phi(x_{1}, x_{2})$. To express such equivalences, we introduce the concept of *value $\phi$-function* similar to the concept of *value expression* [@Saleena2014]. #### Value $\phi$-function A *value $\phi$-function* is an abstraction of a set of equivalent $\phi$-functions (including the extended notion of $\phi$-function). Let $v_{i}$, $v_{j}$ be value numbers and *vpf* be a value $\phi$-function. Then $\phi_{k}(v_{i}, v_{j})$, $\phi_{k}(\emph{vpf}, v_{j})$, $\phi_{k}(v_{i}, \emph{vpf})$, and $\phi_{k}(\emph{vpf}, \emph{vpf})$ are *value $\phi$-functions*. #### Partition A partition at a point represents equivalences that hold in the paths to the point. An equivalence class in the partition has a value number and elements like variables, constant, and value expression. It is also annotated with a value $\phi$-function when necessary. The notation for a partition is similar to that in [@Saleena2014] except that a class can be annotated with value $\phi$-function. Proposed Method =============== Using the concept of value $\phi$-function we propose an iterative data-flow analysis algorithm to compute equivalences at each point in the program. The two main tasks in this algorithm are *join* operation and *transfer function*: *Join* operation. ----------------- A *join* operation detects equivalences that are common in all paths to a join point. The join is conceptually a class-wise intersection of input partitions. Let $C_{1}$ and $C_{2}$ be two classes, one from each input partition. If the classes have same value number then the resulting class $C$ is intersection of $C_{1}$ and $C_{2}$. If the classes have different value numbers, say $v_{1}$ and $v_{2}$ respectively, then common equivalences are found by intersection of $C_{1}$ and $C_{2}$. The common equivalences obtained are actually a merge of different variables, which is indicated by the difference in value numbers and hence class $C$ is annotated with $\phi(v_{1}, v_{2})$. Now if the classes have different value expressions, say $v_{m}+v_{n}$ and $v_{p}+v_{q}$ respectively, the value expressions may be merged to form a resultant value expression say $v_{i}+v_{j}$. Value expressions $v_{m}+v_{n}$ and $v_{p}+v_{q}$ are merged to get $v_{i}+v_{j}$ by recursively merging classes of $v_{m}$ and $v_{p}$ to get class of $v_{i}$ and classes of $v_{n}$ and $v_{q}$ to get class of $v_{j}$ [@Saleena2014]. But merging the value expressions can lead to exponential growth of resulting partition [@Gulwani2007]. We do not merge different value expressions now instead merge them at a point where an expression represented by $v_{i}+v_{j}$ actually occurs in the program. This merge is achieved simply by detecting equivalence of $v_{i}+v_{j}$ with $\phi(v_{1}, v_{2})$ and is done during application of transfer function. #### Example Let us now consolidate the concept of join using an example. Consider the case of applying join on partitions $P_{1} = \{v_{1}, x_{1}, x_{3} | v_{2}, y_{1}, y_{3}, v_{1}+1 | v_{3}, z_{1}, z_{3}\}$ and $P_{2} = \{v_{4}, x_{2}, x_{3} | v_{5}, y_{2}, y_{3} | v_{6}, z_{2}, z_{3}, v_{4}+1 \}$. In the classes with value numbers $v_{1}$ in $P_{1}$ and $v_{4}$ in $P_{2}$ there is only one common variable $x_{3}$ and this will appear in a class in the resulting partition $P_{3}$. Since the two classes in $P_{1}$ and $P_{2}$ have different value numbers $v_{1}$ and $v_{4}$, respectively, the resulting class is annotated with value $\phi$-function $\phi(v_{1}, v_{4})$. The class is assigned a new value number, say $v_{7}$. The resulting class is $| v_{7}, x_{3} : \phi(v_{1}, v_{4}) |$. Now consider the classes with value numbers $v_{2}$ in $P_{1}$ and $v_{6}$ in $P_{2}$. There are no obvious common equivalences in the classes and we don’t merge the different value expressions now. Hence no new class is created. Similar strategies are adopted in detecting common equivalences in other pairs of classes one each from $P_{1}$ and $P_{2}$. The resulting partition $P_{3}$ is $\{v_{7}, x_{3} : \phi(v_{1}, v_{4}) | v_{8}, y_{3} : \phi(v_{2}, v_{5}) | v_{9}, z_{3} : \phi(v_{3}, v_{6})\}$. $P \gets \{ \}$ each pair of classes $C_{i} \in P_{1}$ and $C_{j} \in P_{2}$ $C_{k} \gets C_{i} \cap C_{j}$ set intersection $C_{k} \neq \{\}$ and $C_{k}$ does not have value number $C_{k} \gets C_{k} \cup \{v_{k}, \phi_{b}(v_{i}, v_{j})\}$ $v_{k}$ is new value number $v_{i} \in C_{i}$, $v_{j} \in C_{j}, $ $b$ is join block $P \gets P \cup C_{k}$ Ignore when $C_{k}$ is empty $P$ Note: We define special partition $\top$ such that $\proc{Join}(\top, P) \gets P \gets \proc{Join}(\top, P)$. We assume $\phi$-functions in a join block are transformed to copies and appended to appropriate predecessors of join block. Transfer Function. {#trfn} ------------------ Given a partition $PIN_{s}$, that represents equivalences at *in* point of a statement $s: x \gets e$ the transfer function computes equivalences at its *out* point, denoted $POUT_{s}$. Let *ve* be the value expression of $e$ computed using $PIN_{s}$. If *ve* is present in a class in $PIN_{s}$, then $x$ is just inserted into corresponding class in $POUT_{s}$. Otherwise the transfer function checks whether $e$ could be expressed as a merge of variables represented by a value $\phi$-function *vpf* (as illustrated below). If it is present in a class in $PIN_{s}$ then $x$, *ve* are inserted into corresponding class in $POUT_{s}$. Else a new class is created in $POUT_{s}$ with new value number and $x$, *ve*, *vpf* are inserted into it. For an example, consider processing the statement $w_{3} = x_{3}+ y_{3}$ as shown in code segment in Fig.2. ![Concept of Transfer Function[]{data-label="fig:tf"}](dc-figure2.eps){width="115mm" height="30mm"} Since value expression $v_{7}+v_{8}$ of $x_{3}+y_{3}$ is not in $PIN_{3}$, the transfer function proceeds to check whether $x_{3}+y_{3}$ is actually a merge of variables as follows:\ $x_{3}+y_{3} \equiv v_{7}+v_{8} \equiv \phi(v_{1}, v_{4})+\phi(v_{2}, v_{5}) \equiv \phi(v_{1}+v_{2}, v_{4}+v_{5}) \equiv \phi(v_{3}, v_{6}) $.\ This implies $x_{3}+y_{3}$ is actually a merge of variables, here $p_{1}$ and $q_{2}$. Since neither $v_{7}+v_{8}$ nor $\phi(v_{3}, v_{6})$ are present in $PIN_{3}$, a new class is created in $POUT_{3}$ with new value number say $v_{9}$ and $w_{3}$, $v_{7}+v_{8}$, and $\phi(v_{3}, v_{6})$ are inserted into it. The classes in $PIN_{3}$ are inserted as such into $POUT_{3}$. The resulting partition $POUT_{3}$ is $\{v_{7}, x_{3} : \phi(v_{1}, v_{4}) | v_{8}, y_{3} : \phi(v_{2}, v_{5}) | v_{9}, w_{3}, v_{7}+v_{8} : \phi(v_{3}, v_{6}) \}$. $POUT_{s} \gets PIN_{s}$ $C_{i} \gets C_{i} - \{x\}$ $x \in C_{i}$, a class in $POUT_{s}$ *ve* $\gets \proc{valueExpr}(e)$ *vpf* $\gets \proc{valuePhiFunc}(ve, PIN_{s})$ can be NULL *ve* or *vpf* is in a class $C_{i}$ in $POUT_{s}$ ignore *vpf* when NULL $C_{i} \gets C_{i} \cup \{x, ve\}$ set union $POUT_{s} \gets POUT_{s} \cup \{v_{n}, x, ve :$ *vpf*$\}$ $v_{n}$ is new value number $POUT_{s}$ The is a recursive algorithm to compute value $\phi$-function corresponding to input value expression when possible else it returns NULL. Detect Redundancies. -------------------- Given partition $POUT$ at *out* of statement $x = e$, expression $e$ is detected to be redundant if there exists a variable in the class of $x$ in $POUT$, other than $x$, or the class of $x$ in $POUT$ is annotated with value $\phi$-function. In the example code in Fig.2, consider the case of checking whether $x_{3}+y_{3}$ in the last statement $w_{3} = x_{3}+y_{3}$ is redundant. In the class of $w_{3}$ in $POUT_{3}$ (computed in previous subsection) there are no variables other than $w_{3}$. However the class is annotated with a value $\phi$-function. Hence the expression $x_{3}+y_{3}$ is detected to be redundant. Two program expressions are equivalent at a point iff the iterative data-flow analysis algorithm detects their equivalence. This can be proved by induction on the length of a path in a program. Complexity Analysis =================== Let there be $n$ expressions in a program. The two main operations in this iterative algorithm are join and transfer function. By definitions of and a partition can have $O(n)$ classes. If there are $j$ join points, the total time taken by all the join operations in an iteration is $O(n.j)$. The transfer function involves constructing and then looking up for value expression or value $\phi$-function in the input partition. The transfer function of a statement takes $O(n.j)$ time. In an iteration total time taken by transfer functions is $O(n^{2}.j)$. Thus the time taken by all the joins and transfer functions in an iteration is $O(n^{2}.j)$. In the worst case the iterative analysis takes $n$ iterations and hence the total time taken by the analysis is $O(n^{3}.j)$. Conclusion ========== We presented GVN algorithm using the novel concept of value $\phi$-function which made the algorithm precise and efficient.

--- abstract: 'Observations of transition region emission in solar active regions represent a powerful tool for determining the properties of hot coronal loops. In this Letter we present the analysis of new observations of active region moss taken with the Extreme Ultraviolet Imaging Spectrometer (EIS) on the *Hinode* mission. We find that the intensities predicted by steady, uniformly heated loop models are too intense relative to the observations, consistent with previous work. To bring the model into agreement with the observations a filling factor of about 16% is required. Furthermore, our analysis indicates that the filling factor in the moss is nonuniform and varies inversely with the loop pressure.' author: - 'Harry P. Warren, Amy R. Winebarger, John T. Mariska, George A. Doschek, Hirohisa Hara' title: 'Observation and Modeling of Coronal “Moss” With the EUV Imaging Spectrometer on *Hinode*' --- Introduction ============ Understanding how the solar corona is heated to high temperatures is one of the most important problems in solar physics. In principle, observations of emission from the solar corona should reveal important characteristics of the coronal heating mechanism, such as the time scale and location of the energy release. In practice, however, relating solar observations to physical processes in the corona has proved to be very difficult. One of the more significant obstacles is the complexity of the solar atmosphere, which makes it difficult to isolate and study individual loops, particularly in the cores of solar active regions. One circumstance in which the line of sight confusion can be reduced is observations of coronal “moss.” The moss is the bright, reticulated pattern observed in many EUV images of solar active regions. These regions are the footpoints of high temperature active region loops, and are a potentially rich source of information on the conditions in high temperature coronal loops (see @peres1994 [@berger1999; @fletcher1999; @depontieu1999; @martens2000]). A particularly useful property of the moss is that its intensity is proportional to the total pressure in the coronal loop [@martens2000; @vourlidas2001], which is an important constraint for coronal loop modeling. For steady heating models the pressure and the loop length completely determine the solution up to a filling factor (see, @rosner1978). Recently [@winebarger2007] used this property of the moss in conjunction with a magnetic field extrapolation to infer the volumetric heating rate for each field line in the core of an active region. This allowed them to simulate both the soft X-ray and EUV emission independent of any assumptions about the relationship between the volumetric heating rate and the magnetic field. One limitation of this approach is that the filling factor is left as a free parameter and is adjusted so that the soft X-ray and EUV emission best match the observations. Observations of both the intensity and the electron density in the moss would determine the filling factor and remove this degree of freedom. A more fundemental question is how relevant steady heating models are to coronal heating. Recent work has shown that high temperature coronal emission ($\sim 3$MK) can be modeled successfully with steady heating (e.g., @schrijver2004 [@lundquist2004; @warren2006b]). However, pervasive active region Doppler shifts (e.g., @winebarger2002) and the importance of non-equilibrium effects in lower temperature loops (e.g., @warren2003), suggest an important role for dynamical heating in the solar corona. The purpose of this paper is to address these two issues with new observations from the Extreme Ultraviolet Imaging Spectrometer (EIS) on *Hinode*. EIS has an unprecedented combination of high spatial, spectral, and temporal resolution and provides a unique view of the solar corona. Here we use EIS density sensitive line ratios to infer the density and pressure of the moss and determine the filling factor of high temperature coronal loops. We also use EIS measurements of several emission lines formed near 1MK to test the consistency of steady heating models with moss observations. Modeling EIS Moss Observations ============================== In this section we discuss the application of steady heating models to observations of the moss with EIS. To begin we calculate the plasma emissivites as a function of both temperature and density for ions relevant to EIS using the CHIANTI atomic physics database [e.g., @young2003]. We use the ionization fractions of [@mazzotta1998] and the coronal abundances of [@feldman1992]. The plasma emissivities are related to the observed intensity by the usual expression $$I_\lambda = \frac{1}{4\pi}\int_s\epsilon_\lambda(n_e,T_e)n_e^2\,ds, \label{eq:ints}$$ where $n_e$ and $T_e$ are the electron density and temperature and $s$ is a coordinate along the field line. Note that what we refer to as the emissivity ($\epsilon_\lambda$) is actually the emissivity divided by $n_e^2$. For many strong emission lines this quantity is largely independent of the density. In this context, however, including the density dependence explicitly is important since many of the EIS emission lines have emissivities that are sensitive to the assumed density. Our calculations cover the range $\log\,n_e = 6$–13 and $\log\,T_e=4$–8. To explore the variation of the observed emission with base pressure we have calculated a family of solutions to the hydrodynamic loop equations using a numerical code written by Aad van Ballegooijen (e.g., @schrijver2005). We consider total loop lengths in the range $L = 10$–100Mm and maximum temperatures in the range $\log T_{max}=2.5$–7.5MK, which are typical of active region cores. The loops are assumed to be oriented perpendicular to the solar surface and to have constant cross sections. For consistency with our emissivity calculations we use the more recent radiative loss calculations given in [@brooks2006]. Each solution to the loop equations gives the density and temperature as a function of position along the loop ($n_e(s),T_e(s)$). We interpolate to find the emissivity from the calculated density and temperature in each computational cell. We then integrate the emissivity at heights below 5Mm to determine the total footpoint intensity using Equation \[eq:ints\]. The resulting intensities are displayed in Figure \[fig:calc\] and are generally consistent with earlier work which showed that the moss intensities are proportional to the pressure and independent of the loop length [e.g., @martens2000; @vourlidas2001]. The linear relationship between the intensity and the pressure breaks down for some of the hotter lines for the lowest pressures we’ve considered. For these solutions the peak temperature of formation for the line is closer to the apex temperature and there is some emission from along the loop leg. For each line we perform a fit of the form $I_\lambda = a_\lambda p_0^{b_\lambda}$ to the calculated intensities. Only pressures above $10^{16}$cm$^{-3}$ K are considered in the fits. The exponent, which is also shown in Figure \[fig:calc\], is generally close to 1. One approximation that has been made in earlier theoretical work that is not strictly valid is that the emissivity is independent of the density. Our calculations account for this and are more accurate. These model calculations suggest the following recipe for deriving the loop pressure and filling factor from the observations. The 186.88/195.12Å ratio can be used to infer the pressure. The pressure then can be used to determine the expected intensities in each of the lines. The filling factor is derived from the ratio of the simulated to observed intensity. EIS Moss Observations ===================== To test these model calculations we use observations from EIS on *Hinode*, which was launched 23 September 2006. The EIS instrument produces high resolution solar spectra in the wavelength ranges of 170–210Å and 250–290Å. The instrument has 1 spatial pixels and 0.0223Å spectral pixels. Further details are given in [@culhane2007] and [@korendyke2006]. Despite the fact that *Hinode* was launched close to the minimum in the solar activity cycle there have been several active regions available for observation. For this work we analyzed *Hinode* observations of NOAA active region 10940 from 2007 February 2 at about 10:42 UT. These observations show areas of moss that are relatively free from contamination from other emission in the active region. The observing sequence for this period consisted of stepping the 1 slit over a $256\arcsec\times256\arcsec$ region taking 15s exposures at each position. These data were processed to remove the contribution of CCD background (pedestal and dark current), electron spikes, and hot pixels, and converted to physical units using standard software. We then fit the calibrated data assuming Gaussian profiles and a constant background. The resulting rasters for some of the strongest lines are shown in Figure \[fig:rast\]. To identify the moss in this region we adopt the very simple strategy of looking for the brightest 186.88Å emission. This line is very sensitive to density. The line intensity relative to 195.12Å rises by about a factor of 6 as the densities rises from $10^9$, which is typical of active region loops at this temperature, to $10^{11}$cm$^{-3}$, which is at the high end of what we expect to observe in the moss. The result of using a simple intensity threshold is shown in Figure \[fig:rast\]. For each of the 1416 spatial pixels identified as moss we use the 186.88/195.12Å ratio to determine the base pressure in the loop from the calculation shown in Figure \[fig:calc\]. The base pressure is then used to infer the intensities in all of the emission lines, again using the calculations of this type also shown in Figure \[fig:calc\]. As has been found in many previous studies, the simulated intensities are much higher than what is observed. To reconcile the observed and simulated intensities we introduce a filling factor for each point derived from the 195.12Å intensities. As shown in Figure \[fig:filling\], the distribution of filling factors is approximately Gaussian with a peak at about 16% and standard deviation of about 4%. The mean filling factor we derive here is generally consistent with previous results that have been about 10% [e.g., @porter1995; @martens2000]. We also find that the derived filling factor is nonuniform and inversely proportional to the base pressure. To compare the model with the observation we have made scatter plots of observed intensity versus simulated intensity (including the filling factor) for a number of emission lines. These plots are shown in Figure \[fig:compare\] and indicate a reasonable agreement between the steady heating model and the observations. The agreement is particularly good for 188.23 and 203.82. The simulated 184.54Å intensities are about 20% too high and the simulated 202.04 are about 10% too low. These discrepancies are not alarming considering the uncertainties in the atomic data. There are, however, rather signficant discrepancies in the 275.35Å intensities. The simulated intensities are generally about a factor of 4 larger than what is predicted from the steady heating model. The correlation between the simulated and observed intensities is also poor. It is possible that considering loop expansion or nonuniform heating may provide a better match at these temperatures, but we have not yet performed these simulations. It is also possible that there are errors in the atomic data for this line or that the instrumental calibration is not correct at this wavelength. More extensive analysis, including the observation of other lines formed at similar temperatures, will be required to resolve this issue. Discussion ========== We have presented an initial analysis of active region moss observed with the EIS instrument on *Hinode*. We find that the intensities predicted by steady, uniformly heated loop models are too high and a filling factor is required to bring the simulated intensities into agreement with the observations. The mean filling factor we derive here ($\sim16$%) is similar to that determined from earlier work with steady heating models. Furthermore, we also find that the filling factor in the moss must be nonuniform and varies inversely with the loop pressure. The next step is to use the methodology outlined here to simulate all of the active region emission (including the high temperature loops) and compare with EIS observations. Finally, we emphasize that while we find reasonable agreement between the moss intensities and steady heating, we cannot rule out dynamical heating processes, such as nano-flares [e.g., @cargill1997; @cargill2004]. The X-ray emission observed in this region clearly has a dynamic component [@warren2007b] and it would be surprising if non-equilibrium effects did not play some role in the heating of this active region. Hinode is a Japanese mission developed and launched by ISAS/ JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. 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The simulated intensities include a filling factor derived from the 195.12Å line.[]{data-label="fig:filling"}](f03.eps) 

--- abstract: 'Centralized systems in the Internet of Things—be it local middleware or cloud-based services—fail to fundamentally address privacy of the collected data. We propose an architecture featuring secure multiparty computation at its core in order to realize data processing systems which already incorporate support for privacy protection in the architecture.' author: - | Marcel von Maltitz and Georg Carle\ \ \ \ bibliography: - 'literature.bib' title: | Leveraging Secure Multiparty Computation\ in the Internet of Things --- Introduction ============ Smart environments and smart buildings constitute a vital part of the Internet of Things. In these contexts, sensors are deployed to gather information about the state of the real-world environment. This information, in turn, represents the data foundation for services that influence the environment state, provide insights for inhabitants and interact with them. Examples for these services are public displays, which give statistical information about the building state, monitoring services for maintenance personnel and anomaly detection systems which detect incidents and failures. These and many other services have in common that they do not directly work on the raw data gathered by the sensors. Instead, they use derived aggregated results by computational preprocessing: Public displays show diagrams of statistical data, monitoring and anomaly detection services work with events and alerts gained by rules, machine learning or other types of computation. For mediating the data flow between the sensor platforms—the data sources—and the services—the data consumers—typically a middleware is deployed. Its purpose encompasses collection and storage of raw data, analysis, processing and finally forwarding the obtained results to the data consuming services. This middleware can either be a local part of the smart environment but can also be provided as cloud service. This type of architecture and the corresponding handling of data has severe implications for the privacy of the sensor data: c@fail **)**  **)**  The middleware acts as a third party which gains full access to raw data coming from the sensors. This third party might not even be under control of the administrators of the smart environment and hence untrustworthy. c@fail **)**  **)**  By pushing data to a third party, sources lose insights into how their data is used afterwards. Data processing becomes intransparent for them. c@fail **)**  **)**  Similarly, sources lose control over the usage of their data. Especially, revocation of data requires trust in the data holder to actually obey. c@fail **)**  **)**  Even if trustworthy, the third party is still a high value target for attackers. Privacy Preserving Data Processing ================================== Our vision is to realize the described functionality while fundamentally providing privacy protection on the architectural level. We propose that raw data created by the distributed sources is not collected by a middleware but remains distributed on these sources. This allows secure computations and can make consent and cooperation of the sources a necessity for the execution. Our understanding of privacy and data protection is based on [@Datenschutzschutzziele; @Hansen2015]. They most importantly feature the protection goals of *data minimization*, *unlinkability*, *transparency*, and *intervenability*. Against this background, the positive implications of our approach are as follows: The amount of data in the system is minimized since there are no intermediaries which can also access data. Logically, the derived results are directly transmitted from the sources to the final consumers. The potential for data misuse and unauthorized recombination of data is decreased since data of different sources is not stored at the same logical place in a linkable fashion. Specifically, only making allowed computations technically possible concomitantly realizes purpose binding. The required cooperation of the data sources in turn provides them with information about the ongoing computations and the usage of their data. This constitutes transparency, especially when this feedback is enhanced with meta information about the final consumer. Persisting these insights can additionally realize accountability. Lastly, given the cooperation requirement and aforementioned transparency, they remain in control since they can specifically decide beforehand whether to cooperate and to provide their data for the usage in question or not. Architecture Design =================== The provided vision satisfies several data protection goals which are not yet fulfilled by state-of-the-art architectures. In order to realize this vision technically, the following main challenge has to be addressed: It must be possible to derive computation results from raw data of different sources without sharing this data among them nor handing it to a third party for computation. For this purpose, secure multiparty computation (SMC) [@Yao1982; @Yao1986; @SMCBook] can be employed. Instead of local computations of a third party a secure protocol among the sources is executed [@Canetti1999; @Canetti2001]. Afterwards each only knows its own input and the final output of the computation. All exchanged intermediary data due to technical reasons does not allow recovering other parties inputs. Mathematical foundations for realizing arbitrary functions as SMC invocations are known since the 80’s [@BGW88; @Chaum1988a] but protocol improvements for security and performance [@BristolMPC; @MASCOT] and new applications [@Bonawitz2017] are still current research. For sucessfully applying SMC in smart environment we propose the following architecture: The formerly stated middleware is replaced by a *gateway*. The vital difference is that the gateway does not obtain access to the raw data of the sources. Instead, facing the sources it only fulfills management and orchestration purposes to carry out SMC computations. Towards the consumers, it presents an API which abstracts from SMC and resembles an interface a centralized middleware would provide. #### Robust automated execution of SMC [@vonMaltitz2018b] The main purpose of the gateway is to handle SMC sessions in cooperation with the sources. For this, the gateway must be initially known by them. Similarly, upon connection interruption or due to churn of mobile sources a present gateway has to be redetected. This is realized by a service discovery technology like mDNS [@RFC6762; @RFC6763]. After detection, a setup between the new source and the gateway is performed: The gateway is informed about data and computation protocols provided by the new source. This data constitutes a state about currently obtainable insights about the environment in the form of a metadata directory. Furthermore, the gateway establishes a control channel to the source allowing to prepare and orchestrate SMC sessions. The gateway specifies all aspects of an upcoming session and communicates them to the participating sources: The identity and the connection endpoints of cooperators, the data to be used for computation and the protocol to be executed. The computation itself is monitored by the gateway. On success, the gateway receives the result. If the computation fails, the gateway tries to recover or to fully restart the session. This is hidden from any consumer in any possible cases to achieve service character. #### Data Requests and Access Control The purpose of the gateway towards the consumers is to mimic a standard middleware providing data upon request. Here, the metadata directory provides information to the consumers what data is obtainable at this point in time. This metadata should also abstract from SMC specifics allowing to post requests which already declare the aggregation result, “the average amount of individuals in floor 3.A of the building per hour”. Receiving these requests, the gateway then transforms them into a corresponding SMC session and replies with the result afterwards. Correct representation of data requests supports access control, transparency and intervenability essentially. We assume requests to be authenticated and integrity protected. The gateway is then able to perform access control and plausbility checks when examining the purpose of the request, the identity of the consumer and the type of requested data. During SMC session setup the gateway also transmits the original request of the consumer to each collaborator, consequently realizing request transparency for sources. Additional persisting the requests provides distributed request accountability. Lastly, this information can be evaluated by the sources before executing the computation. Each source can decide individually whether to contribute to the requested computation or not. In case a single source veotes against the computation, it cannot be executed; this is handled as a special, expected error by the gateway and can be addressed accordingly by it. In summary, we deliberately leverage the necessity of cooperation when performing computations to support the mentioned further privacy properties. Conclusion ========== We presented a vision of privacy preserving data processing in dynamic environments. Our design features a management and orchestration middleware for secure multiparty computation which allows application of SMC as an adaptive and robust service. Furthermore, we show how the features of SMC can be complemented in order to fulfill further established privacy protection goals. We see that fundamental innovation in system architecture allows more straightforward addressing of privacy goals. While also raising new challenges to be solved, they provide an alternative approach to establishing privacy as an afterthought in a predetermined system. Acknowledgements ================ This work has been supported by the German Federal Ministry of Education and Research, project DecADe, grant 16KIS0538 and the German-French Academy for the Industry of the Future.

--- abstract: 'We propose and demonstrate a novel method to reduce the pulse width and timing jitter of a relativistic electron beam through THz-driven beam compression. In this method the longitudinal phase space of a relativistic electron beam is manipulated by a linearly polarized THz pulse in a dielectric tube such that the bunch tail has a higher velocity than the bunch head, which allows simultaneous reduction of both pulse width and timing jitter after passing through a drift. In this experiment, the beam is compressed by more than a factor of four from 130 fs to 28 fs with the arrival time jitter also reduced from 97 fs to 36 fs, opening up new opportunities in using pulsed electron beams for studies of ultrafast dynamics. This technique extends the well known rf buncher to the THz frequency and may have a strong impact in accelerator and ultrafast science facilities that require femtosecond electron beams with tight synchronization to external lasers.' author: - 'Lingrong Zhao$^{1,2}$, Heng Tang$^{1,2}$, Chao Lu$^{1,2}$, Tao Jiang$^{1,2}$, Pengfei Zhu$^{1,2}$, Long Hu$^{3}$, Wei Song$^{3}$, Huida Wang$^{3}$, Jiaqi Qiu$^{4}$, Chunguang Jing$^{5}$, Sergey Antipov$^{5}$, Dao Xiang$^{1,2,6*}$ and Jie Zhang$^{1,2*}$' title: 'Femtosecond relativistic electron beam with reduced timing jitter from THz-driven beam compression' --- Ultrashort electron beams with small timing jitter with respect to external lasers are of fundamental interest in accelerator and ultrafast science communities. For instance, such beams are essential for laser and THz-driven accelerators ([@DLARMP; @THzNC; @THzIFEL; @THzNP]) where the beam energy spread and beam energy stability largely depend on the electron bunch length and injection timing jitter, respectively. For MeV ultrafast electron diffraction (UED [@UED3; @UCLA; @THU; @OSAKA; @SJTU; @BNL; @SLAC; @DESY]) where ultrashort electron beams with a few MeV energy are used to probe the atomic structure changes following the excitation of a pump laser, the temporal resolution is primarily limited by the electron bunch length and timing jitter. Similar limitations exist for x-ray free-electron lasers (FEL [@LCLS; @SACLA; @PAL]) too, since the properties of the x-ray pulses depend primarily on that of the electron beams. Therefore, one of the long-standing goals in accelerator and ultrafast science communities is to generate ultrashort electron beams with small timing jitter. Photocathode rf gun is the leading option for producing high brightness ultrashort electron beam for FEL and MeV UED (see, e.g. [@KoreaUED; @LCLSinjector]). Due to space charge effect, the electron beam pulse width is broadened and therefore bunch compression is typically needed to reduce the pulse width. Bunch compression requires first a mechanism to imprint energy chirp (correlation between an electron’s energy and its longitudinal position) and then sending the beam through a dispersive element such that the longitudinal displacement of the electrons is changed in a controlled way for reducing the pulse width. For MeV beam, this is typically achieved by first sending the beam through a rf buncher cavity at zero-crossing phase where the bunch head ($t<0$) is decelerated while the bunch tail ($t>0$) is accelerated. This imprints a negative chirp $h=d\delta/cdt<0$ in the beam longitudinal phase space, where $\delta$ is the relative energy difference of an electron with respect to the reference electron and $c$ is the speed of light. Then the electron beam is sent through a drift after which the electrons at the bunch tail catch up with those at the bunch head, leading to compression in pulse width. Full compression is achieved when $hR_{56}$=-1, where $R_{56}\approx L/\gamma^2$ is the momentum compaction of the drift with length $L$ and $\gamma$ is the Lorentz factor of the electron beam. Recently, sub-10 (rms) fs beams have been produced with this rf buncher technique [@CUCLA; @PRX]. However, the rf phase jitter results in increased beam timing jitter after compression. While THz pulse based time-stamping techniques have been developed to measure and correct the electron beam arrival time jitter [@PRX; @SLACstreaking] for UED, the detector response time limited this shot-to-shot correction technique to low repetition rate. It is highly desired (in particular for those experiments that require long data acquisition time (see, e.g. [@SLACgasphaseCF3I])) if the beam can be compressed without increasing the jitter such that time-stamping technique is not needed. Many efforts have been devoted towards this goal in the past few years. For instance, it has been shown that replacing the rf buncher with a laser-driven THz buncher can be used to compress keV beams while simultaneously keeping the timing jitter below 10 fs [@THzNP; @THzbuncher1; @THzbuncher2], making full use of the fact that the THz pulse is tightly synchronized with the laser. However, with the THz pulse propagating perpendicularly to the electron beam path, a very strong THz source is required for compressing a MeV beam because the interaction length is rather limited. In this Letter, we demonstrate a novel method to compress a relativistic electron beam using a THz pulse with moderate strength. In this technique the THz pulse co-propagates with the electron beam in a dielectric tube and the interaction length is thus greatly increased. The schematic layout of the experiment is shown in Fig. 1. An 800 nm laser is split into three pulses with one pulse (1 mJ) used for producing electron beam in a photocathode rf gun and the other two pulses (4 mJ each) for producing THz radiation (about 1.5 $\mu$J each) through optical rectification in LiNbO$_3$ crystal [@TPFP]. The first THz pulse with vertical polarization is injected into a dielectric tube where it interacts with the electron beam and imprints an energy chirp in beam longitudinal phase space. The beam is then compressed after passing through a 1.4 m drift where the bunch length and arrival time jitter are measured with a second THz pulse that deflects the beam in horizontal direction. The electron beam can be measured either on a screen (P1) before the energy spectrometer or after it (P2). The field pattern in a conventional rf buncher is typically TM01 mode for which the longitudinal field has a very weak dependence on transverse position, ideal for producing energy chirp without increasing beam uncorrelated energy spread. However, efficient excitation of TM01 mode typically requires an input pulse with radial polarization and it is difficult to excite such mode with a linearly polarized THz pulse [@MC]. In our experiment, a vertically polarized THz pulse is directly injected into a dielectric tube (D1) and thus only HEM11 mode is excited. Such a mode has recently been used to deflect electron beam for measuring bunch length and timing jitter [@THzO]. It should be noted that HEM11 mode has longitudinal electric field which varies linearly with transverse offset. This effect may be understood with Panofsky-Wenzel theorem [@PW] which connects the time-dependent angular kick with offset-dependent energy kick. So energy chirp may be produced with this longitudinal electric field when the electron beam passes through the dielectric tube off-axis, allowing THz-driven bunch compression with HEM11 mode. In our experiment D1 is a $L_D$=15 mm long cylindrical quartz tube with inner diameter of 910 $\mu$m and outer diameter of 970 $\mu$m. The outer surface of the tube is gold coated and analysis shows that such structure can support HEM11 mode with frequency at about 0.66 THz. To clearly show the effect of the longitudinal electric field, we used a set of BBO crystals to stack the UV laser pulse for producing a long electron beam with pulse width comparable to the wavelength of the HEM11 mode. Representative beam distributions when it passes through D1 with various offsets are measured at screen P2 and shown in Fig. 2a. Because the beam is deflected in vertical direction and bent in horizontal direction, the horizontal axis on screen P2 becomes the energy axis and vertical axis becomes the time axis. From Fig. 2a, one can see that when the electron beam enters the tube on-axis ($y_0=0$), the beam is only deflected by the THz pulse and the distribution takes a stripe shape. With the electron pulse width comparable to the wavelength of HEM11 mode, the sinusoidal deflection leads to a double-horn distribution. When the beam passes through the tube off-axis, the beam receives time-dependent energy kick from the longitudinal electric field and thus takes a ring shape. Specifically, for $y_0<0$, region A represents the electrons that experience acceleration phase; electrons in region C experience deceleration phase; electrons in regions B and D are at the zero-crossing phases and thus the centroid energy is not changed. As the offset is gradually changed from $-200~\mu$m to $150~\mu$m, the distance between regions A and C first reduces and then increases again after reaching a minimum at $y_0=0$. Furthermore, after the sign of the offset is reversed ($y_0>0$), electrons in region A are now decelerated while those in region C are accelerated, consistent with the fact that the longitudinal field scales linearly with transverse offset. The maximal energy change of the electrons can be found using the distance between A and C in Fig. 2a. Similarly, the maximal transverse energy kick can be found using the distance between B and D. After converting the distance into voltage using the known dispersion, the longitudinal energy kick from longitudinal electric field ($V_{\parallel}$) and transverse energy kick ($V_{\perp}$) from transverse electromagnetic field are shown in Fig. 2b. The transverse kick is independent of the transverse offset and the longitudinal kick scales linearly with the offset. In a separate experiment, the beam distribution at P2 is measured with a fine scan of $y_0$ and the distributions are then superimposed and shown in Fig. 2c where a cone shape (guided with the dotted line) is clearly seen that shows the linear dependence of energy change on transverse offset. A closer look at the rings in Fig. 2a indicates that the transverse kick and longitudinal kick have $\pi/2$ phase difference. For instance, electrons in regions A and C experience on-crest phase for acceleration but zero-crossing phase for deflection. To confirm this, the pulse stacker is removed and now a beam with about 130 fs pulse width is produced. The timing of the THz beam is varied and the beam centroid change as well as energy change are measured simultaneously and the results are shown in Fig. 3 which confirms the $\pi/2$ phase difference in acceleration and deflection. This $\pi/2$ phase difference results in a net shift of the beam centroid divergence during compression and thus can be straightforwardly corrected with a steering magnet. With the group velocity of the THz pulse being about $v_g\approx0.84$c in D1, the interaction window within which the electrons can catch up and interact with the THz pulse in D1 is about $(c/v_g-1)L_D/c\approx10~$ps. This is the main reason that multiple oscillations are observed in Fig. 3 even when a single-cycle THz pulse is used. To measure the time information of the compressed bunch, a second THz pulse is used to deflect the beam for converting time information into spatial distribution. In our first attempt, the second THz pulse is also vertically polarized as the crystals and gratings have the same configurations for the two THz sources. However, it is noted that such a configuration may introduce ambiguity in data interpretation. For instance, when the electron beam passes through both tubes on-axis, but with $\pi$ difference in phase, then the deflection from the first THz pulse is canceled by the second THz pulse. This produces a transverse beam size comparable to that with full compression while apparently no compression occurs here. To allow unambiguous determination of the bunch compression effect, in our experiment a polarization rotation (PR) element consisting of a pair of wire grid polarizers and roof mirrors are used to manipulate the THz polarization. As shown in [@THzO], by changing the path length of the two roof mirrors, a vertically polarized THz pulse can be converted into a horizontally polarized pulse. Then we sent the electron beam through D1 at an offset of 200 $\mu$m, and used a steering magnet to center the beam in the second tube (D2). The beam distributions on screen P1 measured with the two THz pulses on for various time delay between the electron beam and first THz pulse are shown in Fig. 4. In this measurement the timing of the second THz pulse is adjusted such that the beam always rides at the zero-crossing phase of the deflection field. Fig. 4a-d corresponds to the cases when the electron beam is at regions A, B, C, and D in Fig. 2a, respectively. As shown in Fig. 4a and Fig. 4c, the beam is at on-crest phases for acceleration and at the zero-crossing phases for deflection in D1, so the beam is deflected vertically by the first THz pulse and deflected horizontally by the second THz pulse, leading to correlation in horizontal and vertical distribution. The beam is not compressed at these time delays. It should be noted that the streaked beam size is larger than the aperture of D2, so only part of the beam is measured on screen P1. The beam is at one of the zero-crossing phases for acceleration in Fig. 4b. However, at such phase the bunch head is accelerated while the bunch tail is decelerated. As a result, a positive energy chirp is imprinted in the beam phase space, leading to bunch lengthening by roughly a factor of 2. The bunch duration in this case is larger than the dynamic range of the measurement (roughly one quarter of the deflection wavelength), and thus a quasi-flattop distribution is seen for the deflected beam. Fig. 4d shows the distribution when the beam is at the right zero-crossing phase for bunch compression. For this case the beam is shorter than the dynamic range of the measurement and thus the bunch length can be accurately determined. In this measurement the ramping rate of the deflection from the second THz pulse is found to be about 4 $\mu$m/fs. With the transverse beam size and beam centroid fluctuation at screen P1 measured to be about 120 $\mu$m and 16 $\mu$m with the THz off, the resolution of beam temporal profile measurement and the accuracy of beam arrival time measurement are determined to be about 30 fs and 4 fs, respectively. With the strength of the first THz pulse varied to provide the optimal energy chirp, full compression is achieved and the raw beam pulse width in Fig. 4d after converting the beam size to time is measured to be about 41 fs (rms). Subtracting the resolution in quadrature yields a true bunch length of about 28 fs (rms). Under full compression condition, 50 consecutive measurements of the raw beam profile at P1 (with horizontal axis converted into time) with THz buncher off and on are shown in Fig. 5a and Fig. 5d, respectively. The effect of longitudinal compression can be clearly seen. The average bunch length (calculated by subtracting the contribution from intrinsic beam size) before and after THz compression is about 130 fs and 28 fs, respectively. The fluctuation of the beam centroid (black dots in Fig. 5a and Fig. 5d) which represents the time jitter with respect to the second THz pulse is also greatly reduced from about 97 fs to about 36 fs after compression. It should be noted that while the initial jitter before the THz buncher is greatly compressed, the energy stability of the beam still limits the residual timing jitter of the THz buncher scheme to $\Delta t=R_{56}\delta E/E$, where $\delta E/E$ is the relative energy stability of the beam. In this experiment, the beam energy stability is measured to be about $2.5\times10^{-4}$, which together with 3.9 cm momentum compaction from D1 to D2 leads to a residual timing jitter of about 32 fs, in good agreement with the experimental result. For keV UED with high energy stability, such timing jitter is negligible. With full compression, the residual bunch length is limited by the uncorrelated beam energy spread ($\sigma_{\delta}$) to $\sigma_{\delta}R_{56}/E$. In our experiment, the beam uncorrelated energy spread grows in the THz buncher because of the linear dependence of the longitudinal electric field and the finite transverse beam size (about 20 $\mu$m) in D1. The uncorrelated energy spread growth is estimated to be about 0.55 keV which results in a residual bunch length of about 24 fs, consistent with the measured value. The minimal bunch length may be further reduced by focusing the beam to a smaller size. It is also possible to use an additional dielectric tube to cancel the energy spread growth. For instance, after the THz buncher, the beam may be sent through a second tube on-axis. By using a THz pulse with $\pi$ phase difference, the energy spread growth may be effectively canceled with the chirp unchanged. The residual timing jitter may be reduced with improved energy stability, or using a THz pulse with stronger strength [@GV] for lowering the required momentum compaction. Such steps should be able to push both the electron bunch length and arrival time jitter to sub-10 fs regime, opening up new opportunities in ultrafast science and advanced acceleration applications. In conclusion, we have demonstrated a novel method to manipulate relativistic beam phase space for longitudinal compression. By using the longitudinal field in HEM11 mode for imprinting the energy chirp and with the THz pulse co-propagating with the electron beam, the mode is easily excited with a linearly polarized THz pulse and the required THz pulse energy for producing sufficient energy chirp is greatly reduced. In our experiment we have demonstrated significant reduction in both bunch length and arrival time jitter, which may allow one to significantly enhance the temporal resolution of UED. The demonstrated colinear interaction scheme is also of interest for THz-driven beam acceleration that holds potential for downsizing accelerator-based large scientific facilities such as FELs and colliders. We expect this THz-driven beam manipulation method to have wide applications in many areas of researches. 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--- abstract: 'We have developed a highly-tuned software library that accelerates the calculation of quadrupole terms in the Barnes-Hut tree code by use of a SIMD instruction set on the x86 architecture, Advanced Vector eXtensions 2 (AVX2). Our code is implemented as an extension of Phantom-GRAPE software library that significantly accelerates the calculation of monopole terms. If the same accuracy is required, the calculation of quadrupole terms can accelerate the evaluation of forces than that of only monopole terms because we can approximate gravitational forces from closer particles by quadrupole moments than by only monopole moments. Our implementation can calculate gravitational forces about 1.1 times faster in any system than the combination of the pseudoparticle multipole method and Phantom-GRAPE. Our implementation allows simulating homogeneous systems up to 2.2 times faster than that with only monopole terms, however, speed up for clustered systems is not enough because the increase of approximated interactions is insufficient to negate the increased calculation cost by computing quadrupole terms. We have estimated that improvement in performance can be achieved by the use of a new SIMD instruction set, AVX-512. Our code is expected to be able to accelerate simulations of clustered systems up to 1.08 times faster on AVX-512 environment than that with only monopole terms.' author: - 'Tetsushi <span style="font-variant:small-caps;">Kodama</span>' - 'Tomoaki <span style="font-variant:small-caps;">Ishiyama</span>,' title: Acceleration of the tree method with SIMD instruction set --- Introduction ============ Gravitational $N$-body simulations are widely used to study the nonlinear evolution of astronomical objects such as the large-scale structure in the universe, galaxy clusters, galaxies, globular clusters, star clusters and planetary systems. Directly solving $N$-body problems requires the computational cost in proportion to $N^2$ and is unpractical for large $N$, where $N$ is the number of particles. Therefore, many ways to reduce the calculation cost have been developed. One of the sophisticated algorithms is the tree method ([@key-1]) that evaluates gravitational forces with calculation cost in proportion to $N\log{N}$. The tree method constructs a hierarchical oct-tree structure to represent a distribution of particles and approximates the forces from a distant group of particles by the multipole expansion. The opening parameter $\theta$ is used to determine the tradeoff between accuracy and performance. If $l/d < \theta$, forces from a group of particles are approximated by the multipole expansion, where $l$ is the spatial extent of the group and $d$ is the distance to the group. Thus, larger $\theta$ gives higher performance and less accuracy. The tree method is also used with the Particle-Mesh (PM) method ([@key-5]) when the periodic boundary condition is applied. This combination is called as the TreePM method ([@key-13; @key-14; @key-19; @key-20; @key-12; @key-6; @key-21; @key-25]) that calculates the short-range force by the tree method and the long-range force by the PM method. The TreePM method has been widely used to follow the formation and evolution of the large-scale structure in the universe and has been adopted in many recent ultralarge cosmological $N$-body simulations. (e.g., [@key-32]) For collisional $N$-body simulations that require high accuracy, the Particle-Particle Particle-Tree (PPPT) algorithm ([@key-17]) has been developed. In this algorithm, short-range forces are calculated with the direct summation method and integrated with the fourth-order Hermite method ([@key-29]) , and long-range forces are calculated with the tree method and integrated with the leapfrog integrator. The tree method has been combined with other algorithms and used to study various astronomical objects. Yet another way to accelerate $N$-body simulations is the use of additional hardware, for example GRAPE (GRAvity PipE) systems ([@key-22; @key-7; @key-8; @key-33]) and Graphics Processing Units (GPUs) ([@key-24; @key-15; @key-9; @key-18; @key-23]). GRAPEs are special-purpose hardware for gravitational $N$-body simulations and have been used to improve performances of $N$-body algorithms such as the tree ([@key-11]), and the TreePM ([@key-12]). A different approach is utilizing a SIMD (Single Instruction Multiple Data) instruction set. Phantom-GRAPE ([@key-10; @key-2; @key-3]) [^1] is a highly-tuned software library and dramatically accelerates the calculation of monopole terms utilizing a SIMD instruction set on x86 architecture. Quadrupole terms can be calculated by the combination of the pseudoparticle multipole method ([@key-4]) and Phantom-GRAPE for collisionless simulations ([@key-3]). In this method, a quadrupole expansion is represented by three pseudoparticles. However, the pseudoparticle multipole method requires additional calculations such as diagonalizations of quadrupole tensors that may cause substantial performance loss. To address this issue, we have implemented a software library that accelerates the calculation of quadrupole terms by using a SIMD instruction set AVX2 without positioning pseudoparticles. Our code is based on Phantom-GRAPE for collisionless simulations and works as an extension of the original Phantom-GRAPE. When the required accuracy is the same, simulations should become faster by using quadrupole terms than by using only monopole terms because we can increase the opening angle $\theta$. Increasing $\theta$ gives another advantage that we can also reduce the calculation cost of tree traversals. The calculation including quadrupole terms should become further efficient as the length of SIMD registers gets longer than 256-bit (AVX2). Force evaluation is relatively scalable with respect to the length. On the other hand, the time for tree traversals would not be because hierarchical oct-tree structures are used. Thus, in environments such as AVX-512 with the SIMD registers of 512-bit length, the total calculation for tree traversals and force evaluation should be more accelerated in using quadrupole terms with larger $\theta$ than in using the monopole only and smaller $\theta$. This paper is organized as follows. In section \[sec:AVX\], we overview the AVX2 instruction set. We then describe the implementation of our code in section \[sec:imp\]. In section \[sec:acu\] and \[sec:perf\], we show the accuracy and performance, respectively. Future improvement in performance by utilizing AVX-512 is estimated in section \[sec:dis\]. Section \[sec:sum\] is for the summary of this paper. The AVX2 instruction set {#sec:AVX} ======================== The Advanced Vector eXtensions 2 (AVX2) is a SIMD instruction set, which is an improved version of AVX. Dedicated “YMM register” with the 256-bit length is used to store eight single-precision floating-point numbers or four double-precision floating-point numbers. The lower 128-bit of the YMM registers are called “XMM registers”. The number of dedicated registers on a core is 16 in AVX2. Note that differently from AVX, AVX2 supports Fused Multiply-Add (FMA) instructions for floating-point numbers. More precisely, AVX2 support and FMA support are not the same, but many CPUs supporting AVX2 also support FMA instructions. FMA instructions perform multiply-add operations. Without FMA instructions, a calculation $A \times B + C$ is done by two operations, $D = A \times B$ and $D + C$. With FMA instructions, this calculation can be executed in one operation. Therefore, in such situations, FMA instructions can gain the twice higher performance than AVX environment. Modern compilers do not necessarily generate optimized codes with SIMD instructions from a source code written in high-level languages because the detection of concurrency of loops and data dependency is not perfect [@key-3]. To manually assign YMM registers to computational data in assembly-languages and use SIMD instructions efficiently, we partially implemented our code with GCC (GNU Compiler Collection) inline-assembly as original Phantom-GRAPE [@key-10; @key-2; @key-3]. Implementation Details {#sec:imp} ====================== In this section, we show our implementation that accelerates calculations of quadrupole terms in the Barnes-Hut tree code utilizing the AVX2 instructions. Our code is based on Phantom-GRAPE for collisionless simulations and works as an extension of original Phantom-GRAPE ([@key-3]). The quadrupole expansion of the potential at the position $\mbox{\boldmath $r$}_i$ exerted by $n_j$ tree cells is expressed as $$\begin{aligned} \label{eq:pot} \phi_i=&&-\sum_{j=1}^{n_j} \left\{\frac{Gm_j}{\sqrt{|\mbox{\boldmath $r$}_j - \mbox{\boldmath $r$}_i|^2 + \epsilon^2}}+ \right. \nonumber \\ && \left. \frac{G}{2(| \mbox{\boldmath $r$}_j - \mbox{\boldmath $r$}_i|^2 + \epsilon^2)^{5/2}}(\mbox{\boldmath $r$}_j - \mbox{\boldmath $r$}_i)\cdot\mbox{\boldmath $Q$}_j\cdot(\mbox{\boldmath $r$}_j - \mbox{\boldmath $r$}_i) \right\},\end{aligned}$$ where $G, m_j, \mbox{\boldmath $r$}_j, \mbox{\boldmath $Q$}_j$, and $\epsilon$ are the gravitational constant, the total mass of the $j$-th cell, the position of the center of mass of the $j$-th cell, the quadrupole tensor of the $j$-th cell, and the gravitational softening length, respectively. We represent the quadrupole tensor as $$\begin{aligned} \mbox{\boldmath $Q$}_j &=& \left[ \begin{array}{rrr} q_{00} & q_{01} & q_{02} \\ q_{01} & q_{11} & q_{12} \\ q_{02} & q_{12} & q_{22} \\ \end{array} \right] \nonumber \\ &=& \sum_{k=1}^{k_j}m_k\left[ \begin{array}{rrr} 3x_{jk}^2 - r_{jk}^2 & 3x_{jk}y_{jk} & 3x_{jk}z_{jk}\\ 3y_{jk}x_{jk} & 3y_{jk}^2 - r_{jk}^2 & 3y_{jk}z_{jk}\\ 3z_{jk}x_{jk} & 3z_{jk}y_{jk} & 3z_{jk}^2 - r_{jk}^2\\ \end{array} \right], \end{aligned}$$ where $k_j$ is the number of particles in the $j$-th cell, $m_k$ is the mass of the $k$-th particle, $x_k$, $y_k$ and $z_k$ are $x$, $y$ and $z$ component of the position of the $k$-th particle, $x_j$, $y_j$, and $z_j$ are $x$, $y$, and $z$ component of the position of the center of mass of the $j$-th cell, $x_{jk}=x_k-x_j$, $y_{jk}=y_k-y_j$, $z_{jk}=z_k-z_j$, and $r_{jk}=\sqrt{x_{jk}^2 + y_{jk}^2 + z_{jk}^2}$, respectively. Since a quadrupole tensor is symmetric and traceless, five values of $q_{00}, q_{01}, q_{02}, q_{11},$ and $q_{12}$ are needed to memory a quadrupole tensor at least. The calculation of $q_{22}$ is as $$q_{22}=-(q_{00}+q_{11}).$$ However, our code loads the value of $q_{22}$ instead of calculating to avoid redundant calculations of $q_{22}$ of the same cell. Therefore, our code loads the six numbers to memory a quadrupole tensor. The first term in the summation of the equation (\[eq:pot\]) is the monopole term, and the second term is the quadrupole. We rewrite the monopole term as $\phi_{j}^\mathrm{mono}$ and the quadrupole term as $\phi_{j}^\mathrm{quad}$. These are $$\phi_{j}^\mathrm{mono} = \frac{Gm_j}{\hat{r}_{ij}},$$ $$\phi_{j}^\mathrm{quad} = \frac{G}{2\hat{r}_{ij}^5}\mbox{\boldmath $r$}_{ij}\cdot\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij},$$ where $\hat{r}_{ij}=\sqrt{|\mbox{\boldmath $r$}_j - \mbox{\boldmath $r$}_i|^2 + \epsilon^2}$, and $\mbox{\boldmath $r$}_{ij} = \mbox{\boldmath $r$}_j - \mbox{\boldmath $r$}_i$. The gravitational force at the position $\mbox{\boldmath $r$}_i$ is given as follows: $$\label{eq:nabla} \mbox{\boldmath $a$}_i = -\nabla\phi_i.$$ From equation (\[eq:pot\]) and equation (\[eq:nabla\]), $$\label{eq:force} \mbox{\boldmath $a$}_i = -\sum_{j=1}^{n_j}\left(\frac{\phi_{j}^\mathrm{mono} + 5\phi_{j}^\mathrm{quad}}{\hat{r}_{ij}^2}\mbox{\boldmath $r$}_{ij} - \frac{1}{\hat{r}_{ij}^5}\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij}\right).$$ We aim to speed up the calculations of potential given in equation (\[eq:pot\]) and a gravitational force given in equation (\[eq:force\]) with AVX2 instructions. In those equations, the $j$-th cell exerts forces on the $i$-th particle. In this paper, we call them as “$j$-cells”, and “$i$-particles”. Since forces exerted by $j$-cells on $i$-particles are independent of each other, multiple forces can be calculated in parallel. Since the AVX2 instructions compute eight single-precision floating-point numbers in parallel, our code calculates the forces on four $i$-particles from two $j$-cells in parallel as original Phantom-GRAPE ([@key-3]). Structures for the particle and cell data ----------------------------------------- The data assignment of four $i$-particles in YMM registers is the same as original Phantom-GRAPE for collisionless simulations. The data assignment of two $j$-cells in YMM registers is also the same as the assignment of two $j$-particles on original Phantom-GRAPE for collisionless simulations. The details are given in @key-3. Our implementation shares the structures for $i$-particles, the resulting forces, and potentials with original Phantom-GRAPE for collisionless simulations. We define the structures for $j$-cells as shown in List 1. The positions of the center of mass, total masses, and quadrupole tensors of two $j$-cells are stored in the structure `Jcdata`. ``` {#jcdata label="jcdata"} // List 1: Structure for j-cells typedef struct jcdata{ // xm={{x0, y0, z0, m0}, {x1, y1, z1, m1}} float xm[2][4]; /* q={ {q0-00, q0-01, q0-02, 0.0, q1-00, q1-01, q1-02, 0.0}, {q0-11, q0-12, q0-22, 0.0, q1-11, q1-12, q1-22, 0.0} } */ float q[2][8]; } Jcdata, *cJcdata; ``` Macros for inline assembly codes -------------------------------- Original Phantom-GRAPE defines some preprocessor macros expanded into inline assembly codes. We use these macros to write a force loop for calculating gravitational force on four $i$-particles with evaluating quadrupole expansions. Descriptions of the macros used in our code are summarized in Table \[tab:macros\]. The title of Table \[tab:macros\] and the descriptions of the macros except for `VPERM2F128`, `VEXTRACTF128`, `VSHUFPS`, `VFMADDPS`, and `VFNMADDPS` are adapted from @key-3. Operands `reg`, `reg1`, `reg2`, `dest`, and `dst` specify the data in XMM or YMM registers, and `mem` is data in the main memory or the cache memory. The operand named `imm` is an 8-bit number to control the behavior of some operations. More details of the AVX2 instructions are presented in Intel’s website [^2]. [lX]{} Macro & Description\ `VLOADPS(mem, reg)` & Load four or eight packed values in `mem` to `reg`\ `VSTORPS(reg, mem)` & Store four or eight packed values in `reg` to `mem`\ `VADDPS(reg1, reg2, dst)` & Add `reg1` to `reg2`, and store the result to `dst`\ `VSUBPS(reg1, reg2, dst)` & Subtract `reg1` from `reg2`, and store the result to `dst`\ `VMULPS(reg1, reg2, dst)` & Multiply `reg1` by `reg2`, and store the result to `dst`\ `VRSQRTPS(reg, dst)` & Compute the inverse-square-root of `reg`, and store the result to `dst`\ `VZEROALL` & Zero all YMM registers\ `VPERM2F128(src1, src2, dest, imm)` & Permute 128-bit floating-point fields in `src1` and `src2` using controls from `imm` , and store result in `dest`\ `VEXTRACTF128(src, dest, imm)` & Extract 128 bits of packed values from `src` and store results in `dest`\ `VSHUFPS(src1, src2, dest, imm)` & Shuffle packed values selected by `imm` from `src1` and `src2`, and store the result to `dst`\ `PREFETCH(mem)` & Prefetch data on `mem` to the cache memory\ `VFMADDPS(dst, reg1, reg2)` & Multiply eight packed values from `reg1` and `reg2`, add to `dst` and put the result in `dst`.\ `VFNMADDPS(dst, reg1, reg2)` & Multiply eight packed values from `reg1` and `reg2`, negate the multiplication result and add to `dst` and put result in `dst`.\ \[tab:macros\] The title of this table and the descriptions of the macros except for `VPERM2F128`, `VEXTRACTF128`, `VSHUFPS`, `VFMADDPS`, and `VFNMADDPS` are adapted from @key-3. A force loop ------------ The following routine computes the forces on four $i$-particles from $j$-cells. 1. Zero all the YMM registers. 2. \[step1\] Load the $x$, $y$ and $z$ coordinates of four $i$-particles to the lower 128-bit of YMM00, YMM01 and YMM02, and copy them to the upper 128-bit of YMM00, YMM01 and YMM02, respectively. 3. Load the $x$, $y$ and $z$ coordinates of the center of mass and the total masses of two $j$-cells to YMM14. 4. Broadcast the $x$, $y$, and $z$ coordinates of the center of mass of two $j$-cells in YMM14 to YMM03, YMM04, and YMM05, respectively. 5. Subtract YMM00, YMM01 and YMM02 from YMM03, YMM04, and YMM05, then store the results ($x_{ij}, y_{ij}$ and $z_{ij}$) in YMM03, YMM04, YMM05, respectively. 6. \[step6\] Load squared softening lengths to the lower 128-bit of YMM01, and copy them to the upper 128-bit of YMM01. 7. \[step7\] Square $x_{ij}$ in YMM03, $y_{ij}$ in YMM04, $z_{ij}$ in YMM05 and add them to the squared softening lengths in YMM01. It is the softened squared distances $\hat{r}^2_{ij} \equiv r^2_{ij}+\epsilon^2$ between the center of mass of two $j$-cells and four $i$-particles are stored in YMM01. 8. \[step8\] Calculate inverse-square-root for $\hat{r}^2_{ij}$ in YMM01, and store the result $1/\hat{r}_{ij}$ in YMM01. 9. \[step9\] Square $1/\hat{r}_{ij}$ in YMM01 and store the results in YMM00. 10. \[step10\] Broadcast the total masses of two $j$-cells in YMM14 to YMM02. 11. Multiply $1/\hat{r}_{ij}$ in YMM01 by $m_j$ in YMM02 to obtain $\phi_{j}^\mathrm{mono}=m_j/\hat{r}_{ij}$, and store the results in YMM02. 12. Load $q_{00}$, $q_{01}$ and $q_{02}$ of two $j$-cells to YMM08, $q_{11}$, $q_{12}$ and $q_{22}$ of two j-cells to YMM15, respectively. 13. Broadcast the $q_{00}$, $q_{01}$, $q_{02}$, $q_{11}$, $q_{12}$ and $q_{22}$ to YMM06, YMM07, YMM08, YMM13, YMM14, YMM15, respectively. 14. Multiply YMM03, YMM04, and YMM05 by YMM06, YMM07, and YMM08, respectively, and sum them up. The results are $x$-component of $\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij}$, and stored in YMM06. 15. Multiply YMM03, YMM04, and YMM05 by YMM07, YMM13, and YMM14, respectively, and sum them up. The results are $y$-component of $\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij}$, and stored in YMM13. 16. Multiply YMM03, YMM04, and YMM05 by YMM08, YMM14, and YMM15, respectively, and sum them up. The results are $z$-component of $\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij}$, and stored in YMM15. 17. Multiply YMM06, YMM13, and YMM15 by YMM03, YMM04, and YMM05, respectively, and sum them up to calculate $\mbox{\boldmath $r$}_{ij}\cdot\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij}$. The results are stored in YMM07. 18. Square $1/\hat{r}^2_{ij}$ in YMM00 and store the results in YMM08. 19. Multiply $1/\hat{r}^4_{ij}$ in YMM08 by $1/\hat{r}_{ij}$ in YMM01 to calculate $1/\hat{r}^5_{ij}$ and store the results in YMM08. 20. Load 0.5 in YMM14. 21. Multiply $\mbox{\boldmath $r$}_{ij}\cdot\mbox{\boldmath $Q$}_j\cdot\mbox{\boldmath $r$}_{ij}$ in YMM07 by $1/\hat{r}^5_{ij}$ in YMM08, then multiply it by 0.5 in YMM14 to calculate $\phi_{j}^\mathrm{quad}$ and store the results in YMM02. 22. Accumulate $\phi_{j}^\mathrm{mono}$ in YMM02 and $\phi_{j}^\mathrm{quad}$ in YMM07 into $\phi_i$ in YMM09. 23. Load 5 in YMM14. 24. Calculate $\phi_{j}^\mathrm{mono} + 5.0\phi_{j}^\mathrm{quad}$ and store the results in YMM02. 25. Multiply YMM00 by YMM03, YMM04, and YMM05 to calculate $x$, $y$, and $z$ components of the first term of the summation in equation \[eq:force\], then accumulate them into YMM10, YMM11 and YMM12, respectively. 26. Multiply YMM08 by YMM06, YMM13, and YMM15 to calculate $x$, $y$, and $z$ components of the second term of the summation in equation \[eq:force\], then subtract them from YMM10, YMM11, and YMM12, respectively. 27. Return to step \[step1\] until all the j-cells are processed. 28. Perform sum reduction of partial forces and potentials in the lower and upper 128-bits of YMM10, YMM11, YMM12, and YMM09, and store the results in the lower 128-bit of YMM10, YMM11, YMM12, YMM09, respectively. 29. Store forces and potentials in the lower 128-bit of YMM10, YMM11, YMM12, and YMM09 to the structure `Fodata`. List 2 is the function `c_GravityKernel` calculating the forces on four $i$-particles. We changed the order of operations in an actual code a little to make contiguous instructions independently, resulting in improved throughput. The data of $i$-particles and the squared softening length are common for all $j$-cells. However, unlike original Phantom-GRAPE for collisionless system, loading the data of $i$-particles is necessary for each $j$ loop, because the number of SIMD registers of AVX2 is not enough to keep the data over the loop. In step \[step6\] squared softening lengths overwrite $y$-coordinates of $i$-particles in YMM01 and are replaced with $\hat{r}^2_{ij}$ in step \[step7\]. In step \[step9\] $x$-coordinates of $i$-particles in YMM00 are replaced with $1/\hat{r}^2_{ij}$ . In step \[step10\] $z$-coordinates of $i$-particles in YMM02 are replaced with $m_j$. Assuming that one division and one square-root each require 10 floating point operations ([@key-26]), thus one inverse-square-root requires 20 floating point operations. The number of floating point operations needed for the calculation of force exerted by one $j$-cell on one $i$-particle is counted to be 71. According to IntelR 64 and IA-32 Architectures Optimization Reference Manual [^3], the latency of one inverse-square-root (`VRSQRTPS`) is seven. Therefore, if we assume that one inverse-square-root requires seven floating point operations, the total number of floating point operations per interaction is counted to be 58. /* List 2: A force loop which evaluates up to quadrupole term by using AVX2. */ void c_GravityKernel(pIpdata ipdata, pFodata fodata, cJcdata jcdata, int nj){ int j; float five[8] = {5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0}; float half[8] = {0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5}; PREFETCH(jcdata[0]); VZEROALL; for(j = 0; j < nj; j += 2){ // load i-particle VLOADPS(*ipdata->x, XMM00); VLOADPS(*ipdata->y, XMM01); VLOADPS(*ipdata->z, XMM02); VPERM2F128(YMM00, YMM00, YMM00, 0x00); VPERM2F128(YMM01, YMM01, YMM01, 0x00); VPERM2F128(YMM02, YMM02, YMM02, 0x00); // load jcell's coordinate VLOADPS(jcdata->xm[0][0], YMM14); VSHUFPS(YMM14, YMM14, YMM03, 0x00); //xj VSHUFPS(YMM14, YMM14, YMM04, 0x55); //yj VSHUFPS(YMM14, YMM14, YMM05, 0xaa); //zj // r_ij,x -> YMM03 VSUBPS(YMM00, YMM03, YMM03); // r_ij,y -> YMM04 VSUBPS(YMM01, YMM04, YMM04); // r_ij,z -> YMM05 VSUBPS(YMM02, YMM05, YMM05); // eps^2 -> YMM01 VLOADPS(*ipdata->eps2, XMM01); VPERM2F128(YMM01, YMM01, YMM01, 0x00); // r_ij^2 -> YMM01 VFMADDPS(YMM01, YMM03, YMM03); VFMADDPS(YMM01, YMM04, YMM04); VFMADDPS(YMM01, YMM05, YMM05); // 1 / r_ij -> YMM01 VRSQRTPS(YMM01, YMM01); // 1 / r_ij^2 -> YMM00 VMULPS(YMM01, YMM01, YMM00); // phi_p(mj / r_ij) -> YMM02 VSHUFPS(YMM14, YMM14, YMM02, 0xff); // mj VMULPS(YMM01, YMM02, YMM02); /* q00, q01, q02, q11, q12, q22 -> YMM06, 07, 08, 13, 14, 15, respectively */ VLOADPS(jcdata->q[0][0], YMM08); VLOADPS(jcdata->q[1][0], YMM15); VSHUFPS(YMM08, YMM08, YMM06, 0x00); VSHUFPS(YMM08, YMM08, YMM07, 0x55); VSHUFPS(YMM08, YMM08, YMM08, 0xaa); VSHUFPS(YMM15, YMM15, YMM13, 0x00); VSHUFPS(YMM15, YMM15, YMM14, 0x55); VSHUFPS(YMM15, YMM15, YMM15, 0xaa); // q00 * r_ij,x -> YMM06 VMULPS(YMM03, YMM06, YMM06); // YMM06 + q01 * r_ij,y -> YMM06 VFMADDPS(YMM06, YMM04, YMM07); // YMM06 + q02 * r_ij,z -> YMM06 VFMADDPS(YMM06, YMM05, YMM08); // q11 * r_ij,y -> YMM13 VMULPS(YMM13, YMM04, YMM13); // YMM13 + q01 * r_ij,x -> YMM13 VFMADDPS(YMM13, YMM03, YMM07); // YMM13 + q12 * r_ij,z -> YMM13 VFMADDPS(YMM13, YMM05, YMM14); // q22 * r_ij,z -> YMM15 VMULPS(YMM15, YMM05, YMM15); // YMM15 + q02 * r_ij,x -> YMM15 VFMADDPS(YMM15, YMM03, YMM08); // YMM15 + q12 * r_ij,y -> YMM15 VFMADDPS(YMM15, YMM04, YMM14); // calculate drqdr // qdr[0] * r_ij,x -> YMM07 VMULPS(YMM03, YMM06, YMM07); // YMM07 + qdr[1] * r_ij,y -> YMM07 VFMADDPS(YMM07, YMM04, YMM13); // YMM07 + qdr[2] * r_ij,z -> YMM07 VFMADDPS(YMM07, YMM05, YMM15); // 1/(r_ij)^5 -> YMM08 VMULPS(YMM00, YMM00, YMM08); VMULPS(YMM01, YMM08, YMM08); // 0.5 -> YMM14 VLOADPS(half, YMM14); // 1/(r_ij)^5 * drqdr * 0.5 -> YMM07 VMULPS(YMM07, YMM08, YMM07); VMULPS(YMM07, YMM14, YMM07); // phi += phi_p(YMM02) + phi_q(YMM07) VADDPS(YMM02, YMM07, YMM14); VADDPS(YMM14, YMM09, YMM09); // 5.0 -> YMM14 VLOADPS(five, YMM14); // 5.0 * phi_q + phi_p -> YMM02 VFMADDPS(YMM02, YMM07, YMM14); // YMM02 / (r_ij)^2 ->YMM00 VMULPS(YMM02, YMM00, YMM00); // ax, ay, az -> YMM10, YMM11, YMM12 VFMADDPS(YMM10, YMM00, YMM03); VFMADDPS(YMM11, YMM00, YMM04); VFMADDPS(YMM12, YMM00, YMM05); VFNMADDPS(YMM10, YMM08, YMM06); VFNMADDPS(YMM11, YMM08, YMM13); VFNMADDPS(YMM12, YMM08, YMM15); jcdata++; } VEXTRACTF128(YMM10, XMM00, 0x01); VEXTRACTF128(YMM11, XMM01, 0x01); VEXTRACTF128(YMM12, XMM02, 0x01); VEXTRACTF128(YMM09, XMM03, 0x01); VADDPS(YMM10, YMM00, YMM10); VADDPS(YMM11, YMM01, YMM11); VADDPS(YMM12, YMM02, YMM12); VADDPS(YMM09, YMM03, YMM09); VSTORPS(XMM10, *fodata->ax); VSTORPS(XMM11, *fodata->ay); VSTORPS(XMM12, *fodata->az); VSTORPS(XMM09, *fodata->phi); } Application programming interfaces ---------------------------------- List 3 shows the application programming interfaces (APIs) for our code. `g5c_set_nMC` tells our code the number of $j$-cells. `g5c_set_xmjMC` transfer positions, mass and quadrupole tensors of $j$-cells to the array of the structure `Jcdata`. `g5c_calculate_force_on_xMC` transmits coordinates and number of $i$-particles to an array of the structure `Ipdata`, which is defined in the original Phantom-GRAPE ([@key-3]), and calculates the forces and potentials exerted by $j$-cells on the $i$-particles and store the result in the arrays `ai` and `pi`, respectively. List 4 shows a part of C++ code that calculates the forces and potentials of all particles. In this code, we use the modified tree algorithm ([@key-27]), where the particles in a cell that contains $n_{\rm crit}$ or less particles shares the same interaction list. The particles sharing the same interaction list are $i$-particles, the particles in the interaction list are $j$-particles, and the cells in the interaction list are $j$-cells. The functions beginning with `g5_` are the APIs for the original Phantom-GRAPE ([@key-3]), and calculate particle-particle interactions. The functions beginning with `g5c_` are the APIs for our code, and calculate interactions from cells. // List 3: APIs for our code. void g5c_set_xmjMC(int devid, int adr, int nj, double (*xj)[3], double *mj, double (*qj)[6]); void g5c_set_nMC(int devid, int n); void g5c_calculate_force_on_xMC(int devid, double (*x)[3], double (*a)[3], double *p, int ni); ``` {#sample label="sample"} // List 4: Sample code class particle; // Contains particle data class node; // Contains cell data /* a cell that contain particles which share same interactions */ class ilist{ public: int ni; // Number of particles double l; // Cell's length double (*xi)[3]; // Position double (*ai)[3]; // Force double (*pi); // Potential particle*(*pp); // Pointer to particle double cpos[3]; // Cell's center }; class jlist{// contains one j-particle data public: int nj; // Number of particles double (*xj)[3]; // Position double (*mj); // Mass }; class jcell{// contains one j-cell data public: int nj; // Number of cells double (*xj)[3]; // Mass center double (*mj); // Total mass double (*qj)[6]; // Quadrupole tensor }; /* create tree structure and groups of i-particles which share the same interaction list. */ void create_tree(node *, particle *, int, ilist, int, int); /* traverse the tree structure and make lists of j-particles and j-cells. (a interaction list.) */ void traverse_tree(node *, ilist, jlist, jcell, double, int); /* assign or add the values of force and potential in ilist to those in particle class. */ void assign_force_potential(ilist); void add_force_potential(ilist); int n; // number of particles double theta2; // square of theta /* calculate forces and potentials of all particles. */ void calc_force(int n, int nnodes, particle pp[], node *bn, double eps, double theta2, int ncrit){ // Number of groups of i-particles int ni; // index of loop int i, k; create_tree(bn, pp, ni, i_list, n, ncrit); g5_open(); g5_set_eps_to_all(eps); for(i = 0; i < ni; i++){ tree_traversal(bn, i_list, j_list, j_cell, theta2, ncrit); /* calculate forces exerted by j-particles */ g5_set_xmjMC(0, 0, j_list->nj, j_list->xj, j_list->mj); g5_set_nMC(0, j_list->nj); g5_calculate_force_on_xMC(0, i_list[i]->xi, i_list[i]->ai, i_list[i]->pi, i_list[i]->ni ); assign_force_potential(i_list); // calculate forces exerted by j-cells g5c_set_xmjMC(0, 0, j_cell->nj, j_cell->xj, j_cell->mj, j_cell->qj); g5c_set_nMC(0, j_cell->nj); g5c_calculate_force_on_xMC(0, i_list[i]->xi, i_list[i]->ai, i_list[i]->pi, i_list[i]->ni ); add_force_potential(i_list); } g5_close(); } ``` Accuracy {#sec:acu} ======== In this section, we compare the accuracy of forces obtained by utilizing only monopole terms and that obtained by calculating up to quadrupole terms. The detailed discussion about errors of forces in the tree method is given in @key-30, Barnes and Hut (), and @key-28. Figure \[fig:err\_cc65k\] shows the cumulative distribution of relative force errors in particles distributed in a homogeneous sphere (top), a Plummer model (middle), and an exponential disk (bottom), respectively. Relative errors in the forces of particles are given as $$\begin{aligned} \frac{|\mbox{\boldmath $a$}_{\mathrm{TREE}}-\mbox{\boldmath $a$} _{\mathrm{DIRECT}}|}{|\mbox{\boldmath $a$}_{\mathrm{DIRECT}}|} \label{eq:error_acc} ,\end{aligned}$$ where $\mbox{\boldmath $a$}_{\mathrm{TREE}}$ is the force calculated using the tree method, and $\mbox{\boldmath $a$}_{\mathrm{DIRECT}}$ is the force computed using the direct particle-particle method with Phantom-GRAPE for collisionless simulations. We used our implementation to calculate quadrupole terms and original Phantom-GRAPE to calculate monopole terms. The number of particles is 65,536 for all three particle distributions. The top panel of Figure \[fig:err\_cc65k\] (the homogeneous sphere) shows that the result of using quadrupole terms with $\theta=0.65$ has accuracy comparable to that of only monopole terms with $\theta=0.3$. When using quadrupole terms with $\theta=0.75$, most particles have smaller errors than using only monopole terms with $\theta=0.5$ and only a few percent of particles have larger errors. The middle panel (the Plummer model) of Figure \[fig:err\_cc65k\] suggests that about a half of the particles have smaller errors with calculating the quadrupole terms using $\theta=0.4$ than with calculating only monopole terms using $\theta=0.3$. The rest of the particles have slightly larger errors. However, these differences are small and both error distributions agree with each other. The result of using quadrupole terms with $\theta=0.6$ has accuracy comparable to that of only monopole terms with $\theta=0.5$. About a tenth part of particles have larger errors when we calculate the quadrupole terms with $\theta=0.6$ than when we calculate only the monopole terms with $\theta=0.5$. The bottom panel (the exponential disk) of Figure \[fig:err\_cc65k\] shows that the result of using quadrupole terms with $\theta=0.45$ has accuracy comparable to that of only monopole terms with $\theta=0.3$. When using quadrupole terms with $\theta=0.65$, most particles have smaller errors than using only monopole terms with $\theta=0.5$ and only a few percent of particles have larger errors. In a homogeneous system, the net force exerted by particles located at a certain range $r$ does not depend on $r$ because the gravitational force from a particle at $r$ is proportional to $r^{-2}$ and the number of particles at $r$ is proportional to $r^{2}$. The force from distant particles, which is not negligible compared to the force from close particles, can be significantly more accurate by using quadrupole than by using only the monopole. On the other hand, in a clustered system such as a Plummer model and a disk, the gravitational force is dominated by nearby particles for a large fraction of particles. Thus accuracy cannot be significantly improved even if quadrupole terms are used. Therefore, $\theta$ cannot be very large in a clustered system. Figure \[fig:err\_f90\] shows the error at 90% of the particles as a function of $\theta$ and highlights the results described above. The errors when we utilize up to monopole terms and quadrupole terms are roughly proportional to $\theta^{5/2}$ and $\theta^{7/2}$, respectively. This result is consistent with the scaling law of error described in @key-28. When we use only monopole terms, the error is the smallest in the Plummer model because the net force is dominated by the forces from nearby particles, most of which are calculated directly. The error in the disk is the largest because of the anisotropic structure of the disk. If the same $\theta$ is used, calculation of quadrupole terms reduces the error more in the homogeneous sphere than in other models because the force from distant particles can be well approximated by the quadrupole terms and such force constitutes a larger portion of the net force in a homogeneous system than in a clustered system such as the Plummer model and the disk. ![Cumulative distribution of errors in forces of particles with $N=65,536$. From top to bottom, the particle distributions are a homogeneous sphere, a Plummer model and an exponential disk, respectively. []{data-label="fig:err_cc65k"}](p_error_cc65k2.eps "fig:"){width="8cm"} ![Cumulative distribution of errors in forces of particles with $N=65,536$. From top to bottom, the particle distributions are a homogeneous sphere, a Plummer model and an exponential disk, respectively. []{data-label="fig:err_cc65k"}](p_error_plummer65k2.eps "fig:"){width="8cm"} ![Cumulative distribution of errors in forces of particles with $N=65,536$. From top to bottom, the particle distributions are a homogeneous sphere, a Plummer model and an exponential disk, respectively. []{data-label="fig:err_cc65k"}](p_error_disk65k2.eps "fig:"){width="8cm"} ![ Error of 90% of the particles as a function of $\theta$. The squares, circles, and triangles show the result of a homogeneous sphere, a Plummer model, and a disk, respectively. The solid and dashed lines without points show $\theta^{5/2}$ and $\theta^{7/2}$ scaling, respectively. []{data-label="fig:err_f90"}](f90_4.eps){width="8cm"} Performance {#sec:perf} =========== In this section, we compare the performance of our implementation, original Phantom-GRAPE, and the pseudoparticle multipole method when the same force accuracy is imposed. The system we used to measure the performance is shown in Table \[tab:system\]. We used only one core, and Intel Turbo Boost Technology is enabled. Compiler options were -O3 -ffast-math -funroll-loops. Theoretical peak FLOPS of the system per core is 67.2 GFLOPS. The values of $\theta$ when we utilize quadrupole moments are based on the result that we described in section \[sec:acu\]. ---------- --------------------------------------- CPU Intel Xeon E5-2683 v4 2.10GHz Memory 128GB OS CentOS Linux release 7.3.1611 (core) Compiler gcc 4.8.5 20150623 (Red Hat 4.8.5-11) ---------- --------------------------------------- : The system we use to measure the performance. \[tab:system\] Comparison of calculation time when the same accuracy is required ----------------------------------------------------------------- Table \[tab:LTsample\] shows the wall clock time for evaluating forces and potentials of all the particles with $N=4,194,304$. In general, when we utilize quadrupole moments, the time consumed in the tree construction becomes slightly longer because quadrupole tensors of cells are calculated. When we use the pseudoparticle multipole method, the time consumed in the tree construction becomes longer because of the positioning of pseudoparticles. The simulations of the homogeneous sphere with only the monopole moments can be accelerated from 1.23 to 2.20 times faster when we use our code and evaluate quadrupole terms. The simulations of the exponential disk using only the monopole terms with $\theta=0.3$ can be accelerated 1.13 times faster when we use our implementation and set $\theta=0.45$. In other $\theta$ and particle distribution, using the quadrupole terms slows simulations. As described in section \[sec:acu\], using quadrupole terms allows us to use significantly larger $\theta$ than using only the monopole in a homogeneous system, while we can increase $\theta$ moderately in a clustered system. Therefore, more interactions from particles are approximated by quadrupole expansion in a homogeneous system than in a clustered system. Thus, using the quadrupole terms can efficiently accelerate simulations of a homogeneous system. In the clustered system such as the disk and the Plummer model, the number of approximated interactions by using quadrupole terms and larger $\theta$ is not enough to negate the increased calculation cost by computing quadrupole terms. Our implementation is always faster than the combination of pseudoparticle multipole method and Phantom-GRAPE for collisionless simulations by a factor of 1.1 in any condition because calculations such as diagonalizations of quadrupole tensors are unnecessary. Program $\theta$ Particle distribution $T_\mathrm{construct}$\[s\] $T_\mathrm{traverse}$\[s\] $T_\mathrm{force}$\[s\] $T_\mathrm{total}$\[s\] Ratio ---------------- ---------- ----------------------- ----------------------------- ---------------------------- ------------------------- ------------------------- ------- Program $\theta$ Particle distribution $T_\mathrm{construct}$\[s\] $T_\mathrm{traverse}$\[s\] $T_\mathrm{force}$\[s\] $T_\mathrm{total}$\[s\] Ratio monopole 0.3 Homogeneous 1.06 3.03 19.82 23.99 1 pseudoparticle 0.65 Homogeneous 1.67 1.20 8.96 11.91 0.50 quadrupole 0.65 Homogeneous 1.10 1.16 8.53 10.88 0.45 monopole 0.5 Homogeneous 1.06 1.34 7.77 10.26 1 pseudoparticle 0.75 Homogeneous 1.67 0.94 6.53 9.22 0.90 quadrupole 0.75 Homogeneous 1.10 0.92 6.25 8.35 0.81 monopole 0.3 Plummer 2.05 7.79 31.30 41.23 1 pseudoparticle 0.4 Plummer 2.70 6.41 39.41 48.61 1.18 quadrupole 0.4 Plummer 2.11 5.81 36.64 44.65 1.08 monopole 0.5 Plummer 2.05 2.71 10.28 15.13 1 pseudoparticle 0.6 Plummer 2.69 3.22 18.31 24.30 1.61 quadrupole 0.6 Plummer 2.11 2.96 17.11 22.26 1.47 monopole 0.3 Disk 1.45 5.49 20.95 28.01 1 pseudoparticle 0.45 Disk 2.10 3.76 21.11 27.09 0.97 quadrupole 0.45 Disk 1.51 3.52 19.72 24.86 0.89 monopole 0.5 Disk 1.45 2.22 7.96 11.75 1 pseudoparticle 0.65 Disk 2.11 1.98 9.70 13.92 1.18 quadrupole 0.65 Disk 1.51 1.88 9.18 12.67 1.08 : Wall clock time for evaluating forces and potentials of all the particles with $N=4,194,304$. “Monopole” calculates only monopole terms. “Pseudoparticle” calculates quadrupole terms with pseudoparticles. “Quadrupole” calculates quadrupole terms with our implementation. “Homogeneous” is the homogeneous sphere. “Plummer” is the Plummer model. “Disk” is the exponential disk. $T_\mathrm{construct}$, $T_\mathrm{traverse}$, and $T_\mathrm{force}$ are time for tree constructions, tree traverse, force calculation, respectively. $T_\mathrm{total}$ is total time. The column “Ratio” is ratios of the total time to that of using only monopole.[]{data-label="tab:LTsample"} The dependency of calculation time in the number of particles and interactions per second ----------------------------------------------------------------------------------------- Figure \[fig:n\_cc\] shows wall clock time on various $N$ for calculating forces and potentials of particles in the homogeneous sphere (top), the Plummer model (middle), and the exponential disk (bottom), respectively. Solid curves are for small $\theta$, and dashed curves with points are for large $\theta$. Dashed lines without point show $N \log N$ scaling. We can see that the total time to calculate the force and potential of particles is roughly proportional to $N \log N$. However, from $N=65,536$ to $N=131,072$ on the homogeneous sphere, the actual scaling of the total time slightly deviates from the $N\log N$ scaling. From $N=65,536$ to $N=131,072$, the depth level of the tree traversals became deep because of the nature of the hierarchical oct-tree structure. Thus, more part of interactions is approximated with the multipole expansions. Therefore, the total number of particle-particle and particle-cells interactions and the total calculation time deviates slightly from the $N\log N$ scaling. Deviation from $N \log N$ scaling can also be seen on the Plummer model and the disk. However, the deviation is not as obvious as that on the homogeneous sphere. The calculation time can fluctuate by other running processes. As seen in Figure \[fig:ips\], the number of interactions from cells per second is greatly reduced in $N < 262,144$. This slowdown comes from the overhead of storing $i$-particles into the structure named `Ipdata`. Our code, as well as the original Phantom-GRAPE ([@key-3]) stores four $i$-particles into `Ipdata`. Each time a calculation of the net force on four $i$-particles is done, next four $i$-particles are loaded into `Ipdata`. The number of interaction is proportional to $n_i \times n_j$, where $n_i$ is the number of $i$-particles, and the computational cost for storing $i$-particles is proportional to $n_i$. If $N$ becomes fewer, $n_i$ and $n_j$ also become fewer. Therefore, the overhead of storing $i$-particles becomes relatively large compared to the calculation of interactions itself, resulting in the speed down of the calculation of interactions. This behavior is also seen in the original Phantom-GRAPE ([@key-3]), which shows lower performance for smaller $n_i$ and $n_j$. Theoretical peak FLOPS per core of the CPU which we use is 67.2 GFLOPS, however, this value is based on the assumption that the CPU is executing FMA operations all the time. Actually, 36 counts of floating point operations in our code are FMA , and the rest come from non-FMA, add, subtract, multiply, and inverse-square root operations. Therefore, if we count 71 and 58 operations per interaction, theoretical peak FLOPS in our code with Intel Xeon E5-2683 v4 is 50.6 and 54.5 GFLOPS, respectively. From Figure \[fig:ips\], the numbers of interactions from cells per second are $\sim 7\times10^8$ at sufficiently large $N$. For 71 and 58 operations per interaction, the measured performances of our code are 50 and 41 GFLOPS, which correspond to 99% and 75% of the peak. To validate effectiveness of our implementation for astrophysical regimes, we performed three cold collapse simulations. We set the gravitational constant, the total mass of particles, the unit length, the total number of particles, the time step, and the softening length as $G=1$, $M=1$, $R=1$, $N=4,194,304$, $\Delta t=2^{-8}$, $\epsilon=2^{-8}$, respectively. The initial particle distribution was the homogeneous sphere whose radius is a unity, and the initial virial ratio was $0.1$. Three simulations were conducted on the machine shown in Table \[tab:system\] with 30 CPU cores. Differences between the three simulations are $\theta$ and whether quadrupole terms are calculated. Simulation A utilized only monopole terms with $\theta=0.3$. Simulation B and C calculated quadrupole terms and used $\theta=0.4$ and 0.65. Figure \[fig:rho\] shows the radial density profiles of these simulations at $t=10$. Note that we plotted from $R=0.01$, which is about five times of $\epsilon=2^{-8}$. The results of the three simulations agree well each other. The particle distribution is nearly homogeneous at $t<1$. Thus, if we consider accuracy only at $t<1$, we can use $\theta=0.65$ when we calculate quadrupole terms to achieve comparable accuracy with only monopole terms as shown in Figure \[fig:err\_cc65k\]. The collapse occurs around $t=1$, and then a dense flat core forms at $t>1$ as shown in Figure \[fig:rho\]. Therefore, if we take account of accuracy at $t>1$, it is assumed that we should use $\theta=0.4$ rather than $\theta=0.65$ when quadrupole terms are adopted. However, there was little difference in the density profiles. Thus, practically, we might be able to use larger $\theta$ than expected to reduce the calculation cost. The average calculation time per step of these simulations were 3.20 seconds, 3.70 seconds, and 2.05 seconds, for simulation A, B, and C, respectively. Therefore, we can gain 1.56 times better performance by calculating quadrupole terms with $\theta=0.65$ than by calculating only monopole terms of $\theta=0.3$. As another practical astrophysical test, we performed a suite of cosmological $N$-body simulations using the same initial condition. The initial condition consists of $128^3$ dark matter particles in a comoving box of 103 Mpc and the mass resolution is $2.07 \times 10^{10} \, M_{\odot}$. We generated the initial condition at $z=33$ by a publicly available code, MUSIC [^4] [@key-34]. Here, we aim to evaluate the performance of our implementation for the late phase of large scale structure formation. For this reason, we first simulated this initial condition down to $z=1$ by a TreePM code, GreeM [@key-6; @key-21]. Then we identified particles within a spherical region with a radius of 51 Mpc on the box center and added hubble velocities to these particles. We use these particles as the new initial condition of our cosmological test calculations. We simulated the initial condition from $z=1$ to $z=0$ with three different settings in the same manner as the cold collapse simulations shown above, namely, simulation A2 which utilized only monopole terms with $\theta=0.3$, simulation B2 and C2 which calculated quadrupole terms with $\theta=0.4$ and $\theta=0.65$. We also conducted the full box simulation by the TreePM. Figure \[fig:massfunc\] shows the mass functions of dark matter halos at $z=0$, identified by ROCKSTAR phase space halo/subhalo finder [@key-35]. The results of the three tree simulations and TreePM simulation agree well each other and well fitted by a fitting function calibrated by a suite of huge simulations [@key-32]. In the late phase of large scale strcuture formation such as $z<1$, particle distributions are highly inhomogeneous because dense dark matter halos form everywhere, indicating that it should be more reasonable to use $\theta=0.4$ rather than $\theta=0.65$ when quadrupole terms are adopted as discussed in cold collapse simulations. However, the difference of halo mass functions is indistinguishable. Thus, practically, larger $\theta$ than expected might be allowed to reduce the calculation cost. The average calculation time per step of these simulations were 0.415 seconds, 0.441 seconds, and 0.292 seconds, for simulation A2, B2, and C2, respectively. Therefore, these results demonstrate that we can gain 1.42 times better performance by our implementation. These simple tests reinforce the effectiveness of our implementation for some astrophysical targets. ![Wall clock time for calculating forces and potentials of all the particles. From top to bottom, the particle distributions are a homogeneous sphere, a Plummer model and an exponential disk, respectively. Dotted lines show $N\log N$ scaling.[]{data-label="fig:n_cc"}](cc_ndepend6.eps "fig:"){width="8cm"} ![Wall clock time for calculating forces and potentials of all the particles. From top to bottom, the particle distributions are a homogeneous sphere, a Plummer model and an exponential disk, respectively. Dotted lines show $N\log N$ scaling.[]{data-label="fig:n_cc"}](plummer_ndepend5.eps "fig:"){width="8cm"} ![Wall clock time for calculating forces and potentials of all the particles. From top to bottom, the particle distributions are a homogeneous sphere, a Plummer model and an exponential disk, respectively. Dotted lines show $N\log N$ scaling.[]{data-label="fig:n_cc"}](disk_ndepend4.eps "fig:"){width="8cm"} ![ Numbers of interactions from cells per second. Note that particle-particle interactions are excluded. The squares, circles, and triangles show the result of a homogeneous sphere, a Plummer model, and a disk, respectively. Solid curves show the result for smaller $\theta$. Dotted curves show the result for larger $\theta$.[]{data-label="fig:ips"}](ips3.eps){width="8cm"} ![ Radial density profiles of cold collapse simulations at $t=10$. Solid curves without points and with open squares show the results of simulations that utilized only monopole terms with $\theta=0.3$ (simulation A) and up to quadrupole terms with $\theta=0.4$ (simulation B), respectively. Circles show the result of simulation that utilized up to quadrupole terms with $\theta=0.65$ (simulation C). []{data-label="fig:rho"}](rho2.eps){width="8cm"} ![ Mass functions of dark matter halos at $z=0$ obtained in a suite of cosmological test simulations. Three solid curves with symbols are results from our implementation for three different settings. Dashed curve is obtained from the simulation done by the TreePM method [@key-6; @key-21]. Solid curve without symbols denotes a fitting function proposed by @key-32. []{data-label="fig:massfunc"}](massfunc.eps){width="8cm"} Discussion {#sec:dis} ========== In this section, we estimate the performance of our implementation on AVX-512 environment. In AVX-512, the number of SIMD registers is 32, which is twice of AVX2. This number is enough to hold data that are currently needed to load every time the force calculation loop is done. Line 18 to 23 in List 2 are the operations for loading coordinates of $i$-particles. Line 36 and 37 is the operation for loading the gravitational softening length. Line 98 and 107 are the operations for loading constant floating-point numbers, which are necessary to calculate the quadrupole term of equation (\[eq:pot\]) and the gravitational force given in equation (\[eq:force\]), respectively. All data loaded by those operations do not change throughout the entire $j$ loop in the force calculation from Line 16 to 123 in List 2. Therefore, Line 18 to 23, Line 36, 37, 98, and 107 in List 2 can be moved to before the loop. Furthermore, the width of SIMD registers in AVX-512 is 512-bit, which is twice of AVX2. This enables us to remove Line 56 in List 2 because the elements of the quadrupole tensors of two $j$-cells, which are $6 \times 2=12$ elements, can be stored in one register. Without additional instructions, we can replace six VSHUFPS operations from Line 57 to 62 in List 2 to six VPERMPS operations, which permute single-precision floating-point value. The detail of VPERMPS is available in IntelR 64 and IA-32 Architectures Software Developer’s Manual [^5]. Totally, we can reduce the numbers of operations in the force loop from 59 to 48. Furthermore, the AVX-512 instructions can simultaneously calculate 16 single-precision floating-point numbers because of the twice width of the SIMD registers. Overall, we can estimate that the calculation of quadrupole terms becomes $59/48\times2=2.46$ times faster in AVX-512 than AVX2. The calculation of monopole terms will be twice faster in AVX-512 than AVX2 because of the twice width of the SIMD registers. It is difficult to gain speed up in other parts such as the tree construction and the tree traversal because hierarchical oct-tree structures are used. Therefore, we assume that the calculation time for tree construction and tree traversal does not change on AVX-512 environment compared to that of AVX2 environment. Table \[tab:AVX-512\] is the estimated ratios of the time for calculating forces to that of using only the monopole on AVX-512 environment. Our implementation gives 1.08 times faster using the quadrupole terms with $\theta=0.4$ than using only monopole terms with $\theta=0.3$ for the Plummer model, and 1.02 times faster with $\theta=0.65$ than that of $\theta=0.5$ with using monopole terms only in the disk. Program $\theta$ Particle distribution Ratio ------------ ---------- ----------------------- ------- Program $\theta$ Particle distribution Ratio monopole 0.3 Homogeneous 1 quadrupole 0.65 Homogeneous 0.44 monopole 0.5 Homogeneous 1 quadrupole 0.75 Homogeneous 0.77 monopole 0.3 Plummer 1 quadrupole 0.4 Plummer 0.93 monopole 0.5 Plummer 1 quadrupole 0.6 Plummer 1.26 monopole 0.3 Disk 1 quadrupole 0.45 Disk 0.79 monopole 0.5 Disk 1 quadrupole 0.65 Disk 0.98 : Estimated ratios of the time for calculating forces and potentials to that of using only the monopole when we assume that the force calculation part is implemented with AVX-512. “Monopole” calculates only monopole terms. “Quadrupole” calculates quadrupole terms with our implementation. “Homogeneous” is the homogeneous sphere. “Plummer” is the Plummer model. “Disk” is the exponential disk. []{data-label="tab:AVX-512"} Summary {#sec:sum} ======= We have developed a highly-tuned software library to accelerate the calculations of quadrupole term with the AVX2 instructions on the basis of original Phantom-GRAPE ([@key-3]). Our implementation allows simulating homogeneous systems such as the large-scale structure of the universe up to 2.2 times faster than that with only monopole terms. Also, our implementation shows 1.1 times higher performance than the combination of the pseudoparticle multipole method and Phantom-GRAPE. Further improvement of the performance is estimated when we implement our code with the new SIMD instruction set, AVX-512. On AVX-512 environment, our code is expected to be able to accelerate simulations of clustered system up to 1.08 times faster than that with only monopole terms. Our implementation will be more useful as the length of the SIMD registers gets longer. 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--- abstract: 'We study a recently proposed kinetic exchange opinion model (Lallouache et. al., Phys. Rev E 82:056112, 2010) in the limit of a single parameter map. Although it does not include the essentially complex behavior of the multiagent version, it provides us with the insight regarding the choice of order parameter for the system as well as some of its other dynamical properties. We also study the generalized two-parameter version of the model, and provide the exact phase diagram. The universal behavior along this phase boundary in terms of the suitably defined order parameter is seen.' author: - 'Krishanu Roy Chowdhury, Asim Ghosh, Soumyajyoti Biswas and Bikas K. Chakrabarti' bibliography: - '&lt;your-bib-database&gt;.bib' title: 'Kinetic exchange opinion model: solution in the single parameter map limit' --- Introduction {#sec:1} ============ Dynamics of opinion and subsequent emergence of consensus in a society are being extensively studied recently [@ESTP; @Stauffer:2009; @Castellano:RMP; @Galam:1982; @Liggett:1999; @Sznajd:2000; @Galam:2008; @Sen:opin]. Due to the involvement of many individuals, this type of dynamics in a society can be treated as an example of a complex system, thus enabling the use of conventional tools of statistical mechanics to model it [@Hegselman:2002; @Deffuant:2000; @Fortunato:2005; @forunato; @Toscani:2006]. Of course, it is not possible to capture all the diversities of human interaction through any model of this kind. But often it is our interest to find out the global perspectives of a social system, like average opinion of all the individuals regarding an issue, where the intricacies of the interactions, in some sense, are averaged out. This is similar to the approach of kinetic theory, where the individual atoms, although following a deterministic dynamics, are treated as randomly moving objects and the macroscopic behaviors of the whole system are rather accurately predicted. Indeed, there have been several attempts to realize the human interactions in terms of kinetic exchange of opinions between individuals [@Hegselman:2002; @Deffuant:2000; @Toscani:2006]. Of course, there is no conservations in terms of opinion. Otherwise, this is similar to momentum exchange between the molecules of an ideal gas. These models were often studied using a finite confidence level, i.e., agents having opinions close to one another interact. However, in a recently proposed model [@Lallouache:2010], unrestricted interactions between all the agents were considered. The single parameter in the model described the ‘conviction’ which is a measure of an agent’s tendency to retain his opinion and also to convince others to take his opinion. It was found that beyond a certain value of this ‘conviction parameter’ the ‘society’, made up of $N$ such agents, reaches a consensus, where majority shares similar opinion. As the opinion values could take any values between \[-1:+1\], a consensus means a spontaneous breaking of a discrete symmetry. There have been subsequent studies to generalize this model, where the ‘conviction parameter’ and ‘ability to influence’ were taken as independent parameters [@Sen:2010]. In that two-parameter version, similar phase transitions were observed. However, the critical behaviors in terms of the usual order parameter, the average opinion, were found to be non-universal. There have been other extensions in terms of a phase transition induced by negative interactions [@bcs], an exact solution in a discrete limit [@sb], the effect of non-uniform conviction and update rules in these discrete variants [@nuno], a generalized map version [@asc:2010], a percolation transitions in a square lattices [@akc] and the effect of bounded confidence [@ps] in these models. In the present study, we investigate the single parameter map version of the model, also proposed in Ref.[@Lallouache:2010]. Although the original model is difficult to tackle analytically, in this mean field limit, it can simply be conceived as a random walk. Using standard random walk statistics, several static and dynamical quantities have been calculated. We show that the fraction of extreme opinion behaves like the actual order parameter for the system, and the average opinion shows unusual behavior near critical point. The critical behavior of the order parameter and its relaxation behavior near and at the critical point have been obtained analytically which agree with numerical simulations. Model and its map version ========================= Let the opinion of any individual ($i$) at any time ($t$) is represented by a real valued variable $O_i(t)$ ($-1\leq O_i<1$). The kinetic exchange model of opinion pictures the opinion exchange between two agents like a scattering process in an ideal gas. However, unlike ideal gas, there is no conservation of the total opinion. This is similar to the kinetic exchange models of wealth redistribution, where of course the conservation was also present [@CC-CCM]. The (discrete time) exchange equations of the model read $$O_i(t+1)=\lambda O_i(t)+\lambda \epsilon O_j(t), \label{exchange}$$ and a similar equation for $O_j(t+1)$, where $O_i(t)$ is the opinion value of the $i$-th agent at time $t$, $\lambda$ is the ‘conviction parameter’ (considered to be equal for all agents for simplicity) and $\epsilon$ is an annealed random number drawn from a uniform and continuous distribution between \[0:1\], which is the probability with which $i$ and $j$ interact (see [@Lallouache:2010]). Note that the choice of $i$ and $j$ are unrestricted, making the effective interaction range to be infinite. The opinion values allowed are bounded between the limits $-1\le O_i(t)\le +1$. So, whenever the opinion values are predicted to be greater (less) than +1 (-1) following Eq. (\[exchange\]), it is kept at +1 (-1). This bound, along with Eq. (\[exchange\]), defines the dynamics of the model. This model shows a symmetry breaking transition at a critical value of $\lambda$ ($\lambda_c\approx 2/3$). The critical behaviors were studied using the average opinion ${O_a}=|\sum\limits_{i=1}^NO_i(t\to \infty)|/N$ [@bcc]. An alternative parameter was also defined in Ref.[@Lallouache:2010], which is the fraction of agents having extreme opinion values. This quantity also showed critical behaviors at the same transition point. The model in its original form is rather difficult to tackle analytically (it can be solved in some special limits though [@sb]). However, as it is a fully connected model, a mean field approach would lead to the following evolution equation for the single parameter opinion value (cf. [@Lallouache:2010]) $$\label{map} O(t+1)=\lambda(1+\epsilon)O(t).$$ This is, in fact, a stochastic map with the bound $|O(t)|\le 1 $. For all subsequent discussions, whenever an explicit time dependence of a quantity is not mentioned, it denotes the steady state value of that quantity and a subscript $a$ denotes the average over the randomness (i.e., ensemble average). As we will see from the subsequent discussions, this map can be conceived as a random walk with a reflecting boundary. As in the case of the multiagent version, the distribution of $\epsilon$ does not play any role in the critical behavior. We have considered two distributions, one is continuous in the interval \[0:1\] and the other is 0 and 1 with equal probability. Both of these give similar critical behavior. We also briefly discuss the two-parameter model, where the ‘conviction’ of an agent and the ability to convince others were taken as two independent parameters [@Sen:2010]. In that context, the map would read $$O(t+1)=(\lambda+\mu\epsilon)O(t), \label{mapp}$$ where $\mu$ is the parameter determining an agent’s ability to influence others. As before, $|O(t)|\le 1$. Results ======= Random walk picture ------------------- One can study the stochastic map in Eq. (\[map\]) by describing it in terms of random walks. Writing $X(t)=\log({O(t)})$ (for all subsequent discussions we always take $O(t)$ to be positive), Eq. (\[map\]) can be written as $$X(t+1)=X(t)+\eta, \label{rw}$$ where, $\eta(t)=\log[\lambda(1+\epsilon)]$. As is clear from the above equation, it actually describes a random walk with a reflecting boundary at $X=0$ to take the upper cut-off of $O(t)$ into account. Depending upon the value of $\lambda$, the walk can be biased to either ways and is unbiased just at the critical point. As one can average independently over these additive terms in Eq. (\[rw\]), this gives an easy way to estimate the critical point [@Lallouache:2010]. An unbiased random walk would imply $\langle \eta\rangle=0$ i.e., $$\int\limits_0^1\log[\lambda_c(1+\epsilon)] d\epsilon=0$$ giving $\lambda_c=e/4$, where we have considered an uniform distribution of $\epsilon$ in the limit \[0:1\]. This estimate matches very well with numerical results of this and earlier works [@Lallouache:2010]. ![The average return time $T$ of ${O}(t)$ to 1 in the map described in Eq. (\[map\]) is plotted with $(\lambda-\lambda_c)$. It shows a divergence with exponent 1 as is predicted from Eq. (\[returntime\])[]{data-label="time-map-conti"}](time-map-conti.eps){width="9cm"} ![ $O_a$ is plotted with $\lambda$. The data points are results of numerical simulations, which fits rather well with the solid line predicted from Eq. (\[averageO\]), with $k=0.7$[]{data-label="order-map-conti"}](order-map-conti.eps){width="9cm"} In order to guess the $\lambda$ dependence of ${O_a}$ in the ordered region, we first estimate the “average return time” $T$ (return time is the time between two successive reflections from $X=0$) as a function of bias of the walk. For this uniform distribution of $\epsilon$, the average position to which the walker goes following a reflection from the barrier is $(\lambda+1)/2$. The average amount of contribution in each step is given by $\int\limits_o^1\log[\lambda(1+\epsilon)]d\epsilon=\log(\lambda/\lambda_c)$. This, in fact, is a measure of the bias of the walk, which vanishes linearly with $(\lambda-\lambda_c)$ as $\lambda\to \lambda_c$. So, in this map picture, one would expect that on average by multiplying this $\lambda/\lambda_c$ factor $T$ times (i.e., adding $\log(\lambda/\lambda_c)$ $T$ times in the random walk picture), $O(t)$ would reach 1 from $(\lambda+1)/2$. Therefore, $$\frac{\lambda +1}{2}\left( \frac{\lambda}{\lambda_c} \right)^T=1,$$ giving $$T=- \frac{\log\lambda}{\log\lambda-\log\lambda_c} \approx - \frac{\log\lambda}{\lambda-\lambda_c} \label{returntime}$$ for $\lambda\to \lambda_c$. Clearly, the average return time diverges near the critical point obeying a power law: $T \sim (\lambda-\lambda_c)^{-1}$. In Fig. \[time-map-conti\] we have plotted this average return time as a function of $\lambda$. The power-law divergence agrees very well with the prediction. The steady state average value of $X(t)$ i.e., $X_a$ (and correspondingly $O_a$) is expected to be proportional to $\sqrt{T}$ in steps of $\log\lambda$: $${X_a}\sim \sqrt{T}\log\lambda = k\sqrt{T}\log\lambda,$$ where $k$ is a constant. This gives $${O_a}= \exp[-{k}|\log\lambda|^{3/2} (\lambda-\lambda_c)^{-1/2}]. \label{averageO}$$ The above functional form fits quite well (see Fig. \[order-map-conti\]) with the numerical simulation results near the critical point. It may be noted that the numerical results for the kinetic opinion exchange Eq. (\[map\]) also fits quite well with this expression (Eq. (\[averageO\])). ![The average condensation fraction (probability that $O=1$) $\rho_a$ is plotted with $(\lambda-\lambda_c)$. A linear fit in the log-log scale gives the growth exponent 1, as predicted from Eq. (\[condf\]). Inset shows the variation of $\rho$ with $\lambda$.[]{data-label="one-map-conti"}](one-map-conti.eps){width="9cm"} We note that ${O_a}$ increases from zero at the critical point and eventually reaches 1 at $\lambda=1$. But its behavior close to critical point cannot be fitted with a power-law growth usually observed for order-parameters. Such peculiarity in the critical behavior of ${O_a}$ compels us to exclude it as an order parameter though it satisfies some other good qualities of an order parameter. Instead, we consider the average ‘condensation fraction’ $\rho_a$ as the order parameter. In the multi-agent version, it was defined as the fraction of agents having extreme opinion values i.e., -1 or +1. In this case it is defined as the probability that $O(t)=1$. We denote this quantity by $\rho (t)$. As is clear from the definition, one must have $$\rho_a\sim \frac{1}{T}, \label{condf}$$ where, $T$ is the return time of the walker. As $T\sim (\lambda-\lambda_c)^{-1}$, $\rho_a\sim (\lambda-\lambda_c)^{\beta} $ with $\beta=1$. This behavior is clearly seen in the numerical simulations (see Fig. \[one-map-conti\]). Also, the relaxation time shows a divergence as the critical point is approached. We argue that there is a single relaxation time scale for both $O(t)$ and $\rho (t)$. So we calculate the divergence of relaxation time for $O(t)$ and numerically show that the results agree very well with the relaxation time divergence for both $O(t)$ and $\rho (t)$. Consider the subcritical regime where the random walker is biased away from the reflector (at the origin) and would have a probability distribution for the position of the walker as $$p(X)=\frac{A}{\sqrt{t}}\exp[-B(X-vt)^2)/t],$$ where $v\sim 1/T\sim (\lambda-\lambda_c)$ is the net bias and constants $A$, $B$ do not depend on $t$. One can therefore obtain the probability distribution $P$ of $O$ using $p(X)dX=P(O)dO$, $$P(O)= \frac{A}{\sqrt{t}}\frac{1}{O}\exp[-B(\log{O}-vt)^{2}/{t}].$$ Hence $$\begin{aligned} {O_a}(t) &=& \int_{0}^{1}OP(O)dO , \nonumber \\ &=& \frac{A}{\sqrt{t}}\int_{0}^{1}\exp[-B(\log{O}-vt)^{2}/{t}]dO \nonumber \\ &\sim& \frac{A}{\sqrt{t}}\exp(-Bv^{2}{t}), \label{relaxtime}\end{aligned}$$ in the long time limit, giving a time scale of relaxation $\tau\sim v^{-2} \sim (\lambda-\lambda_c)^{-2}$. We have fitted the relaxation of ${O_a}(t)$, obtained numerically, with an exponential decay and found $\tau$. As can be observed from Fig. \[relax-map-conti\] it shows a clear divergence close to critical point with exponent 2. ![The average relaxation time for ${O_a}(t)$ is plotted with $\lambda$. This shows a prominent divergence as the critical point is approached. In the inset, the relaxation time is is plotted in the log-log scale against $(\lambda-\lambda_c)$. The exponent is 2 as is expected from Eq. (\[relaxtime\])[]{data-label="relax-map-conti"}](relax-map-conti.eps){width="9cm"} ![The time dependence of both ${O_a}(t)$ and $\rho (t)$ are plotted at the critical point in the log-log plot. The linear fit shows a time variation of the form $t^{-\delta}$ with $\delta=1$, as is expected from Eq. (\[relaxtime\]).[]{data-label="delta-map-conti"}](delta-map-conti.eps){width="9cm"} We have obtained the relaxation time of ${\rho_a}(t)$ and it also shows similar divergence. Note that at $\lambda=\lambda_c$, $v=0$ and it follows from Eq. (\[relaxtime\]) that $O_a(t)\sim t^{-1/2}$. This behavior is also confirmed numerically (see Fig. \[delta-map-conti\]). The average condensation fraction $\rho_a(t)$ too follows this scaling, giving $\delta=1/2$ (as order parameter relaxes as $t^{-\delta}$ at critical point). ![The variation of $O_a$ and $\rho_a$ are plotted against the external field $h$ at the critical point $\lambda=\lambda_C$. The linear fit in log-log scale shows $\delta^{\prime}=1$.[]{data-label="field-map-conti"}](field-map-conti.eps){width="9cm"} We have also investigated the effect of having an external field linearly coupled with $O(t)$. In the multiagent scenario, this can have the interpretation of the influence of media. The map equation now reads, $$O(t+1)=\lambda(1+\epsilon_t)O(t)+h O(t),$$ where $h$ is the field (constant in time). We have studied the response of ${O_a}$ and ${\rho_a}$ at $\lambda=\lambda_c$ due to application of small $h$. We find that (see Fig. [\[field-map-conti\]]{}) both grows linearly with $h$. One expects the order parameter to scale with external field at the critical point as ${\rho_a}\sim h^{1/\delta^{\prime}}$. In this case $\delta^{\prime}=1$. Random walk with discrete step size ----------------------------------- One can simplify the random walk mentioned above and make it a random walk with discrete step sizes. This can be done by considering the distribution of $\epsilon$ to be a double delta function, i.e., $\epsilon=1$ or $0$ with equal probability. This will make $\eta(t)$ in Eq. (\[map\]) to be $\log\lambda$ or $\log(2\lambda)$ with equal probability. Below critical point, both steps are in negative direction (away from reflector) and consequently taking the walker to $-\infty$. Exactly at critical point ($\lambda=\lambda_c$) the step sizes become equal and opposite i.e., $\log\lambda_c=-\log(2\lambda_c)$ giving $\lambda_c= 1/\sqrt2$. Above critical point, one of the steps is positive and the other is negative. However, the magnitudes of the steps are different. This unbiased walker (probability of taking positive and negative steps are equal) with different step sizes can approximately be mapped to a biased walker with equal step size in both directions. To do that consider the probability $p(x,t)$ that the walker is at position $x$ at time $t$. One can then write the master equation $$p(x,t+1)=\frac{1}{2}p(x+a,t)+\frac{1}{2}p(x+a+b,t),$$ where $a=\log \lambda$ and $b=\log 2$. Clearly, $$\frac{\partial p(x,t)}{\partial t}=\left(a+\frac{b}{2}\right)\frac{\partial p(x,t)}{\partial x}+\left(\frac{a^2}{2}+\frac{ab}{2}+\frac{b^2}{4}\right)\frac{\partial^2p(x,t)}{\partial x^2}. \label{diff1}$$ Now the master equation for the usual biased random walker can be written as $$p(x,t+1)= p^{\prime}p(x+a^{\prime},t)+q^{\prime}p(x-a^{\prime},t),$$ where $p^{\prime}$ and $q^{\prime}$ denote respectively the probabilities of taking positive and negative steps ($p^{\prime}+q^{\prime}=1$) and $a^{\prime}$ is the (equal) step size in either direction. The differential form of this equation reads $$\frac{\partial p(x,t)}{\partial t}=(p^{\prime}-q^{\prime})a^{\prime}\frac{\partial p(x,t)}{\partial x}+\frac{a^{\prime2}}{2}\frac{\partial^2p(x,t)}{\partial x^2}. \label{diff2}$$ Comparing these equations (\[diff1\]) and (\[diff2\]), one gets $$\begin{aligned} p^{\prime} &=& \frac{1}{2}\left[1+\frac{a+b/2}{a^{\prime}}\right] \nonumber \\ q^{\prime} &=& \frac{1}{2}\left[1-\frac{a+b/2}{a^{\prime}}\right] \nonumber \\ a^{\prime}&=& -\sqrt{(\log(\lambda)-\log(\lambda_c))^2+(\log(\lambda_c))^2}.\end{aligned}$$ Therefore, as $\lambda\to \lambda_c$, the bias $(p^{\prime}-q^{\prime})\sim (\lambda-\lambda_c)/a^{\prime}$. These are consistent with the earlier calculations where we have taken the bias to be proportional to $(\lambda-\lambda_c)$. To check if this mapping indeed works, we have simulated a biased random walk with above mentioned parameters and found it to agree with the original walk (see Fig. \[comp-map-conti\]). ![The comparison of the biased walk with equal step size with the original walk is shown. Reasonable agreement is seen for a wide range of $\lambda$ values.[]{data-label="comp-map-conti"}](comp-map-conti.eps){width="9cm"} Similar to the approach taken for the continuous step-size walk, one can find the return time of the walker. This time the walker is exactly located at $\lambda$ after it is reflected from the barrier. The return time again diverges as $(\lambda-\lambda_c)^{-1}$. Also ${O_a}(t)$ will have a similar form upto some prefactors. Condensation fraction will increase linearly with $(\lambda-\lambda_c)$ close to the critical point. All the other exponents regarding the relaxation time, time dependence at the critical point and dependence with external field are same as before. This shows that the critical behavior is universal with respect to changes in the distribution of $\epsilon$. Two parameter map ----------------- As the multi-agent model was generalized in a two-parameter model [@Sen:2010], one can also study the map version of that two-parameter model. It would read $${O}(t+1)=(\lambda+\mu\epsilon){O}(t). \label{map2}$$ As before, one can take $\log$ of both sides and in similar notations $$X(t+1)=X(t)+\log(\lambda+\mu\epsilon).$$ This can also be seen as a biased random walk. In fact, one can write the above equation as $$X(t+1)=X(t)+\log[\lambda(1+\epsilon^{\prime})],$$ where $\epsilon^{\prime}=(\mu/\lambda)\epsilon$. This effectively changes the limit of the distribution of the stochastic parameter. Here $\epsilon^{\prime}$ is distributed between $0$ and $\mu/\lambda$. One can then do the earlier exercise with this version as well. If one makes $\epsilon$ discrete, this is again a walk with unequal step sizes, which can again be mapped to a biased walk with equal step sizes. Therefore, in either case, some pre-factors will change, but the critical behavior will be the same as before. The critical behavior is, therefore, universal when studied in terms of the proper order parameter (condensation fraction). ![The phase diagram for the two-parameter map as predicted from Eq. (\[phdia\]) (upper line). The phase diagram for the multi-agent version ($\mu_c=2(1-\lambda_c)$) is also plotted for comparison (lower line).[]{data-label="ph-dia"}](ph-dia.eps){width="9cm"} For uniformly distributed $\epsilon$ (in the range \[0:1\]), one can get the expression for the phase boundary from the equation $$\int\limits_0^1\log(\lambda+\mu\epsilon)d\epsilon=0,$$ which gives $$\log(\lambda_c+\mu_c)+\frac{\lambda_c}{\mu_c}\log\left(\frac{\lambda_c+\mu_c}{\lambda_c}\right)=1. \label{phdia}$$ Of course, this gives back the $\lambda_c=e/4$ limit when $\lambda=\mu$. The phase boundary is plotted in Fig. \[ph-dia\]. It also agrees with numerical simulations. A map with a natural bound -------------------------- In the maps mentioned above, the upper (and lower) bounds are additionally provided with the evolution dynamics. Here we study a map where this bound occurs naturally. We intend to see its effect on the dynamics. Consider the following simple map $$O(t+1)=\tanh[\lambda(1+\epsilon)O(t)]. \label{tanh}$$ Due to the property of the function, the bounds in the values of $O(t)$ are specified within this equation itself. This map shows a spontaneous symmetry breaking transition as before. This function appears in mean field treatment of Ising model, but of course without the stochastic parameter. ![The divergence of relaxation time is shown for the map described by Eq. (\[tanh\]) for both sides of the critical point $\lambda_c=e/4$. The inset shows that the exponent is 2 as is argued in the text.[]{data-label="tanh-map"}](tanh-map.eps){width="9cm"} Numerical simulations show that the $O_a$ behaves as $(\lambda-\lambda_c)^{1/2}$. An analytical estimate of $\lambda_c$ can be made by linearizing the map for small values of $O(t)$, which is valid only at critical point. Of course, after linearization the map is the same as the initial single parameter map, giving $\lambda_c=e/4$. The relaxation of $O_a(t)$ at $\lambda=\lambda_c$ behaves as $t^{-1/2}$ as before. These are also seen from numerical simulations. However, apart from the critical point, the map is strictly non-linear. Hence the results for its linearized version do not hold except for critical point. Also $\rho=0$ here always. To check the divergence of the relaxation time at the critical point, it is seen that it follows $\tau\sim (\lambda-\lambda_c)^{-2}$. Of course, in the deterministic version (mean-field Ising model), the exponent is $1$. But as this map has stochasticity, the time exponent is doubled (see Fig. \[tanh-map\]) (i.e., the relaxation time is squared as happens for random motion as opposed to ballistic motion). Summary and conclusions ======================= In this paper we have studied the simplified map version of a recently proposed opinion dynamics model. The single parameter map was proposed in Ref.[@Lallouache:2010] and the critical point was estimated. Here we study the critical behavior in details as well as propose the two-parameter map motivated from [@Sen:2010]. The phase diagram is calculated exactly. The maps can be cast in a random walk picture with reflecting boundary. Then using the standard random walk statistics, some steady state as well as dynamical behaviors are calculated and these are compared with the corresponding numerical results. It is observed that the usual parameter of the system, i.e. the average value of the opinion ($O_a$) in the steady state, does not follow any power-law scaling (see Eq. (\[averageO\])). In fact, it is the condensation fraction, or in this case the probability ($\rho_a$) that the opinion values touch the limiting value, turns out to be the proper order parameter, showing a power-law scaling behavior near critical point. The dynamical behaviors of the two quantities ($O_a(t)$ and $\rho_a(t)$) are similar. We have calculated the power-law relaxation of these quantities at the critical point, which compares well with simulations. Also, the divergence of the relaxation time on both sides of criticality shows similar behavior. We have also studied the effect of an “external field” (representing media or similar external effects) in these models. At critical point, both the quantities grow linearly with the applied field. The average fluctuation in both of these quantities show a maximum near the critical point. These theoretical behavior fits well with the numerical results. 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--- abstract: | In many Dark Matter (DM) scenarios, the annihilation of DM particles can produce gamma rays with a continuum spectrum that extends up to very high energies of the order of the electroweak symmetry breaking scale (hundreds of GeV).\ Astrophysical structures supposed to be dynamically dominated by DM, such as dwarf Spheroidal Galaxies, Galaxy Clusters (the largest ones in the local Universe being mostly observable from the northern hemisphere) and Intermediate Mass Black Holes, can be considered as interesting targets to look for DM annihilation with Imaging Atmospheric Cherenkov Telescopes (IACTs). Instead, the center of our Galaxy seems to be strongly contaminated with astrophysical sources.\ The 17m Major Atmospheric Gamma-ray Imaging Cherenkov (MAGIC-I) Telescope, situated in the Canary island of La Palma (2200 m a.s.l.), is best suited for DM searches, due to its unique combination of high sensitivity and low energy threshold among current IACTs which can potentially allow to provide clues on the high energy end, and possibly peak, of the gamma-ray DM-induced spectrum constrained at lower energies with the Fermi Space Telescope.\ The recent results achieved by MAGIC-I for some of the best candidates, as well as the DM detection prospects for the MAGIC Phase II, are reported. author: - '\' title: | Search for Dark Matter signatures with MAGIC-I\ and prospects for MAGIC Phase-II --- MAGIC, Dark Matter, Dwarf Galaxies Introduction ============ Nowadays there are compelling experimental evidences for a large non-baryonic component of the matter density of the Universe at all observed astrophysical scales, such as galaxies, galaxy clusters and cosmic background radiation [@bertone]. The so-called Dark Matter (DM) makes its presence known through gravitational effects and it could be made of so far undetected relic particles from the Big Bang. In the Cold Dark Matter cosmological scenario ($\Lambda$CDM) about $80\%$ of the matter of our Universe is believed to be constituted by cold, neutral, non-baryonic, weakly-interacting massive particles (WIMPs) [@komatsu]. Although plenty of experimental and theoretical efforts have taken place so far and despite recent exciting and controversial results which can be interpreted as possible DM detection [@DAMA] [@PAMELA] [@ATIC], the nature of DM has not yet been clarified.\ Among the huge plethora of cold DM candidates proposed in literature, the best motivated ones are related to the Super Symmetrical (SUSY) and Unified Extra Dimensional (UED) extensions of the Standard Model of particle physics (see [@bertone] and references therein). In the widely studied Minimal Supersymmetric extension of the Standard Model (MSSM) the lightest neutralino ($\chi~\equiv~\chi_{1}^{0}$), a linear combination of the neutral superpartners of the $W^{3}$, $B^{0}$ and the neutral Higgs bosons ($H_{1}^{0}$, $H_{2}^{0}$), is the most studied candidate. If the neutralino is the lightest SUSY particle (LSP) and R-parity is conserved then it must be stable and it can represent an excellent cold DM candidate with a relic density compatible with the WMAP bounds and a mass at the GeV-TeV scale.\ The most relevant neutralino interaction for the purposes of indirect DM searches is the self annihilation in fermion-antifermion pairs, gauge bosons pairs and final states containing Higgs bosons. The subsequent hadronization results in a gamma-ray power-law spectrum with a sharp cutoff at the neutralino mass (expected to be between 50 GeV and several TeV). A direct annihilation in gamma rays (such as $\chi\chi\rightarrow~Z^{0}\gamma$ or $\gamma\gamma$) provides line emissions but those processes are loop-suppressed. WMAP relic density measurements provide an upper limit to the total neutralino cross section of the order of $\langle \sigma v \rangle\sim10^{-26}$ cm$^{3}$ s$^{-1}$, which implies that the neutralino is an extremely low interacting particle.\ Recently it has been pointed out that the Internal Bremsstrahlung (IB) process may boost the gamma-ray yield of the neutralino self-annihilation at the higher energies by up to four orders of magnitude, even for neutralino masses considerably below the TeV scale [@IB]. This discovery represents a very important issue for the indirect DM search, particularly for the IACTs which are sensitive to the energy range most affected by the gamma-ray flux enhancement due to the IB process. Moreover, the IB introduces features in the gamma-ray spectrum that potentially allow an easier discrimination between a DM source and the standard astrophysical sources located in the vicinity, whose spectrum is usually a featureless power law.\ The DM is believed to be structured as smooth halos with several clumps down to very small scales (the size of the Earth or less, depending on the models). Since the expected gamma ray flux from DM annihilation is proportional to the square of the DM density, any DM density enhancement, due to the presence of substructures (expected to be present in any DM halo [@kuhlen]) and possibly to adiabatic compression of the DM in the innermost regions of the halos [@prada], can provide boost factors up to two orders of magnitude. Expected gamma-ray flux from DM self-annihilation ================================================= The gamma-ray flux from DM particle self annihilations can be factorized into a contribution called the astrophysical factor $J(\Psi)$ and a contribution called the particle physics factor $\Phi^{PP}$ $$\Phi(E>E_{0}) = J(\Psi) \cdot \Phi^{PP}(E>E_{0}) , \label{eqn:gammaflux}$$ where $E_{0}$ is the energy threshold of the detector and $\Psi$ is the angle under which the observation is performed.\ The astrophysical factor can be written as $$J(\Psi_{0}) = \frac{1}{4\pi} \int_{V} d \Omega \int_{l.o.s.} d \lambda[ \rho^{2} \ast B_{\theta_{r}} (\theta)] , \label{eqn:APfactor}$$ where $\Psi_{0}$ denotes the direction of the target. The first integral is performed over the spatial extension of the source, the second one over the line-of-sight variable $\lambda$. The DM density $\rho$ is convoluted with a Gaussian function $B_{\theta_{r}}(\theta)$ in order to consider the telescope angular resolution ($\sim0.1^{\circ}$), where $\theta=\Psi-\Psi_{0}$ is the angular distance with respect to the center of the object.\ The particle physics factor can be expressed as a product of two terms. The first one depends only on the DM candidate mass and cross section, whereas the second term depends on the annihilation gamma-ray spectrum and must be integrated above the energy threshold $E_{0}$ of the telescope $$\Phi^{PP}(>E_{0})= \frac{\langle \sigma v_{\chi \chi} \rangle}{2 m_{\chi}^{2}} \int_{E_{0}}^{m_{\chi}} S(E) dE , \label{eqn:PPfactor}$$ where $\langle \sigma v_{\chi \chi} \rangle$ is the total averaged cross section times the relative velocity of the particles, $m_{\chi}$ is the DM particle mass, and $S(E)$ is the resulting gamma-ray annihilation spectrum.\ The indirect search for DM is nowadays affected by large uncertainties in the flux prediction which put serious hindrances to the estimation of the observability: on the one hand, the astrophysical factor uncertainties can raise up to two orders of magnitude, on the other hand, the allowed parameter space for the mass and the annihilation cross section of the DM particle spans many orders of magnitude giving rise to flux estimations which can differ up to six orders of magnitude (or even more). Interesting astrophysical objects for indirect DM searches ========================================================== Since the gamma-ray flux is proportional to the square of the DM density (see eq. \[eqn:APfactor\]), a relevant question concerning the indirect search for DM annihilation products is where to look for *hot DM spots* in the sky.\ In the past, the Galactic Center (GC) was considered the best option. However, this is a very crowded region, which makes it difficult to discriminate between a possible gamma-ray signal due to DM annihilation and that from other astrophysical sources. WHIPPLE, CANGAROO and especially H.E.S.S. and MAGIC-I [@MAGICGC] have already carried out detailed observations of the GC and all of them reported a point-like emission spatially close to Sgr A$^{\ast}$ location. Only very massive neutralino of the order of 10-20 TeV could explain the results, for which the gamma-ray yield is expected to be 2-3 orders of magnitude lower than the measured flux [@HESSGCDM].\ Very promising targets with high DM density in relative proximity to the Earth (less than 100 kpc) are the dwarf Spheroidal (dSph) satellite galaxies of the Milky Way. These galaxies are believed to be the smallest (size $\sim1$ kpc), faintest (luminosities 10$^2$–10$^8$ L$_{\odot}$) astronomical objects whose dynamics are dominated by DM, with a DM halo of the order of 10$^5$–10$^9$ M$_{\odot}$, very high mass–to–light ratios (up to $\sim10^3$ M$_\odot/$L$_\odot$) [@gilmore] and no expected astrophysical gamma-ray sources located in the vicinity.\ Clusters of galaxies are the largest and most massive gravitationally bound systems in the universe, with radii of the order of the Mpc and total masses around 10$^{14}$–10$^{15}$ M$_{\odot}$. These systems are thought to host enormous amounts of DM, which should gravitationally cluster at their center and present numerous local substructures which could lead to a significant boost in the flux.\ Another interesting DM target scenario is represented by the so-called intermediate mass black holes (IMBHs). The model described in ref. [@bertoneIMBH] shows that studying the evolution of super massive black holes, a number of IMBHs do not suffer major merging and interaction with baryons along the evolution of the Universe. DM accretes on IMBH in a way that the final radial profile is spiky so that the IMBHs could be bright gamma-ray emitters. These targets could be related to the unidentified Fermi sources [@FERMI]. The IACT technique and the MAGIC Telescopes =========================================== When the primary VHE gamma rays reach the top of the Earth atmosphere, they produce positron-electron pairs which then emit energetic gamma rays via Bremsstrahlung. The secondary gamma rays in turn emit positron-electron pairs giving rise to the so-called electromagnetic cascade where highly relativistic particles cause a flash ($\sim3$ ns) of UV-blue Cherenkov light which propagates in a cone with an opening angle of $\sim1^{\circ}$. The resulting circle of projected light, at 2000 m a.s.l., has a radius of about 130 m. The light is collected by a reflective surface and focused onto a multipixel camera which records the shape of the image produced by the shower which has an elliptical shape pointing to the center of the camera. Since Cherenkov light is emitted also by charged particles produced in atmospheric showers induced by charged isotropic cosmic rays, an image reconstruction algorithm [@hillas] is used in order to recover the energy and the direction of the primary particle and to determine whether it was more likely a hadron or a photon, allowing the rejection of more than $99\%$ of the background.\ Among all the IACTs, the MAGIC-I Telescope, located on the canary island of La Palma ($28.8^{\circ}$N, $17.9^{\circ}$W, 2200 m a.s.l.), is the largest single dish facility in operation (see [@baixeras] for detailed descriptions). The 17m diameter tessellated reflector of the telescope consists of 964 $0.5\times$$0.5$ m$^{2}$ diamond-milled aluminium mirrors, mounted on a light weight frame of carbon fiber reinforced plastic tubes. The MAGIC-I camera has a field-of-view of $3.5^{\circ}$ and consists of 576 enhanced quantum efficiency photomultipliers (PMTs). The analog signals recorded by the PMTs are transported via optical fibers to the trigger electronics and are read out by a 2GSamples/s FADC system. The collection area reaches a maximum value of the order of $10^{5}$ m$^{2}$ and the trigger energy threshold is about 60 GeV for gamma rays at zenith angles (ZA) below $30^{\circ}$.\ A second 17m diameter telescope (MAGIC-II) is currently in opening operation phase. The stereoscopic observation of the sky will bring a significant improvement of the shower reconstruction (especially for the incoming direction) and of the background rejection and consequently a better angular ($\sim20\%$) and energy ($\sim40\%$) resolutions, a lower energy threshold ($\sim30\%$) and a $\sim$2–3 times higher sensitivity (see [@MAGIC2] and [@M2simulations] and references therein for more details).\ MAGIC-I observations for DM searches and prospects for MAGIC Phase II ===================================================================== Besides the GC [@MAGICGC], MAGIC-I has observed two of the most promising DM targets: the dSph Draco [@DRACO] and the ultra faint dSph Willman 1 [@WILLMAN1]. Both objects, together with other very interesting sources, as Segue 1 [@SEGUE1][^1] and the large clusters of galaxies Perseus and Coma, are well observable from the MAGIC site at low ZA which assure the lowest reachable energy threshold. Draco observation ----------------- Draco is a dSph galaxy accompanying the Milky Way at a galactocentric distance of about 82 kpc. From a kinematical analysis of a sample of 200 stellar line–of–sight velocities it was possible to infer the DM profile: the result of the fit, assuming a Navarro-Frenk-White (NFW) smooth profile [@nfw], indicates a virial mass of the order of 10$^9$ M$_{\odot}$, with a corresponding mass–to–light ratio of M/L$\sim200$ M$_\odot/$L$_\odot$ [@DRACOprofile]. With 7.8 hours of observation performed in 2007 a 2$\sigma$ upper flux limit on steady emission of $1.1 \times 10^{-11}$ photons cm$^{-2}$ s$^{-1}$ was found, under the assumption of a generic annihilation spectrum without cutoff and a spectral index of -1.5 for photon energies above 140 GeV. For different mSUGRA model parameters using the benchmark points defined by Battaglia et al. [@DRACObenchmarks] and for other models, the gamma-ray spectrum expected from neutralino annihilations was computed. Assuming these underlying spectra and a smooth DM density profile as suggested in [@DRACOprofile2], the upper limits on the integrated flux above 140 GeV were calculated and compared to the experimental ones. As one can see from fig. \[fig:DRACOresults\], the resulting upper limits are at least three orders of magnitude higher than expected by mSUGRA without enhancements. ![ Draco observation: thermally averaged neutralino annihilation cross section as a function of the neutralino mass for the chosen mSUGRA models [@DRACObenchmarks] after renormalization to the relic density. The red boxes indicate the experimental flux upper limits, displayed in units of $\langle \sigma v \rangle$, assuming a smooth Draco halo as suggested in [@DRACOprofile2]. []{data-label="fig:DRACOresults"}](icrc0629_fig01){width="3in"} ------- ----------- --------- ------------- ------- ------------- ---------- ------------------------------------- -------------------------------- BM $m_{1/2}$ $m_0$ $\tan\beta$ $A_0$ $sign(\mu)$ $m_\chi$ $\langle\sigma v_{\chi\chi}\rangle$ $\Phi^{PP}(>100)$ \[GeV\] \[GeV\] \[GeV\] \[cm$^3$/s\] \[cm$^3$ GeV$^{-2}$ s$^{-1}$\] $I'$ 350 181 35 0 $+$ 141 $3.62\times10^{-27}$ $7.55\times10^{-34}$ $J'$ 750 299 35 0 $+$ 316 $3.19\times10^{-28}$ $1.23\times10^{-34}$ $K'$ 1300 1001 46 0 $-$ 565 $2.59\times10^{-26}$ $6.33\times10^{-33}$ $F^*$ 7792 22100 24.1 17.7 $+$ 1926 $2.57\times10^{-27}$ $5.98\times10^{-34}$ ------- ----------- --------- ------------- ------- ------------- ---------- ------------------------------------- -------------------------------- BM $\Phi^{model}$ $\Phi^{u.l.}$ $B^{u.l.}$ ------- ---------------------- ---------------------- ------------------- $I'$ 2.64$\times10^{-16}$ $9.87\times10^{-12}$ $3.7\times10^{4}$ $J'$ 4.29$\times10^{-17}$ $5.69\times10^{-12}$ $1.3\times10^{5}$ $K'$ 2.32$\times10^{-15}$ $6.83\times10^{-12}$ $2.9\times10^{3}$ $F^*$ 2.09$\times10^{-16}$ $7.13\times10^{-12}$ $3.4\times10^{4}$ : Willman 1 observation: comparison of estimated integral flux above 100 GeV for the chosen benchmark models and the upper limit in the integral flux $\Phi^{u.l.}$ above 100 GeV coming from MAGIC-I data in units of photons cm$^{-2}$ s$^{-1}$. On the rightmost column, the corresponding upper limit on the boost factor $B^{u.l.}$ required to match the two fluxes is calculated. []{data-label="tab:fluxresults"} Willman 1 observation --------------------- Willman 1 dSph galaxy is located at a distance of 38 kpc in the Ursa Major constellation. It represents one of the least massive satellite galaxies known to date (M$\sim5 \times10^5$ M$_{\odot}$) with a very high mass–to–light ratio, M/L$\sim500-700$ M$_\odot/$L$_\odot$, making it one of the most DM dominated objects in the Universe. Following ref. [@WILLMAN1profile], its DM halo was parametrized with a NFW profile.\ The observation of Willman 1 took place in 2008 for a total amount of 15.5 hours. No significant gamma-ray emission was found above 100 GeV, corresponding to 2$\sigma$ upper flux limits on steady emission of the order of $10^{-12}$ photons cm$^{-2}$ s$^{-1}$, taking into account a subset of four slightly modified Battaglia mSUGRA benchmark models, as defined by Bringmann et al. [@WILLMAN1benchmarks] (see table \[tab:benchmarks\]). These benchmark models represent each a different interesting region of the mSUGRA parameter space, namely the *bulk* ($I^{\prime}$), the *coannihilation* ($J^{\prime}$), the *funnel* ($K^{\prime}$) and the *focus* ($F^{\ast}$) point regions, and they include for the first time the contribution of IB process in the computation of the cross sections and spectra.\ A comparison with the measured flux upper limit and the fluxes predicted assuming the underlying mSUGRA benchmark spectra and the chosen Willman 1 density profile was computed. The results are summarized in table \[tab:fluxresults\]. Although the boost factor upper limits seem to show that a DM detection could still be far (the most promising scenario, $K^{\prime}$, being three orders of magnitude below the sensitivity of the telescope), it is important to keep in mind the large uncertainties in the DM profile and particle physics modeling that may play a crucial role in detectability. In particular the possible presence of substructures in the dwarf, which is theoretically well motivated, may increase the astrophysical factor and therefore the flux of more than one order of magnitude. Furthermore, since the parameter space was not fully scanned, it is likely that there are models of neutralino with higher $\Phi^{PP}$. Prospects for MAGIC Phase II ---------------------------- The use of a more advanced detector like MAGIC Telescopes, already in opening operation phase, with a much increased combination of energy threshold, energy resolution and flux sensitivity [@M2simulations], could favour possible scenarios of DM detection or at least the exclusion of parts of the mSUGRA parameter space. Nonetheless, while all other current IACTs can only cover SUSY models with a large IB contribution due to their higher energy threshold, MAGIC Telescopes will explore a much larger region of DM annihilation models where the peak of the emission is at lower energy.\ Indeed, it has been shown by Bringmann et al. [@WILLMAN1benchmarks] that in case of the observation of Draco and Willman 1, MAGIC Telescopes performances, and in particular those of CTA (Cherenkov Telescope Array, a new generation IACT currently in the design phase [@CTAsimulations]), are very close to allow constraining the mSUGRA parameter space, with the lowest predicted boost upper limits of the order of $\sim10$. These results are of course strengthened once all the already mentioned uncertainties for the gamma-ray DM annihilation flux are taken into account. [99]{} G. Bertone et al., *Phys. Rept.*, 405:279-390, 2005. E. Komatsu et al., *Astrophys. J. Suppl.*, 180:330-376, 2009. R. Bernabei et al., *Eur. Phys. J.*, C56:333-355, 2008. O. Adriani et al., *Nature*, 458:607-609, 2009. J. Chang et al., *Nature*, 456:362-365, 2008. T. Bringmann et al., *JHEP*, 01:049, 2008. M. Kuhlen et al., arXiv:astro-ph/0805.4416. F. Prada et al., *Phys. Rev. Lett.*, 93:241301, 2004. J. Albert et al., *Astrophys. J.*, 638:L101-L104, 2006. F. Aharonian et al., *Phys. Rev. Lett.*, 97:221102, 2006. G. Gilmore et al., *Astrophys. J.*, 663:948-959, 2007. G. Bertone, *Phys. Rev.*, D72:103517, 2005. A.A. Abdo et al., arXiv:astro-ph/0902.1340. A.M. Hillas, *Proc. of the 19th ICRC*, p445, 1985. C. Baixeras et al., *Nucl. Instrum. Meth.*, A518:188192, 2004. J. Cortina et al., “Technical Performance of the MAGIC Telescopes”, *these proceedings*. P. Colin et al., “Performance of the MAGIC Telescopes in stereoscopic mode”, *these proceedings*. J. Albert et al., *Astrophys. J.*, 679:428-431, 2008. J. Albert et al., arXiv:astro-ph/0810.3561. M. Geha et al., arXiv:astro-ph/0809.2781. J.F. Navarro et al., *Astrophys. J.*, 490:493-508, 1997. M.G. Walker et al., arXiv:astro-ph/0708.0010. M. Battaglia et al., *Eur. Phys. J.*, C33:273-296, 2004. M.A. Sanchez-Conde et al., *Phys. Rev.*, D76:123509, 2007. L.E. Strigari et al., *Astrophys. J.*, 678:614-620, 2008. T. Bringmann et al., *JCAP*, 01, 016, 2009. *http://www.mpi-hd.mpg.de/hfm/CTA/*. [^1]: The ultra faint dSph Segue 1 has been observed by MAGIC-I during the beginning of 2009. The data analysis is ongoing.

--- abstract: 'We present 11.7-micron observations of nine late-type dwarfs obtained at the Keck I 10-meter telescope in December 2002 and April 2003. Our targets were selected for their youth or apparent IRAS 12-micron excess. For all nine sources, excess infrared emission is not detected. We find that stellar wind drag can dominate the circumstellar grain removal and plausibly explain the dearth of M Dwarf systems older than 10 Myr with currently detected infrared excesses. We predict M dwarfs possess fractional infrared excess on the order of $L_{IR}/L_{*}\sim 10^{-6}$ and this may be detectable with future efforts.' author: - 'Peter Plavchan, M. Jura, & S. J. Lipscy$^{\dagger}$' title: 'Where Are The M Dwarf Disks Older Than 10 Million Years?' --- INTRODUCTION ============ Until $\sim$20 years ago, our knowledge of planets and their formation had been limited to our own solar system, with planets around other stars being the subject of fiction and hypothesis. With recent advances in instrumentation and detection capabilities, we are now confirming the existence of planetary systems around other stars. In 1983, the InfraRed Astronomical Satellite (hereafter IRAS) offered the first evidence of dusty debris, or debris disks, orbiting other stars [@zuckerman01]. The dust is heated by the incident stellar radiation and re-emits this radiation in the infrared, which we can then detect as an “excess” of infrared flux. The first extra-solar planets were indirectly detected around a millisecond pulsar in 1993 by analysis of the pulsar timing [@wolszczan92]. Finally, the first jovian planets around stars like our sun were discovered in 1995 through the characteristic “wobble” or radial velocity variations induced in the star [@mayor95]. Studying the evolution of these systems and their characteristics – both directly and indirectly – allow us to begin to answer fundamental questions about the existence of life in the universe. M Dwarfs are the lowest mass, size, luminosity and temperature main sequence stars which comprise $\sim\!\!70\%$ of the total number of stars in our galaxy [@mathioudakis93]. If planetary systems exist around M Dwarfs, they could represent the most abundant planetary systems in the galaxy. Thus, determining or constraining the abundance of exo-planetary systems around M Dwarfs is important. About $15\%$ of nearby main-sequence stars exhibit far-infrared excess characteristic of cold circumstellar dust, or debris disks analogous to our Kuiper belt [@habing01]. However, most M-type stars do not currently have detected infrared excesses despite several targeted and blind surveys [@song02; @weinberger04; @zuckerman01; @liu04; @fajardo00]. Reported 12$\mu$m, 20$\mu$m and 25$\mu$m IRAS and ISO (Infrared Space Observatory) excess emission candidates have been largely demonstrated as false-positives when followed-up by smaller aperture ground-based observations [@zuckerman01; @song02; @fajardo00; @aumann91]. We have performed a search for 12$\mu$m infrared excess around candidate M Dwarfs indicative of extrasolar zodiacal dust. The presence of such warm debris disks, analogous to the zodiacal dust in our own inner solar system, could be an indirect marker for the presence of parent bodies – rocky planetesimals, asteroids and comets that can generate dust. We present selection criteria for our targets in Section 2, and we present our observations in Section 3. In Section 4, we present an analysis of the observations using stellar atmosphere models to estimate photospheric contributions to the observed infrared flux. We present a model for circumstellar grain removal for M dwarfs in Section 5 to explain our results and the dearth of M dwarfs older than 10 Myr with observed infrared excess. In section 6, we discuss the implications of this model and we present our conclusions in Section 7. SAMPLE SELECTION ================ We constructed our sample from two selection criteria – youth and apparent 12$\mu$m excess. Our target list is given in Table 1. Youth ----- @spangler01 have found that for solar-type stars, the infrared excess varies as $t_{*}^{-1.76}$ where $t_{*}$ is the age of the star. They present a simple model for asteroidal destruction to explain this result. Extrapolating from the infrared excess produced by our own zodiacal debris, if we can identify M type stars which are $\sim$0.1 the age of the Sun ($t_{*}\sim500$ Myr) or younger, any existing infrared excess might be detectable with ground-based facilities. Similarly, Liu et al. (2004) estimated that the dust mass, ${M_{d}}$, varies as $t_{*}^{-0.5}$ to $t_{*}^{-1}$ from debris disk masses estimated from sub-mm data, with a simple unweighted fit giving ${M_{d}} \propto t^{-0.7 \pm 0.2}$. Finally, from the lunar impact record one can infer that the amount of debris and resulting infrared excess can be significantly higher for young solar analogs [@chyba91]. See @decin03 for a re-examination of the time-dependence of the IR excess amplitude, which leads to a different conclusion than @spangler01. In order to identify young, nearby M-type stars, we have used the results for the ${\beta}$ Pic moving group from @zuckerman01b and the “rapid” rotators from the survey by @gizis02 to identify three targets. The most obvious candidate in the ${\beta}$ Pic moving group is GJ 3305 [@song02] since AU Mic (GJ 803) is also member of this moving group and an M-type dwarf with an imaged debris disk [@kalas04; @liu04b]. The estimated age of GJ 3305 is 12 Myr, or 3 ${\times}$ 10$^{-3}$ $t_{\odot}$. From @gizis02, GJ 3304 and GJ 3136 are “rapid rotator” stars with $v\sin i$ = 30 km s$^{-1}$, so these stars are likely younger than the Hyades and therefore about 0.1 the age of the Sun [@terndrup00]. Apparent 12$\mu$m Excess ------------------------ With the release of the 2MASS All Sky Survey, we cross-correlated this catalog with the IRAS Point Source and Faint Source Catalogues, similar and analogous to the work of @fajardo00 . We used the volume complete 676-source M Dwarf sample of @gizis02 and the 278-source dwarf K and M-type sample identified in @mathioudakis93. There is moderate overlap between these two samples. We selected targets with unusually red K-\[12$\mu$m\] colors. We eliminated sources exhibiting a false color excess attributable to IRAS beam confusion, such as binaries or relatively bright nearby companions within 1’. Additionally, with the exception of HU Del (GJ 791.2), known binary sources were discarded. Our most promising six targets were observed. With the exception of GJ 585.1, a K5 dwarf, the remaining five targets are M-type. OBSERVATIONS ============ We present 11.7$\mu$m observations of nine late-type dwarfs obtained at the Keck I 10-meter telescope in December 2002 and April 2003. Table 1 lists the target stars, their distance, 11.7 ${\mu}$m observed photometry, 12 ${\mu}$m IRAS photometry, and derived 3-$\sigma$ upper-limits on the infrared excess from synthetic spectral fitting. Table 2 lists the observations and integration times. Figures 1 and 2 shows spectral fits, observations and available literature photometry. These three targets selected for their youth – GJ 3136, GJ 3304 and GJ 3305 – were observed at Keck at 11.7$\mu$m using the Long Wavelength Spectrograph (LWS) in imaging mode in December 2002 [@jones93]. On-source integrations times for all three targets were two minutes. Standard IDL routines were used to reduce the data, and we wrote IDL routines to flux-calibrate the data. For the particular observations in December 2002, it was discovered later by Varoujan Gorjian [@gorjian03 private comm.] and confirmed by Keck staff that a screw that adjusts a mirror in the LWS detector had come loose and fallen out. This allowed some movement of the optics as a function of telescope orientation during the time of our observations. However, we observed standards within 1’ of our targets for GJ 3304 and GJ 3305, and all three targets and standards were observed at airmasses $\leq\!1.2$. Nevertheless, this potentially adversely affected the calibrations used in our observations, especially for GJ 3136, introducing a systematic error. Thus, we treated the variance in the calibration of our standards as systematic errors rather than random, and adjusted our resulting uncertainties in our observations accordingly. The net result was to roughly double our final calibration uncertainties. The six targets selected for their apparent IRAS 12$\mu$m excesses were observed at Keck using LWS at 11.7$\mu$m in April 2003, using the same reduction techniques described above. On source integration times ranged from 2 to 5 minutes. In the cases of G 203-047 and HU Del, target acquisition and identification was performed with NIRC (Keck Near Infra-Red Camera) K-band imaging [@matthews94]. ANALYSIS ======== Photospheric flux was estimated by fitting PHOENIX NextGen stellar atmosphere models [@hauschildt99] to available UBVRIJHK$_{s}$ photometry, taking into account effective bandpasses and photometric uncertainties in finding the minimum $\chi^{2}$ in temperature and normalization. The JHK$_{s}$ photometry were obtained from 2MASS All-Sky Survey and converted to flux densities based on the calibration by @cohen03, outlined in the 2MASS Explanatory Supplement [@cutri03 ch VI.4a]. UBV and R-I photometry, when available, were obtained from the Gliese catalogue. R magnitudes, when available, were obtained from the USNO-A V2.0 catalogue queried through Vizier [@ochsenbein00; @gliese79; @gliese91; @monet98]. Correcting USNO-R to Landolt-R magnitudes did not affect our model SED fits. The IRAS 12${\mu}$m flux densities were color-corrected assuming blackbody emission, following the procedures outlined in the IRAS Explanatory Supplement [@beichman88 ch.VI.C.3]. The adopted color correction factors were chosen depending on the effective temperature $T_{eff}$ of each star, with a typical value of $\sim\!\!1.41$. PHOENIX Nextgen models were used to fit the observed photometry rather than a blackbody, since the stellar SEDs are very different from that of a blackbody [@song02; @mullan89]. We assumed $log\:g = 4.5$ and solar metallicity, as is typical for main-sequence stars in the solar neighborhood [@dantona97; @drilling00]. Six of our nine targets are previously identified as flare stars, which primarily affect the U and B magnitudes during flaring. However, the exclusion of U and B band magnitudes in our SED fits did not alter the derived temperatures for our targets. Uncertainties in the predicted model flux at 11.7$\mu$m are derived from the resulting variance of the spectral fit as a function of the temperature ($\pm 100K$) and effective normalization ($\pm 5\%$) uncertainties around the minimum reduced $\chi^{2}$ fit. We find that the spectral models and photometric data are self consistent with one another, and consistent with no detected infrared excess at $11.7{\mu}m$ for all nine targets. Our derived effective temperatures are in agreement with published spectral types when available. We obtain an overall reduced $\chi^{2}$ of the model fits to the data of 0.95, and on average the 3-$\sigma$ upper limits correspond to a ratio of $F_{d,\nu}/F_{*,\nu} \sim30\%$ at 11.7$\mu$m, excluding GJ 4068. This ratio limit is typical for current ground-based mid-infrared observational capabilities; we are limited in our analysis by our photometric quality rather than model-fitting uncertainties. Comments on the discrepancies between our ground-based observations and IRAS flux measurements. ----------------------------------------------------------------------------------------------- IRAS did not detect GJ 3304, GJ 3305 and GJ 3136 in the Point Source and Faint Source Catalogues, but as one can see in Figure 1 our measurements at 11.7$\mu$m are consistent with the predicted photospheric emission. For GJ 4068, GJ 507.1, HU Del and GJ 585.1, we note that the color-corrected IRAS 12$\mu$m measurements are inconsistent with our 11.7$\mu$m measurements with a significance of $\sim$5.5-, 3-, 2-, and 2.5-$\sigma$ respectively. The IRAS 12$\mu$m measurements are also inconsistent with the predicted photospheric emission at the $\sim$9-, 2.5-, 2-, and 3-$\sigma$ respectively. It is unlikely that these discrepancies are due to the presence of Si emission in the IRAS 12$\mu$m bandpass, which would be missed by the LWS 11.7$\mu$m bandpass [@metchev04; @gaidos04]. For HU Del and GJ 4068 we suspect that this inconsistency is due to beam confusion from nearby sources, located $\sim$1.0 and $\sim$2.1 arc-minutes away, respectively. For GJ 507.1 and GJ 585.1, we suggest that the inconsistencies are probably due to statistical fluctuations and the “Malmquist-bias” for stars near the detection threshold [@song02]. For G 203-047 and GJ 729, we note that the color-corrected IRAS 12$\mu$m measurements and our 11.7$\mu$m measurements are both consistent with one another and the predicted model photospheric emission. Although G 203-047 and GJ 729 possessed unusually red K-\[12$\mu$m\] colors in our initial sample, this was not indicative of a true excess with subsequent modeling of their cool photospheres. For GJ 4068, the flux error is dominated by systematic error in our calibration rather than statistical uncertainties in the detection itself. Thus, we report a detection rather than an upper-limit. For HU Del, we note that it is a spectroscopic binary, which may account for the model spectral fit appearing to be slightly inconsistent with the literature photometry. Finally, for GJ 585.1, the only K dwarf in our sample, our 11.7$\mu$m measurement is above the predicted model photosphere at the 1.5$\sigma$ level, and may warrant further observations at longer wavelengths. However, given that this is only a 1.5$\sigma$ excess, we do not believe it is likely to be real. MODELING OF CIRCUMSTELLAR PROCESSES =================================== In section 5.1, we review the current observational knowledge of M dwarf circumstellar environments. In section 5.2, we contrast observations with a steady-state application of P-R grain removal for M Dwarfs. In section 5.3 we present a model for stellar wind drag grain removal for M dwarfs. In section 5.4, we discuss the current knowledge of M dwarf stellar winds. In section 5.5, we then evaluate the relevance of stellar wind drag in M dwarf debris disks and we present a model for their evolution. In section 5.6, we discuss other dust removal mechanisms. The Observational Dearth of M Dwarf Debris Disks ------------------------------------------------ Circumstellar disks are common around M Dwarfs younger than $\sim$5 Myr; the inner parts of these disks appear to dissipate within 5-10 Myr, and any remaining disks are relatively rare in systems older than 10 Myr. AU Mic (GJ 803) and GJ 182 are the only two M-type dwarfs older than 10 Myrs with an infrared or sub-mm excess, confirmed with ground-based observations, that is indicative of a debris disk [@song02; @liu04; @zuckerman01; @fajardo00]. This lack of late-type dwarfs older than 10 Myrs with an infrared excess does not appear to be a selection effect due to IRAS detection limits but rather an age dependent phenomenon [@song02; @song03 private comm.]. @weinberger04 propose that the lack of warm infrared excesses associated with the inner disks of K and M Dwarfs in the TW Hya association implies rapid planet formation within 5-10 Myrs, which in turn depletes available parent bodies for dust replenishment. For AU Mic, @liu04 argue that the lack of warm (T $\sim$ 200K) dust in the inner region of the circumstellar disk suggests the presence of an unseen planetary companion, also indicative of rapid planet formation within $\sim\!\!10$ Myr. In support of this rapid planet formation theory, it is known that known that primordial material to potentially form planets is common around low-mass stars and brown dwarfs with ages less than a few million years [@liu03; @klein03; @pascucci03; @lada00; @beckwith90; @osterloh95]. Hen 3-600 (TWA 3, a multiple star system) and TWA 7 are M-type pre-main sequence members of the $\sim$5-10 Myr old TW Hydrae Association [@zuckerman01]. Coku Tau 4, AA Tau and CD $-40^{\circ}8434$ are further examples of T Tauri type pre-main sequence M Dwarfs with observed infrared excesses [@metchev04; @quillen04; @hartmann95]. However, recent simulations of terrestrial planet formation around solar type stars take on average 10-50 Myrs to accrete into the analogs of our solar system terrestrial planets. This process includes large-scale impacts – like the one that is theorized to have formed Earth’s moon – lasting on the time-scale of $\sim$100 Myrs [@agnor99] and thus providing parent bodies to generate extrasolar zodiacal clouds on these timescales. The arguments of @weinberger04 [@song02; @liu04] lead us to conclude that there potentially exists a different debris disk evolution timescale for K and M Dwarfs, and herein we propose such a mechanism. Grain Removal from Poynting-Robertson Drag ------------------------------------------ The dust we observe in a debris disk is replenished by the grinding (destruction) of parent bodies [@zuckerman01], and removed under the action of Poynting-Robertson and other forces. Thus, the infrared excess can be related to the destruction rate of the parent bodies. In a simple model, we derive from @chen01 and @jura04 a relation between L$_{IR}$ and the rate at which mass is being removed from parent bodies and converted into dust, ${\dot {M_{d}}}$: $${\dot M_{d}}\,c^{2}\;=\;C_{0}\,L_{IR}$$ where $C_{0}$ is a numerical constant of order unity, $L_{IR} \approx (\nu L_{\nu})_{IR}$ is the luminosity of the dust infrared excess and $c$ is the speed of light. @chen01 used $C_{0}$ = 4, where $C_{0}$ depends upon the assumed initial dust distribution relative to the inner radius at which the dust sublimates. We have used the steady-state assumption that ${\dot {M_{d}}} = M_{dust}/t_{removal}$ and Equations (4) and (5) in @chen01 to arrive at Equation (1) above. Note that in the derivation of @chen01, the dominant grain removal process is assumed to be Poynting-Robertson (P-R) drag ($t_{removal} = t_{PR}$). For M-type dwarfs, the characteristic grain removal time-scales from Poynting-Roberston drag are significantly longer than for earlier type dwarfs due to the relatively lower luminosities in late-type dwarfs. Thus, contrary the observational dearth of M-type debris disks, we might expect a large abundance of dust around these stars. Grain Removal from Corpuscular Stellar Wind Drag ------------------------------------------------ We propose that the corpuscular (proton) stellar wind drag, hereafter stellar wind drag, in late-type dwarfs serves as the dominant mechanism for grain removal in these stars, in contrast to the P-R drag cited for earlier-type debris disk evolution. An analogous and more detailed derivation applied to red giants can be found in @jura04. The stellar wind drag force is caused primarily by the proton flux from the solar wind impacting dust grains and the resulting anisotropic recoil; the effect is analogous to the Poynting-Robertson drag [@burns79; @gustafson94; @holmes03]. In our own solar system, the solar wind drag has been measured to be on the order of 30% of the Poynting-Robertson drag force, varying in strength between 20% and 43% [@gustafson94]. @holmes03 approximates a value of 1/3. To first order in $v_{orb}/c$ and $v_{orb}/v_{sw}$ – assuming each of these quantities is $\ll 1$ as in @gustafson94, where $v_{orb}$ is the grain orbital velocity of a grain assumed to be in a circular orbit, and where $v_{sw}$ is the proton stellar wind velocity assumed to be entirely radial – we can write the ratio of drag times $t_{pr}/t_{sw}$ in terms of the magnitude of force ratios: $$\frac{t_{pr}}{t_{sw}} = \left|\frac{\vec{F_{sw}}\cdot\hat{\theta}}{\vec{F_{PR}}\cdot\hat{\theta}}\right| = {\frac{c}{v_{sw}}}\times \left|\frac{\vec{F_{sw}}\cdot\hat{r}}{\vec{F_{rad}}\cdot\hat{r}}\right|$$ where $F_{sw}$, $F_{rad}$, and $F_{PR}$ correspond to the solar wind pressure, radiation pressure and Poynting-Robertson drag forces respectively, and $\hat{\theta}$ and $\hat{r}$ correspond to the azimuthal and radial unit vectors in spherical coordinates. We note that while the radiative stellar wind pressure can be dominated by the radiation pressure, the drag force ratio is enhanced by a factor of c/v$_{sw}$ in favor of the corpuscular drag force. We then write: $$\frac{t_{pr}}{t_{sw}} = \frac{c}{v_{sw}}\times \frac{Q_{sw}}{Q_{pr}}\left(\frac{\dot M_{sw}v_{sw}}{L_{*}/c}\right) = \frac{Q_{sw}}{Q_{pr}}\frac{\dot M_{sw} c^{2}}{L_{*}}$$ where properties such as grain size, density and distance from the star drop out from the above expression, and where $Q_{sw}/Q_{pr}$ is the ratio of coupling coefficients. For our own Solar System, taking $\dot M_{sw} \sim 2\times10^{-14}$ M$_{\odot}$ yr$^{-1}$ and assuming $Q_{sw}/Q_{pr} = 1$, Equation (3) evaluates to 37%. Equation (3) presents a first-order expression to evaluate the relative importance of the P-R and stelar wind drags; more detailed expressions for the drag forces can be found in [@gustafson94]. We deduce from Equation (3) that stellar wind drag can dominate P-R drag due to the lower M-dwarf luminosities relative to the Sun. We write a more general expression for the inferred parent body destruction rate in cgs units, $\dot M_{d}$, by combining Equation (1) and Equation (3): $$\dot M_{d} = \left(\frac{C_{0}\,L_{IR}}{c^{2}}\right) \times \left(1+\frac{\dot M_{sw}c^{2}}{L_{*}}\right)$$ where we have assumed for simplification and clarity above that $Q_{sw}/Q_{pr} = 1$ and we note again that L$_{IR}$ represents the luminosity of the infrared excess due to dust. We can write the above equation in terms of the observed fractional infrared excess as: $$\frac{L_{IR}}{L_{*}} = \frac{\dot M_{d} c^{2}}{ C_{0} (L_{*} + \dot M_{sw} c^{2})}$$ Equations (4) and (5) limit to the following expressions when stellar wind drag is the dominant grain removal mechanism: $$\dot M_{d} = \frac{C_{0} \dot M_{sw} L_{IR}}{L_{*}}$$ $$\frac{L_{IR}}{L_{*}} = \frac{\dot M_{d} }{ C_{0} \dot M_{sw}}$$ Scaling the P-R drag timescale (see Equation (5) in @chen01) by $t_{sw}/t_{pr}$ from the right hand side of Equation (3), we derive the grain removal time-scale for stellar-wind drag: $$t_{sw} = \frac{4\pi a \rho_{dust} D^{2}}{3 Q_{sw} \dot M_{sw}}$$ where D is the distance of the grain from the star. Late-Type Dwarf Stellar Winds ----------------------------- Empirical and semi-empirical arguments have shown that late-type dwarf wind mass loss rates can exceed the solar wind mass loss rate of $\sim\!2\times10^{-14}$ M$_{\odot}$ yr$^{-1}$ by factors ranging from $\sim$10 [@mullan92; @fleming95; @wargelin01; @wargelin02; @wood01; @wood02] to a proposed $\sim$10$^{4}$ in the case of the M dwarf flare-star YZ Cmi [@mullan92]. This enhanced stellar wind mass-loss is expected due to the hotter coronae and increased magnetic activity in active late-type dwarfs. Recent models suggest that the mass-loss rates from late-type dwarfs are a few times $10^{-12}$ M$_{\odot}$ yr$^{-1}$ [@wargelin01; @lim96; @vanden97]; this predicted mass loss rate is still a factor of $\sim$100 times the solar value. The task of remotely observing such winds (directly or indirectly) around other solar-type dwarfs has been a substantial challenge. The measurements of wind rates were made indirectly from “astrospheric” absorption high-resolution Hubble Space Telescope spectra of Ly$\alpha$ absorption lines – neutral interstellar hydrogen heated by the presence of a wind and the resulting interaction with the local interstellar medium. From observations of G and K dwarfs, @wood02 derived the relationship that $\dot m_{sw} \propto F_{X}^{1.15\pm0.20}$ from correlating observed stellar wind mass-loss rates with X-ray activity, where F$_{X}$ is the X-ray flux at the surface of the star and $\dot m_{sw}$ is the mass-loss rate per unit surface area. This relation implies a mass loss rate, $\dot M_{sw}$, $\sim$1000 times that of the solar wind for the particular case of AU Mic [@wood02; @huensch99]. With substantial uncertainty, @wood02 derive $\dot M_{sw} \approx \dot M_{\odot} (t/t_{\odot})^{-2.00\pm0.52}$ from stellar rotational velocity evolution and correlation with F$_{X}$. @wood02 caution that this relation may not accurately extrapolate for stars younger than $\sim$300 Myr, but the mass-loss will generally decrease with time from some maximum value. Solar wind mass-loss rates as high as 10$^{3}$ times its present value have been hypothesized to account for the required luminosity and corresponding initial solar mass [@sackmann03; @wood02], but such a large enhancement in the mass-loss rate may not extend to young M-type dwarfs [@lim96; @vanden97]. ### Proxima Cen No one has detected a wind from the nearest M Dwarf, Proxima Cen, but ascribing an appropriate upper-limit is a matter of debate. @wood01 placed an upper limit on $\dot M_{sw}$ of 0.2 times the solar wind mass-loss rate, contrary to the arguments of @wargelin01 [@lim96; @vanden97]. However, the assumptions of @wood01 about the intrinsic stellar L$\alpha$ line profile are cited as controversial in @wargelin02. From X-ray observations of charge exchange between the wind of Proxima Cen and neutral ISM gas, @wargelin02 instead place a 3-$\sigma$ upper-limit constraint of 14 times the solar wind mass-loss rate, $\dot M_{sw}$. More accurate estimates and measurements of M dwarf wind mass-loss rates will resolve these differences and uncertainties. The Relevance of Stellar Wind Drag for M Dwarf Debris Disk Evolution -------------------------------------------------------------------- It seems plausible that M Dwarf wind mass-loss rates can exceed the solar value by at least a factor of 10. Consequently, scaling from Equation (3), we find that stellar wind drag dominates P-R drag in M dwarfs. We can now estimate $L_{IR}/L_{*}$ when stellar wind drag is the dominant grain removal mechanism. From Equation (7), using $\dot M_{sw} = \dot M_{\odot} (t/t_{\odot})^{-2.00\pm0.52}$ from @wood02, estimating $\dot M_{d} = 4\times 10^{6} \left(t_{*}/t_{\odot}\right)^{-1.76\pm0.2}$ g/s, and setting $C_{0}=4$, we calculate: $$\frac{L_{IR}}{L_{*}}\sim 9\times10^{-7} \; \left(\frac{t_{*}}{t_{\odot}}\right)^{0.24\pm0.6}$$ Equation (9) predicts that the frequency of M dwarf debris disks older than $\sim$10 Myr is roughly independent of age. Furthermore, the predicted infrared excess ratio is below typical observational limits [@metchev04; @spangler01; @liu04]. For our estimate of $\dot M_{d}$ used above, we inferred the time-dependence from @spangler01 [@liu04], and we adopted a proportionality constant of $C_{0}L_{IR}/c^{2}=C_{0}L_{*}f_{d}/c^{2} = 4\times 10^6$g/s from Equation (1). For this proportionality constant for $\dot M_{d}$, we again set $C_{0}=4$, $L_{*}=L_{\odot}$ and we inferred $f_{d} = 2.5 \times10^{-7}$ at $t=t_{\odot}$ from Figure 2 in @spangler01. This estimate for $\dot M_{d}$ is consistent with the dust replenishment rate for the zodiacal cloud, $\dot M_{d} \sim\!3\times 10^{6}$ g/s [@fixsen02]. Other Dust Removal Mechanisms ----------------------------- Stellar wind drag can dominate P-R drag in the evolution of M dwarf debris disks when $t_{pr}/t_{sw}>1$, which occurs when $\dot M_{sw}/ \dot M_{\odot}$ 3$\times L_{*}/L_{\odot}$. We can now put this result in context with the role of collisions and the role of grain blowout, both through radiation pressure and stellar wind pressure. Dust grain blowout due to radiation pressure is irrelevant for M dwarfs, due to the lower luminosity and longer peak wavelength of radiation ($L_{*}\sim10^{-1\;.. -3}L_{\odot}$, $\lambda_{peak}>1\mu m$). We present a simplistic spherical grain model to explain this result. Assuming the absorption coupling coefficient $Q_{a}=1$ and the average grain density $\rho_{dust} = 2.5$ g cm$^{-3}$, we calculate from Equation (2) in @chen01 a radiative blowout grain size radius, a$_{blowout} \sim 0.1\mu$m for a M0 dwarf, with $L_{*} = 0.1 L_{\odot}$, $M_{*} = 0.5 M_{\odot}$. Because $2\pi a_{blowout} / \lambda_{peak} < 1$ for all M Dwarfs, our assumption that $Q_{a} = 1$ is invalid and Equation (2) in @chen01 is not applicable. When $2\pi a / \lambda < 1$, we write from Equation 7.7 in @spitzer: $$Q_{a}(a,\lambda)= -8\pi \; \mbox{Im}\left(\frac{n^2-1}{n^2+2}\right)\frac{a}{\lambda}$$ where n is the complex index of refraction that is material-dependent. Equation (10) evaluates to $Q_{a}(a,\lambda)\sim a/\lambda$ for various silicates at wavelengths of $\sim$1$\mu$m [@dorschner95; @ossenkopf92]. We integrate $Q_{a}(a,c/\nu)L_{\nu}$ over $\nu$ for a stellar blackbody spectrum and derive $Q_{a}L_{*}\sim L_{*}\; a/\lambda_{peak}$. The ratio of the radiation pressure to the gravitational attraction is then independent of grain size: $$\beta \equiv \frac{F_{rad}}{F_{grav}} \approx \frac{3L_{*}}{16\pi c G M_{*} \rho_{dust} \lambda_{peak}}$$ where G is Newton’s gravitational constant, and c is the speed of light. Taking $\rho_{dust} = 2.5$ g cm$^{-3}$,$L_{*}=0.1L_{\odot}$, $M_{*}=0.5M_{\odot}$, and $\lambda_{peak} = 1 \mu m$, we calculate $\beta \sim 0.05$. We conclude that $\beta<<1$ for all M dwarfs – radiation pressure is insufficient to overcome gravitational attraction in expelling orbiting grains of any size. The grain morphology for these small grains can be important relative to the blackbody approximation, and a more detailed calculation of the effective cross section to evaluate the relevance of radiative blowout has been done by @saija03. The authors similarly conclude that the radiative pressure for small amorphous grains orbiting a 2700K M dwarf is negligible. @krist05 suggest that stellar wind pressure may play a role in the dissipation of dust around AU Mic. We compute the stellar wind blowout radius to be: $$a_{blowout} = \frac{3Q_{sw} \dot M_{sw} v_{sw}}{16\pi G M_{*} \rho_{dust}}$$ where $v_{sw}$ is the stellar wind velocity, and any grains smaller than $a_{blowout}$ will be expelled under the action of stellar wind pressure. This can be derived directly by taking the ratio of the stellar wind and gravitational forces and setting them equal to 1, or by multiplying Equation (2) in @chen01 by $|\vec{F_{sw}}\cdot\hat{r} / \vec{F_{rad}}\cdot\hat{r}|$ (see Equations (2) and (3)). For a 0.5M$_{\odot}$ M0 dwarf with a solar mass-loss rate and solar wind velocity of $\sim$400km/s [@feldman77], assuming $Q_{sw}=1$ and $\rho_{dust}=2.5$g cm$^{-3}$, the corresponding blowout radius is insignificant at $2\times$10$^{-4}\mu m$. However, for a 0.5M$_{\odot}$ M0 dwarf with a mass-loss rate of $\sim$10$^{3} \dot M_{\odot}$, the corresponding blowout radius would be 0.2$\mu$m, increasing to 0.9$\mu$m for a 0.1M$_{\odot}$ late-M dwarf with a mass-loss rate of $\sim$10$^{3} \dot M_{\odot}$. Thus, the relevance of stellar wind blowout increases for later spectral types, but is only relevant for extremely young or active M Dwarfs with potentially unrealistic mass-loss rates. The validity of our assumption that $Q_{sw}=1$ for these grain sizes is also uncertain. @dominik03 suggest that all the known debris disk systems are collision-dominated and not P-R drag dominated. This relies on the assumption that dust grains can be ground down by collisions until they are blown out by radiation pressure on a time-scale short compared to a drag time-scale. This assumption does not hold for M Dwarf debris disks, since dust grains are not blown out by radiation pressure. While M dwarf debris disks will be collisionally processed on short time-scales, stellar wind drag or grain growth is likely to still be the dominant grain removal process. This modification predicts an excess of small grains for M dwarf debris disks relative to those around earlier-type stars and hence a relatively blue disk color. DISCUSSION ========== Some Implications of Stellar Wind Drag for M Dwarfs --------------------------------------------------- The time-dependence and steady-state assumptions of Equation (9) may be oversimplified, and the normalization is similarly uncertain by at least a factor of two. Nonetheless, stellar wind drag offers an explanation for the lack of M-type debris disks older than 10 Myrs identified in @song02 [@fajardo99; @mamajek04]. For the late-type dwarfs with confirmed infrared excesses in @metchev04, the estimated disk masses derived from the @chen01 model should be adjusted higher by the ratio of $t_{pr}/t_{sw}$. For the late-type dwarfs in the TW Hydrae association, the removal of grains by the stellar wind drag offers an alternative explanation to rapid planet formation put forth by @weinberger04. The circumstellar disks and forming planetesimals may still exist, but the infrared excess emission can be effectively “surpressed” by the stellar wind drag relative to the effects of the Poynting-Robertson drag. When gas is still present in the disk, the dynamics are more complicated and our model does not apply. AU Mic: An Application of the Model ----------------------------------- AU Mic is the only M-dwarf with an imaged circumstellar disk and infrared excess older than 10 Myr. We propose that stellar wind drag offers an alternative explanation for the lack of warm dust grains observed near AU Mic. The observed substructure and turnover in the slope of the radial surface brightness profile are naturally explained by the presence of a planetesimal disk in the inner $\sim$35 AU @metchev05 [@krist05; @liu04b]. This inner disk could be produced by the stochastic destruction of parent-bodies, the subsequent generation of dust, and the removal of the dust by stellar wind drag within a time short compared to the $\sim$12 Myr age of AU Mic. If circumstellar gas is present in the AU Mic disk, the physical interpretation becomes more complicated and our model does not apply. We can estimate grain removal time-scales from Equation (8). We assume a grain size of a = 0.5$\mu$m [@metchev05], a grain density of $\rho_{s} = 2.5$ g cm$^{-3}$, and $Q_{sw} = Q_{pr} \sim 1$. Since these assumptions dictate the grain opacity, which is unknown to within an order of magnitude, our estimates are similarly uncertain. For AU Mic, we could take $\dot M_{sw}\sim$10$^{3} \dot M_{\odot}$ from the X-ray luminosity [@wood02; @huensch99]. However, this is a factor of 5-10 higher than the maximum $\dot M_{sw}$ inferred from modeling M dwarf winds [@wargelin01; @lim96; @vanden97]. Thus, our estimate for $\dot M_{sw}$ is also uncertain by an order of magnitude, and we instead adopt $\dot M_{sw} = 10^2 \dot M_{\odot}$ for this calculation. If $\dot M_{sw} = 10^{2} \dot M_{\odot}$, then stellar wind drag removes grains smaller than 0.5$\mu$m in the inner 35AU of the AU Mic disk within $\sim$$4\times10^4$yr. Similarly, grains smaller than 0.5$\mu$m will be removed out to 250AU in $\sim$2Myr. We deduce that at least the inner portion of the AU Mic debris disk must have been replenished by collisions of parent bodies. Neglecting stellar wind drag for comparison and assuming $L_{*}=0.13L_{\odot}$ [@metchev05], P-R drag would remove grains smaller than 0.5$\mu$m out to $\sim$35 AU in $\sim$8 Myr. If $\dot M_{sw} = 10^{3} \dot M_{\odot}$, we calculate a corresponding stellar wind blowout radius of 0.2$\mu$m from Equation (12), where we assume $Q_{sw}=1$, $\rho_{dust} = 2.5$ g cm$^{-3}$, $M_{*}=0.5M_{\odot}$, and $v_{sw}=v_{sw,\odot}=400$km/s. This is consistent with the minimum grain radius of 0.3$\mu$m derived observationally by @metchev05 at a distance of 50-60AU. This implies there will be a relatively large number of grains less than $1\mu$m in size in the AU Mic disk, since such grains are too large to be blown out by stellar wind pressure and radiation pressure is negligible. This is consistent with the observed overall blue color of the AU Mic disk relative to the colors of disks around earlier-type stars [@metchev05; @krist05]. Equation (9) predicts that $L_{IR}/L_{*}\sim 2 \times 10^{-7}$ for AU Mic, although we caution against applying Equation (9) to stars as young as AU Mic. @liu04 measure $L_{IR}/L_{*}\sim 6 \times 10^{-4}$, a factor of $3\times10^{3}$ higher, from infrared and sub-mm observations. This implies that the AU Mic disk is not in a steady-state; leftover primordial material or a recent major planetesimal collision contributing to the observed dust excess could be considered “transient” effects. CONCLUSIONS =========== We do not observe any excess 11.7$\mu$m emission for nine late-type dwarfs, selected for their youth or apparent IRAS 12$\mu$m excess. We find that stellar wind drag can dominate the Poynting-Robertson effect in grain removal from late-type dwarf debris disks, and this offers an explanation for the dearth of known M Dwarf systems older than 10 Myrs with infrared excesses. We predict that M Dwarf debris disks older than 10 Myr will have a roughly equal age distribution, with $L_{IR}/L_{*}\sim10^{-6}$. Future efforts, such as the Spitzer Space Telescope, may be successful in detecting these systems. Acknowledgements ---------------- The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. Thanks to Michael Liu, Ben Zuckerman, Inseok Song, Paul Kalas, Alycia Weinberger, Christine Chen, Stanimir Metchev and Thayne Currie for their insightful conversations and comments. This work is supported by NASA. 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Extrasolar Zodiacal Dust\ \ ----------- ------------------- ------------------------ ---------------------------------- --------------------------------------- ---------------------------------- ----------------------------------- *Name* *$T_{eff}^{(1)}$* *D$^{(2)}\!\!$* *$F_{11.7\mu m}\!\!$ obs.$\!\!$* *$F_{IRAS\:12\mu m}\!\!^{(3)}$$\!\!$* *$F_{11.7\mu m\!\!}$ mod.$\!\!$* *$F_{d,\nu}/F_{*,\nu}^{(4)}\!\!$* (K) (pc) (mJy) (mJy) (mJy) GJ 3136 3200 14.3 $\pm$ 2.9 43 $\pm$ 8 – 48 $\pm$ 2 $<$ 0.56 GJ 3305 3700 29.8 $\pm$ 0.3$^{(5)}$ 106 $\pm$ 19 – 103 $\pm$ 2 $<$ 0.54 GJ 3304 3100 10.0 $\pm$ 1.7 50 $\pm$ 9 – 61 $\pm$ 2 $<$ 0.54 GJ 507.1 3600 19.8 $\pm$ 4.3 98 $\pm$ 3 166 $\pm$ 23 109 $\pm$ 3 $<$ 0.09 GJ 585.1 4200$^{(6)}$ 25.6 $\pm$ 1.1 62 $\pm$ 5 106 $\pm$ 18 53 $\pm$ 2 $<$ 0.24 G 203-047 3200 7.3 $\pm$ 0.5 122 $\pm$ 5 135 $\pm$ 15 125 $\pm$ 4 $<$ 0.12 GJ 729 3100 2.97 $\pm$ 0.02 373 $\pm$ 8 392 $\pm$ 74 382 $\pm$ 14 $<$ 0.06 GJ 4068 3100 18.2 $\pm$ 4.3 22 $\pm$ 12$^{(7)}$ 102 $\pm$ 8 30 $\pm$ 1 $<$ 1.64 HU Del 3000 8.9 $\pm$ 0.2 55 $\pm$ 3 89 $\pm$ 16 59 $\pm$ 3 $<$ 0.16 ----------- ------------------- ------------------------ ---------------------------------- --------------------------------------- ---------------------------------- ----------------------------------- \ $^{1}$ From reduced $\chi^{2}$ fit of PHOENIX NextGen model spectra to nearest 100 K.\ $^{2}$ Inferred from parallax obtained through SIMBAD and Vizier unless otherwise noted.\ $^{3}$ Color-corrected as described in section 2.3, taken from the IRAS Faint Source Catalogue and Point Source Catalogue where available.\ $^{4}$ 3-$\sigma$ upper limits inferred from 11.7$\mu$m observations as a function of frequency, $\nu$.\ $^{5}$ @zuckerman01b; Inferred from SIMBAD parallax for primary companion, 51 Eri.\ $^{6}$ Model spectra fitted to $\pm 200 K$. The best fit temperature is consistent with this source’s previously identified spectral type, K5.\ $^{7}$ The error quoted for this particular flux measurement is dominated by systematic error in our calibration rather than statistical uncertainties in the detection itself. Thus we report a detection rather than an upper-limit. Table 2. Journal of 11.7$\mu$m Observations\ \ ----------- ------------------- --------------- -- ** *Date Observed* *Integration* *Name* *(HST, night of)* *Time (sec)* GJ 3136 12/20/2002 120 GJ 3305 12/20/2002 120 GJ 3304 12/20/2002 120 GJ 507.1 4/23/2003 240 GJ 585.1 4/23/2003 300 G 203-047 4/23/2003 300 GJ 729 4/23/2003 144 GJ 4068 4/23/2003 120 HU Del 4/23/2003 120 ----------- ------------------- --------------- -- \ \ \ \ \

--- abstract: 'The characterisation of biomarkers and endophenotypic measures has been a central goal of research in psychiatry over the last years. While most of this research has focused on the identification of biomarkers and endophenotypes, using various experimental approaches, it has been recognised that their instantiations, through computational models, have a great potential to help us understand and interpret these experimental results. However, the enormous increase in available neurophysiological and neurocognitive as well as computational data also poses new challenges. How can a researcher stay on top of the experimental literature? How can computational modelling data be efficiently compared to experimental data? How can computational modelling most effectively inform experimentalists? Recently, a general scientific framework for the generation of executable tests that automatically compare model results to experimental observations, SciUnit, has been proposed. Here we exploit this framework for research in psychiatry to address the challenges mentioned above. We extend the SciUnit framework by adding an experimental database, which contains a comprehensive collection of relevant experimental observations, and a prediction database, which contains a collection of predictions generated by computational models. Together with appropriately designed SciUnit tests and methods to mine and visualise the databases, model data and test results, this extended framework has the potential to greatly facilitate the use of computational models in psychiatry. As an initial example we present ASSRUnit, a module for auditory steady-state response deficits in psychiatric disorders.' author: - | Christoph Metzner, Tuomo Mäki-Marttunen,\ Bartosz Zurowski and Volker Steuber bibliography: - 'mybib.bib' title: 'Modules for Automated Validation and Comparison of Models of Neurophysiological and Neurocognitive Biomarkers of Psychiatric Disorders: ASSRUnit - A Case Study' --- Introduction ============ Psychiatric nosology, for centuries widely untouched by findings from (clinical) neuroscience is at the beginning of a transformation process [@Friston2017] towards an interactive evolution of diagnostic and biological categories. This change of focus stems from the hope that biomarkers and endophenotypic measures show a better correspondence with genetic alterations identified by large genome-wide association studies [@Meyer2006] and promises to more readily shed light on the mechanisms underlying these disorders and to facilitate the discovery of novel therapeutic interventions [@Siekmeier2015]. Naturally, a lot of effort has been put into the translation of these measures into practice using human studies [@Perlis2011] as well as animal models [@Markou2009]. Computational approaches also have gained significantly more attention over the last years and this has led to the emergence of ’Computational Psychiatry’ as a novel multidisciplinary and integrative discipline (see for example [@Montague2012; @Wang2014; @Friston2014; @Corlett2014; @Stephan2014; @Adams2016]). This emergence can be attributed to three main factors: First, the above mentioned increase in experimental studies has provided a wealth of neuroscientific (including neurochemical, molecular, anatomic, and neurophysiological) data which are essential to build computational models. Second, methodological and infrastructural advances, such as the various atlases, databases and online tools from the Allen Brain Institute (*http://brain-map.org/*) or the BRAIN initiative (*https://www.braininitiative.nih.gov/*), have made it possible to analyze and process this enormous amount of data. Third, the increase in computing power of high performance computers as well as standard personal computers has made it possible (and affordable) to build and use models of increasingly high computational complexity. Therefore, the rapid growth of the field of computational psychiatry comes as no surprise. However, in order to fully exploit the potential that computational modeling offers, we have to identify systemic weaknesses in current approaches and take a look at other disciplines that use computational models (and have used them for much longer than psychiatry) and even look at disciplines, like software development, which face similar challenges. At the core of computational modeling lies the concept of validation, i.e. the rigorous comparison of model predictions against experimental findings. Furthermore, for a model to be useful and provide a true contribution to knowledge, the validation has to use sound criteria and the experimental observations need to sufficiently characterize the phenomenon the model tries to reproduce. Hence, in order to develop a computational model scientists need to have an in-depth understanding of the current, relevant experimental data, the current state of computational modeling in the given area and the state-of-the-art of statistical testing, to choose the appropriate criteria with which the model predictions and experimental observations will be compared [@Gerkin2013; @Sarma2016]. In a field where both the number of experimental and computational studies grows rapidly, as is the case for psychiatry, this becomes more and more impracticable. Furthermore, the increase in modeling and experimental studies has made it harder for reviewers not only to judge whether a new model adequately replicates the full range of experimental observations but also how it compares to competing models. Again, also reviewers need an in-depth knowledge of the modeling and experimental literature as well as profound statistical knowledge. Finally, since computational modeling tries to generate predictions which can be experimentally tested, experimental neuroscientists must be able to extract and assess predictions from a rapidly growing body of computational models, a task which is also becoming more and more impracticable. The problems described above are not unique to the field of computational psychiatry but occur in all scientific areas that use computational models. Furthermore, building a computational model is in the end a software development project of sort. Omar et al. [@Omar2014] have therefore proposed a framework for automated validation of scientific models, SciUnit, which is based on unit testing, a technique commonly used in software development. SciUnit addresses the problems mentioned above by making the scope (i.e. the set of observable quantities that it can generate predictions about) of the model explicit and by allowing its validity (i.e. the extent to which its predictions agree with available experimental observations of those quantities) to be automatically tested [@Omar2014]. In this paper, we propose to adopt this framework for the computational psychiatry community and to collaboratively build common repositories of computational models, tests, test suites and tools. As a case in point, we have implemented a Python module (*ASSRUnit*) for auditory steady-state response (ASSR) deficits in schizophrenic patients, which are based on observations from several experimental studies ([@Krishnan2009; @Vierling2008; @Kwon1999])and we demonstrate how existing computational models ([@Metzner2016; @Beeman2013; @Vierling2008; @Metzner2017]) can be validated against these observations and compared with each other. The SciUnit Framework ===================== The module we present here is based on the general SciUnit framework for the validation of scientific models against experimental observations [@Omar2014] (see Figure \[Fig:SciUnit-Scheme\]). ![Schematic representation of the SciUnit framework. Models can be tested against experimental observations using specific tests. These tests incorporate an experimental observation and interface with the model through capabilities. Tests can be grouped into so-called test suites. The execution of a test produces a score, which describes how well the model captures the experimental observations. SciUnit also provides methods to visualize the resulting score(s), for example in a table.[]{data-label="Fig:SciUnit-Scheme"}](SciUnit-Overview-Schematic){width="\textwidth"} In SciUnit models declare and implement so-called capabilities, which the validation tests then use to interact with those models. By a capability of the model, we mean the ability of the model to describe certain biological phenomena that are possible to assess using physical quantities. Furthermore, the declaration and implementation of capabilities are separated, which allows to test two different models that share the *same capabilities* on the *same experimental observations* using the *same test*. Tests then take the model, use its capabilities to generate data and compare these data to the experimental observations which are linked to the test and create a score. This score, which can simply be a Boolean (pass/fail) or another more complex score type, describes if and to which extent the model data and the experimental observation(s) match. Before we describe the actual implementations of capabilities, models, tests and scores in our framework for ASSRs in schizophrenia, we first start with a summary of the experimental observations we included in the database and then we describe the computational models which were realized. The *ASSRUnit* Module ===================== The structure of the ASSRUnit module proposed here is shown schematically in Figure \[Fig:Scheme\]. As outlined earlier, there are three main functionalities the proposed module aims to provide: 1) To provide a simple way of getting an overview of the experimental literature, 2) To provide an easy and flexible way to automatically test computational models against experimental observations, 3) To provide an automated way of generating predictions from computational models. Functionality 1 is fully covered by the experimental database and its methods to query the database and visualize the results. Functionality 2 is provided by linking both the experimental database as well as the computational models to the SciUnit tests that cover the relevant experimental obervations. The only action required from the user is, if the computational model has not yet been included into the model repository of the module, to provide an interfacing Python class for the model which implements all the required capabilities. Note that the model itself does not have to be written in Python, it only has to be executable from shell. Once the model is included, the SciUnit framework allows for automated testing and the visualization methods provided in the proposed module allow for a comprehensive and clear presentation of the results. Functionality 3 can be achieved by a set of SciUnit tests and capabilites that, instead of covering experimental observations, cover experiments that have not yet been performed. By running the computational models with these tests, the module can be used to generate new predictions from the models, which can then be used to populate a prediction database similar to the experimental database. The module is available on GitHub: *https://github.com/ChristophMetzner/ASSRUnit* ![Schematic representation of the proposed framework highlighting the three main functions: 1) Overview of experimental observations. 2) Validation of computational models. 3) Creation of a predictions database. At its core lies the SciUnit module, which provides the infrastructure for the automated validation of the computational models. In particular, through a set of suitable tests, the computational models can be compared against experimental observations queried from the experimental database. Another set of tests, the so-called prediction tests, are then employed to extract predictions from the computational models, thus populating the predictions database. []{data-label="Fig:Scheme"}](SciUnit-Paper-Schematic-2){width="\textwidth"} Experimental Observations Database ---------------------------------- In patients suffering from schizophrenia oscillatory deficits in general and ASSR deficits in particular have been extensively studied using electroencephalography (EEG) and magnetoencephalography (MEG) (e.g. [@Kwon1999; @Vierling2008; @Krishnan2009; @Light2006; @Zhang2016; @Hamm2015; @Brenner2003; @Spencer2009b; @Spencer2008; @Spencer2012; @OConnell2015; @Mulert2011]). Here, we focus on three of these studies looking at entrainment deficits in the gamma and beta range. Kwon et al. [@Kwon1999] used a click train paradigm to study ASSRs at $20$, $30$, and $40$Hz in schizophrenic patients using EEG and found a prominent reduction of power at the driving frequency for $40$Hz drive, an increase in power at the driving frequency during $20$Hz drive and no changes for $30$Hz drive. Furthermore, they found small changes of power at certain harmonic/subharmonic frequencies, namely, an increase of power at $20$Hz for $40$Hz drive and a decrease of power at $40$Hz for $20$Hz drive. Vierling-Claassen et al. [@Vierling2008] reproduced these findings using the same paradigm with MEG. Krishnan et al. [@Krishnan2009] used a slightly different paradigm, which employed amplitude-modulated tones instead of click trains, and tested a wide range of driving frequencies from $5$ to $50$Hz. They found reduction of power at the driving frequency in the gamma range (i.e. at $40$, $45$ and $50$Hz) and no changes at other frequencies. Furthermore, they did not find any changes of power at harmonic or subharmonic frequencies. The experimental database is realized as a nested Python dictionary, with an entry for each study included. Each study entry consists of two entries, which describe the study observations, one in a quantitative way and the other in a qualitative way. We have included the qualitative description because often either computational models do not allow for a strict quantitative comparison with experimental data or publications of experimental studies do not provide enough detail on the results, and in these cases, only a qualitative comparison is possible. \[Tab:Experiments\] [lccccccc]{}& & & Harmonic & & Subharmonic\ Drive& $40$Hz & $30$Hz & $20$Hz& & $20$Hz & &$40$Hz\ Kwon/Vierling & $\downarrow$& -& $\uparrow$& &$\downarrow$ & & $\uparrow$\ Krishnan & $\downarrow$& -& -& &- & &-\ Together with the database, *ASSRUnit* provides basic methods to query and visualize the content of the database. These methods include commands to retrieve all studies or observations in the database and a method to display an overview of the results for the whole database or for certain studies or observations. Finally, the meta-data associated with each study (for example, the number of participants, the modality, the patient group, etc.) can also be retrieved and displayed. Prediction Database ------------------- The prediction database is also implemented as a nested Python dictionary. Similar to the experimental observation database, methods that retrieve and visualize the content of the database are included in *ASSRUnit*. Models, Capabilities, Tests and more ------------------------------------ #### Models In order to demonstrate the flexibility of the proposed framework, we included three different neural models of ASSR deficits. The first model is based on a biophysically detailed model of primary auditory cortex by Beeman [@Beeman2013]. It has recently been used to study ASSR deficits by our group [@Metzner2016]. The model was implemented using the neural simulator GENESIS [@Bower1992; @Bower1998]. Not only is this model a good example of a biophysically detailed model of ASSR deficits, its inclusion also demonstrates how models that are not written in Python can be used. The second model is a reimplementation of the model of Beeman in NeuroML2, a simulator-independent markup language to describe neural network models developed by the NeuroML project [@Cannon2014], which is featured in the open source brain model database [@Gleeson2012]. We included this model to demonstrate the ability of the proposed framework to incorporate state-of-the-art tools and databases for the design, implementation and simulation of network models. The last model we included is the simple model presented by Vierling-Claassen et al. [@Vierling2008]. The model is a simple network of two populations of theta neurons. We reimplemented the model in Python (for more details on the model and the replication see [@Metzner2017]). The model was included first of all to demonstrate that the framework is not limited to biophysically detailed models but can also be used with simpler, more abstract models. Additionally, the inclusion of the model demonstrates the simplest way of including a model, implementing the model in Python. This might not be the most common scenario, but since it is the simplest, we included it here. We do not discuss the models in more detail here, since they have been described elsewhere [@Beeman2013; @Metzner2016; @Vierling2008; @Metzner2017]. Furthermore, our focus lies on the framework with which to use, validate and compare models not on the models themselves. The three models mentioned above are included into the SciUnit framework by wrapper classes that implement the necessary capabilities and make the models available to the tests. One important thing to note here is that, since we are dealing with models of neurofunctional deficits found in individuals with a particular disorder, a ’model’ as used in the module always means two configurations of a computational model, one representing the control configuration and one the disorder configuration. Therefore, all wrapper classes take two sets of parameters as an argument describing the necessary parameters for the two configurations, respectively. In addition to the standard model classes, we also implemented a second version of the model classes, which simulates a certain number $n$ of simulations, instead of a single one, where each simulation differs in background noise. This allows for assessing the robustness of the results. #### Capabilities Table \[Tab:Experiments\] summarizes the experimental observations included in the module at this stage. All observations are similar in nature: the power value of the EEG/MEG at a certain frequency in response to auditory entrainment at a certain frequency. Therefore, the only capability necessary for a model to produce output that can be compared to these observations is a method that produces the power at a certain frequency X of a simulated EEG/MEG signal in response to drive at a frequency Y. This capability, *ProduceXY*, is included in ASSRUnit and all models must implement it. #### Tests and Scores The five tests we implemented, examine the five observations summarized in Table \[Tab:Experiments\] individually. Furthermore, we implemented one prediction test, which tests 10Hz power at 10Hz drive. For the sake of simplicity, the test scores implemented so far are simple Boolean scores, indicating whether a model output fails or passes a test, that is whether the difference between model output for the control and the ’schizophrenia-like’ network matches the experimental observation. In case of the model classes implementing sets of outputs, simply the mean difference is compared to the experimental observations. For the prediction test we have chosen a RatioScore instead of a Boolean, which returns the ratio of the power for the ’schizophrenia-like’ configuration and the power for the control configuration. #### Visualization, Statistics, Additional Data In addition to the main features of the SciUnit framework for the analysis and comparison of the models, we use the fact that SciUnit allows to pass additional data, beyond the test scores, to provide a class that offers tools for the visualization of the results. This class includes functions to display the test results in a table, plot the results from a set of model outputs as a box plot, and perform and visualize a student’s t-test of the differences between control and ’schizophrenia-like’ networks. Next, we describe three different use cases, which show how the proposed module can be used for different purposes by experimentalists, modelers and reviewers. Use Case I: Overview of the Experimental Literature --------------------------------------------------- The first use case demonstrates how the experimental database can be used to get a comprehensive overview of the current experimental literature related to a neurophysiological or neurocognitive biomarker, in our case ASSR deficits in patients suffering from schizophrenia. Figure \[Fig:ListStudies\] shows that with two simple commands one can retrieve the names of all studies and all observations present in the database. These names will have to be used for all further queries of the database. ![Display all studies and all observations included in the database[]{data-label="Fig:ListStudies"}](list_studies_observations){width="\textwidth"} Figure \[Fig:ExpOverview1\] a) then shows how to get a complete overview of all observations of all studies in the database. As we can see in Figures \[Fig:ExpOverview0\] \[Fig:ExpOverview1\] b), simply adding the parameter *meta=true*, to the command, will additionally output the meta-data associated with each study. This contains information on the subjects, modality etc. The overview command presents the data in a simple table and can be used to see which studies provided which observation and what the results were. However, as we can already see for our small demonstration database containing only three studies, this is likely to become big and therefore hard to fully grasp. By explicitly stating the studies and/or the observations one is interested in, one can reduce the complexity of the table and get a clear and simple overview, as depicted in Figure \[Fig:ExpOverview2\]. Note that in the examples, we have only used the qualitative description of the observations, the same functionality also applies to the quantitative descriptions. The functionality described here, along with more examples, can be explored in an accompanying Python notebook (*Example\_Experimental\_Database.ipynb* in *https://github.com/ChristophMetzner/ASSRUnit/Code/*). {width="\textwidth"} ![ By setting the *meta* flag to *True*, additional information on the studies are displayed.[]{data-label="Fig:ExpOverview1"}](display_meta2){width="\textwidth"} ![The *experimental\_overview* command allows for querying for specific studies and observations using the names retrieved with the *get\_studies* and *get\_observations* commands.[]{data-label="Fig:ExpOverview2"}](specific_studies_and_observations){width="\textwidth"} This simple querying functionality allows the user to get a quick, clean and comprehensive overview of the experimental literature, to identify observations that are supported by many studies (see in our case the reduction of gamma power for stimulation at gamma frequency) but also to detect controversial findings. Furthermore, the display of the associated meta-data allows to check for example whether identified common observations extend over different modalities and post-processing techniques, and also whether controversial findings might be explained by differences in the experimental setup or other related aspects. In the future, it will also be possible to look at more than one database and compare the same observations across different patient groups to highlight commonalities and differences between disorders. Use Case II: Model Comparisons ------------------------------ While our first use case only exploited the experimental database, we now show the additional benefits of joining experimental and modeling data. #### Simple model comparison By creating tests, based on the model capabilities, and grouping them into test suites, we can easily compare models against experimental data and against each other. Figure \[Fig:ModelComparisons\] demonstrates how we can use the module to create two different models along with several tests, then run the models to produce the data relevant for the tests and then judge the model outputs against experimental data and display the result together. Note that in this context we use the term model as the *in silico* instantiation of a theoretical/conceptual model. Two different models therefore, do not necessarily have to use different model implementations but might simply differ in parameters. #### Advanced modeling data and visualization As already described in the Methods section, the model classes do not only contain the standard methods that implement the necessary capabilities, but also contain so-called ’...\_plus’ methods which generate additional model data. Together with the methods from the visualization class, this additional model data can be used to better understand the model behavior, to judge the robustness of findings and to statistically analyze model output (see Figure \[Fig:ModelPlusMethods\]). Use Case III: Overview over Model Predictions --------------------------------------------- Finally, we show how predictions can be generated from existing models (see Figure \[Fig:ModelPlusMethods\]). In order to generate the predictions, a set of prediction tests along with prediction capabilities, that is, capabilities the models must have in order for the model to generate the relevant data needs to be created. For demonstration purposes, we have chosen to implement a single, simple prediction test. Since in *ASSRUnit* so far, we have only looked at experimental observations and computational models that cover gamma and beta range entrainment, the first test simply generates a prediction how, in a given model, power in the alpha band (here at 10Hz) differs between the control network and the schizophrenia-like network at 10Hz drive. Note that this prediction test has been studied in the experimental literature, which means that it could have already been included in the experimental database and therefore does not represent a true prediction. However, we have chosen to include it for the purpose of demonstration. \[Fig:Tests4040\] Discussion ========== #### The potential role of the framework within computational psychiatry The use of computational approaches has seen a significant increase over the last decades in almost all areas of medicine and life sciences. Especially in psychiatry it has become clear that the complex and often polygenic nature of psychiatric disorders might only be understood with the help of computational models [@Adams2016; @Wang2014; @Friston2014; @Corlett2014; @Stephan2014; @Montague2012; @Siekmeier2015]. Naturally, the number of computational models in the field of psychiatry has also increased significantly over the last years and it has been argued that *in silico* instantiations of biomarkers and endophenotypes are a crucial step towards an understanding of underlying disease mechanisms [@Siekmeier2015]. While this large increase in modeling studies shows the importance of computational methods in the field, it also raises several issues that, impede the community to exploit these approaches to their full potential. In order for a computational model to be a substantial contribution to knowledge it has to adequately instantiate experimental observations, correctly implement the mathematical equations of the model and generate experimentally testable predictions. The approach presented here, addresses two of these three requirements, namely, the instantiation of experimental observations and the generation of testable predictions. While correctness of the code is an equally important requirement, it was out of scope of the current work, since it very strongly depends on the type of computational model and on the programming language used to implement the model. Nevertheless, the approach presented here offers significant benefits for, not only the computational psychiatry community, but for the psychiatry community as a whole, while imposing little additional efforts for the users and contributors. It gives modelers a tool to query experimental observations on neurophysiological and neurocognitive biomarkers, and therefore, helps them to include the current relevant experimental data into their modeling efforts. It further enables them to validate their modeling output against experimental observations during model construction and to demonstrate the performance of their model, both, with respect to the experimental literature and with respect to other competing models. In addition to the benefits it offers the modelers, it also enables experimentalists to quickly gain insight into the current state of modeling and to extract experimentally testable predictions from the models. Last but not least, it offers a tool to reviewers which allows them to judge a newly proposed model by making explicit its performance against experimental data and competing models. The concept of automated code testing and validation has been successfully applied in computer science for many years now, however, it is only slowly finding its way into the computational branches of scientific fields. SciUnit attempts to satisfy this demand by providing a simple, flexible yet powerful framework to address the above-mentioned issues. The computational neuroscience community has started to adopt this framework for the automatic validation of single neuron models (NeuronUnit, [@Gerkin2014]). To the best of our knowledge, we are not aware of any similar efforts in the field of psychiatry. Since schizophrenia is a polygenic, multi-factorial and very heterogeneous disorder, it has been argued that the usefulness of biomarkers and endophenotypes lies in their potential to dissect the disorder into subtypes, which might even be linked more closely to findings on the genetic level [@Meyer2006; @Perlis2011; @Markou2009]. The proposed *ASSRUnit* module together with computational models of biomarkers/endophenotypes and specifically designed test suites could strongly facilitate this process by providing mechanistic links between neurophysiological or neurocognitive biomarkers and changes at the synaptic,cellular and/or network level. #### Future directions for ASSRUnit The presented *ASSRUnit* module can be easily extended and modified by others to fit their needs (for example to include more specialized visualization tools). Our efforts for establishing *ASSRUnit* as a widely used tool will focus on three main areas: 1) We aim to cover the majority of existing experimental studies with our experimental database in the future. Furthermore, we hope to convince experimentalists to provide more detailed experimental data or to ideally create database entries themselves. 2) We also aim to cover the majority of current computational models that describe the cortical circuitry responsible for the ASSR. Again, we hope to encourage modelers to actively contribute to *ASSRUnit*. 3) We aim to extend our set of prediction tests, and thus, our prediction database. The most straightforward extension, in our view, is to include information on phase-locking in addition to pure power in certain frequency bands. Several studies report, additionally to a reduction in gamma power, a reduction in the phase-locking factor for patients suffering from schizophrenia (for example [@Kwon1999; @Brenner2003; @Light2006; @Vierling2008; @Krishnan2009]). These observations can very easily be incorporated into the existing module, simply by including the experimental observations into the database, adding the necessary capabilities to the model classes and by adding the appropriate tests that link the experimental observations to the model capabilities. Furthermore, the changes in oscillatory activity upon auditory stimulation are not limited to the gamma and the beta range for schizophrenic patients, but also extend to lower frequency bands such as alpha, theta and delta. For example, Brockhaus-Dumke and colleagues find reduced phase-locking in the alpha and theta band for schizophrenic patients in an auditory paired-click paradigm [@Brockhaus2008], and Ford et al. find a reduction of phase-locking in the delta and theta range for schizophrenic patients in an auditory oddball task [@Ford2008]. Abnormalities in these frequency bands have also been found in many other paradigms outside of the auditory system (see [@Basar2013]). To the best of our knowledge, ASSRs to entrainment stimuli in the theta and delta range have not been looked at in schizophrenia. Therefore, *ASSRUnit* could be used to generate predictions in these frequency ranges as demonstrated in use case III. However, an inclusion of the above-mentioned observations together with computational models explaining these deficits is not straightforward, because either the paradigms are different from the ones used to elicit ASSRs and/or the mechanisms underlying the effect are different, and therefore the computational models, are substantially different to models of ASSRs. Therefore, these deficits are better explored in separate modules solely focusing on each paradigm/deficit. However, it would be very interesting to ’co-explore’ computational models that have the capabilities to explain both, ASSR gamma/beta band and delta/theta/alpha phase-locking, deficits. Such an analysis could highlight interactions between different mechanisms underlying different symptoms/biomarkers. Another very interesting and promising extension of the current module would be to include data and models from different psychiatric disorders, since schizophrenia is not the only disorder where patients show entrainment deficits. Wilson et al. [@Wilson2007], explored gamma power adolescents with psychosis and found reductions compared to normally developing controls. Their patient group consisted of patients suffering from schizophrenia and also from schizoaffective disorder and bipolar disorder. Interestingly, these disorders show overlapping symptoms, neurobiological substrates and predisposing gene loci. Other studies have also found reduced power and phase-locking in the gamma range in patients with bipolar disorder [@ODonnell2004; @Spencer2008; @Rass2010]. The presented module is perfectly suited to highlight commonalities and differences across disorders and to link those to mechanistic explanations via different theoretical/computational models. #### Other modules beyond ASSRUnit The approach presented here, combining an experimental database with a collection of models, tests, prediction tests and a resulting predictions database, can be readily applied to a number of other neurophysiological biomarkers of schizophrenia as well as other psychiatric disorders. In patients suffering from schizophrenia a dysfunction of the auditory system has long been suspected. In fact, a large number of biomarkers and endophenotypes for schizophrenia, other than ASSR deficits, involve auditory processing. Several alterations of event-related potentials (ERPs) such as mismatch negativity (MMN), N100, and P50 have been described in the literature (see also [@Siekmeier2015; @Shi2007]). MMN is a negative component of the auditory evoked potential, which is evoked by an alteration in a repetitive sequence of auditory stimuli. MMN seems to be specific to schizophrenia because patients suffering from other psychiatric disorders (for example bipolar disorder and major depression) show normal MMN [@Umbricht2003]. Auditory MMN is likely to be generated in primary and secondary auditory cortices and is therefore very similar to stimulus-specific adaptation (SSA) properties of single neurons in auditory cortex [@Nelken2007] (although not identical; see efor example [@Farley2010; @vonderBehrens2009]). Several models explaining mechanisms underlying MMN/SSA have been proposed (for example [@Mill2011; @Nelken2014]). The P50 potential, a small positive deflection of the EEG signal at around 50ms after the onset of an auditory stimulus, is often reduced to the second of a pair of stimuli. However, this so-called P50 reduction is markedly reduced in schizophrenic patients (for example [@Braff2007]). Another important measure of sensory gating is the pre-pulse inhibition (PPI) of the auditory startle reflex (i.e. the phenomenon in which a weaker prestimulus inhibits the reaction to a subsequent strong startling stimulus). As P50 reduction, PPI is also reduced in schizophrenic patients, although these phenotypes do not seem to correlate [@Braff2007]. Again, computational models exploring the mechanisms underlying PPI have been developed (for example [@Schmajuk2005; @Ramirez2012; @Leumann2001], and Moxon et al. have investigated the dopaminergic modulation of the P50 auditory-evoked potential and its relationship to sensory gating in schizophrenia [@Moxon2003]. The N100 (also called N1) is a negative component of the EEG signal occurring approximately 100ms after stimulus onset. This negative deflection is again reduced in schizophrenic patients (reviewed in [@Rosburg2008; @Rissling2010; @Javitt2008]) and these deficits have also been modeled [@Ventouras2000]). Although we here concentrated on biomarkers and deficits in the auditory cortex, our approach is well adaptable to brain circuits outside of the auditory system. Working memory deficits are probably one of the most robust and best described cognitive deficits in schizophrenic patients (reviewed in [@Piskulic2007; @Lee2005]). Patients show a decrease in working memory capacity, i.e. the capacity to maintain, manipulate and use information online for a relatively short period of time, across a broad range of paradigms. Again, several theoretical and computational models have been proposed, aiming to provide mechanistic descriptions of the underlying mechanisms (for example [@Compte2000; @Durstewitz2000; @Wang2004; @Singh2006; @Wang2001; @Cano2012]). All these deficits and alterations along with the mentioned computational models could be integrated into a module similar to the proposed ASSRUnit module. Such a unified framework would be of great benefit for the study of schizophrenia pathology due to the diversity of symptoms, biomarkers, and experimental observations linked to the mental disease. Conclusion ========== We have proposed a framework for automated validation and comparison of computational models of neurophysiological and neurocognitive biomarkers of psychiatric disorders. The approach builds on SciUnit, a Python framework for scientific model comparison. As case in point, we used this framework to develop *ASSRUnit*, a module comprising an experimental observations data base, computational models, capabilities, tests/test suites and visualization functions for ASSR response deficits in schizophrenia. Our approach will facilitate the development, validation and comparison of computational models of neurophysiological and neurocognitive biomarkers of psychiatric disorders by making the scope of models explicit and by making it easy for the user to assess a model’s validity and to compare a model against competing models. Furthermore, it is easy to use, straightforward to extend to more experimental observations, computational models and analyses and, ready to apply to other biomarkers. Therefore, the adoption of the proposed framework could be of great use for modelers, reviewers and experimentalists in the field of computational psychiatry.

--- abstract: 'Researchers, technology reviewers, and governmental agencies have expressed concern that automation may necessitate the introduction of added displays to indicate vehicle intent in vehicle-to-pedestrian interactions. An automated online methodology for obtaining communication intent perceptions for 30 external vehicle-to-pedestrian display concepts was implemented and tested using Amazon Mechanic Turk. Data from 200 qualified participants was quickly obtained and processed. In addition to producing a useful early-stage evaluation of these specific design concepts, the test demonstrated that the methodology is scalable so that a large number of design elements or minor variations can be assessed through a series of runs even on much larger samples in a matter of hours. Using this approach, designers should be able to refine concepts both more quickly and in more depth than available development resources typically allow. Some concerns and questions about common assumptions related to the implementation of vehicle-to-pedestrian displays are posed.' author: - | Lex Fridman Bruce Mehler Lei Xia Yangyang Yang Laura Yvonne Facusse Bryan Reimer\ \ bibliography: - 'walk.bib' title: | To Walk or Not to Walk: Crowdsourced Assessment of\ External Vehicle-to-Pedestrian Displays --- Introduction {#sec:introduction} ============ The introduction of semi-automated and automated driving technologies into the vehicle fleet is often seen as having the potential to decrease the overall frequency and severity of crashes and bodily injury [@national2017federal]. At the same time, there is some concern that the transition from manually controlled to technology controlled vehicles could have unintended consequences. One such concern is in the area of communication of intent between automated vehicles and shared road users, particularly pedestrians [@keferbock2015strategies; @lagstrom2015avip; @lundgren2017will; @matthews2015intent; @mirnig2017three]. One perspective asserts that human driven vehicle-pedestrian communication often involves hand and body gestures, as well as eye contact (or avoidance of), when vehicles and pedestrians come together in interactions such as those occurring at crosswalks where miscommunication can easily elevate risk. The question is then posed as to what will replace these forms of communication when a human is no longer actively driving the vehicle? One approach might be to explore technologies such as the Wi-Fi application proposed in [@anaya2014vehicle] or other Vehicle-to-Entity (V2X) communications that alert pedestrians of potential conflict situations. However, most proposed solutions focus on external vehicle displays as replacements for human-to-human visual engagement. Google drew attention to this approach by filing a patent for messaging displays for a self-driving vehicle that included the concept of electronic screens mounted on the outside of the vehicle using images such as a stop sign or text saying “SAFE TO CROSS” [@urmson2015pedestrian]. Drive.ai, a self-driving technology start-up, released an illustration of a roof-mounted display screen concept that combined an image of a pedestrian on a cross walk and the words “Safe to Cross” [@knight2016selfdriving]. Matthews and Chowdhary [@matthews2015intent] describe an LED display for an autonomous vehicle that might display messages such as “STOP” or “PLEASE CROSS” when a pedestrian is encountered. Mirnig and colleagues [@mirnig2017three] briefly describe several visual display strategies that they describe as being informed from human-robot interaction principles. Automotive manufacturers have also proposed design visions such as the Mercedes F015 concept car using lighted displays on the front grill and a laser projection of an image of a crosswalk on the roadway in front of car [@keferbock2015strategies]. A Swedish engineering company has proposed a lighted grill design that “smiles” at pedestrians to indicate they have been detected and it is safe to cross in front of the vehicle [@peters2016selfdriving]. Early Stage Design Assessment ----------------------------- Careful and extensive testing vehicle-to-pedestrian communication concepts under real-world conditions and with a broad demographic sampling would seem to be indicated before a design is put into general use due the potentially safety critical implications of miscommunication. Given the inherent costs of real-world validation testing, efficient methods for early stage concept assessment are highly desirable for narrowing in on designs that are promising and setting aside those less likely to prove out. Further, early stage methods that make it practical to test a large number of minor design variations increase the probability of elucidating subtle considerations that may lead to optimized implementations. Wizard of Oz approaches to assessing how pedestrians might interact with automated vehicles and various external design concepts have been reported [@habibovic2016evaluating; @lagstrom2015avip] and [@doric2016novel] have described a virtual reality based pedestrian simulator. While these methods are less intensive than full scale field testing, they still require significant effort and strategic choices need to be made in selecting concepts to test at this level. To gather data on the extent to which pedestrians might be uncomfortable and uncertain about whether they should cross in front of a vehicle if they were unable to make eye contact with the driver (which was presumed to be more likely in vehicles under autonomous control), Lagstr[ö]{}m and Lundgren [@lagstrom2015avip] presented participants with a set of five photographs ranging from an image of the person in the driver’s seat holding onto the steering wheel and looking into the camera to one where the individual in the “driver’s” seat appeared to be asleep. Participants were asked to image that: You are walking through a city center and are just about to cross an unsignalized zebra crossing. A car has just stopped and you look into the car before passing the crossing, you see what is shown on the picture. How do you feel about crossing the road? Participants then were asked for each to respond for each image whether they would cross immediately and made ratings of their likely emotional reactions. Lagström and Lundgren interpreted the responses as supporting a concern that there may be a risk of misinterpretation on the part of pedestrians that observing a “driver” occupied by activities such as reading or sleeping as indicating that the vehicle was not about to move. They note that this might be a wrong interpretation for an automated vehicle and thus some indication of whether a vehicle is in autonomous mode and what its intentions may be desirable. The present study employed a data gathering approach that is conceptually similar to Lagström and Lundgren’s in that it presented participants with multiple pictures of a vehicle and asked if it was safe to cross in front of the vehicle. However, there were two key differences. First, the goal of the assessment was to evaluate design concepts for communication from the vehicle to a pedestrian whether the pedestrian should cross or not. Second, an automated online presentation methodology was employed that supported efficient presentation of a large number of images across a larger sample of participants. Methods ======= Amazon Mechanical Turk (MTurk) was used for data collection. MTurk is an internet based, integrated task presentation and participant compensation system. It provides access to a large potential participant pool at modest cost per participant. It has been reported that MTurk has good performance, especially on psychology and other social sciences research, since participants are diverse and more representative of a non-college population than traditional samples [@buhrmester2011amazon; @paolacci2010running]. Participants who take frequently part in MTurk tasks are commonly referred to as “Turkers”. One of the challenges of constructing a statistically meaningful MTurk experiment is the ability to filter out any responses by Turkers that were not made with their full attention on the task and representative of a “best effort”. We used two types of filtering: (1) accepted only select Turkers with a proven track record on MTurk (see Participants subsection) and (2) the insertion of “catch” stimuli for which there is a “correct” answer (see Stimuli subsection). A second challenge of setting up a successful MTurk experiment is making to scalable to hundreds or thousands of participants. To this end we implemented a Python framework that created, configured, and served the stimuli in randomized order on a HTML front-end. An asynchronous Javascript (Ajax) communication channel stored the responses in a PostgreSQL database through a PHP-managed backend. This framework allowed for robust, concurrent collection of the dataset underlying this work in just a few hours. Moreover, it allowed for efficient validation of the result and possible future scaling of the number of Turkers and stimuli. Participants ------------ To take part, participants had to be experienced Turkers with a minimum of 1,000 previous HITs (a measure of previous experience where a HIT represents a single, self-contained task that an individual can work on, submit an answer, and collect a reward for completing) and a 98% or higher positive review rating. Data collection continued until 200 Turkers completed the full experiment by providing a response to each of the 30 stimuli. According to tracking of IP addresses, the majority of participants were from the USA and India, and approximately matched the distribution reported by [@ross2009turkers] where 57% of Turkers were from USA and 32% were from India. Compensation was at approximately \$15/hour based on a conservative estimate of a pace necessary to complete the full experiment. This rate is above the compensation of \$2-3/hour commonly provided on MTurk. Stimuli / communication design elements --------------------------------------- The base photograph used to create the stimuli (see [Fig. \[fig:base\]]{}) was of a late model passenger sedan on a one-way urban street approaching an uncontrolled intersection/crosswalk (no traffic light). Under the lighting conditions the driver is not visible. ![The driver was not visible in the base image used to create the designs due to lighting and reflection angles.[]{data-label="fig:base"}](images/designs/base.png){width="\columnwidth"} The stimuli were created by superimposing each of the 30 designs onto the base image. Every design had an animated element in that it was either flashing or playing through a sequence of animation frames. [Fig. \[fig:four-designs\]]{} shows illustrative snapshots of four of the designs. A video of all 30 final stimuli is provided as supplementary material. The size of the stimuli presented to each Turker was 1280 pixels wide and 720 pixels tall. Turkers with screen resolutions below this size were automatically detected and could not participate in the experiment. ![Four of the designs tested (out of 30 total in [Fig. \[fig:designs-all\]]{}) shown in cropped, close-up views. Presentation to participants used the entire image shown in [Fig. \[fig:base\]]{} at a 1280 pixels wide and 720 pixels tall resolution. These 4 designs performed significantly better than the other 26 at communicating their intent as shown in [Fig. \[fig:plot-all\]]{}.[]{data-label="fig:four-designs"}](images/designs/paper_small_pretty.jpg){width="\columnwidth"} “Catch” stimuli were created that, instead of a design, showed instructions on what to respond (e.g., Yes, No). Only responses provided by Turkers who passed these catch stimuli were included in the resulting dataset. Given the filtering in the Turker selection, 100% of the Turkers who completed the entire experiment responded to the catch stimuli correctly. {width="\textwidth"} Procedure --------- Participants were presented with introductory text informing them that they would be presented with a series of images of a vehicle approaching a cross walk. They were to imagine that they were a pedestrian viewing the approaching vehicle and decide if it was safe to cross. Response options were: Yes, No, and Not Sure. The presentation order for the images was randomly shuffled for each participant to control for order effects. Each of the stimuli was animated on screen indefinitely until the Turker provided a response. Response timing information was recorded, but analysis did not reveal any meaningful patterns or correlation between designs and response dynamics. {width="\textwidth"} Results ======= A few of the concepts showed a high degree of match between the designers’ intent and participants’ interpretation. For the examples shown in Figure 2, designs 1 and 7 received a high percentage of responses that it was safe to walk; designs 2 and 10 received a high percentage of responses that it was not safe to walk. Participants’ interpretation of the communication intent on the part of the vehicle for each of the 30 designs is shown in Figure 3; the proportion of the sample that rated each design as indicating it was safe to walk is colored coded in green, were unsure about the intent in yellow, and interpreted the message as indicating they should not walk is shown in red. It can be observed in Figure 3 that the two designs intended to communicate that pedestrians should not walk (2 and 10; images in Figure 2) match relatively well with participants’ interpretations. The degree of successful communication of designer intent was much more varied for the designs intended to indicate that it was safe to walk; the interpretation ratings of these designs are broken out in Figure 4. The responses for concepts the designers thought would be ambiguous are shown in Figure 5. Discussion ========== The external vehicle display concepts intended to communicate vehicle intent to pedestrians were developed by a team of graduate design students (Xia, Yang, and Facusse) as part of a course project. As external advisors and collaborators, the remaining authors attempted to guide the students understanding of the potential need for external vehicle displays, while minimize the amount of input on the designs. The advisors, focused on the design of the web based assessment methodology, technical aspects of the MTurk implementation, and empirical data collection to provide the students with a data set that could be used to evaluate assumptions about design elements. The ability to “risk” resources in testing such a large number of design concepts was only practical through the use of a relatively low cost and low time intensive prototyping methodology such as was explored here. A strong focus of this work was allowing the graduate design students flexibility to consider the advantages and disadvantages of various design approach and gain rapid consumer facing feedback on the appropriateness of their decisions. Twenty of the designs (1, 7, 11-24, 26-30) were created with the intent of communicating that it is safe to walk without explicitly intended ambiguity. Of these, six of the designs obtained an 80% or greater match and none of the designs showed universal agreement. For 8 of the 20 “walk” designs, more than half the participants found the message unclear or misinterpreted the message as “don’t walk”. The 2 “don’t walk” designs faired generally better, although the interpretive match was still not universal – which is of particular concern for a safety critical communication in which nearly, of not 100%, correct interpretation is needed. Clarity and unambiguity will be critical if external communication displays are to achieve the goal of building psychological trust between human and machine [@keferbock2015strategies]. The presence of uncertainty and misinterpretation with all of the designs tested suggests some potential concern around the concept of “needing” to employ external communication signals in automated vehicles intended for public roadways beyond those already used in non-automated vehicles (e.g., turn signals, brake lights, and vehicle kinematic cues). Lagström and Lundgren ([@paolacci2010running]; see also [@lundgren2017will] present a substantive series of small studies that document pedestrians’ desire to understand a driver / vehicle’s intent, and explore a creative design concept involving a row “movable” light bar elements at the top of the front windshield to communicate several messages (e.g. “I’m about to yield.”, “I’m about to start.”). After training in the intended meaning of the messages, all 9 participants in the final test phase were able to correctly report the intention of all of the messages except for the message intended to indicate that the vehicle was in automated mode. What is not clear is how an untrained population would interpret the messaging and whether the net result over time would be greater comfort with automated vehicles and an overall safety benefit for pedestrians. In another very detailed study, Clamann and colleagues [@clamann2017evaluation] tested a variety of designs including a mock automated van with a prominently mounted, large LCD display employing what would appear to be relatively apparent walk/don’t walk graphics (walking figure with and without a diagonal line across the image). It was concluded that while a large number of participants felt that additional displays will be needed on automated vehicles, most appeared to ignore the displays and rely on legacy behaviors such as gap estimation and inferring vehicles’ approaching speed (collectively kinematics) in making decisions on whether or not to cross the road. In an interview [@lefrance2016pedestrians], Clamann observed that the displays tested were “as effective as the current status quo of having no display at all.” The senior researchers on this paper have, as part of a different project, been involved in extensive observation of pedestrian-vehicle interactions [@toyoda2017understanding]. During these observations, we have increasingly developed the impression that pedestrians may take their primary communication cues from overt vehicle kinematics more often than actually depending on eye-contact or body gestures to make judgements about vehicle intent and that multiple attributes may be used to predict intent. As such, vehicle systems may be developed to be responsive to pedestrian movements. Thus, we see it as still an open research question as to whether new external displays are necessarily a priori answer to improving communication of intent. We are in full agreement with Clamann [@lefrance2016pedestrians] that careful, detailed evaluations need to be carried out to make sure that displays and signals work as intended before they are standardized, mandated or released in any production fashion. It is clear, that unanticipated consequences can easily occur if a pedestrian in the dilemma zone (stepping off the curb into the flow of traffic) pauses for even a moment to perceive, read or interoperate the intent of external communication devices. As such, while benefits of external vehicle displays could easily improve the communication of intent in a “trained” or “habituated” population, without nearly ubiquitous understanding risks could easily increase. Furthermore, a transition period during which a mixed population of vehicles with and without communication devices, and a mixed set of educated and non-educated pedestrians could be detrimental to short term safety making the societal hurdles to successful adoption of a new technology more difficult. Conclusion {#sec:conclusion} ========== Experience implementing the assessment methodology described here in MTurk demonstrates that this approach can be applied in a cost effective manner for identify design concepts that may be appropriate for more detailed development and testing. Since a relatively large number of elements or minor variations can be tested through a series of MTurk runs in a matter of hours (as opposed to weeks or months for focus groups or experimental simulation or field testing), designers should be able to refine concepts both more quickly and in more depth than available development resources typically allow. Factors that are often difficult to explore during design phases (e.g. culture, demographic, prior mental model, etc.) can be factored in early in the process. It is worth noting that there is nothing in this method of early stage design development that is limited to the messaging application explored in this study; it should be equally applicable to work on other design elements such as interior interface icons, graphics, gages, and other forms of information presentation in automotive, consumer electronics, advertising and other domains. Limitations =========== As noted, the majority of participants were from the USA and India, so it is unknown the extent to which the findings for specific design elements generalize to other regions. A single vehicle type and setting were assessed. The Turker sample was motivated to pay attention to details of the images and presumably not distracted (e.g. talking on a phone, etc.), thus correct detection of communication intent may have been greater than might be obtained under real-world conditions and may represent something approximating best case evaluations. With these considerations in mind, the methodology explored here is likely to be most useful for rapidly identifying designs or design elements that are promising for further investigation as opposed to use for late stage validation. Acknowledgment ============== Authors Xia, Yang, and Facusse would like to thank their course instructor, Professor Leia A Stirling, for the guidance on human centered design and research methodology. Support for this work was provided by the Toyota Class Action Settlement Safety Research and Education Program. The views and conclusions being expressed are those of the authors, and have not been sponsored, approved, or endorsed by Toyota or plaintiffs’ class counsel.

--- abstract: 'We discuss the impact of recent Belle data on our description of the pion transition form factor based on the assumption that a perturbative formalism and a nonperturbative one can be matched in a physically acceptable manner at a certain hadronic scale $Q_{0}$. We discuss the implications of the different parameters of the model in comparing with world data and conclude that within experimental errors our description remains valid. Thus we can assert that the low $Q^2$ nonperturbative description together with an additional $1/Q^2$ term at the matching scale have a strong influence on the $Q^2$ behavior up to very high values of $Q^2$ .' author: - 'S. Noguera' - 'V. Vento' title: Model analysis of the world data on the pion transition form factor --- New data of the pion transition form factor ($\pi TFF$) from the Belle collaboration have just appeared [@Uehara:2012ag]. These data, above 10 GeV$^2$, are smaller in magnitude than the previous BABAR data [@Aubert:2009mc], which generated considerable excitement. The question to unveil is the scale of asymptotia. BABAR data, taken at face value, implied that asymptotic QCD behavior lies at much higher $Q^2$ than initially expected [@Lepage:1980fj; @Chernyak:1983ej]. Belle data seem to lower that scale. We show here that our scheme can accomodate easily all data without changing the physical input. At the time of the BABAR data we developed a formalism to calculate the $\pi TFF$ [@Noguera:2010fe], which consists of three ingredients: * i*) a low energy description of the $\pi TFF$; * ii*) a high energy description of the $\pi TFF$; * iii*) a matching condition between the two descriptions at a scale $Q_{0}$ characterizing the separation between the two regimes. For the low energy description we took a parametrization of the low energy data to avoid model dependence at $Q_{0}$. The high energy description of the $\pi TFF$, defined by the pion Distribution Amplitude ($\pi DA$), contains Quantum Chromodynamic (QCD) evolution from $Q_{0}$ to any higher $Q$, a mass cut-off to make the formalism finite, and an additional $1/Q^2$ term which leads to modifications of the matching condition. Let us recall some aspects of the formalism. The high energy description, to lowest order in perturbative QCD, for the transition form factor in the process $\pi^{0}\rightarrow\gamma\,\gamma^{\ast}$ in terms of the pion distribution amplitude ($\pi DA$), is given by $$Q^{2}F(Q^{2})=\frac{\sqrt{2}f_{\pi}}{3}\int_{0}^{1}\frac{dx}{x+\frac{M^{2}% }{Q^{2}}}\phi_{\pi}(x,Q^{2}). \label{tff_R}$$ We follow the proposal of Polyakov [@Polyakov:2009je] and Radyushkin [@Radyushkin:2009zg] and introduce a cutoff mass $M$ to make the expression finite. $Q^{2}=-q^{2},$ $q_{\mu}$ is the momentum of the virtual photon, $\phi_{\pi}\left( x,Q^{2}\right) $ is $\pi DA$ at the $Q^{2}$ scale and $f_\pi =0.131$ GeV. In this expression, the $Q^{2}$ dependence appears through the QCD evolution of the $\pi DA$. Despite the fact that several models reproduce the low energy data, in order to have a model independent expression for the form factor at low virtualities, we adopted a monopole parametrization of the $\pi TFF$ in the low energy region as $$F^{LE}\left( Q^{2}\right) =\frac{F\left( 0\right) }{1+a\frac{Q^{2}}% {m_{\pi^{0}}^{2}}}\,. \label{tff_EXP}$$ with $F\left( 0\right) =0.273(10)$ GeV$^{-1}$ and $a=0.032\left( 4\right) $ [@Nakamura:2010zzi], determined from the experimental study of $\pi^{0}\rightarrow\gamma\,e^{+}\,e^{-}$ [@Amsler:2008zzb]. ![We show the result for the transition form factor in our formalism for $M=0.690$ GeV, $a=0.032$ and $C_{3}=2.98\,10^{-2}$ GeV$^{3}$ and defining the matching point at $Q_{0}=1$ GeV (solid line). The band region results from the indeterminacy in $\Delta a = \pm 0.004$. The lower plot shows the detailed behavior for low virtuality. Data are taken from CELLO [@Behrend:1990sr], CLEO [@Gronberg:1997fj], BABAR [@Aubert:2009mc] and Belle [@Uehara:2012ag].[]{data-label="Fig1"}](Fig1a.eps) -1.3cm ![We show the result for the transition form factor in our formalism for $M=0.620$ GeV, $a=0.032$ and the value of $C_{3}= 1.98\,10^{-2}$ GeV$^3$ corresponding to $20\%$ of the contribution at the matching point at $Q_{0}=1$ GeV (solid line). The band region gives the variation of the results due in $ \pm 10 \%$ in the contribution of higher twist. The lower plot shows the detailed behavior for low virtuality. Data are taken from CELLO [@Behrend:1990sr], CLEO [@Gronberg:1997fj], BABAR [@Aubert:2009mc] and Belle [@Uehara:2012ag].[]{data-label="Fig2"}](Fig2a.eps) -1.3cm Additional power corrections can be introduced in Eq. \[tff\_R\] by adding to the lowest order calculation a term proportional to $Q^{-2},$ $$Q^{2}F(Q^{2})=\frac{\sqrt{2}f_{\pi}}{3}\int_{0}^{1}\frac{dx}{x+\frac{M^{2}% }{Q^{2}}}\phi_{\pi}(x,Q^{2})+\frac{C_{3}}{Q^{2}}. \label{tff_R_T3}%$$ Using a constant $\pi$ DA the matching condition becomes [@Noguera:2010fe], $$\frac{\sqrt{2}f_{\pi}}{3}ln\frac{Q_{0}^{2}+M^{2}}{M^{2}}+\frac{C_{3}}% {Q_{0}^{2}}=\frac{F\left( 0\right) \,Q_{0}^{2}}{1+a\frac{Q_{0}^{2}}% {m_{\pi^{0}}^{2}}}, \label{cont3}%$$ with $Q_{0}=1$ GeV. This equation allows to determine $M,$ once we have fixed the value of $C_{3}.$ ![We show the result for the transition form factor in our formalism for $M=0.690$ GeV, $a=0.032$ and the value of $C_{3}= 2.98\,10^{-2}$ GeV$^3$ corresponding to $30\%$ of the contribution at the matching point at $Q_{0}=1$ GeV (solid line).The lower plot shows the detailed behavior for low virtuality.The dotted curve represents the higher twist contribution.Data are taken from CELLO [@Behrend:1990sr], CLEO [@Gronberg:1997fj], BABAR [@Aubert:2009mc] and Belle [@Uehara:2012ag].[]{data-label="Fig3"}](Fig3a.eps) -1.3cm We analyze here the sensitivity of the data to the various parameters involved. We keep as close as possible to our previous fit analyzing the data with respect to small variations in the low virtuality parameter $a$ and in the higher twist parameter $C_3$. In Fig.\[Fig1\] we show the effect of the precision in the determination of the monopole parametrization. We see that as $a$ increases from $0.032$ to $0.036$, i.e. within the error bars, the $\pi TFF$ decreases. The sensitivity to $C_3$ is shown in Fig. \[Fig2\] and we note that as the value of $C_3$ increases from $C_{3}=0.99\,10^{-2}$GeV$^{3}$ , which corresponds to a $10\%$ contribution to the form factor at $Q_0$, to $2.98\,\ 10^{-2}$ GeV$^{3}$ , which corresponds to a $30\%$ contribution, again the value of the $\pi TFF$ decreases. Thus a small increase in $a$ and $C_3$ moves our result toward the Belle data. Finally, in Fig. \[Fig3\] we plot the better fit ($\chi^2/dof=1.21$) taking into account all the world data which corresponds to $a=0.032$ with the $C_3$ term at the $30\%$ value. We stress that there is no strong correlation between $a$ and $C_3$ as long as $a$ is kept within its experimental error bars. Thus the fit is quite stable with respect to the parameters of the low energy model. The fit to the data is excellent with a very small variation of the $1/Q^2$ contribution at $Q_0$ from previous fit, i.e. from $20\%$ to $30\%$. It must be said, before entering the discussion of this fit, that in our previous work [@Noguera:2010fe] we pointed out that the average value of the highest energy data points of BABAR were too large, a conclusion reached also by other analyses [@Mikhailov:2009sa; @Dorokhov:2009zx]. In Fig. \[Fig3\] we show not only the fit for $30\%$ contribution of $C_3/Q^2$ at $Q_0$, but its behavior for higher values of $Q^2$. As can be seen, also stressed in our previous work, this contribution is small in size. However, and this an important outcome of our analysis, it is instrumental in fixing the initial slope at the matching point, which determines, after evolution, the high energy behavior of the form factor. In our opinion the Belle data confirm the BABAR result that the $\pi TFF$ crosses the asymptotic QCD limit. This limit is well founded under QCD assumptions, but nothing is known of how this limit is reached, if from above or from below. BABAR and Belle data suggest that the limit is exceeded around $10-15$ GeV. Our calculation is consistent with this result. The necessary growth of the $\pi TFF$ between $5-10$ GeV to achieve this crossing is in our case an indication of nonperturbative behavior and $C_3/Q^2$ contribution at low virtuality. The determination of the crossing point is a challenge for any theoretical model and therefore, the precise experimental determination of it is of relevance. Many models fail to achieve this crossing because their pion DA is defined close to its asymptotic form. The pion DA can be expressed as a series in the Gegenbauer polynomials, $$\phi_{\pi}\left( x,Q^{2}\right) =6x\left( 1-x\right) \left( 1+\sum_{n\left( even\right) =2}^{\infty}a_{n}\left( Q^{2}\right) \,C_{n}^{3/2}\left( 2x-1\right) \right)$$ We can compare different models by looking at the values of the coefficients of the expansion $a_{n}\left(Q^{2}\right)$. In our case, at $Q^{2}=1$ GeV$^{2}$ many $a_{n}$ coefficients are significant, but we focus our attention in a few terms: $a_{2}=0.389,$ $a_{4}=0.244$ and $a_{6}=0.179$. At $Q^{2} =4$ GeV$^{2}$ we obtain the values $a_{2}=0.307,$ $a_{4}=0.173$ and $a_{6}=0.118$, which are close to those obtained by Polyakov [@Polyakov:2009je]. Consistently, our result for the $\pi TFF$ is similar to that obtained in ref. [@Polyakov:2009je]. At $Q^{2}=5.76$ GeV$^{2}$ we obtain $a_{2}=0.292,$ $a_{4}=0.161$ and $a_{6}=0.108$, which are very different from those of ref. [@Bakulev:2012nh]. These author use for their fit BABAR data for the $\eta TFF$ [@BABAR:2011ad], together with the pion data. It is therefore not a surprise that these authors come to a different conclusion, namely, that the Belle and the BABAR data cannot be reproduced to the same level of accuracy within the Light Cone Sum Rules approach [@Bakulev:2011rp]. However, in an extension of the ideas developed in the present paper to the $\eta$ case studied in ref. [@Noguera:2011fv] looking at the state $\left\vert q\right\rangle =\frac{1}{2}\left( \left\vert u\,\bar{u}\right\rangle +\left\vert d\,\bar{d}\right\rangle \right)$ a very different structure of the $a_{n}$ coefficients to that of the pion arises. At $Q^{2}=1$ GeV$^{2},$ the values of the coefficients are $a_{2}=0.134$ and $a_{4}=0.352$ or, equivalently, at $Q^{2} = 5.76$ GeV$^{2}$ we have$\ a_{2}=0.101$ and $a_{4}=0.232.$ Therefore, that study does not supports the combined use of both data sets We have developed a formalism to describe the $\pi TFF$ on all experimentally accessible range, and hopefully beyond. The formalism is based on a two energy scale description. The formulation in the low energy scale is nonperturbative, while that of the high energy scale is based on perturbative QCD. The two descriptions are matched at an energy scale $Q_{0}$ called hadronic scale [@Traini:1997jz; @Noguera:2005cc]. We stress the crucial role played by the nonperturbative input at the level of the low energy description. It is an important outcome of this calculation the role played by the $1/Q^2$ power correction term in determining the slope of the data at high $Q^2$, despite the fact that they do almost not contribute to the value of the $\pi TFF$ . We have used a flat $\pi$ DA, i.e. a constant value for all $x$ [@Radyushkin:2009zg; @Polyakov:2009je], which with our normalization becomes $\phi(x)=1$. Our choice has been motivated by chiral symmetry [@Noguera:2010fe]. Model calculations, Nambu-Jona-Lasinio (NJL) [@Anikin:1999cx; @Praszalowicz:2001wy; @RuizArriola:2002bp; @Courtoy:2007vy] and the “spectral” quark model [@RuizArriola:2003bs], give a constant $\pi$ DA. The $\pi TFF$ calculated in these models, however, overshoots the data [@Broniowski:2009ft], emphasizing the importance of QCD evolution. The calculation shown proves that the BABAR and Belle results can be accommodated in our scheme, which only uses standard QCD ingredients and low energy data. Moreover, at the light of our results, we confirm that at $40$ GeV$^2$ we have not yet reached the asymptotic regime which will happen at higher energies. We would like to thank A. V. Pimikov and M. V. Polyakov for useful comments.This work has been partially funded by the Ministerio de Economía y Competitividad and EU FEDER under contract FPA2010-21750-C02-01, by Consolider Ingenio 2010 CPAN (CSD2007-00042), by Generalitat Valenciana: Prometeo/2009/129, by the European Integrated Infrastructure Initiative HadronPhysics3 (Grant number 283286). [99]{} S. Uehara [*et al.*]{} \[The Belle Collaboration\], arXiv:1205.3249 \[hep-ex\]. B. Aubert [*et al.*]{} \[The BABAR Collaboration\], Phys. Rev. D [**80**]{}, 052002 (2009) \[arXiv:0905.4778 \[hep-ex\]\]. G. P. Lepage and S. J. Brodsky, Phys. Rev. D **22** (1980) 2157. V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. **112** (1984) 173. S. Noguera and V. Vento, Eur. Phys. J. A [**46**]{}, 197 (2010) \[arXiv:1001.3075 \[hep-ph\]\]. 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A. P. Bakulev, S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, Phys. Rev. D [**84**]{}, 034014 (2011) \[arXiv:1105.2753 \[hep-ph\]\]. S. Noguera and S. Scopetta, Phys. Rev. D [**85**]{} (2012) 054004 \[arXiv:1110.6402 \[hep-ph\]\]. M. Traini, A. Mair, A. Zambarda and V. Vento, Nucl. Phys. A **614** (1997) 472. S. Noguera and V. Vento, Eur. Phys. J. A **28** (2006) 227 \[arXiv:hep-ph/0505102\]. I. V. Anikin, A. E. Dorokhov and L. Tomio, Phys. Lett. B [**475**]{} (2000) 361 \[hep-ph/9909368\]. M. Praszalowicz and A. Rostworowski, Phys. Rev. D **64** (2001) 074003 \[arXiv:hep-ph/0105188\]. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D **66** (2002) 094016 \[arXiv:hep-ph/0207266\]. A. Courtoy and S. Noguera, Phys. Rev. D **76** (2007) 094026 \[arXiv:0707.3366 \[hep-ph\]\]. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D **67** (2003) 074021 \[arXiv:hep-ph/0301202\]. W. Broniowski and E. R. Arriola, arXiv:0910.0869 \[Unknown\].

--- abstract: 'Performance of triple GEM prototypes has been evaluated by means of a muon beam at the H4 line of the SPS test area at CERN. The data from two planar prototypes have been reconstructed and analyzed offline with two clusterization methods: the center of gravity of the charge distribution and the micro Time Projection Chamber ($\mu$TPC). GEM prototype performance evaluation, performed with the analysis of data from a TB, showed that two-dimensional cluster efficiency is above 95% for a wide range of operational settings. Concerning the spatial resolution, the charge centroid cluster reconstruction performs extremely well with no magnetic field: the resolution is well below 100 $\mu m$ . Increasing the magnetic field intensity, the resolution degrades almost linearly as effect of the Lorentz force that displaces, broadens and asymmetrizes the electron avalanche. Tuning the electric fields of the GEM prototype we could achieve the unprecedented spatial resolution of 190 $\mu m$ at 1 Tesla. In order to boost the spatial resolution with strong magnetic field and inclined tracks a $\mu$TPC cluster reconstruction has been investigated. Such a readout mode exploits the good time resolution of the GEM detector and electronics to reconstruct the trajectory of the particle inside the conversion gap. Beside the improvement of the spatial resolution, information on the track angle can be also extracted. The new clustering algorithm has been tested with diagonal tracks with no magnetic field showing a resolution between 100 $\mu m$ and 150 $\mu m$ for the incident angle ranging from 10$^\circ$ to 45$^\circ$. Studies show similar performance with 1 Tesla magnetic field. This is the first use of a $\mu$TPC readout with a triple GEM detector in magnetic field. This study has shown that a combined readout is capable to guarantee stable performance over a broad spectrum of particle momenta and incident angles, up to a 1 Tesla magnetic field.' author: - | R. Farinelli$^a$$^b$ \*,M. Alexeev$^c$, A. Amoroso$^c$, F. Bianchi$^c$, M. Bertani$^d$, D. Bettoni$^a$, N. Canale$^a$$^b$, A. Calcaterra$^d$, V. Carassiti$^a$, S. Cerioni$^d$, J. Chai$^c$, S. Chiozzi$^a$, G. Cibinetto$^a$, A. Cotta Ramusino$^a$, F. Cossio$^c$, F. De Mori$^c$, M. Destefanis$^c$, T. Edisher$^d$, F. Evangelisti$^a$, L. Fava$^c$, G. Felici$^d$, E. Fioravanti$^a$, I. Garzia$^a$$^b$, M. Gatta$^d$, M. Greco$^c$, D. Jing$^d$, L. Lavezzi$^c$$^e$, C. Leng$^c$, H. Li$^c$, M. Maggiora$^c$, R. Malaguti$^a$, S. Marcello$^c$, M. Melchiorri$^a$, G. Mezzadri$^a$$^b$, G. Morello$^d$,S. Pacetti$^f$, P. Patteri$^d$, J. Pellegrino$^c$, A. Rivetti$^c$, M. D. Rolo$^c$, M. Savrie’$^a$$^b$, M. Scodeggio$^a$$^b$, E. Soldani$^d$, S. Sosio$^c$, S. Spataro$^c$, L. Yang$^c$.\  \ $^a$ INFN - Sezione di Ferrara, $^b$ University of Ferrara,$^c$ INFN - Sezione di Torino,\ $^d$ INFN - Sezione di Frascati, Physics dept., $^e$ IHEP, Beijing, $^f$ University of Perugia.\ E-mail address: rfarinelli@fe.infn.it (R. Farinelli), \*Corresponding author. title: | Development and Test of a $\mu$TPC Cluster Reconstruction for a Triple GEM Detector\ in Strong Magnetic Field --- GEM, CGEM, $\mu$TPC, magnetic field, gas detector. Introduction ============ detectors are instruments used to reveal the charged particles that pass throught them. The primary electron generated by the ionization of the gas are amplified and the electrical signal is collected to measure the spatial position of the particles. Compared to the first technology of gas detector, the Micro Pattern Gas Detector (MPGD) bypass the limits of the diffusion using a pitch/cell size of few hundred $\mu m$ that improves granularity and rate capability [@chinese]. Among the MPGD family, the Gas Electron Multiplier (GEM) technology invented by F.Sauli in 1997 [@sauli] exploits 50 $\mu m$ kapton foils covered on both sides by 5 mm of copper, and pierced with 50 $\mu m$ holes to amplify the electron signal by applying an electric field up to 10$^5$ kV/cm between the two faces. The signal is then readout by means of external strips of pads. A triple GEM detector is built by three GEM foils to exploit three multiplication stages. Fig. \[gemm\] shows a schematic example. The electric fields between the electrodes and the gas mixture also determine the performance of the detector [@gem1; @gem2]. The state of the art of the GEMs reports a spatial resolution below 100 $\mu m$ weighting the charge distribution if no magnetic field is present, otherwise the charge information is not enought to measure the position efficiently. A study on another MPGD detector, the MicroMegas, introduced the idea to use the time information to measure the position in presence of low magnetic field [@omegas]. The aim of this work is to demonstrate that the triple GEM technology can achieve spatial resolution of about 100-150 $\mu m$ in presence of high magnetic field. Detector geometry and working contributions =========================================== In a triple GEM the five electrodes (cathode, three GEM and anode) define four gaps, namely “conversion” and “drift”, between the cathode and the first GEM foil, “transfer 1”, “transfer 2”, within the GEMs, and “induction” where the signal is induced on the readout plane. The avalanche diffusion and the charge collection depend largely on the value of the electric fields within those gaps. The gain that this detector can achieve is about 10$^4$-10$^5$ [@gain]. The electrical signal is collected by a segmented anode with two views with 650 $\mu m$ pitch strips. The strips geometry has been optimized to minimize the overlap between the strips [@jagged]. The charge is collected by the strips as function of the time and it is used to reconstruct the signal. Only the electrons generated within the cathode and the first GEM are amplified three times so these electrons dominates the measurement the position of the charge particle. The gas mixture studied in this work are Argon-CO$_2$ (70:30) and Argon-iC$_4$H$_{10}$ (90:10)[^1]. These are composed by a streamer (Argon) in order that the multiplication can take place if an high electric field is present, and a quencher (CO$_2$ or iC$_4$H$_{10}$) to reduce the number of seconday electron generated and to prevent the discharge and to operate the detector safely. An electric field of 3 kV/cm between the GEMs and 5 kV/cm between the anode and the third GEM are used to efficiently extract the signal [@gem1] meanwhile in the drift gap is used a value of 1.5 kV/cm. This particular field, named drift field, is responsable of the drift and diffusion properties of the primary electron generated in this region. ![Efficiency measurements of a triple GEM as function of the detector gain. The results show an efficiency plateau above the 95% for the bidimensional reconstruction for a gain higher that 4000 in both gas mixtures.[]{data-label="eff"}](efficicency.png){width="2.5in"} ![Spatial resolution of an analague (black) and digilat (red) readout as function of the number of fired strip. CC after a certain cluster size reeaches a resolution of 80 $\mu m$ meanwhile the digital readout degrades linearly with the cluster size.[]{data-label="digi"}](ana_vs_digi.png){width="2.5in"} ![Optimization of the CC for two triple GEM with 3 and 5 mm conversion gap in Argon-iC$_4$H$_{10}$ gas mixture as function of the drift field. A 1 Tesla magnetic field is present. As the field increases the electronic avalanche spread decreasees and the resolution improves up to 190 $\mu m$.[]{data-label="ariso"}](ariso_drift_scan.png){width="2.5in"} Experimental setup ================== Two protoypes of triple GEM with an active area of 10x10 cm$^2$ and with 3 and 5 mm drift gap and 2 mm in the others gaps have been characterized with a muon beam of about 150 GeV/c momentum located at the H4 line at CERN within the RD51 Collaboration [@RD51]. These prototypes have been placed within a magnet dipole orthogonal to the beam, Goliath, that can reach 1.5 Tesla in both polarities. A tracking system of four triple GEM, two behind and two below the prototypes, is used to extract the spatial resolutions of the prototypes and the efficiency. The readout electronic is based on the APV25 hybrid and the Scalar Readout System [@APV25] that allow to acquire the information about the charge and the time from each strip by sampling the charge each 25 ns. Reconstruction algorithms measure the position throught the clusterization of the contiguous fired strip with a charge above the threshold of 1.5 fC. To measure the position associated to each cluster two algorithm are used. The Charge Centroid (CC) or center of gravity method exploits the charge values of the readout strips and performe a weighted average of the strip positions and their charge. The micro Time Projection Chamber ($\mu$TPC) method exploit the single strip time information to perform a local track reconstruct of the charge particle in few mm drift gap. Each fired strip give two coordinates, one is the strip position and the other (perpendicular to the strip plane) can be reconstructed from the time measurement using the drift velocity of the electron and the Lorentz angle if the magnetic field is present. These values can be extracted from the literature, Garfield simulation or even from direct measurement from the detector [@omegas; @gas; @gem2]. The $\mu$TPC associates to each fired strip a bidimensional point and the ensemble of the strips are lineary fitted to measure the track. The value that correspond to the middle of the gap is associated to the measured position. Results ======= The voltage difference between the two faces of the GEMs determine the electric field then the detector gain. The first study performed is focused to the research of the operative point, While the gain increase the charge collected increases and therefore the number of fired strips. From experimental data in Fig. \[eff\] at a gain of 4000 the detector reaches the efficiency plateau for both the studied gas mixtures. Despite to previous reconstruction techniques, as the digital readout, the CC reaches its best performance if the cluster size is higher than 2.5 (Fig. \[digi\]). In this configuration, without magnetic field and with orthogonal tracks, the electronic avalanche has a gaussian profile and it is not significantly affected by the diffusion effects. The spatial resolution achieved is 80 $\mu m$. It is not possible to use the $\mu$TPC in this configuration. Reconstruction in magnetic field -------------------------------- The study proceed with the effects of the magnetic field on the performance of the prototypes. As the magnetic field increases, up to 1 Tesla, the Lorentz force acts on the electrons and it increases the dimension of the avalanche. The charge distribution collected at the anode is no more gaussian and this leads to a degradation of the performance of the CC to 300 $\mu m$ in Argon-CO$_2$ and 380 $\mu m$ in Argon-iC$_4$H$_{10}$. A strong dependence between the effectiveness of the CC and the drift field has been found since the direction of the electron is described by the Lorentz angle then by the drift field. The performance of the prototypes has been studied as function of this field from 0.5 to 2.5 kV/cm because in this range the Lorentz angle is extremely variable in Argon-iC$_4$H$_{10}$ (Fig. \[ariso\]). The behavior of the spatial resolution copy the one of the Lorentz angle and gives its best performance at 2.5kV/cm where the angle has a smaller value. At this value it reaches the 190 $\mu m$ with the prototype with 3 mm of drift gap because the diffusion effect are less than the 5 mm prototype. Each reasult discussed until now has been measured with a beam orthogonal to the prototypes. The introduction of not orthogonal tracks leads to another complication for the CC that degrades as the incidence angle vary from the orthogonal one (Fig. \[b0\]). The CC gives its best performance if the primary electrons are concentrated around the line that describe their drift to the cathode. As the incident angle increase the region where the primary are generated is larger and the signal collected at the anode is no more gaussian. ![CC and $\mu$TPC spatial resolution as function of the incident angle with Argon-iC$_4$H$_{10}$ gas mixuters. For orthogonal track (0$^\circ$) CC is the best algorithm but as the angle increses the $\mu$TPC reaches 100-150 $\mu m$ spatial resolution. Similar results are obtained with Argon-C0$_2$.[]{data-label="b0"}](angleB0.png){width="2.5in"} ![CC and $\mu$TPC as function of the incident angle in 1 Tesla magnetic field and Argon/CO$_2$ gas mixture. Similar results are obtain in Argon-iC$_4$H$_{10}$.[]{data-label="b1"}](agnleB1.png){width="2.5in"} ![Optimization of the spatial resolution of the $\mu$TPC algorithm as function of the drift field in Argon/CO$_2$.[]{data-label="tpc"}](optTPC.png){width="2.5in"} The $\mu$TPC ------------ This degradation of the CC makes the introduction of the $\mu$TPC of fundamental importance. Although in presence of orthogonal track and no magnetic field this algorithm is not efficent, as the incident angle increases the resolution improves and it reaches a plateu between 100 and 150 $\mu m$. Fig. \[b0\] shows the experimental results of the CC and $\mu$TPC as function of the angle (0$^\circ$ corresponds to the orthogonal tracks). The two algorithm are anti-correlated and together give a stable performance. The best results have been obtained with the 5 mm drift gap prototype because as the gap is greater as the number of fired strips is higher and this helps the temporal reconstruction. Let’s examine now the performance results measured in 1 Tesla magnetic field and different incident angle. The Lorentz force gives a preferential direction to the drift of the electron and the combination of the Lorentz angle and the incident angle has to be taken into account. It is named focusing effect when the Lorentz angle is close to the incident angle because in this configuration the generated primary electrons drift along the same line of the charge particle and the spatial distribution at the anode is concentrate in few strips. If the incident angle departs from the Lorentz one then the defocusing effect take place and counterwise to the focusing effect the electron avalanche is projected on more strips. In the high defocusing region the diffusion effect worsens the $\mu$TPC algorithm. In Fig. \[b1\] is shown the behavior of the CC and $\mu$TPC as function of the incident angle in magnetic field. The performance are similar at the case without magnetic field but in this case che maximum focusing point coincide with the Lorentz angle: 20$^\circ$ in Argon-CO$_2$, 26$^\circ$ in Argon-iC$_4$H$_{10}$. Here the CC returns its best performance, results comparable to the ones without magnetic field and orthogonal tracks. As the angle departs from this point as the CC degrades and the $\mu$TPC reaches a stable blehavior around 130-150 $\mu m$. The $\mu$TPC algorithm is the most efficient and as the CC it can be optimize with a drift field study. Conterwise the CC case where the only one parameter to influence it is the Lorentz angle, here it has to be included the dependence from the temporal resolution and the drift velocity [@omegas]. A drift field scan has been performed and a value of 1.25kV/cm has been found to improve the $\mu$TPC performance (Fig. \[tpc\]). Conclusion ========== The performances of a triple GEM in magnetic field have been measured and optimized throught several test beam. The detector amplifies the signal created in the ionization of the gas and measure the signal charge and time with a segmented anode. This information are used to reconstruct the charge particle impact position with two algorithm, CC and $\mu$TPC, that are totally anti-correlated. The combination of these two allow to achieve a stable spatial resolution with result of about 120 $\mu m$ that goes beyond the currect state of the art for this technology in high magnetic field. The studied technology is of considerable interest in the scientific comunity because its mechanical properties, its electrical stability and the possibility the be shaped the desider form and the studied described in this report can be extended to the large area detector needed in the high energy physics experiments. [1]{} Chinese Physics C, Vol. 38, Num 9, Sept 2014, 422-426 F. Sauli, *Nucl. Instrum. Methods A*, 386:531, 1997 S. Bachmann et al., *Nucl. Instrum. Methods A*, 438:376, 1999 A. Peisert and F. Sauli, Drift Diffusion of Electrons in Gases, CERN 84-08, 1984 M. Iodice, *JINST*, 9 C01017, 2014 M. Ziegler, P. Sievers and U. Straumann, *Nucl Instrum. Methods A*, 471:260, 2001 I. Garzia et al., *PoS TIPP2014*, 292, 2014 cern.ch/RD51-Public/ M.J. French, et al., *Nucl. Instrum Methods A*, 466:359, 2001 Y. Assran and A. Sharma, *arXiv*, 1110.6761, [^1]: from now on Argon-CO$_2$ (70:30) and Argon-iC$_4$H$_{10}$ (90:10) will be refered as Argon-CO$_2$ and Argon-iC$_4$H$_{10}$

--- --- Research highlight 1 Research highlight 2 [00]{}

--- abstract: 'A litany of research has been published claiming strong solar influences on the Earth’s weather and climate. Much of this work includes documented errors and false-positives, yet is still frequently used to substantiate arguments of global warming denial. This manuscript reports on a recent study by Badruddin & Aslam (2014), hereafter BA14, which claimed a highly significant ($p=1.4\times10^{-5}$) relationship between extremes in the intensity of the Indian monsoon and the cosmic ray flux. They further speculated that the relationship they observed may apply across the entire tropical and sub-tropical belt, and be of global importance. However, their statistical analysis—and consequently their conclusions—were wrong. Specifically, their error resulted from an assumption that their data’s underlying distribution was Gaussian. But, as demonstrated in this work, their data closely follow an ergodic chaotic distribution biased towards extreme values. From a probability density function, calculated using a Monte Carlo sampling approach, I estimate the true significance of the BA14 samples to be $p=0.91$.' address: 'Postboks 1047 Blindern, 0316, Oslo, Norway' author: - 'Benjamin A. Laken' bibliography: - 'B\_refs.bib' title: 'Reply to ‘Influence of cosmic ray variability on the monsoon rainfall and temperature’: a false-positive in the field of solar—terrestrial research' --- Monsoon ,Solar Variability ,Cosmic ray flux ,Statistics Introduction {#intro} ============ @Bad14, hereafter BA14, recently reported a solar—terrestrial link between the cosmic ray (CR) flux and the Indian Monsoon, which they suggested may have implications of global importance and support so-called ‘Cosmoclimatology’ [@Svensmark07]. This work demonstrates the way in which their findings were erroneous. BA14 based their claims on highly significant statistical relationships obtained from composite (epoch-superposed) samples. Specifically, they examined linear changes in monthly neutron monitor counts, analysed over $m=5$ month periods (during the months of May–September) from two samples, each comprised of $n=12$ years of monthly resolution data: ie. composites from two matrices of $n \times m$ elements. The composites—which are vectors of the matrices averaged in the $n$-dimension—are respectively referred to as the ‘Drought’ and ‘Flood’ samples (which I shall also denote here as [**D**]{} and [**F**]{}), and represent the years of weakest and most intense monsoon precipitation respectively, recorded from 1964–2012. These data are shown in Figure  \[fig:composites\], with the May–September periods of the composites emphasised: at first glance, it is true that these data show a linear change during the period highlighted, and also show anti-correlated between the [**D**]{} and [**F**]{} samples. Specifically, BA14 evaluated the Pearson’s correlation coefficients ($r$-values) of the [**D**]{} and [**F**]{} samples, and used a standard two-tailed Student’s t-test (which assumes a Gaussian distribution) to test their probability ($p$) values. They obtained values of $r=-0.95$ ($p=0.01$) for [**D**]{}, and $r=0.99$ ($p=1\times10^{-3}$) for [**F**]{}. Cumulatively, the $p$-value value was $1.4\times10^{-5}$: i.e. such a result should occur by chance only $1 / 71942$ times. BA14 interpreted these results to mean the lowest (and highest) precipitation volumes recording during the Indian monsoon period correspond to statistically significant decreases (and increases) in CR flux. ![Reproduction of the composite samples of BA14, showing the monthly-resolution pressure-adjusted neutron monitor count rate (units: counts min.$^{-1} \times 10^{3}$) from Oulu station (65.05$^{o}$ N, 25.47$^{o}$ E, 0.8 GV) occurring during 12 years of Indian monsoon ‘Drought’ (D) and ‘Flood’ (F) conditions. Composite means (in the matrix $m$-dimension) are plotted, with error ranges shown as $\pm$1 standard error of the mean (SEM) value. The period of May–September, selected by BA14 for Pearson’s correlation analysis, has been emphasised in the plots.[]{data-label="fig:composites"}](Figs/Composite_samples2.pdf) From these apparently highly significant CR flux changes, BA14 concluded that a solar—monsoon link exists, and operates via a theoretical CR flux cloud connection. They speculated that this connection impacts the monsoon in the following manner: increases in the CR flux enhance low cloud, rainfall, and surface evaporation, and also consequently decrease temperature (and vice versa). They further speculated that their findings may be expanded to the whole tropical and sub-tropical belt, and as a result may impact temperatures at a global scale. However, the significance of the [**D**]{} and [**F**]{} samples—and consequently the conclusions—of BA14 are wrong. This error resulted from the assumption of a Gaussian data distribution, which is not true of their data, as I shall demonstrate. Moreover, of broader interest beyond the BA14 study is a recognition of a litany of fallacious solar—terrestrial studies: many of which have been re-examined in detail [e.g. by @Pittock78; @Pittock09; @Farrar00; @Krist00; @Damon04; @Sloan08; @Calogovic10; @Benestad09; @Laken12; @Laken13ERL]. False-positives within this field are of particular concern, as they contribute to a politically-motivated global warming denial movement. Providing material for groups intending to affect policy, such as the Heartland Institute’s Nongovernmental International Panel on Climate Change (NIPCC) or the Centre for Study of Carbon Dioxide and Global Change [@Dunlap10]. Encouragingly though, a recent shift to open-access, and highly-repeatable workflows offers an opportunity for rapid communal development (and cross-checking) across a broad range of fields, including solar–terrestrial studies: at minimum, such approaches can more effectively facilitate the peer-review process, and enhance the quality and reliability of future publications. To illustrate this, this manuscript is supported by an accompanying iPython Notebook [@iPY], enabling users of the open-source software to easily check, repeat, and alter the analysis. This notebook (and all accompanying data) are openly available from figshare [@Laken15]. Analysis {#anal} ======== The CR flux oscillates between high and low values as solar activity progresses from minimum to maximum during the $\sim$11-year solar Schwabe cycle. Consequently, over the 5-month timescales with which the BA14 study was concerned, the CR flux spends relatively little time at stable values. As a result, the population of $r-$values which can be derived from 5-month composites of these data are ergodic and biased towards extreme values. Using a Monte Carlo (MC) sampling approach, I have constructed 100,000 composites of equal dimensions to the original [**D**]{} and [**F**]{} samples from the neutron monitor data, and obtained $r$-values over May–September periods: I note that the May–September restriction is not strictly required, as in reality the only requirement is that the MC-samples span an identical time-period (5-months) to the original samples. I refer to these data as $H_{0}$ samples, as, by drawing these data randomly, they represent tests of the null hypothesis [for more details on this method applied to solar—terrestrial studies see @Laken13]. A probability density function (PDF) of these data are presented in Figure  \[fig:density\]. For comparison, a normalised Gaussian distribution—assumed by BA14—is also shown (dashed line). ![Probability density function (PDF) of $r$-values drawn from 100,000 null hypothesis ($H_{0}$) composites (from Monte Carlo samples of the Oulu neutron monitor data). For comparison, a normalised Gaussian distribution (with a mean and standard deviation of $7.1\times10^{-1}$ and $6.5\times10^{-1}$) is plotted on the dashed line: BA14 wrongly assumed the data possessed this distribution, and consequently, this was the source of their error. A logistical map, which predicts chaotic distributions (given in Equation  \[eq:log\]), is plotted on the solid line. A 4$^{th}$ order polynomial fit to the $H_{0}$ population is plotted on the dotted line: this fit can be used to calculate the probability ($p$) of a given $r$-value.[]{data-label="fig:density"}](Figs/Density_models.pdf) I have used two methods to model the PDF values: Firstly, a 4$^{th}$ order polynomial fit to the $H_{0}$ samples (shown on Figure  \[fig:density\] as the dotted line), of the function $0.03625x^{4} - 0.0002797x^{3} - 0.0037x^{2} + 0.0007109x + 0.01964$. And secondly, analytically using a Logistic map [as introduced by @May76], which predicts the distribution of chaotically oscillating data (shown as the solid black line in Figure  \[fig:density\]). The formula for this is given in Equation  \[eq:log\] [@Ruelle89], where $p$ is the probability, and $u$ is the variable (in this case $r$-values). $$\label{eq:log} p(u)=\frac{1}{\pi\sqrt{1-u^{2}}}$$ The distribution of $r$-values appear to follow the Logistic map to a high-degree, indicating that the solar-cycle is oscillating chaotically. Indeed, the chaotic nature of the solar cycle has been well described [e.g. @Mundt91; @Krem94; @Rozelot95; @Char01; @Hans10; @Hans13]. Disagreement between the Logistic map and the PDF occurs at the most extreme values, where $r$&lt;$-0.9$ or $r$&gt;$0.9$. $$\label{eq:pval} p=1-(0.03625x^{4} - 0.0002797x^{3} - 0.0037x^{2} + 0.0007109x + 0.01964)$$ As the polynomial fit accurately follows the PDF, it can be readily used to estimate the $p$-value associated with a given $r$-value using Equation  \[eq:pval\]. From this, I calculate that the [**D**]{} and [**F**]{} samples possess $p$-values of 0.954 and 0.948 respectively, resulting in a cumulative $p$-value of 0.91, i.e. a chance of occurring under the null hypothesis of $1 / 1.1$. This result has a $p$-value four orders of magnitude larger than that estimated by the Student’s t-test approach of BA14, and is virtually guaranteed by chance. Consequently, I conclude that the high $r$-values obtained in the BA14 composites do not support a relationship between extremes in Indian precipitation during the monsoon and co-temporal changes in the CR flux, but instead they are simply among the most commonly obtained values based on this sampling approach. Discussion {#diss} ========== The Cosmics Leaving OUtdoor Droplets (CLOUD) experiment at CERN has demonstrated that ion-mediated nucleation may lead to enhancements in aerosol formation of 2–10 times neutral values under specific laboratory conditions—low temperatures characteristic of the upper-troposphere, and with low concentrations of amines and organic molecules—however, this effect is absent under conditions more closely representing the lower troposphere [@Kirkby11; @Almeida13]. Despite this, even if we assume that a significant nucleation of new aerosol particles form with solar activity, climate model experiments (which include aerosol microphysics schemes) have found that this would still not result in a significant change in either concentrations of cloud condensation nuclei or cloud properties. This is because the majority of the newly formed particles are effectively scavenged by pre-existing larger aerosols [@Pierce09; @Snow11; @Dunne12; @Yu12]. These conclusions are supported by satellite and ground-based observations [e.g. @Erlykin09; @Kulmala10; @Laken12Jclim; @Benestad13; @KT13]. For these and additional reasons, the IPCC AR5 concluded that the CR flux has played no significant role in recent global warming [@Boucher13]. The numerous pitfalls into which solar—terrestrial studies in particular may fall, were lucidly outlined nearly 40-years ago by @Pittock78. Despite this, many studies with improper statistical methods, black-box approaches, and ad-hoc hypotheses still frequently appear. This problem is prominent within the field of solar—terrestrial studies. Consequently, the literature is replete with cases of demonstrated false-positives [e.g. @Friis91; @Marsh00; @Shaviv03; @Scafetta08; @Svensmark09; @Dragic11], many of which have been (and continue to be) used as the basis for claims behind global warming denial [e.g. such as in @Idso09; @Idso13], immediately making cases such as the one described in this manuscript a serious prospect in need of address. Acknowledgements {#ack} ================ I would like to thank Dr. Beatriz González-Merino (Instituto de Astrofísica de Canarias), Dr. Jaša Čalogović (University of Zagreb), and Professors Frode Stordal and Joseph H. LaCase (University of Oslo), for helpful discussions. I also acknowledge the Python and iPython project (http://ipython.org). Data sources: Prof. Ilya Usoskin and the Sodankyla Geophysical Observatory for the Oulu neutron monitor data (http://cosmicrays.oulu.fi), and the Indian Institute of Tropical Meteorology (www.tropmet.res.in) for the monthly precipitation data (which, in this work, were taken directly from BA14). References {#refs} ==========

--- author: - Tim Kaler - William Kuszmaul - 'Tao B. Schardl' - Daniele Vettorel bibliography: - 'allpapers.bib' title: | Cilkmem: Algorithms for Analyzing the Memory\ High-Water Mark of Fork-Join Parallel Programs[^1] --- [^1]: MIT Computer Science and Artificial Intelligence Laboratory. Supported by NSF Grants CCF 1314547 and CCF 1533644. William Kuszmaul is supported by a Fannie & John Hertz Foundation Fellowship; and by a NSF GRFP Fellowship.

--- abstract: | We calculate the superconformal index for $\mathcal{N}\!=\!6$ Chern-Simons-matter theory with gauge group $U(N)_k\times U(N)_{\!-\!k}$ at arbitrary allowed value of the Chern-Simons level $k$. The calculation is based on localization of the path integral for the index. Our index counts supersymmetric gauge invariant operators containing inclusions of magnetic monopole operators, where latter operators create magnetic fluxes on 2-sphere. Through analytic and numerical calculations in various sectors, we show that our result perfectly agrees with the index over supersymmetric gravitons in $AdS_4\times S^7/\mathbb{Z}_k$ in the large $N$ limit. Monopole operators in nontrivial representations of $U(N)\times U(N)$ play important roles. We also comment on possible applications of our methods to other superconformal Chern-Simons theories. --- \ [ **** ]{} [Seok Kim]{} *Theoretical Physics Group, Blackett Laboratory,\ Imperial College, London SW7 2AZ, U.K.*\ *&* *Institute for Mathematical Sciences,\ Imperial College, London SW7 2PG, U.K.*\ E-mail: [s.kim@imperial.ac.uk]{} Introduction ============ An important problem in AdS/CFT [@Maldacena:1997re] is to understand the Hilbert spaces of both sides. The string or M-theory is put on global AdS, and the dual conformal field theory (CFT) is radially quantized. Partition function encodes the information on Hilbert space. In particular, one has to understand the spectrum of strongly interacting CFT, which is in general difficult, to use the dual string/M-theory to study various phenomena in conventional gravity. With supersymmetry, one can try to circumvent this difficulty by considering quantities which contain possibly less information than the partition function but do not depend on (or depend much more mildly on) the coupling constants controlling the interaction. This has been considered in the context of AdS/CFT. If a superconformal theory has *continuous* parameters, one can construct a function called the superconformal index which does not depend on changes of them [@Kinney:2005ej; @Bhattacharya:2008zy]. The general structure of the superconformal index was investigated in 4 dimension [@Kinney:2005ej], and then in 3, 5, 6 dimensions [@Bhattacharya:2008zy]. See also [@Romelsberger:2005eg]. In all these cases, the superconformal index is essentially the Witten index [@Witten:1982df] and acquires nonzero contribution only from states preserving supersymmetry. The superconformal index was computed for a class of SCFT$_4$ in [@Kinney:2005ej; @Nakayama:2005mf], including the $\mathcal{N}\!=\!4$ Yang-Mills theory. Similar quantity called the elliptic genus was also studied in 2 dimensional SCFT [@Schellekens:1986yi]. The latter index played a major role in understanding supersymmetric black holes in AdS$_3$/CFT$_2$ [@Strominger:1996sh]. In this paper, we study the superconformal index in AdS$_4$/CFT$_3$. Recently, based on the idea of using superconformal Chern-Simons theory [@Schwarz:2004yj] to describe low energy dynamics of M2-branes, and after the first discovery of a class of $\mathcal{N}\!=\!8$ superconformal Chern-Simons theories [@Bagger:2006sk; @Gustavsson:2007vu], $\mathcal{N}\!=\!6$ superconformal Chern-Simons theory with gauge group $U(N)_k\times U(N)_{-k}$ has been found and studied, where the integers $k$ and $-k$ denote the Chern-Simons levels associated with two gauge groups [@Aharony:2008ug]. This theory describes the low energy dynamics of $N$ parallel M2-branes placed at the tip of $\mathbb{C}^4/\mathbb{Z}_k$, and is proposed to be dual to M-theory on $AdS_4\times S^7/\mathbb{Z}_k$. See [@Benna:2008zy; @Hosomichi:2008jb; @Bandres:2008ry; @Aharony:2008gk] for further studies of this theory. Various tests have been made for this proposal. For instance, studies of the moduli space [@Aharony:2008ug] and $D2$ branes [@Mukhi:2008ux; @Aharony:2008ug], chiral operators [@Aharony:2008ug; @Hanany:2008qc], higher derivative correction in the broken phase [@Hosomichi:2008ip; @Baek:2008ws] (see also [@Alishahiha:2008rs]), integrability and nonsupersymmetric spectrum [@integ] have been made. Many of these tests, perhaps except the last example above, rely on supersymmetry in some form. One of the most refined application of supersymmetry to test this duality so far would be the calculation of the superconformal index in the type IIA limit [@Bhattacharya:2008bja]. See also [@Choi:2008za; @Dolan:2008vc]. The superconformal index was calculated in the ’t Hooft limit, in which $N$ and $k$ are taken to be large keeping $\lambda=\frac{N}{k}$ finite. The authors of [@Bhattacharya:2008bja] argue that $\lambda$ can be regarded as a continuous parameter in the ’t Hooft limit. The index calculated in the free Chern-Simons theory, $\lambda\rightarrow 0$, is expected to be the same as the index in the opposite limit $\lambda\gg 1$ from the continuity argument. M-theory can be approximated by type IIA supergravity on $AdS_4\times\mathbb{CP}^3$ in the latter limit, where $\mathbb{CP}^3$ appears as the base space in the Hopf fibration of $S^7/\mathbb{Z}_k$ [@Aharony:2008ug]. The index over multiple type IIA gravitons perfectly agreed with the gauge theory result [@Bhattacharya:2008bja]. It is tempting to go beyond the ’t Hooft limit and calculate the full superconformal index for any value of $k$ (and $N$) for this theory. This is the main goal of this paper. The general index is expected to capture the contribution from states carrying Kaluza-Klein (KK) momenta along the fiber circle of $S^7/\mathbb{Z}_k$, or $D0$ brane charges from type IIA point of view. As $k$ increases, the radius of the circle decreases as $\frac{1}{k}$, and the energy of the KK states would grow. So even when $k$ is very large that weakly coupled type IIA string theory is reliable, our index captures nonperturbative correction to [@Bhattacharya:2008bja] from heavy $D0$ branes. The gauge theory dual to the KK-momentum is argued to be appropriate magnetic flux on $S^2$ [@Aharony:2008ug]. The gauge theory operators creating magnetic fluxes are called the monopole operators, or ’t Hooft operators [@Aharony:2008ug; @Berenstein:2008dc; @Klebanov:2008vq; @Borokhov:2002ib]. These operators are not completely understood to date. Just like the ordinary partition function at finite temperature, the superconformal index for a radially quantized SCFT admits a path integral representation. In this paper we calculate the index from this path integral for the $\mathcal{N}\!=\!6$ Chern-Simons theory on Euclidean $S^2\times S^1$. The index in the sector with nonzero KK-momentum is given by integrating over configurations carrying nonzero magnetic fluxes, which will turn out to be fairly straightforward. Therefore, lack of our understanding on ’t Hooft operators will not cause a problem for us. In fact, monopole operators have been most conveniently studied in radially quantized theories [@Borokhov:2002ib]. Our computation is based on the fact that this path integral is ‘supersymmetric,’ or has a fermionic symmetry. As a Witten index, the superconformal index acquires contribution from states preserving a particular pair of supercharges which are mutually Hermitian conjugates. Calling one of them $Q$, which is nilpotent, $Q^2=0$, the fermionic symmetry of the integral is associated with $Q$. An integral of this kind can be computed by localization. See [@Witten:1992xu] and related references therein for a comprehensive discussion. A simple way of stating the idea is that one can deform the integrand by adding a $Q$-exact term $QV$ in the measure, for any gauge invariant expression $V$, without changing the integral. For a given $V$, one can add $tQV$ to the action $S\!\rightarrow\!S+tQV$ where $t$ is a continuous parameter. With a favorable choice of $V$ as will be explained later, $t$ can be regarded as a continuous coupling constant of the deformed action admitting a ‘free’ theory limit as $t\rightarrow\infty$. As already mentioned in [@Kinney:2005ej], it suffices for the deformed action to preserve only a subset of the full superconformal symmetry, involving $Q$ and symmetries associated with charges with which we grade the states in the index. We only take advantage of a nilpotent symmetry rather than full superconformal symmetry. As in [@Kinney:2005ej; @Sundborg:1999ue; @Aharony:2003sx; @Aharony:2005bq], our result is given by an integral of appropriate unitary matrices. We use our superconformal index to provide a nontrivial test of the $\mathcal{N}\!=\!6$ AdS/CFT proposal. The readers may also regard it as subjecting our calculation to a test against known results from gravity, if they prefer to. In the large $N$ limit, still keeping $k$ finite, our index is expected to be the index over supersymmetric gravitons of M-theory at low energy. In the sectors with one, two and three units of magnetic fluxes (KK-momenta), we provide analytic calculations or evaluate the unitary matrix integral numerically, up to a fairly nontrivial order, to see that the two indices perfectly agree. Similar comparisons with gravitons in dual string theories are made in other dimensions, e.g. for the 2 dimensional elliptic genus index [@de; @Boer:1998ip] and also for the 4 dimensional index in $\mathcal{N}\!=\!4$ Yang-Mills theory [@Kinney:2005ej]. We will find that monopole operators in nontrivial representations of $U(N)\times U(N)$, beyond those studied in [@Aharony:2008ug; @Berenstein:2008dc; @Klebanov:2008vq], play crucial roles for the two indices to agree. An interesting question would be whether our index captures contribution from supersymmetric black holes beyond the low energy limit. In AdS$_3$/CFT$_2$, the contribution to the elliptic genus index from BTZ black holes is calculated and further discussed [@Strominger:1996sh]. See also [@Mandal:2007ug] for a recent study of elliptic genus beyond gravitons. However, in 4 dimension, it has been found that the index does not capture the contribution from supersymmetric black holes [@Kinney:2005ej] in the large $N$ limit, possibly due to a cancelation between bosonic and fermionic states. In this paper, following [@Kinney:2005ej], we consider the large $N$ limit in which chemical potentials are set to order 1 (in the unit given by the radius of $S^2$). The situation here is somewhat similar to $d\!=\!4$ in that a deconfinement phase transition at order $1$ temperature like [@Sundborg:1999ue; @Aharony:2003sx; @Aharony:2005bq] is not found. However, more comment is given in the conclusion. The methods developed in this paper can be applied to other superconformal Chern-Simons theories. $\mathcal{N}\!=\!5$, $\mathcal{N}\!=\!4$ Chern-Simons-matter theories and the gravity duals of some of them are studied in [@Hosomichi:2008jb; @Aharony:2008gk] and [@Gaiotto:2008sd], respectively. There is an abundance of interesting superconformal Chern-Simons theories with $\mathcal{N}\!\leq\!3$ supersymmetry. [@Gaiotto:2007qi] provides a basic framework. For example, some $\mathcal{N}\!=\!3,2$ theories are presented in [@Jafferis:2008qz] with hyper-Kähler and Calabi-Yau moduli spaces. Comments on possible applications of our index to these theories are given in conclusion. The rest of this paper is organized as follows. In section 2 we summarize some aspects of $\mathcal{N}\!=\!6$ Chern-Simons theory and the superconformal index. We also set up the index calculation and explain our results. In section 3 we consider a large $N$ limit and compare our result with the index of M-theory gravitons. Section 4 concludes with comments and further directions. Most of the detailed calculation is relegated to appendices A and B. Appendix C summarizes the index over M-theory gravitons. Superconformal index for $\mathcal{N}=6$ Chern-Simons theory ============================================================ The theory ---------- The action and supersymmetry of $\mathcal{N}=6$ Chern-Simons-matter theory are presented and further studied in [@Aharony:2008ug; @Benna:2008zy; @Hosomichi:2008jb; @Bandres:2008ry]. The Poincare and special supercharges form vector representations of $SO(6)$ R-symmetry, or equivalently rank 2 antisymmetric representations of $SU(4)$ with reality conditions: $$Q_{IJ\alpha}=\frac{1}{2}\epsilon_{IJKL}\bar{Q}^{KL}_\alpha\ ,\ \ S^{IJ}_\alpha=\frac{1}{2}\epsilon^{IJKL}\bar{S}_{KL\alpha}\ ,$$ where $I,J,K,L\!=\!1,2,3,4$ and $\alpha\!=\!\pm$. Under radial quantization, the special supercharges are Hermitian conjugate to the Poincare supercharges: $S^{IJ\alpha}=(Q_{IJ\alpha})^\dag$, $\bar{S}_{IJ}^{\alpha}=(\bar{Q}^{IJ}_\alpha)^\dag$. There are two (Hermitian) gauge fields $A_\mu$, $\tilde{A}_\mu$ for $U(N)\times U(N)$. The matter fields are complex scalars and fermions in ${\bf 4}$ and ${\bf \bar{4}}$ of $SU(4)$, respectively. We write them as $C_I$ and $\Psi^I_\alpha$. They are all in the bifundamental representation $({\bf N},{\bf \bar{N}})$ of $U(N)\times U(N)$. In this paper we are interested in the superconformal index associated with a special pair of supercharges. We pick $Q\equiv Q_{34-}$ and $S\equiv S^{34-}$ without losing generality. For our purpose, it is convenient to decompose the fields in super-multiplets of $d\!=\!3$, $\mathcal{N}\!=\!2$ supersymmetry generated by $Q_\alpha\equiv Q_{34\alpha}$. Writing the matter fields as $C_I=(A_1,A_2,\bar{B}^{\dot{1}},\bar{B}^{\dot{2}})$ and $\Psi^I\sim(-\psi_2,\psi_1,-\bar{\chi}^{\dot{2}},\bar{\chi}^{\dot{1}})$, they group into 4 chiral multiplets as $$(A_a,\psi_{a\alpha})\ \ {\rm in}\ ({\bf N}, {\bf \bar{N}})\ ,\ \ (B_{\dot{a}},\chi_{\dot{a}\alpha})\ \ {\rm in} \ ({\bf \bar{N}}, {\bf N})\ ,$$ where $a=1,2$ and $\dot{a}=\dot{1},\dot{2}$ are doublet indices for $SU(2)\times SU(2)\subset SU(4)$ commuting with $Q_\alpha$. The global charges of the fields and supercharges are presented in Table 1. $h_1,h_2,h_3$ are three Cartans of $SO(6)$ in the ‘orthogonal 2-planes’ basis, $\frac{1}{2}(h_1\!\pm\!h_2)$ being the Cartans of the above $SU(2)\times SU(2)$. $j_3$ is the Cartan of $SO(3)\subset SO(3,2)$. $\epsilon$ is the energy in radial quantization, or the scale dimension of operators. $h_4$ is the baryon-like charge commuting with the $\mathcal{N}\!=\!6$ superconformal group $Osp(6|4)$. \[charges\] $$\begin{array}{c|ccc|cc|c} \hline{\rm fields}&h_1&h_2&h_3&j_3&\epsilon& h_4\\ \hline(A_1,A_2)&(\frac{1}{2},-\frac{1}{2})&(\frac{1}{2},-\frac{1}{2})& (-\frac{1}{2},-\frac{1}{2})&0&\frac{1}{2}&\frac{1}{2}\\ (B_{\dot{1}},B_{\dot{2}})&(\frac{1}{2},-\frac{1}{2})&(-\frac{1}{2}, \frac{1}{2})&(-\frac{1}{2},-\frac{1}{2})&0&\frac{1}{2}&-\frac{1}{2}\\ (\psi_{1\pm},\psi_{2\pm})&(\frac{1}{2},-\frac{1}{2})&(\frac{1}{2}, -\frac{1}{2})&(\frac{1}{2},\frac{1}{2})&\pm\frac{1}{2}&1&\frac{1}{2}\\ (\chi_{\dot{1}\pm},\chi_{\dot{2}\pm})&(\frac{1}{2},-\frac{1}{2})& (-\frac{1}{2},\frac{1}{2})&(\frac{1}{2},\frac{1}{2})&\pm\frac{1}{2}&1 &-\frac{1}{2}\\ \hline A_\mu,\tilde{A}_\mu&0&0&0&(1,0,-1)&1&0\\ \lambda_{\pm},\tilde\lambda_\pm&0&0&-1&\pm\frac{1}{2}&\frac{3}{2}&0\\ \sigma,\tilde\sigma&0&0&0&0&1&0\\ \hline Q_\pm&0&0&1&\pm\frac{1}{2}&\frac{1}{2}&0\\ S^\pm&0&0&-1&\mp\frac{1}{2}&-\frac{1}{2}&0\\ \hline \end{array}$$ The Lagrangian is presented, among others, in [@Aharony:2008ug; @Benna:2008zy]. It is convenient to introduce auxiliary fields $\lambda_\alpha,\sigma$ and $\tilde\lambda_\alpha,\tilde\sigma$ which form vector multiplets together with gauge fields. We closely follow the notation of [@Benna:2008zy]. The action is given by $$\mathcal{L}=\mathcal{L}_{CS}+\mathcal{L}_{m}\ ,$$ where the Chern-Simons term is given by[^1] $$\label{cs-action} \mathcal{L}_{CS}=\frac{k}{4\pi} {\rm tr}\left(A\wedge dA-\frac{2i}{3}A^3 +i\bar\lambda\lambda-2D\sigma\right)-\frac{k}{4\pi} {\rm tr}\left(\tilde{A}\wedge d\tilde{A} -\frac{2i}{3}\tilde{A}^3 +i\bar{\tilde\lambda}\tilde\lambda-2\tilde{D}\tilde\sigma\right)$$ and $$\begin{aligned} \label{matter-action} \hspace*{-0.5cm}\mathcal{L}_{m}&=& {\rm tr}\left[\frac{}{}\right.\!\! -D_\mu\bar{A}^aD^\mu A_a-D_\mu\bar{B}^{\dot{a}} D_\mu B_{\dot{a}}-i\bar\psi^a\gamma^\mu D_\mu\psi_a-i\bar\chi^{\dot{a}} \gamma^\mu D_\mu\chi_{\dot{a}}\nonumber\\ &&\hspace{0.5cm} -\left(\sigma A_a-A_a\tilde\sigma\right) \left(\bar{A}^a\sigma-\tilde\sigma\bar{A}^a\right) -\left(\tilde\sigma B_{\dot{a}}-B_{\dot{a}}\sigma\right) \left(\bar{B}^{\dot{a}}\tilde\sigma-\sigma\bar{B}^{\dot{a}} \right)\nonumber\\ &&\hspace{0.5cm} +\bar{A}^aDA_a-A_a\tilde{D}\bar{A}^a-B_{\dot{a}}D\bar{B}^{\dot{a}} +\bar{B}^{\dot{a}}\tilde{D}B_{\dot{a}}\nonumber\\ &&\hspace{0.5cm}-i\bar\psi^a\sigma\psi_a+i\psi_a\tilde\sigma\bar\psi^a +i\bar{A}^a\lambda\psi_a+i\bar\psi^a\bar\lambda A_a -i\psi_a\tilde\lambda\bar{A}^a-iA_a\bar{\tilde\lambda}\bar\psi^a\nonumber\\ &&\hspace{0.5cm}+i\chi_{\dot{a}}\sigma\bar\chi^{\dot{a}} -i\bar\chi^{\dot{a}}\tilde\sigma\chi_{\dot{a}} -i\chi_{\dot{a}}\lambda\bar{B}^{\dot{a}} -iB_{\dot{a}}\bar\lambda\bar\chi^{\dot{a}} +i\bar{B}^{\dot{a}}\tilde\lambda\chi_{\dot{a}} +i\bar\chi^{\dot{a}}\bar{\tilde\lambda}B_{\dot{a}}\!\left.\frac{}{}\right] +\mathcal{L}_{\rm sup}\ .\end{aligned}$$ $\mathcal{L}_{\rm sup}$ contains scalar potential and Yukawa interaction obtained from a superpotential $$W=-\frac{2\pi}{k}\epsilon^{ab} \epsilon^{\dot{a}\dot{b}}{\rm tr}(A_aB_{\dot{a}}A_bB_{\dot{b}})$$ where the fields $A_a,B_{\dot{a}}$ in the superpotential are understood as chiral superfields $A_a+\sqrt{2}\theta\psi_a+\theta^2F_{A_a}$ and $B_{\dot{a}}+\sqrt{2}\theta\chi_{\dot{a}}+\theta^2F_{B_{\dot{a}}}$. Integrating out the auxiliary fields, one can easily obtain the expressions for $\sigma,\tilde\sigma,\lambda,\tilde\lambda$ in terms of the matter fields. The $\mathcal{N}\!=\!2$ supersymmetry transformation under $Q_\alpha\!=\!Q_{34\alpha}$ can be obtained from the superfields. Most importantly, $$\begin{aligned} Q_\alpha\psi_{a\beta}\!&\!=\!&\!\sqrt{2}\epsilon_{\alpha\beta} \partial_{\bar{A}^a}\bar{W}\ ,\ \ Q_\alpha\chi_{\dot{a}\beta}=\sqrt{2}\epsilon_{\alpha\beta} \partial_{\bar{B}^{\dot{a}}}\bar{W}\nonumber\\ Q_\alpha\bar\psi^a_\beta\!&\!=\!&\!-\sqrt{2}i (\gamma^\mu)_{\alpha\beta}D_\mu\bar{A}^a +\sqrt{2}i\epsilon_{\alpha\beta}\left(\tilde\sigma\bar{A}^a- \bar{A}^a\sigma\right)\nonumber\\ Q_\alpha\bar\chi^{\dot{a}}_\beta\!&\!=\!&\!-\sqrt{2}i (\gamma^\mu)_{\alpha\beta}D_\mu\bar{B}^{\dot{a}} +\sqrt{2}i\epsilon_{\alpha\beta}\left(\sigma\bar{B}^{\dot{a}} -\bar{B}^{\dot{a}}\tilde\sigma\right)\nonumber\\ Q_\alpha \lambda_\beta\!&\!=\!&\!-\sqrt{2}i\left[\frac{}{}\! (\gamma^\mu)_{\alpha\beta} (D_\mu\sigma+i\star F_\mu)+\epsilon_{\alpha\beta}D\right] \nonumber\label{gaugino-susy}\\ Q_\alpha\bar\lambda_\beta\!&\!=\!&\!0\label{gaugino-susy}\end{aligned}$$ where $\epsilon^{012}=1$. Following [@Benna:2008zy], we choose $(\gamma^\mu)_\alpha^{\ \beta}=(i\sigma^2,\sigma^1,\sigma^3)$ so that $(\gamma^\mu)_{\alpha\beta}=(-1,-\sigma^3,\sigma^1)$. We will be interested in the Euclidean version of this theory. The action and supersymmetry transformation can easily be changed to the Euclidean one by Wick rotation, i.e. by replacements $x^0=-ix^0_E$, $A_0=i(A_E)_0$, etc. Note that $D$, playing the role of Lagrange multiplier in (\[cs-action\]), (\[matter-action\]), should be regarded as an imaginary field. After Wick rotation one obtains $\gamma_E^\mu=(-\sigma^2,\sigma^1,\sigma^3)$. The spinors, say, $\psi_a$ and $\bar\psi^a$ are no longer complex conjugates to each other. The notation of [@Benna:2008zy] naturally lets us regard them as independent chiral and anti-chiral spinors $\psi_\alpha$ and $\bar\psi_{\dot\alpha}$ in Euclidean 4 dimension, reduced down to $d\!=\!3$. Indeed, upon identifying $\sigma=A_3$ etc., the kinetic term plus the coupling to $\sigma,\tilde\sigma$ can be written as $$i\bar\psi^{a\alpha}(\gamma^\mu_E)_\alpha^{\ \beta} D_\mu\psi_{a\beta}+\bar\psi^{a\alpha}D_3\psi_{a\alpha}\equiv -\bar\psi^a_\alpha(\bar\sigma^\mu)^{\alpha\beta}D_\mu\psi_{a\beta}$$ in 4 dimensional notation, where $D_3\psi_a\equiv-i\sigma\psi_a+i\psi_a\tilde\sigma$ and $(\bar\sigma^\mu)^{\alpha\beta}\equiv i\epsilon^{\alpha\gamma} (\gamma_E^\mu,-i)_\gamma^{\ \beta}=(1,i\sigma^3,-i\sigma^1,i\sigma^2 )^{\alpha\beta}$. In our computation in appendices, it will be more convenient to choose a new $SO(3)$ frame for spinors so that $$\bar\sigma^\mu=(1,-i\vec\sigma)=(1,-i\sigma^1,-i\sigma^2,-i\sigma^3)\ .$$ To avoid formal manipulations in the main text being a bit nasty, this change of frame will be assumed only in appendices. Similar rearrangement can be made for $\chi_{\dot{a}},\bar\chi^{\dot{a}}$. In Euclidean theory, $Q_\alpha\lambda_\beta$ in (\[gaugino-susy\]) is given by $$Q_\alpha \lambda_\beta=-\sqrt{2}i\left[\frac{}{}\! (\gamma^\mu_E)_{\alpha\beta} (D_\mu\sigma-\star F_\mu)+\epsilon_{\alpha\beta}D\right]\ .$$ Configurations preserving two supercharges $Q_\alpha$ are described by the Bogomolnyi equations $(\star F)_\mu=D_\mu\sigma$ and $D=0$. The first one is the BPS equation for magnetic monopoles in Yang-Mills theory, with a difference that $\sigma$ is a composite field here. See [@Hosomichi:2008ip] for related discussions. We shall shortly deform the theory with a $Q$-exact term. $\sigma$ will not be a composite field then. A conformal field theory defined on $\mathbb{R}^{d+1}$ can be radially quantized to a theory living on $S^d\times\mathbb{R}$, where $\mathbb{R}$ denotes time. The procedures of radial quantization are summarized in appendix A. See also [@Bhattacharyya:2007sa; @Grant:2008sk] for related discussions. In the radially quantized theory, one can consider configurations in which nonzero magnetic flux is applied on spatial $S^2$. From the representations of matter fields under $U(N)\times U(N)$, one finds that ${\rm tr}F={\rm tr}\tilde{F}$ should be satisfied. The Kaluza-Klein momentum in the dual M-theory along the fiber circle of $S^7/\mathbb{Z}_k$ is given by $$\label{kk-momentum} P=\frac{k}{4\pi}\int_{S^2}{\rm tr}F= \frac{k}{4\pi}\int_{S^2}{\rm tr}\tilde{F}\ \in~\frac{k}{2}~\mathbb{Z}$$ in the gauge theory [@Aharony:2008ug]. This, via Gauss’ law constraint, turns out to be proportional to $h_4$ in Table 1 [@Aharony:2008ug]. The superconformal index and localization ----------------------------------------- The superconformal symmetry of this theory is $Osp(6|4)$, whose bosonic generators form $SO(6)\times SO(3,2)$. Its Cartans are given by five charges: $h_1,h_2,h_3$ and $\epsilon,j_3$ in $SO(2)\times SO(3)\subset SO(3,2)$. Some important algebra involving our special supercharges is $$Q^2=S^2=0\ ,\ \ \{Q,S\}=\epsilon-h_3-j_3\ .$$ The first equation says $Q,S$ are nilpotent, while the second one implies the BPS energy bound $\epsilon\!\geq\!h_3\!+\!j_3$. The special supercharges $Q,S$ are charged under some Cartans. From , four combinations $h_1,h_2$, $\epsilon\!+\!j_3$, $\epsilon\!-\!h_3\!-\!j_3$ commute with $Q,S$. The last is nothing but $\{Q,S\}$. The superconformal index for a pair of supercharges $Q,S$ is given by [@Bhattacharya:2008bja; @Bhattacharya:2008zy] $$\label{superconf-index} I(x,y_1,y_2)={\rm Tr}\left[(-1)^Fe^{-\beta^\prime\{Q,S\}} e^{-\beta(\epsilon+j_3)}e^{-\gamma_1h_1-\gamma_2h_2}\right]$$ where $x\equiv e^{-\beta}$, $y_1\equiv e^{-\gamma_1}$, $y_2\equiv e^{-\gamma_2}$. $F$ is the fermion number. The charges we use to grade the states in the index should commute with $Q$ and $S$ [@Kinney:2005ej]. As a Witten index, this function does not depend on $\beta^\prime$ since it gets contribution only from states annihilated by $Q$ and $S$. The above index admits a path integral representation on $S^2\times S^1$, where the radius of the last circle is given by the inverse temperature $\beta+\beta^\prime$. Had the operator inserted inside the trace been $e^{-(\beta+\beta^\prime)\epsilon}$, the measure of the integral would have been given by the Euclidean action with the identification $\epsilon=-\frac{\partial}{\partial\tau}$ with Euclidean time $\tau$. The insertion of $(-1)^F$ would also have made all fields periodic in $\tau\sim\tau\!+\!(\beta\!+\!\beta^\prime)$. The actual insertion $(-1)^Fe^{-(\beta+\beta^\prime)\epsilon-(\beta- \beta^\prime)j_3+\beta^\prime h_3-\gamma_1 h_1-\gamma_2h_2}$ twists the boundary condition: alternatively, this twist can be undone by replacing all time derivatives in the action by $$\partial_\tau\rightarrow\partial_\tau- \frac{\beta-\beta^\prime}{\beta+\beta^\prime}j_3 +\frac{\beta^\prime}{\beta+\beta^\prime}h_3 -\frac{\gamma_1}{\beta+\beta^\prime}h_1-\frac{\gamma_2}{\beta+ \beta^\prime}h_2\ ,$$ leaving all fields periodic. The generators of Cartans assume appropriate representations depending on the field they act on. The angular momentum $j_3$ is given for each mode after expanding fields with spherical harmonics, or with the so called monopole spherical harmonics [@Wu:1976ge] if nontrivial magnetic field is applied on $S^2$. In actual computation we will often formulate the theory on $\mathbb{R}^3$ in Cartesian coordinates, with $r=e^\tau$ (see appendix A and also [@Bhattacharyya:2007sa; @Grant:2008sk]). Change in derivatives on $\mathbb{R}^3$ due to the above twist is $$\vec\nabla\rightarrow\vec\nabla+\frac{\vec{r}}{r^2} \left(-\frac{\beta-\beta^\prime}{\beta+\beta^\prime}j_3 +\frac{\beta^\prime}{\beta+\beta^\prime}h_3 -\frac{\gamma_1}{\beta+\beta^\prime}h_1-\frac{\gamma_2} {\beta+\beta^\prime}h_2\right)\ .$$ From now on our derivatives are understood with this shift, hoping it will not cause confusion. Insertion of a $Q$-exact operator $\{Q,V\}$, for any gauge-invariant operator $V$, to the superconformal index (\[superconf-index\]) becomes zero due to the $Q$-invariance of the Cartans appearing in (\[superconf-index\]) and the periodic boundary condition for the fields due to $(-1)^F$ [@Cecotti:1981fu].[^2] From the nilpotency of $Q$, the operator $e^{-t\{Q,V\}}$ takes the form $1+Q(\cdots)$ for a given $V$ and a continuous parameter $t$. Therefore, in the path integral representation, we may add the $Q$-exact term to the action $S\rightarrow S+t\{Q,V\}$ without changing the integral. The parameter $t$ can be set to a value with which the calculation is easiest.[^3] In particular, by suitably choosing $V$, setting $t\rightarrow+\infty$ may be regarded as a semi-classical limit with $t$ being $\hbar^{-1}$. This semi-classical or Gaussian ‘approximation’ then provides the exact result since the integral is $t$-independent. We choose to deform the action of the $\mathcal{N}\!=\!6$ Chern-Simons theory by a $Q$-exact term which looks similar to the $d\!=\!3$ $\mathcal{N}\!=\!2$ ‘Yang-Mills’ action as follows. Using the gaugino superfield (Euclidean) $$\mathcal{W}_\alpha(y)\sim-\sqrt{2}i\lambda_\alpha(y)+2D(y)\theta_\alpha+ \left(\gamma^\mu\theta\right)_{\alpha}\left(D_\mu\sigma-\star F_\mu\right)(y)+\sqrt{2}\theta^2(\gamma^\mu D_\mu\bar\lambda(y))_\alpha$$ with $y^\mu=x^\mu+i\theta\gamma^\mu\bar\theta$, we add $$t\{Q,V\}=\left.\frac{1}{g^2}\int d^3x\ r \mathcal{W}^\alpha\mathcal{W}_\alpha\right|_{\theta^2\bar\theta^0} \ \ \ \ \ \ ({\rm taking}\ t=\frac{1}{g^2}~)$$ to the original action. Let us provide supplementary explanations. The multiplication of $r$ in the integrand makes this term scale invariant: it is crucial to have this symmetry since it will be our time translation symmetry after radial quantization. Of course translation symmetry on $\mathbb{R}^3$ is broken, which does not matter to us. It is also easy to show that the above deformation is $Q$-exact. Taking the coefficient of $\theta^2$ is equivalent to $\partial_{\theta^-}\partial_{\theta^+}$, which in turn is related to $Q_\alpha$ by $Q_\alpha=\partial_\alpha-i(\gamma^\mu\bar\theta)_\alpha\partial_\mu$. However, since we are keeping terms with $\bar\theta^0$, $\partial_\alpha$ is effectively $Q_\alpha$. Furthermore, $y^\mu$ can be replaced by $x^\mu$ for the same reason. Therefore the added term indeed takes the form $Q_-Q_+(\cdots)$ with $y^\mu\rightarrow x^\mu$ everywhere. Finally, note that we do not add a term of the form $\int d^2\bar\theta\ \bar{\mathcal{W}}_\alpha\bar{\mathcal{W}}^\alpha$. Expanding in components, one finds $$\label{exact-deform} \Delta S=t\{Q,V\}=\frac{1}{2g^2}\int_{1\leq r\leq e^{\beta\!+\! \beta^\prime}}\!\!\!d^3x\ r\left[\frac{}{}\!\!\left(\star F_\mu-D_\mu\sigma \right)^2-D^2+\lambda^\alpha(\sigma^\mu)_{\alpha\beta} D_\mu\bar\lambda^\beta\right]$$ where $\sigma^\mu=(1,-i\sigma^3,i\sigma^1,-i\sigma^2)$ in the basis of [@Benna:2008zy], or $(1,i\vec\sigma)$ in the basis we use in the appendix, and $D_3\bar\lambda^\beta=-i\sigma\bar\lambda^\beta+ i\bar\lambda^\beta\sigma$. We already turned $D$ to an imaginary field during Wick rotation, which makes $-D^2$ positive. Everything goes similarly for the other vector multiplet $\tilde{A}_\mu,\tilde\sigma,\tilde\lambda_\alpha$. Some 1-loop study has been made for ‘ordinary’ Yang-Mills Chern-Simons theories [@Kao:1995gf]. Due to various differences in our construction, we will obtain very different results. Calculation of the index ------------------------ Having set up the localization problem in the previous subsection, we first find saddle points in the limit $g\rightarrow 0$, and then compute the 1-loop determinants around them. All fields are subject to the periodic boundary condition along $S^1$, or the radial direction: working with fields in $\mathbb{R}^3$, the equivalent boundary condition is $$\label{boundary} \Psi(r=e^{\beta})=e^{-\beta\Delta_\Psi}\Psi(r=1)\ ,$$ where $\Delta_\Psi$ is the scale dimension of the field $\Psi$. See appendix A and [@Bhattacharyya:2007sa; @Grant:2008sk]. The saddle point equations, which can be deduced either from (\[exact-deform\]) or from the supersymmetry transformation, are given by $$\label{saddle-eqn} \star F_\mu=D_\mu\sigma\ ,\ \ D=0\ ,\ \ \star \tilde{F}_\mu=D_\mu\tilde\sigma\ ,\ \ \tilde{D}=0\ .$$ Note $\sigma,\tilde\sigma$ are no longer composite fields. From the supersymmetry transformation of matter fermions, we find that only $A_a=B_{\dot{a}}=0$ satisfies the above boundary condition. All fermions are naturally set to zero. An obvious solution for the fields $F_{\mu\nu},\sigma$ and $\tilde{F}_{\mu\nu},\tilde\sigma$ is Dirac monopoles in the diagonals $U(1)^N\times U(1)^N\subset U(N)\times U(N)$: $$\star F_\mu=\frac{x_\mu}{2r^3} {\rm diag}(n_1,n_2,\cdots,n_N)\ ,\ \ \star\tilde{F}_\mu=\frac{x_\mu}{2r^3}{\rm diag}(\tilde{n}_1,\tilde{n}_2, \cdots,\tilde{n}_N)\ ,$$ together with $$\sigma=-\frac{1}{2r}{\rm diag}(n_1,n_2,\cdots,n_N)\ ,\ \ \tilde\sigma=-\frac{1}{2r}{\rm diag}(\tilde{n}_1,\tilde{n}_2, \cdots,\tilde{n}_N)\ .$$ Since we are considering the region $1\leq r\leq e^\beta$ in $\mathbb{R}^3$ excluding the origin $r=0$, this solution is regular. The spherical symmetry of the solution implies that there is no twisting in derivatives. Note that all fields satisfy the boundary condition (\[boundary\]) with $\Delta_\sigma=\Delta_{\tilde\sigma}=1$ and $\Delta_{F}=\Delta_{\tilde{F}}=2$. The coefficients $n_i$ and $\tilde{n}_i$ ($i=1,2,\cdots,N$) have to be integers since the diagonals of $\int_{S^2}F$ and $\int_{S^2}\tilde{F}$ are $2\pi$ times integers.[^4] Apart from the above Abelian solutions, we could not find any other solutions satisfying the boundary condition. For instance, although the governing equation is the same, non-Abelian solutions like the embedding of $SU(2)$ ’t Hooft-Polyakov monopoles are forbidden since the boundary condition is not met. There still is a possibility that deformation of derivatives from twisting might play roles, which we have not fully ruled out. Anyway, we shall only consider the above saddle points and find agreement with the graviton index, which we regard as a strong evidence that we found all relevant saddle points. To the above solution, one can also superpose holonomy zero modes along $S^1$ as follows. Since the solution is diagonal, turning on constant $A_\tau,\tilde{A}_\tau$ diagonal in the same basis obviously satisfies (\[saddle-eqn\]). In terms of fields normalized in $\mathbb{R}^3$, this becomes $$A_r=\frac{1}{(\beta\!+\!\beta^\prime)r} {\rm diag}(\alpha_1,\alpha_2,\cdots,\alpha_N) \ ,\ \ \tilde{A}_r=\frac{1}{(\beta\!+\!\beta^\prime)r}{\rm diag} (\tilde\alpha_1,\tilde\alpha_2,\cdots,\tilde\alpha_N)$$ where we insert factors of $\beta\!+\!\beta^\prime$ for later convenience. Taking into account the large gauge transformation along $S^1$, the coefficients $\alpha_i$, $\tilde\alpha_i$ are all periodic: $$\alpha_i\sim\alpha_i+2\pi\ ,\ \ \tilde\alpha_i\sim\tilde\alpha_i+2\pi\ \ \ (i=1,2,\cdots,N).$$ This holonomy along the time circle is related to the Polyakov loop [@Aharony:2003sx]. The full set of saddle points is parametrized by integer fluxes $\{n_i,\tilde{n}_i\}$ and the holonomy $\{\alpha_i,\tilde\alpha_i\}$. The analysis around the saddle point in which all $n_i$, $\tilde{n}_i$ vanish was done in [@Bhattacharya:2008bja]. We provide the 1-loop analysis around all saddle points. We explain various ingredients in turn: classical contribution, gauge-fixing and Faddeev-Popov measure, 1-loop contribution, Casimir-like energy shift and finally the full answer. We first consider the ‘classical’ action. Plugging in the saddle point solution to the action, the classical action proportional to $g^{-2}$ is always zero, as expected since our final result should not depend on $g$. Since the classical action itself involves two classes of terms, one of order $\mathcal{O}(g^{-2})$ from $Q$-exact deformation and another $\mathcal{O}(g^0)$ from original action, one should keep the latter part of the classical action to do the correct 1-loop physics. This comes from the Chern-Simons term. To correctly compute it, one has to extend $S^2\times S^1$ to a 4-manifold $\mathcal{M}_4$ bounding it, and use $\frac{1}{4\pi}\int_{S^2\times S^1}{\rm tr} A\wedge F= \frac{1}{4\pi}\int_{\mathcal{M}_4}{\rm tr}F\wedge F$. If one chooses a disk $D_2$ bounded by $S^1=\partial D_2$ and take $\mathcal{M}_4=S^2\times D_2$, one finds $$\frac{1}{4\pi}{\rm tr}\int_{S^2\times D_2}F\wedge F= \frac{1}{2\pi}{\rm tr}\int_{D_2}F\cdot\int_{S^2} F= \frac{1}{2\pi}\int_{S^1}A\cdot\int_{S^2}F=\sum_{i=1}^Nn_i\alpha_i\ .$$ Taking into account the second gauge field $\tilde{A}_\mu$ at level $-k$, the exponential of the $\mathcal{O}(g^0)$ part of the classical action is a phase given by $$\label{phase} e^{ik\sum_{i=1}^N(n_i\alpha_i-\tilde{n}_i\tilde\alpha_i)}\ .$$ The exponent is linear in $\alpha_i,\tilde\alpha_i$. Apart from these linear terms, there are no quadratic terms in the holonomy (around any given value) in the rest of the classical action. Thus Gaussian approximation is not applicable and they should be treated exactly. We shall integrate over them after applying Gaussian approximation to all other degrees. Before considering 1-loop fluctuations, we fix the gauge. Following [@Aharony:2003sx] we choose the Coulomb gauge, or the background Coulomb gauge for the components of fluctuations coupled to the background. In the calculation around the saddle point where all fields are zero, the Faddeev-Popov determinant computed following [@Aharony:2003sx] is that for the $U(N)\times U(N)$ unitary matrices with eigenvalues $\{e^{i\alpha_i}\},\{e^{i\tilde\alpha_i}\}$ $$\prod_{i<j}\left[2\sin\left(\frac{\alpha_i\!-\!\alpha_j}{2}\right)\right]^2 \prod_{i<j}\left[2\sin\left(\frac{\tilde\alpha_i\!-\!\tilde\alpha_j}{2} \right)\right]^2\ .$$ In the saddle point with nonzero fluxes, the flux effectively ‘breaks’ $U(N)\times U(N)$ to an appropriate subgroup. For instance, with fluxes $\{3,2,2,0,0\}$ on $U(1)^5\subset U(5)$, $U(5)$ is broken to $U(1)\times U(2)\times U(2)$. As explained in appendix B, the Faddeev-Popov measure for the saddle point with flux is the unitary matrix measure for the *unbroken subgroup* of $U(N)\times U(N)$: $$\prod_{i<j;n_i=n_j} \left[2\sin\left(\frac{\alpha_i\!-\!\alpha_j}{2}\right)\right]^2 \prod_{i<j;\tilde{n}_i=\tilde{n}_j} \left[2\sin\left(\frac{\tilde\alpha_i\!-\!\tilde\alpha_j}{2}\right)\right]^2$$ where the restricted products keep a sine factor for a pair of eigenvalues in the same unbroken gauge group only. The fluctuations of fields with nonzero quadratic terms in the action can be treated by Gaussian approximation. The result consists of several factors of determinants from matter scalars, fermions and also from fields in vector multiplets, which is schematically $$\label{schematic-det} \frac{\det_{\psi_a,\chi_a}\det_{\lambda}\det_{\tilde\lambda}} {\det_{A_a,B_a}\det_{A_\mu,\sigma}\det_{\tilde{A}_\mu,\tilde\sigma}}\ .$$ The fields in $U(N)\times U(N)$ vector multiplets will turn out to provide nontrivial contribution. We first consider the 1-loop determinant from matter fields, namely $A_a,B_{\dot{a}},\psi_a,\chi_{\dot{a}}$ and their conjugates. In this determinant, the index over the so-called ‘letters’ play important roles.[^5] To start with, we consider the basic fields $A_a,\bar{B}^{\dot{a}},\psi_a,\bar\chi^{\dot{a}}$ in ($N$,$\bar{N}$) representation of the gauge group and pick up the $ij$’th component, where $i$ ($j$) runs over $1,2,\cdots,N$ and refers to the fundamental (anti-fundamental) index of first (second) $U(N)$. In the quadratic Lagrangian, these modes couple to the background magnetic field with charge $n_i\!-\!\tilde{n}_j$. The index over letters in this component is given by $$\label{letter-bif} f^+_{ij}(x,y_1,y_2)=x^{|n_i\!-\!\tilde{n}_j|}\left[ \frac{x^{1/2}}{1-x^2} \left(\!\sqrt{\frac{y_1}{y_2}}+\!\sqrt{\frac{y_2}{y_1}}\right)- \frac{x^{3/2}}{1-x^2} \left(\!\sqrt{y_1y_2}+\!\frac{1}{\sqrt{y_1y_2}}\right)\right]\equiv x^{|n_i\!-\!\tilde{n}_j|}f^+(x,y_1,y_2)\ .$$ See appendix B.1 for the derivation. Similarly, the index over the $ij$’th component of letters in ($\bar{N}$,$N$) representation is given by $$\label{letter-antibif} f^-_{ij}(x,y_1,y_2)=x^{|n_i\!-\!\tilde{n}_j|} \left[\frac{x^{1/2}}{1-x^2} \left(\!\sqrt{y_1y_2}+\!\frac{1}{\sqrt{y_1y_2}}\right)- \frac{x^{3/2}}{1-x^2} \left(\!\sqrt{\frac{y_1}{y_2}}+\!\sqrt{\frac{y_2}{y_1}}\right) \right]\equiv x^{|n_i\!-\!\tilde{n}_j|}f^-(x,y_1,y_2)\ .$$ There is no dependence on the regulator $\beta^\prime$, as expected. Again see appendix B.1 for details. With these letter indices, the 1-loop determinant for given $\alpha_i,\tilde\alpha_i$ is given by $$\prod_{i,j=1}^N\exp\left[\sum_{n=1}^\infty\frac{1}{n}\left( f^+_{ij}(x^n,y_1^n,y_2^n)e^{in(\tilde\alpha_j\!-\!\alpha_i)}+ f^-_{ij}(x^n,y_1^n,y_2^n)e^{in(\alpha_i\!-\!\tilde\alpha_j)}\right) \right]\ .$$ The expression $\exp\left[\sum_{n=1}^\infty\frac{1}{n}f(x^n)\right]$ above, sometimes called the Plethystic exponential of a function $f(x)$, appears since we count operators made of identical letters [@Benvenuti:2006qr]. Note that the determinant around the saddle point in which all fluxes are zero, $n_i=\tilde{n}_i=0$, reduces to the result obtained in [@Bhattacharya:2008bja] using combinatoric methods in the free theory. When all $n_i,\tilde{n}_i$ are zero, our letter indices $f^\pm_{ij}$ all reduces to $f^\pm$, which are exactly the letter indices obtained in [@Bhattacharya:2008bja]. We also consider the determinant from fields in vector multiplets. Here, the letter index over the $ij$’th component of the adjoint fields $A_\mu,\lambda_\alpha,\sigma$ is given by $$\label{letter-adj1} f^{{\rm adj}}_{ij}(x)=-(1-\delta_{n_in_j})x^{|n_i\!-\!n_j|} =\left\{\begin{array}{ll} 0&{\rm if}\ n_i=n_j\\-x^{|n_i\!-\!n_j|}&{\rm if}\ n_i\neq n_j\end{array} \right.\ ,$$ and similarly for the $ij$’th component of the fields $\tilde{A}_\mu,\tilde\lambda_\alpha,\tilde\sigma$ one finds $$\label{letter-adj2} \tilde{f}^{{\rm adj}}_{ij}(x)=-(1-\delta_{\tilde{n}_i\tilde{n}_j}) x^{|\tilde{n}_i\!-\!\tilde{n}_j|} =\left\{\begin{array}{ll}0&{\rm if}\ \tilde{n}_i=\tilde{n}_j\\ -x^{|\tilde{n}_i-\tilde{n}_j|}&{\rm if}\ \tilde{n}_i\neq\tilde{n}_j \end{array}\right.\ .$$ The full 1-loop determinant from adjoint fields is again given by the same exponential: $$\prod_{i,j=1}^N\exp\left[\sum_{n=1}^\infty\frac{1}{n}\left( f^{{\rm adj}}_{ij}(x^n)e^{-in(\alpha_i-\alpha_j)}+ \tilde{f}^{{\rm adj}}_{ij}(x^n)e^{-in(\tilde\alpha_i-\tilde\alpha_j)} \right) \right]\ .$$ For the trivial vacuum with no fluxes, all $f^{\rm adj}_{ij}$ and $\tilde{f}^{\rm adj}_{ij}$ are zero that the adjoint determinant is simply 1. This is consistent with the result in [@Bhattacharya:2008bja], where the vector multiplets including gauge fields played no roles. See appendix B.2 for the derivation. When evaluating the determinant in appendix B, one encounters an overall factor $$\exp\left[-\beta\epsilon_0\right]\ ,\ \ \ {\rm where}\ \ \ \ \epsilon_0\equiv\frac{1}{2}{\rm tr}\left[(-1)^F(\epsilon+j_3)\right]\ .$$ This is similar to the ground state energy traced over all modes, twisted by $j_3$ basically because we are only considering charges commuting with $Q,S$ in our index. Although this is not just energy due to $j_3$, we slightly abuse terminology and call this quantity Casimir energy. With appropriate regulator respecting supersymmetry, this is given by $$\label{casimir} \epsilon_0=\sum_{i,j=1}^N|n_i\!-\!\tilde{n}_j|-\sum_{i<j}|n_i\!-\!n_j| -\sum_{i<j}|\tilde{n}_i\!-\!\tilde{n}_j|\ .$$ See appendix B.3 for the derivation and some properties of $\epsilon_0$. In particular, $\epsilon_0$ is non-negative and becomes zero if and only if the two sets of flux distributions $\{n_i\}$, $\{\tilde{n}_i\}$ are identical. Some features of this energy shift related to AdS/CFT is discussed in the next section. Finally we integrate over the modes $\alpha_i$ and $\tilde\alpha_j$ with all factors explained above in the measure. The result for a given saddle point labeled by $\{n_i\},\{\tilde{n}_i\}$ is $$\label{exact-index} \boxed{\begin{gathered} \hspace*{-0.8cm} I(x,y_1,y_2)= x^{\epsilon_0}\int\frac{1}{\rm (symmetry)} \left[\frac{d\alpha_id\tilde\alpha_i}{(2\pi)^2}\right] \prod_{\substack{i<j;\\ n_i=n_j}} \left[2\sin\left(\frac{\alpha_i\!-\!\alpha_j}{2}\right)\right]^2 \prod_{\substack{i<j;\\ \tilde{n}_i=\tilde{n}_j}} \left[2\sin\left(\frac{\tilde\alpha_i\!-\!\tilde\alpha_j}{2}\right) \right]^2\\ \hspace{2cm}\times e^{ik\sum_{i=1}^N(n_i\alpha_i\!-\! \tilde{n}_i\tilde\alpha_i)}\! \prod_{i,j=1}^N\!\exp\left[\sum_{n=1}^\infty\frac{1}{n}\left( f^+_{ij}(x^n,y_1^n,y_2^n)e^{in(\tilde\alpha_j\!-\!\alpha_i)}+ f^-_{ij}(x^n,y_1^n,y_2^n)e^{in(\alpha_i\!-\!\tilde\alpha_j)}\right) \right]\\ \hspace{-2.3cm} \times\!\prod_{i,j=1}^N\!\exp\left[\sum_{n=1}^\infty\frac{1}{n} \left(f^{{\rm adj}}_{ij}(x^n)e^{-in(\alpha_i\!-\!\alpha_j)}+ \tilde{f}^{{\rm adj}}_{ij}(x^n)e^{-in(\tilde\alpha_i\!-\!\tilde\alpha_j)} \right)\right] \end{gathered}}$$ with (\[letter-bif\]), (\[letter-antibif\]), (\[letter-adj1\]), (\[letter-adj2\]), (\[casimir\]) for the definitions of various functions. The symmetry factor on the first line divides by the factor of identical variables among $\{\alpha_i\},\{\tilde\alpha_i\}$ according to ‘unbroken’ gauge group. For the $U(5)\!\rightarrow\!U(1)\times U(2)\times U(2)$ example above, this factor is $\frac{1}{1!\times 2!\times 2!}$. The full index is the sum of (\[exact-index\]) for all flux distributions $\{n_i\}$, $\{\tilde{n}_i\}$. Apart from the first phase factor on the second line, the integrand is invariant under the overall translation of $\alpha_i,\tilde\alpha_i$. Therefore, the integral vanishes unless $\sum_i n_i=\sum_i\tilde{n}_i$. This is of course a consequence of a decoupled $U(1)$ as explained in [@Aharony:2008ug]. The KK-momentum, or $\frac{k}{2}$ times the baryon-like charge, for states counted by the above index is given by (\[kk-momentum\]). From the structure of this integral and the letter indices, it is also easy to infer that the energy of the states contributing to this index is bounded from below by $$\label{energy-bound} \epsilon\geq\frac{k}{2}\sum_{i=1}^N n_i=\frac{k}{2} \sum_{i=1}^N \tilde{n}_i$$ if the two flux distributions $\{n_i\}$, $\{\tilde{n}_i\}$ are identical. If the two distributions are different, the energy is strictly larger than this bound. We discuss this in the next section. There is a unifying structure in the integrand of the above index if one combines the matrix integral measure on the first line to the last line. Note that the measure can be written as $$\prod_{\substack{i<j;\\ n_i=n_j}}\! \left[2\sin\!\left(\!\frac{\alpha_i\!-\!\alpha_j}{2}\!\right) \right]^2\!\!\prod_{\substack{i<j;\\ \tilde{n}_i=\tilde{n}_j}}\! \left[2\sin\!\left(\!\frac{\tilde\alpha_i\!-\!\tilde\alpha_j}{2}\! \right)\right]^2\!=\prod_{i\neq j}\exp\! \left[-\sum_{n=1}^\infty\frac{1}{n}\left( \delta_{n_in_j}e^{-in(\alpha_i\!-\!\alpha_j)}+ \delta_{\tilde{n}_i\tilde{n}_j}e^{-in(\tilde\alpha_i\!-\!\tilde\alpha_j)} \right)\right]\ .\nonumber$$ As in [@Sundborg:1999ue; @Aharony:2003sx; @Kinney:2005ej], this provides a 2-body repulsive effective potentials between $\alpha_i$’s and $\tilde\alpha_i$’s in the same unbroken gauge group. From the form of adjoint letter indices in (\[letter-adj1\]), (\[letter-adj2\]), the above measure combines with the last line and become $$\prod_{i\neq j}\exp \left[-\sum_{n=1}^\infty\frac{1}{n}\left( x^{n|n_i\!-\!n_j|}e^{-in(\alpha_i\!-\!\alpha_j)}+ x^{n|\tilde{n}_i\!-\!\tilde{n}_j|}e^{-in(\tilde\alpha_i\!-\!\tilde\alpha_j)} \frac{}{}\!\right)\right]\ .$$ Therefore, in the presence of fluxes, there are repulsive 2-body potentials between all pairs with the strength $\frac{1}{n}$ weakened to $\frac{1}{n}x^{n|n_i\!-\!n_j|}$ and $\frac{1}{n}x^{n|\tilde{n}_i\!-\!\tilde{n}_j|}$, by factors of $x$. Large $N$ limit and index over gravitons ======================================== We further analyze the gauge theory index we obtained in the previous section in the large $N$ limit. The limit we take is $N\rightarrow\infty$ while keeping the chemical potentials at order 1. Among other motivations, this setting lets us study the low energy spectrum which can be compared to that from supergravity. In this limit, only $\mathcal{O}(1)$, namely $\mathcal{O}(N^0)$, numbers of $U(1)^N$’s in each $U(N)$ have nonzero magnetic flux. This is because states with more than $\mathcal{O}(1)$ U(1)’s filled with nonzero fluxes have energies bigger than $\mathcal{O}(1)$ from (\[energy-bound\]) and are suppressed in the large $N$ limit we take. This implies that there always exist an $U(N\!-\!\mathcal{O}(1))\times U(N\!-\!\mathcal{O}(1))$ part in the integral over holonomies $\{\alpha_i,\tilde\alpha_i\}$. Let us call this unbroken gauge group $U(N_1)\times U(N_2)$, where $N_1$ and $N_2$ denote numbers of $U(1)$ with zero fluxes. In the large $N$ limit, there is a well-known way of calculating this part of the integral [@Sundborg:1999ue; @Aharony:2003sx]. We first introduce $$\rho_n=\frac{1}{N_1}\sum_i e^{-in\alpha_i}\ ,\ \ \chi_n=\frac{1}{N_2}\sum_i e^{-in\tilde\alpha_i}$$ for nonzero integers $n$, where the summations are over $N_1$ and $N_2$ $U(1)$ indices, respectively. In the large $N_1,N_2$ limit, the integration over $\alpha_i,\tilde\alpha_i$ belonging to $U(N_1)\times U(N_2)$ becomes $$\prod_{n=1}^\infty\left[N_1^2d\rho_nd\rho_{-n}\right] \left[N_2^2d\chi_nd\chi_{-n}\right]\ .$$ The integrand containing $\rho_n,\chi_n$ is given by $$\begin{aligned} &&\hspace{-1.5cm}\exp\left[-\sum_{n=1}^\infty\frac{1}{n} \left(N_1^2\rho_n\rho_{-n}\!+\!N_2^2\chi_n\chi_{-n}\!-\! N_1N_2f^+(x^n,y_1^n,y_2^n) \rho_n\chi_{-n}\!-\!N_1N_2f^-(x^n,y_1^n,y_2^n)\rho_{-n}\chi_n\right)\right] \\ &&\hspace{-1.5cm}\times\exp\left[N_1\!\sum_{n=1}^\infty\frac{1}{n} \rho_n\left(\sum_{i=1}^{M_2}x^{n|\tilde{n}_i|}f^+(\cdot^n) e^{in\tilde\alpha_i}\!-\!\sum_{i=1}^{M_1}x^{n|n_i|}e^{in\alpha_i} \!\right)\!+\!\frac{1}{n} \rho_{-n}\!\left(\sum_{i=1}^{M_2}x^{n|\tilde{n}_i|}f^-(\cdot^n) e^{-in\tilde\alpha_i}\!-\!\sum_{i=1}^{M_1}x^{n|n_i|}e^{-in\alpha_i} \!\right)\!\right]\nonumber\\ &&\hspace{-1.5cm}\times\exp\left[N_2\!\sum_{n=1}^\infty\frac{1}{n} \chi_n\!\left(\sum_{i=1}^{M_1}x^{n|n_i|}f^-(\cdot^n) e^{in\alpha_i}\!-\!\sum_{i=1}^{M_2}x^{n|\tilde{n}_i|} e^{in\tilde\alpha_i}\!\right)\!+\! \frac{1}{n} \chi_{-n}\!\left(\sum_{i=1}^{M_1}x^{n|n_i|}f^+(\cdot^n) e^{-in\alpha_i}\!-\!\sum_{i=1}^{M_2}x^{n|\tilde{n}_i|} e^{-in\tilde\alpha_i}\!\right)\!\right] \nonumber\end{aligned}$$ where $M_1\equiv N-N_1$ and $M_2\equiv N-N_2$ are numbers (of order $1$) of $U(1)$’s in two gauge groups with nonzero fluxes, and $\cdot\ {}^n$ denotes taking $n$’th powers of all arguments $x,y_1,y_2$. The integral of $\rho_n,\chi_n$ is Gaussian, where the first line (with $N_1=N_2=N$) is the one encountered in [@Bhattacharya:2008bja]. After this integration, one obtains $$\label{large-N-gaussian} I^{(0)}\exp\left[\frac{1}{2}\sum_{n=1}^\infty\frac{1}{n} V^T(\cdot\ {}^n)M(\cdot\ {}^n)V(\cdot\ {}^n)\right]$$ where $$V=\left(\begin{array}{c}\displaystyle{ \sum_{i=1}^{M_2}x^{|\tilde{n}_i|}f^+ e^{i\tilde\alpha_i}\!-\!\sum_{i=1}^{M_1}x^{|n_i|}e^{i\alpha_i}}\\ \displaystyle{\sum_{i=1}^{M_1}x^{|n_i|}f^-e^{i\alpha_i}\!-\! \sum_{i=1}^{M_2}x^{|\tilde{n}_i|}e^{i\tilde\alpha_i}}\\ \displaystyle{\sum_{i=1}^{M_2}x^{|\tilde{n}_i|}f^- e^{-i\tilde\alpha_i}\!-\!\sum_{i=1}^{M_1}x^{|n_i|}e^{-i\alpha_i}}\\ \displaystyle{\sum_{i=1}^{M_1}x^{|n_i|}f^+e^{-i\alpha_i}\!-\! \sum_{i=1}^{M_2}x^{|\tilde{n}_i|}e^{-i\tilde\alpha_i}} \end{array}\right)\ \ ,\ \ M=\frac{1}{1-f^+f^-}\left(\begin{array}{cccc} &&1&f^-\\&&f^+&1\\1&f^+&&\\f^-&1&&\end{array}\right)$$ and $$\label{IIA-index} I^{(0)}=\prod_{n=1}^\infty\det\left[M(x^n,y_1^n,y_2^n)\frac{}{}\! \right]^{\frac{1}{2}}=\prod_{n=1}^\infty\frac{(1-x^{2n})^2} {(1-x^ny_1^n)(1-x^ny_1^{-n})(1-x^ny_2^n)(1-x^ny_2^{-n})}\ .$$ The factor $I^{(0)}$ was computed in [@Bhattacharya:2008bja]. Since the second factor becomes $1$ (from $V=0$) if there are no fluxes in the saddle point, this is a generalization of the large $N$ result of [@Bhattacharya:2008bja]. We now turn to the remaining part of the holonomy integral in (\[exact-index\]) apart from $I^{(0)}$. The integral over $M_1\!+\!M_2$ variables $\alpha_i,\tilde\alpha_i$ including the second factor in (\[large-N-gaussian\]) can be written as $$\begin{aligned} &&\hspace{-0.7cm} x^{\epsilon_0}\!\int_0^{2\pi}\!\!\! \frac{1}{{\rm (symmetry)}}\left[\frac{d\alpha}{2\pi}\right]\! \left[\frac{d\tilde\alpha}{2\pi}\right]\!\! \prod_{\substack{i,j;\\n_i\!=\!n_j}}\! \left(2\sin\frac{\alpha_i\!-\!\alpha_j}{2}\right)^2\! \prod_{\substack{i,j;\\ \tilde{n}\!=\!\tilde{n}_j}}\! \left(2\sin\frac{\tilde\alpha_i\!-\!\tilde\alpha_j}{2}\right)^2\! e^{ik(\sum n_i\alpha_i-\sum\tilde{n}_i\tilde\alpha_i)}\\ &&\hspace{-0.7cm}\times \exp\left[\sum_{i=1}^{M_1}\sum_{j=1}^{M_2}\frac{1}{n} {\bf f}^{\rm bif}_{ij}(x^n,y_1^n,y_2^n,e^{in\alpha},e^{in\tilde\alpha})\! +\!\sum_{i,j=1}^{M_1}\frac{1}{n} {\bf f}^{\rm adj}_{ij}(x^n,y_1^n,y_2^n,e^{in\alpha})\!+\! \sum_{i,j=1}^{M_2}\frac{1}{n}\tilde{\bf f}^{\rm adj}_{ij}(x^n,y_1^n,y_2^n, e^{in\tilde\alpha})\right]\nonumber\end{aligned}$$ where $$\begin{aligned} {\bf f}^{\rm bif}_{ij}&=& \left(x^{|n_i\!-\!\tilde{n}_j|}-x^{|n_i|\!+\!|\tilde{n}_j|}\right) \left(f^+e^{i(\tilde\alpha_j\!-\!\alpha_i)}+ f^-e^{i(\alpha_i\!-\!\tilde\alpha_j)}\right)\nonumber\\ {\bf f}^{\rm adj}_{ij}&=&\left[-(1-\delta_{n_in_j})x^{|n_i\!-\!n_j|}+ x^{|n_i|\!+\!|n_j|}\right]e^{-i(\alpha_i\!-\!\alpha_j)}\nonumber\\ \tilde{\bf f}^{\rm adj}_{ij}&=& \left[-(1-\delta_{\tilde{n}_i\tilde{n}_j}) x^{|\tilde{n}_i\!-\!\tilde{n}_j|}+x^{|\tilde{n}_i|\!+\!|\tilde{n}_j|}\right] e^{-i(\tilde\alpha_i\!-\!\tilde\alpha_j)}\ ,\label{eff-letter-index}\end{aligned}$$ and the symmetry factor again divides by the permutation symmetry of identical variables $\alpha_i,\tilde\alpha_i$, depending on the gauge symmetry unbroken by fluxes. Recall that $\epsilon_0$ is Casimir energy like quantity which can be nonzero in the background with nonzero flux. The above integral can be factorized as follows. To explain this, we decompose nonzero fluxes $\{n_i\}$, $\{\tilde{n}_i\}$ into positive and negative ones $\{n^+_i: n^+_i>0, i=1,2,\cdots,M^+_1\}$, $\{n^-_i:n^-_i<0,i=1,2,\cdots,M^-_1\}$ and similarly $\{\tilde{n}^+_i: i=1,2,\cdots,M^+_2\}$, $\{\tilde{n}^-_i: i=1,2,\cdots,M^-_2\}$. Having a look at the indices in (\[eff-letter-index\]), one can observe that none of these functions get contribution from modes connecting two $U(1)$’s with one positive and one negative flux. This simply follows from $x^{|n_i\!-\!\tilde{n}_j|}\!=\!x^{|n_i|\!+\!|\tilde{n}_j|}$, $x^{|n_i\!-\!n_j|}\!=\!x^{|n_i|\!+\!|n_j|}$ and $x^{|\tilde{n}_i\!-\!\tilde{n}_j|}\!=\!x^{|\tilde{n}_i|\!+\!|\tilde{n}_j|}$ for pairs of fluxes with different signs. Furthermore, as explained in appendix B.3, the Casimir energy also factorizes into contributions coming from modes connecting positive fluxes or negative fluxes only. This proves a complete factorization of the integrand and the pre-factor into two pieces, each of which depending only on fluxes $\{n_i^+\},\{\tilde{n}^+_i\}$ and $\{n^-_i\},\{\tilde{n}^-_i\}$, respectively. Due to the overall translational invariance of $\alpha_i,\tilde\alpha_i$ and factorization, the integral is nonzero only if $$\sum_{i=1}^{M^+_1}n^+_i=\sum_{i=1}^{M^+_2}\tilde{n}^+_i\ \ ,\ \ \sum_{i=1}^{M^-_1}n^-_i=\sum_{i=1}^{M^-_2}\tilde{n}^-_i\ ,$$ namely the total positive and negative fluxes over two gauge groups match separately. We now write the expression for the full large $N$ index, summing over all saddle points. Since $\frac{k}{2}$ times the total number of fluxes is the Kaluza-Klein momentum along the Hopf fiber circle of $S^7/\mathbb{Z}_k$, we grade the summation with the chemical potential $y_3$ as $y_3^{\frac{k}{2}\sum_{i=1}^{M_1} n_i}$ (or $y_3^{\frac{k}{2}\sum_{i=1}^{M_2} \tilde{n}_i}$). The large $N$ index is $$I_{N=\infty}(x,y_1,y_2,y_3)=I^{(0)}(x,y_1,y_2,y_3) I^{(+)}(x,y_1,y_2,y_3)I^{(-)}(x,y_1,y_2,y_3)\ ,$$ where $I^{(0)}$ is given by (\[IIA-index\]), and $$\label{gauge-pos-index} \boxed{\begin{gathered} \hspace*{-3.5cm}I^{(+)}(x,y_1,y_2,y_3)=\sum_{M_1,M_2=0}^\infty\ \sum_{\substack{n_1\geq\cdots\geq n_{M_1}>0\\ \tilde{n}_1\geq\cdots \geq \tilde{n}_{M_2}>0}} y_3^{\frac{k}{2}\sum n_i}x^{\sum|n_i\!-\!\tilde{n}_j| -\sum_{i<j}|n_i\!-\!n_j|-\sum_{i<j}|\tilde{n}_i\!-\!\tilde{n}_j|}\\ \hspace*{-1.2cm}\times\int_0^{2\pi}\frac{1}{{\rm (symmetry)}} \left[\frac{d\alpha}{2\pi}\right]\left[\frac{d\tilde\alpha}{2\pi}\right] e^{ik(\sum n_i\alpha_i\!-\!\sum\tilde{n}_i\tilde\alpha_i)} \prod_{\substack{i,j;\\n_i\!=\!n_j}}\left[2\sin\frac{\alpha_i\!-\!\alpha_j}{2} \right]^2\prod_{\substack{i,j;\\ \tilde{n}_i\!=\!\tilde{n}_j}} \left[2\sin\frac{\tilde\alpha_i\!-\!\tilde\alpha_j}{2}\right]^2\\ \hspace*{0.1cm}\times\exp\left[\sum_{i=1}^{M_1}\sum_{j=1}^{M_2}\frac{1}{n} {\bf f}^{\rm bif} _{ij}(x^n,y_1^n,y_2^n,e^{in\alpha},e^{in\tilde\alpha})+ \sum_{i,j=1}^{M_1}{\bf f}^{\rm adj}_{ij}(x^n,y_1^n,y_2^n,e^{in\alpha})+ \sum_{i,j=1}^{M_2}\tilde{\bf f}^{\rm adj}_{ij}(x^n,y_1^n,y_2^n, e^{in\tilde\alpha})\right] \end{gathered}}$$ with definitions for various functions given by (\[eff-letter-index\]). The last factor $I^{(-)}(x,y_1,y_2,y_3)$, which is a summation of saddle points with negative fluxes only, takes a form similar to $I^{(+)}$ with signs of $n_i,\tilde{n}_i$ flipped. The signs of these integers appear only in $y_3^{\frac{k}{2}\sum n_i}$ and $e^{ik(\sum n_i\alpha_i-\sum\tilde{n}_i\tilde\alpha_i)}$. The sign flip in the first factor can be undone by replacing $y_3$ by $1/y_3$, and that in the second factor can be undone by changing integration variables from $\alpha,\tilde\alpha$ to $-\alpha,-\tilde\alpha$. The latter change affects ${\bf f}^{\rm bif}_{ij}$ by the exchange $f^+\leftrightarrow f^-$, which can be achieved by changing $\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}$ and $\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}$. Collecting all, one finds that $$I^{(-)}(x,y_1,y_2,y_3)=I^{(+)}(x,y_1,1/y_2,1/y_3)= I^{(+)}(x,1/y_1,y_2,1/y_3)\ .$$ Since the knowledge of $I^{(+)}$ would be enough to obtain the full index, we will mainly consider this function in the rest of this section. We want to compare the above result with the index over supersymmetric gravitons in $AdS_4\times S^7/\mathbb{Z}_k$. As shown in appendix C, the index of multiple gravitons also split into three parts, $I_{\rm mp}=I^{(0)}_{\rm mp} I^{(+)}_{\rm mp}I^{(-)}_{\rm mp}$, essentially because gravitons with positive and negative momenta do not mutually interact, even without the ‘statistical interaction’ for identical particles. It was shown in [@Bhattacharya:2008bja] that $I^{(0)}_{\rm mp}=I^{(0)}$. For the gauge theory and gravity indices to agree, one has to show $I^{(+)}I^{(-)}=I^{(+)}_{\rm mp}I^{(-)}_{\rm mp}$, or $\frac{I^{(+)}}{I^{(+)}_{\rm mp}}=\frac{I^{(-)}_{\rm mp}}{I^{(-)}}$. Left hand side and right hand side can be Taylor-expanded in $y_3^{\frac{1}{2}}$ and $y_3^{-\frac{1}{2}}$, respectively, together with positive power expansions in $x$. The only way this equation can hold is both sides being a constant, which is actually $1$. Thus, one only has to show $$I^{(+)}(x,y_1,y_2,y_3)=I^{(+)}_{\rm mp}(x,y_1,y_2,y_3)$$ to check the agreement of the indices in gauge theory and gravity. By definition, saddle points for $I^{(+)}$ carry positive fluxes only. Since a sequence of non-decreasing positive integers can be represented by a Young diagram, we will sometimes represent positive $\{n_i\},\{\tilde{n}_i\}$ by a pair of Young diagrams $Y$ and $\tilde{Y}$, where the lengths of $i$’th rows are $n_i$ and $\tilde{n}_i$. We denote by $d(Y)=d(\tilde{Y})$ the total number of boxes in the Young diagram. The summation in $I^{(+)}$ can be written as $$\begin{aligned} \label{young-expand} &&\hspace{-0.7cm}I^{(+)}(x,y_1,y_2,y_3)= \sum_{Y,\tilde{Y}: d(Y)=d(\tilde{Y})} y_3^{\frac{k}{2}d(Y)}I_{Y\tilde{Y}}(x,y_1,y_2)\\ &&\hspace{-0.6cm} =1+y_3^{\frac{k}{2}}I_{\Yboxdim4pt\yng(1)~\yng(1)}+ y_3^k\left(I_{\Yboxdim4pt\yng(2)~\yng(2)}\!+\! I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}\!+\! 2I_{\Yboxdim4pt\yng(2)~\yng(1,1)}\right)\!+\! y_3^{\frac{3k}{2}}\left(I_{\Yboxdim4pt\yng(3)~\yng(3)}\!+\! I_{\Yboxdim4pt\yng(2,1)~\yng(2,1)}\!+\! I_{\Yboxdim4pt\yng(1,1,1)~\yng(1,1,1)}\!+\! 2I_{\Yboxdim4pt\yng(3)~\yng(2,1)}\!+\! 2I_{\Yboxdim4pt\yng(3)~\yng(1,1,1)}\!+\! 2I_{\Yboxdim4pt\yng(2,1)~\yng(1,1,1)}\right)\!+\!\cdots\nonumber\end{aligned}$$ where we used $I_{Y\tilde{Y}}=I_{\tilde{Y}Y}$, which we do not prove here but can be checked by suitable redefinitions of integration variables in (\[gauge-pos-index\]). We did not manage to analytically prove $I^{(+)}=I^{(+)}_{\rm mp}$ generally. Below we provide nontrivial analytic and numerical checks of this claim in various sectors: we consider the sectors in which the total number of positive fluxes $\sum_{i}n_i^+=\sum_i\tilde{n}^+$ is $1$, $2$ and $3$. One KK-momentum: analytic tests ------------------------------- We analytically prove the agreement between gauge theory and gravity indices in the sector with unit KK-momentum. This amounts to comparing the coefficients of $y_3^{\frac{k}{2}}$ in $I^{(+)}$ and $I^{(+)}_{\rm mp}$. The gravity result is simply $$\label{grav-1-flux} I_{k}^{\rm sp}(x,y_1,y_2)=\oint\frac{d\sqrt{y_3}}{2\pi i\sqrt{y_3}} y_3^{-\frac{k}{2}}I^{\rm sp}(x,y_1,y_2,y_3)\ ,$$ namely index over single graviton with $k$ (i.e. minimal) units of KK-momentum. The contour for $\sqrt{y_3}$ integration is the unit circle in the complex plane. On the gauge theory side, the result comes from one saddle point with fluxes given by $n\!=\!\tilde{n}\!=\!1$: from the general formula (\[gauge-pos-index\]) one obtains $$\label{gauge-1-flux} I_{\Yboxdim4pt\yng(1)~\yng(1)}= \int_0^{2\pi}\frac{d\alpha d\tilde\alpha}{(2\pi)^2} e^{ik(\alpha\!-\!\tilde\alpha)}\exp\left[ \sum_{n=1}^\infty\frac{1}{n}\left((1-x^{2n})\left(f^+(\cdot^n) e^{in(\tilde\alpha\!-\!\alpha)}+f^-(\cdot^n) e^{in(\alpha\!-\!\tilde\alpha)}\!\frac{}{}\right)+2x^{2n} \right)\right]\ .$$ The ‘effective letter index’ $(1-x^2)\left(f^+e^{i(\tilde\alpha\!-\!\alpha)}+f^- e^{i(\alpha\!-\!\tilde\alpha)}\!\frac{}{}\right)+2x^{2}$ in the exponential is $$\label{eff-1-flux} \hspace{-1cm}\left[x^{\frac{1}{2}}\left(\!\sqrt{\frac{y_1}{y_2}}\!+\! \sqrt{\frac{y_2}{y_1}}\right)\!-\!x^{\frac{3}{2}}\left(\!\sqrt{y_1y_2}\!+\! \frac{1}{\sqrt{y_1y_2}}\right)\right]e^{i(\tilde\alpha\!-\!\alpha)} \!+\!\left[x^{\frac{1}{2}}\left(\!\sqrt{y_1y_2}\!+\! \frac{1}{\sqrt{y_1y_2}}\right)\!-\!x^{\frac{3}{2}} \left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}} \right)\right]e^{i(\alpha\!-\!\tilde\alpha)}+2x^2\ .$$ Note that $f^{\pm}$ have $\frac{1}{1-x^2}$ factors, coming from many derivatives acting on fields, and take the form of infinite series in $x$. The factor $(1\!-\!x^2)$ cancels these derivative factors and lets the effective index be a finite series. Defining an integration variable $z\equiv e^{i(\tilde\alpha\!-\!\alpha)}$ in (\[gauge-1-flux\]), and after exponentiating (\[eff-1-flux\]), one obtains $$\label{1-flux-gauge} I_{\Yboxdim4pt\yng(1)~\yng(1)}= \oint\frac{dz}{(2\pi i)z}z^{-k} \frac{\left(1-x\sqrt{xy_1y_2}z\right)\left(1-x\sqrt{\frac{x}{y_1y_2}}z\right) \left(1-x\sqrt{\frac{xy_1}{y_2}}z^{-1}\right) \left(1-x\sqrt{\frac{xy_2}{y_2}}z^{-1}\right)} {\left(1-\sqrt{\frac{xy_1}{y_2}}z\right)\left(1-\sqrt{\frac{xy_2}{y_1}}z\right) \left(1-\sqrt{xy_1y_2}z^{-1}\right) \left(1-\sqrt{\frac{x}{y_1y_2}}z^{-1}\right)(1-x^2)^2}\ .$$ Using the relation (\[relation\]), and identifying the integration variable as $z=\sqrt{y_3}$, the above result can be rewritten as $$I_{\Yboxdim4pt\yng(1)~\yng(1)}= \oint\frac{d\sqrt{y_3}}{(2\pi i)\sqrt{y_3}}y_3^{-\frac{k}{2}} \left(I^{\rm sp}(x,y_1,y_2,y_3)+\frac{1-x^2+x^4}{(1-x^2)^2}\right)\ .$$ Since the second term in the integrand does not survive the contour integral, this is exactly the gravity expression (\[grav-1-flux\]), proving the agreement in this sector. Before proceeding to more nontrivial examples, let us explain a bit more on the above index. As stated in the previous section, the flux provides a lower bound to the energy of states. In the sector with unit flux, we can actually find from the integrand of (\[1-flux-gauge\]) that \#($\sqrt{x}$) in a term is always larger than or equal to \#($z$). We can actually arrange the terms in Taylor expansion of the integrand so that the number \#($\sqrt{x}$)$-$\#($z$) ascends. The lowest order terms come from the first two factors in the denominator containing $\sqrt{x}z$, for which this number is $0$. The index for these states is $$\label{1-flux-gauge-low} \oint\frac{dz}{(2\pi i)z}z^{-k} \frac{1}{\left(1-\sqrt{\frac{xy_1}{y_2}}z\right) \left(1-\sqrt{\frac{xy_2}{y_1}}z\right)}= x^{\frac{k}{2}}\left(y_1^{\frac{k}{2}}y_2^{-\frac{k}{2}}+ y_1^{\frac{k}{2}\!-\!1}y_2^{-\frac{k}{2}\!+\!1}+\cdots+ y_1^{-\frac{k}{2}}y_2^{\frac{k}{2}}\right)\ .$$ From Table 1, the two factors in the integrand originate from the gauge theory letters $\bar{B}^{\dot{2}}$ and $\bar{B}^{\dot{1}}$ in s-waves. The operators made of these letters form a subset of chiral operators studied in [@Aharony:2008ug; @Hanany:2008qc]. The operator took the form of $k$’th product of $\bar{B}^{\dot{a}}$, multiplied by a ’t Hooft operator in the (Sym($\bar{\bf N}^k$),Sym(${\bf N}^k$)) representation to make the whole operator gauge invariant. Since no fermionic letters can contribute in the lowest energy sector due to their larger dimensions than scalars, the above index equals to the partition function. This is not true any more as one goes beyond lowest energy as fermionic letters start to enter. This aspect in the lowest energy sector will continue to appear with more fluxes below. The full spectrum of these chiral operators, preserving specific $\mathcal{N}\!=\!2$ supersymmetry, has been studied in [@Aharony:2008ug] by quantizing the moduli space [@Kinney:2005ej; @Mandal:2006tk; @Benvenuti:2006qr]. We expect our result to be identical to the result in [@Aharony:2008ug], which we check explicitly for the case with two fluxes in the next subsection. Two KK-momenta: analytic and numerical tests -------------------------------------------- Monopole operators which have been studied in the context of $\mathcal{N}\!=\!6$ Chern-Simons theory are in the conjugate representations of the two $U(N)$ gauge groups, such as (Sym($\bar{\bf N}^k$),Sym(${\bf N}^k$)) in the previous subsection or more general examples studied in [@Berenstein:2008dc; @Klebanov:2008vq]. In our analysis, these are related to the saddle points in which two flux distributions $\{n_i\}$ and $\{\tilde{n}_i\}$ are the same. In the sector with two fluxes, two of the four saddle points $\Yboxdim8pt\yng(2)~\yng(2)~$ and $\Yboxdim8pt\yng(1,1)~\yng(1,1)$ are in this category, while the other two $\Yboxdim8pt\yng(2)~\yng(1,1)$ and $\Yboxdim8pt\yng(1,1)~\yng(2)$ are not.Let us call the former kind of flux distributions as ‘equal distributions.’ Incidently, the way of having equally distributing given amount of fluxes to many $U(1)$ factors is the same as the way of distributing same amount of momenta (in units of $\frac{k}{2}$) to multiple gravitons, each carrying positive KK-momenta. For instance, the first of the above two distributions maps to giving two units of KK-momenta to a single graviton, while the second maps to picking two gravitons and giving one unit of momentum to each. This might let one suspect that there could be some relation between the index from a saddle point with equal distribution and the multi-graviton index with the corresponding momentum distribution. What we find below empirically says that this is true up to a certain order in the $x$ expansion. However, as we go beyond certain energy, it will turn out that only the total sum over all saddle points with equal distributions equals the total multi-graviton index. As one goes beyond an even higher energy threshold, the saddle points with unequal saddle points start to appear which add to the saddle points with equal distributions to correctly reproduce the graviton index. The saddle points with unequal flux distributions provide examples in which monopole operators contribute to the energy, or the scaling dimension, of the whole gauge invariant operator. Monopole operators with vanishing scale dimensions are studied in [@Aharony:2008ug; @Berenstein:2008dc; @Klebanov:2008vq] in $\mathcal{N}\!=\!6$ Chern-Simons theory, while in general this does not have to be the case. See [@Borokhov:2002ib; @Klebanov:2008vq] as well as recent works [@Imamura:2009ur; @Gaiotto:2009tk] on monopole operators in $\mathcal{N}\!=\!4$ and $\mathcal{N}\!=\!3$ theories. Let us present various checks that we did in this sector. In this subsection we write $p\equiv\sqrt{y_1y_2}$ and $r=\sqrt{\frac{y_1}{y_2}}$ to simplify the formulae. Let us define the following functions $$\begin{aligned} f(x,p,r,z) &=&\sqrt{x}z(r+r^{-1})+\sqrt{x}z^{-1}(p+p^{-1})-x\sqrt{x}z(p+p^{-1})- x\sqrt{x}z^{-1}(r+r^{-1})\nonumber\\ F(x,p,r,z)&=&\exp\left(\sum_{n=1}^\infty\frac{1}{n}f(x^n,p^n,r^n,z^n)\right) \nonumber\\ &=&\frac{(1-x\sqrt{x}pz)(1-x\sqrt{x}p^{-1}z)(1-x\sqrt{x}rz^{-1}) (1-x\sqrt{x}r^{-1}z^{-1})} {(1-\sqrt{x}rz)(1-\sqrt{x}r^{-1}z)(1-\sqrt{x}pz^{-1}) (1-\sqrt{x}p^{-1}z^{-1})}\ ,\nonumber\end{aligned}$$ which should be familiar from our analysis in the previous subsection, and also define $$\begin{aligned} f_m(x,p,r,z)&=&x^mf(x,p,r,z)\nonumber\\ F_m(x,p,r,z)&=&\exp\left(\sum_{n=1}^\infty\frac{1}{n}f_m(x^n,p^n,r^n,z^n)\right) \nonumber\\ &=&\frac{(1-x^{m\!+\!1}\sqrt{x}pz)(1-x^{m\!+\!1}\sqrt{x}p^{-1}z) (1-x^{m\!+\!1}\sqrt{x}rz^{-1})(1-x^{m\!+\!1}\sqrt{x}r^{-1}z^{-1})} {(1-x^m\sqrt{x}rz)(1-x^m\sqrt{x}r^{-1}z)(1-x^m\sqrt{x}pz^{-1}) (1-x^m\sqrt{x}p^{-1}z^{-1})}\nonumber\end{aligned}$$ Using these functions, $$f_{|n_i\!-\!\tilde{n}_j|}(x,p,r,e^{i(\alpha_i\!-\!\tilde\alpha_j)})+ f_{|n_i\!-\!\tilde{n}_j|\!+\!2}(x,p,r,e^{i(\alpha_i\!-\!\tilde\alpha_j)})+ \cdots+f_{n_i\!+\!\tilde{n}_j\!-\!2}(x,p,r,e^{i(\alpha_i\!-\!\tilde\alpha_j)})$$ for $n_i\!\neq\!\tilde{n}_j$ is what we called ${\bf f}^{\rm bif}_{ij}$ in the previous section. To compare $I^{(+)}$ with $I^{(+)}_{\rm mp}$ in the sector with two units of fluxes, i.e. at the order $y_3^{k}$, we write the integral expressions for indices from four saddle points. Getting rid of a trivial integral corresponding to the decoupled $U(1)$, one obtains the following expressions. $$\begin{aligned} \label{saddle-1} I_{\Yboxdim4pt\yng(2)~\yng(2)}&=&\frac{1}{2\pi i}\oint\frac{dz}{z} z^{-2k}\frac{F(x,p,r,z)F_2(x,p,r,z)}{(1-x^4)^2}\end{aligned}$$ where $z\equiv e^{i(\tilde\alpha\!-\!\alpha)}$, $$\begin{aligned} \label{saddle-2} I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)} &=&\oint\frac{dz}{(2\pi i)z}z^{-2k} \oint\frac{dadb}{(2\pi i)^2ab}\left(1-\frac{a^2}{2}-\frac{1}{2a^2}\right) \left(1-\frac{b^2}{2}-\frac{1}{2b^2}\right)\nonumber\\ &&\hspace{1cm}\times\frac{F(x,p,r,zab)F(x,p,r,xab^{-1}) F(x,p,r,za^{-1}b)F(x,p,r,za^{-1}b^{-1})} {(1-x^2)^4(1-x^2a^2)(1-x^2a^{-2})(1-x^2b^2)(1-x^2b^{-2})}\end{aligned}$$ where $e^{i\alpha_1}=z^{-1/2}a$, $e^{i\alpha_2}=z^{-1/2}a^{-1}$, $e^{i\tilde\alpha_1}=z^{1/2}b$, $e^{i\tilde\alpha_2}=z^{1/2}b^{-1}$, and $$\label{saddle-3} I_{\Yboxdim4pt\yng(2)~\yng(1,1)}=I_{\Yboxdim4pt\yng(1,1)~\yng(2)} =x^2\oint\frac{dz}{(2\pi i)z}z^{-2k}\oint\frac{da}{(2\pi i)a} \left(1-\frac{a^2}{2}-\frac{1}{2a^2}\right)\frac{F_1(x,p,r,za) F_1(x,p,r,za^{-1})}{(1-x^4)^2(1-x^2)^2(1-x^2a^2)(1-x^2a^{-2})}$$ where $e^{i\alpha_1}\!=\!z^{-1/2}a$, $e^{i\alpha_2}\!=\!z^{-1/2}a^{-1}$, $e^{i\tilde\alpha}\!=\!z^{1/2}$ for $I_{\Yboxdim4pt\yng(2)~\yng(1,1)}$ , and $e^{i\alpha}\!=\!z^{-1/2}$, $e^{i\tilde\alpha_1}\!=\!z^{1/2}a$, $e^{i\tilde\alpha_2}\!=\!z^{1/2}a^{-1}$ for $I_{\Yboxdim4pt\yng(1,1)~\yng(2)}$ . All contour integrals here and below are along the unit circle on the complex plane. The graviton index with two units of momenta is given by $$\label{graviton-2-flux} \left.I^{(+)}_{\rm mp}(x,p,r)\frac{}{}\right|_{y_3^{k}} =I_{2k}(x,p,r)+\frac{1}{2}I_k(x^2,p^2,r^2)+\frac{1}{2}I_k(x,p,r)^2$$ The first term is from single graviton with two units of momenta, while the last two terms are from two identical gravitons, each with unit momentum. We expect the sum of four contributions (\[saddle-1\]), (\[saddle-2\]), (\[saddle-3\]) to equal (\[graviton-2-flux\]). We first study the lowest energy states in this sector analytically. As in the previous section, we arrange the Taylor expansions of the integrands in ascending orders in \#($\sqrt{x}$)$-$\#($z$). From the behaviors of the functions $F,F_1,F_2$, one finds that unequal distributions do not contribute to the lowest energy sector. The two equal distributions contribute with $\epsilon\!=\!k$ as $$I_{\Yboxdim4pt\yng(2)~\yng(2)}\rightarrow \oint\frac{dz}{(2\pi i)z}z^{-2k} \frac{1}{\left(1-\sqrt{\frac{xy_1}{y_2}}z\right) \left(1-\sqrt{\frac{xy_2}{y_1}}z\right)}= x^{k}\left(y_1^{k}y_2^{-k}+y_1^{k\!-\!1}y_2^{-k\!+\!1}+\cdots+ y_1^{-k}y_2^{k}\right)$$ and $$\begin{aligned} \label{2-low-gauge} I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}&\rightarrow& \oint\frac{dz}{(2\pi i)z}z^{-2k} \oint\frac{d\nu}{(2\pi i)\nu}\oint\frac{d\mu}{(2\pi i)\mu} (1-\mu\nu)(1-\mu\nu^{-1})\\ &&\times\frac{1}{(1-tz\mu)(1-uz\mu)(1-tz\mu^{-1})(1-uz\mu^{-1}) (1-tz\nu)(1-uz\nu)(1-tz\nu^{-1})(1-uz\nu^{-1})}\nonumber\end{aligned}$$ where we redefined integration variables and the chemical potentials as $a^2=\mu\nu$ and $b^2=\mu\nu^{-1}$, $x^{\frac{1}{2}}r=t$, $x^{\frac{1}{2}}r^{-1}=u$. The two factors on the first line in the integrand come from the $U(2)\times U(2)$ measure, which we can effectively change to $(1\!-\!a^2)(1\!-\!b^2)$ since rest of the integrand is invariant under $a\rightarrow a^{-1}$ and $b\rightarrow b^{-1}$, separately. From the results in appendix C, the two graviton indices at the lowest energy are given by (upon identifying $z=\sqrt{y_3}$) $$I_{2k}\rightarrow\oint\frac{dz}{(2\pi i)z} z^{-2k}\frac{1}{(1-tz)(1-uz)} \stackrel{\rm lowest}{=}I_{\Yboxdim4pt\yng(2)~\yng(2)}$$ and $$\begin{aligned} \label{2-low-grav} \frac{1}{2}I_{k}(\cdot^2)+\frac{1}{2}I_k(\cdot)^2\!&\!\rightarrow\!&\! \frac{1}{2}\oint\frac{dz}{(2\pi i)z}z^{-2k}\frac{1}{(1-t^2z^2)(1-u^2z^2)}\\ &&\hspace{1cm} +\frac{1}{2}\oint\frac{dz_1dz_2}{(2\pi i)^2z_1z_2}z_1^{-k}z_2^{-k} \frac{1}{(1-tz_1)(1-uz_1)(1-tz_2)(1-uz_2)}\nonumber\\ &=&\oint\frac{dz}{(2\pi i)z}z^{-2k}\left[\!\frac{}{}\right. \frac{1}{2(1-t^2z^2)(1-u^2z^2)}\nonumber\\ &&\hspace{1cm}+\frac{1}{2}\oint\frac{d\nu}{(2\pi i)\nu}\frac{1} {(1-tz\nu)(1-uz\nu)(1-tz\nu^{-1})(1-uz\nu^{-1})} \left.\frac{}{}\!\right]\nonumber\end{aligned}$$ where we changed to variables $z_1=z\nu$, $z_2=z\nu^{-1}$ on the last line. Keeping the $z$ integral, we contour-integrate $\mu$ and $\nu$ in (\[2-low-gauge\]) and (\[2-low-grav\]) to compare them. After some algebra, one finds $$\frac{1}{(1-t^2z^2)(1-u^2z^2)(1-tuz^2)}+{\rm\ const.}$$ for both indices, where the last constant is $0$ and $-1$ for gauge theory and gravity, respectively, which does not survive the remaining $z$ integration. This shows the agreement of the indices from gauge theory and gravity at the lowest energy with two fluxes. The indices from two saddle points separately agree with two corresponding gravity indices, as explained before. This result can also be obtained by quantizing the moduli space [@Aharony:2008ug], restricted to the fields $\bar{B}^{\dot{1}}$ and $\bar{B}^{\dot{2}}$. Since we are not aware of any further way of treating the integral analytically, we compare the two indices order by order after expanding in $x$. Firstly, for $k\!=\!1$, we find a perfect agreement to $\mathcal{O}(x^{9})$ terms that we checked, as follows. We find $$\begin{aligned} I_{\Yboxdim4pt\yng(2)~\yng(2)}\!&=&\! x(r^2+1+r^{-2})+x^2(p+p^{-1})(r^3+r^{-3}) +x^3\left[(p^2+p^{-2})(r^4\!+\!r^{-4})-2(r^2+r^{-2})\right]\nonumber\\ &&\!+x^4\left[(p^3+p^{-3})(r^5+r^{-5})+(p+p^{-1})(r+r^{-1})\right] \nonumber\\ &&\!+x^5\left[(p^4+p^{-4})(r^6+r^{-6})+(r^4-2r^2-4-2r^{-2}+r^{-4})\right] \nonumber\\ &&\!+x^6\left[\left(p^5+p^{-5}\right)(r^7+r^{-7})+ \left(p+p^{-1}\right)(-2r^3+2r\cdots)\right]\nonumber\\ &&\!+x^7\left[(p^6+p^{-6})(r^8+r^{-8})+(p^2+p^{-2})(r^2+r^{-2})+ (r^4-3+r^{-4})\right]\nonumber\\ &&\!+x^8\left[(p^7+p^{-7})(r^9+r^{-9})+(p+p^{-1})(r^5-2r^3-2r\cdots)\right] \nonumber\\ &&\!+x^9\left[(p^8\!+\!p^{-8})(r^{10}\!+\!r^{-10})+(p^2\!+\!p^{-2}) (\!-2r^4\!+\!2r^2\!+\!3\cdots) +(\!-r^4\!+\!5r^2\!+\!6\cdots)\right]\!+\!\mathcal{O}(x^{10})\nonumber\end{aligned}$$ $$\begin{aligned} I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}\!&=&\! x(r^2+1+r^{-2})+x^2(p+p^{-1})(r^3+r+r^{-1}+r^{-3})\nonumber\\ &&\!+x^3\left[(p^2+p^{-2})(2r^4+r^2+1+r^{-2}+2r^{-4}) +(r^4+r^{-4})\right]\nonumber\\ &&\!+x^4\left[(p^3+p^{-3})(2r^5+r^3+r\cdots)+(p+p^{-1})(r^5-r^3-r\cdots)\right] \nonumber\\ &&\!+x^5\left[(p^4+p^{-4})(3r^6+r^4+r^2+1\cdots)+(p^2+p^{-2})(r^6-r^4\cdots) +(r^6-r^4+3\cdots)\right]\nonumber\\ &&\!+x^6\left[\left(p^5\!+\!p^{-5}\right)(3r^7\!+\!r^5\!+\!r^3\!+\!r\cdots)+ \left(p^3\!+\!p^{-3}\right)(r^7\!-\!r^5\cdots)+ (p\!+\!p^{-1})(r^7\!+\!r^3\cdots)\right]\nonumber\\ &&\!+x^7\left[(p^6+p^{-6})(4r^8+r^6+r^4+r^2+1\cdots)+(p^4+p^{-4}) (r^8-r^6\cdots)\right.\nonumber\\ &&\!\left.\hspace{2cm}+(p^2+p^{-2})(r^8-r^4-r^2-2\cdots)+ (r^8-r^4-3r^2-1\cdots)\right]\nonumber\\ &&\!+x^8\left[(p^7+p^{-7})(4r^9+r^7+r^5+r^3+r\cdots)+(p^5+p^{-5})(r^9-r^7\cdots) \right.\nonumber\\ &&\!\left.\hspace{2cm}+(p^3+p^{-3})(r^9-r^5\cdots) +(p+p^{-1})(r^9-r^5+r^3+4r\cdots)\right]\nonumber\\ &&\!+x^9\left[(p^8+p^{-8})(5r^{10}+r^8+r^6+r^4+r^2+1\cdots)+(p^6+p^{-6}) (r^{10}-r^8\cdots)\right.\nonumber\\ &&\!\left.\hspace{0.5cm}+(p^4\!+\!p^{-4})(r^{10}\!-\!r^6\cdots)+ (p^2\!+\!p^{-2})(r^{10}\!+\!r^4\!-\!1\cdots)+ (r^{10}\!+\!r^4\!-\!2r^2\!-\!3\cdots)\right]\!+\!\mathcal{O}(x^{10})\nonumber\end{aligned}$$ $$\begin{aligned} I_{\Yboxdim4pt\yng(2)~\yng(1,1)}=I_{\Yboxdim4pt\yng(1,1) ~\yng(2)}&=&x^5-x^6\left(p+p^{-1}\right)\left(r+r^{-1}\right)+ x^7\left[(p^2+p^{-2})+(r^2+3+r^{-2})\right]\nonumber\\ &&-x^8\left(p\!+\!p^{-1}\right)\left(r\!+\!r^{-1}\right)-x^9 \left[(p^2\!+\!p^{-2})(r^2\!+\!1\!+\!r^{-2})\!+\!2(r^2\!+\!r^{-2})\right] \!+\!\mathcal{O}(x^{10})\nonumber\end{aligned}$$ and on the gravity side we find $$\begin{aligned} I_2(x,p,r)\!&=&\!x(r^2+1+r^{-2})+x^2(p+p^{-1})(r^3+r^{-3})+ x^3\left[(p^2+p^{-2})(r^4+r^{-4})-(r^2+r^{-2})\right]\nonumber\\ &&\!+x^4(p^3\!+\!p^{-3})(r^5\!+\!r^{-5})\!+\! x^5\left[(p^4\!+\!p^{-4})(r^6\!+\!r^{-6})\!-\!(r^2\!+\!r^{-2}) \right]\!+\!x^6(p^5\!+\!p^{-5})(r^7\!+\!r^{-7})\nonumber\\ &&\!+x^7\left[(p^6+p^{-6})(r^8+r^{-8})-(r^2+r^{-2})\right]+ x^8(p^7+p^{-7})(r^9+r^{-9})\nonumber\\ &&\!+x^9\left[(p^8+p^{-8})(r^{10}+r^{-10}) -(r^2+r^{-2})\right]+\mathcal{O}(x^{10})\nonumber\end{aligned}$$ $$\begin{aligned} &&\hspace{-1.5cm} \frac{1}{2}I_1(x^2,p^2,r^2)+\frac{1}{2}I_1(x,p,r)^2\nonumber\\ &=&\!x(r^2\!+\!1\!+\!r^{-2})+x^2(p\!+\!p^{-1})(r^3\!+\!r\cdots) +x^3\left[(p^2\!+\!p^{-2})(2r^4\!+\!r^2\!+\!1\cdots)\!+\! (r^4\!-\!r^2\cdots)\right]\nonumber\\ &&\!+x^4\left[(p^3+p^{-3})(2r^5+r^3+r\cdots)+(p+p^{-1})(r^5-r^3\cdots)\right] \nonumber\\ &&\!+x^5\left[(p^4+p^{-4})(3r^6+r^4+r^2+1\cdots)+(p^2+p^{-2})(r^6-r^4\cdots)+ (r^6-r^2+1\cdots)\right]\nonumber\\ &&\!+x^6\left[(p^5\!+\!p^{-5})(3r^7\!+\!r^5\!+\!r^3\!+\!r\cdots)+ (p^3\!+\!p^{-3})(r^7\!-\!r^5\cdots) +(p\!+\!p^{-1})(r^7\!-\!r^3\cdots)\right]\nonumber\\ &&\!+x^7\left[(p^6+p^{-6})(4r^8+r^6+r^4+r^2+1\cdots)+ (p^4+p^{-4})(r^8-r^6\cdots)\right.\nonumber\\ &&\!\left.\hspace{2cm} +(p^2+p^{-2})(r^8-r^4\cdots)+(r^8+2+r^{-8})\right]\nonumber\\ &&\!+x^8\left[(p^7+p^{-7})(4r^9+r^7+r^5+r^3+r\cdots)+ (p^5+p^{-5})(r^9-r^7\cdots)\right.\nonumber\\ &&\!\left.\hspace{2cm} +(p^3+p^{-3})(r^9-r^5\cdots)+(p+p^{-1})(r^9-r^3\cdots)\right]\nonumber\\ &&\!+x^9\left[(p^8+p^{-8})(5r^{10}+r^8+r^6+r^4+r^2+1\cdots)+ (p^6+p^{-6})(r^{10}-r^8\cdots)\right.\nonumber\\ &&\!\left.\hspace{1cm}+(p^4\!+\!p^{-4})(r^{10}\!-\!r^6\cdots)+ (p^2\!+\!p^{-2})(r^{10}\!-\!r^4\cdots) +(r^{10}\!+\!3\!+\!r^{-10})\right]+\mathcal{O}(x^{10})\ .\nonumber\end{aligned}$$ The terms in ‘$\cdots$’ take negative powers of $r$, and can be completed from the fact that the expression in any parenthesis has $r\rightarrow r^{-1}$ invariance. From this one can check $$I_{\Yboxdim4pt\yng(2)~\yng(2)}+ I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}+ I_{\Yboxdim4pt\yng(2)~\yng(1,1)}+ I_{\Yboxdim4pt\yng(1,1)~\yng(2)} =I_2(x,p,r)+\frac{1}{2}I_1(x^2,p^2,r^2)+\frac{1}{2}I_1(x,p,r)^2 +\mathcal{O}(x^{10})\nonumber$$ for $k\!=\!1$. We also found perfect agreement for $k\!=\!2$ for all terms that we calculated. To reduce execution time for numerical calculation, space of presenting our result and most importantly to reduce possible typos, we set $r\!=\!p\!=\!1$ and keep $x$ only. We find $$\begin{aligned} I_{\Yboxdim4pt\yng(2)~\yng(2)}\!&=&\! 5x^2+4x^3+0x^4+8x^5-2x^6+4x^7+4x^8+4x^9-2x^{10}+ 8x^{11}+5x^{12}+\mathcal{O}(x^{13})\nonumber\\ I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}\!&=&\! 6x^2\!+\!12x^3\!+\!18x^4\!+\!16x^5\!+\!29x^6\!+\!28x^7\!+\!32x^8\!+\! 44x^9\!+\!29x^{10}\!+\!72x^{11}\!+\!31x^{12}\!+\!\mathcal{O}(x^{13}) \nonumber\\ I_{\Yboxdim4pt\yng(2)~\yng(1,1)}\!=\! I_{\Yboxdim4pt\yng(1,1)~\yng(2)}\!&=&\! x^8-4x^9+10x^{10}-16x^{11}+11x^{12}+\mathcal{O}(x^{13})\nonumber\end{aligned}$$ and $$\begin{aligned} I_4(x)\!&\!=\!&\!5x^2+4x^3+2x^4+4x^5+2x^6+4x^7+2x^8+4x^9+2x^{10} +4x^{11}+2x^{12}+\mathcal{O}(x^{13})\nonumber\\ \frac{I_2(x^2)\!+\!I_2(x)^2}{2}\!&\!=\!&\! 6x^2\!+\!12x^3\!+\!16x^4\!+\!20x^5\!+\!25x^6\!+\!28x^7\!+\!36x^8\!+\!36x^9 \!+\!45x^{10}\!+\!44x^{11}\!+\!56x^{12}\!+\!\mathcal{O}(x^{13})\nonumber\end{aligned}$$ which proves $$I_{\Yboxdim4pt\yng(2)~\yng(2)}+ I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}+ I_{\Yboxdim4pt\yng(2)~\yng(1,1)}+ I_{\Yboxdim4pt\yng(1,1)~\yng(2)} =I_4(x)+\frac{1}{2}I_2(x^2)+\frac{1}{2}I_2(x)^2 +\mathcal{O}(x^{13})\nonumber$$ for $k=2$. One might think that theories with $k\!=\!1,2$ are somewhat special since we expect enhancement of supersymmetry to $\mathcal{N}\!=\!8$ [@Aharony:2008ug]. To ensure that the agreement has nothing to do with this property, we also check the case with $k=3$. We find $$\begin{aligned} \hspace*{-0cm}I_{\Yboxdim4pt\yng(2)~\yng(2)}\!&\!=\!&\! 7x^3+4x^4+0x^5+8x^6-2x^7+4x^8+4x^9+4x^{10}-2x^{11}+8x^{12}+0x^{13}+ \mathcal{O}(x^{14})\nonumber\\ \hspace*{-0cm}I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}\!&\!=\!&\! 10x^3\!+\!16x^4\!+\!20x^5\!+\!20x^6\!+\!31x^7\!+\!32x^8\!+\!36x^9\!+\! 40x^{10}\!+\!49x^{11}\!+\!52x^{12}\!+\!40x^{13}\!+\!\mathcal{O}(x^{14}) \nonumber\\ \hspace*{-0cm}I_{\Yboxdim4pt\yng(2)~\yng(1,1)}\!=\! I_{\Yboxdim4pt\yng(1,1)~\yng(2)}\!&\!=\!&\! x^{11}-4x^{12}+10x^{13}+\mathcal{O}(x^{14}) \nonumber\end{aligned}$$ and $$\begin{aligned} I_6(x)\!&\!=\!&\! 7x^3+4x^4+2x^5+4x^6+2x^7+4x^8+2x^9+4x^{10}+2x^{11}+4x^{12}+2x^{13}+ \mathcal{O}(x^{14})\nonumber\\ \frac{I_3(x^2)\!+\!I_3(x)^2}{2}\!&\!=\!&\! 10x^3\!+\!16x^4\!+\!18x^5\!+\!24x^6\!+\!27x^7\!+\!32x^8\!+\!38x^9\!+\! 40x^{10}\!+\!47x^{11}\!+\!48x^{12}\!+\!58x^{13}\!+\!\mathcal{O}(x^{14}) \nonumber\end{aligned}$$ proving $$I_{\Yboxdim4pt\yng(2)~\yng(2)}+ I_{\Yboxdim4pt\yng(1,1)~\yng(1,1)}+ I_{\Yboxdim4pt\yng(2)~\yng(1,1)}+ I_{\Yboxdim4pt\yng(1,1)~\yng(2)} =I_6(x)+\frac{1}{2}I_3(x^2)+\frac{1}{2}I_3(x)^2 +\mathcal{O}(x^{14})\nonumber$$ for $k=3$. In these examples, the two saddle points with equal distributions start to deviate from their ‘corresponding’ graviton indices at two orders higher than the lowest energy. The saddle points with unequal distributions start to enter at $2k+2$ orders higher than the lowest level. The $2k$ comes from the energy shifts in the letter indices (\[eff-letter-index\]), while $2$ comes from the Casimir energy shift $\epsilon_0=2$ for the saddle points $\Yboxdim8pt\yng(2)~\yng(1,1)$ and $\Yboxdim8pt\yng(1,1)~\yng(2)$. See Table 2 in appendix B.3. Three KK-momenta: numerical tests --------------------------------- We consider the case with $k\!=\!1$ only. The gauge theory indices are given by $$\begin{aligned} \label{3-equal-distribute} I_{\Yboxdim4pt\yng(3)~\yng(3)}&=&x^{\frac{3}{2}} \left(4+4x+0x^2+8x^3-4x^4+8x^5+2x^6+4x^7+0x^8+\mathcal{O}(x^9)\ \right)\\ I_{\Yboxdim4pt\yng(2,1)~\yng(2,1)}&=&x^{\frac{3}{2}} \left(6+20x+24x^2+28x^3+64x^4+34x^5+34x^6+166x^7-32x^8+\mathcal{O}(x^9) \ \right)\nonumber\\ I_{\Yboxdim4pt\yng(1,1,1)~\yng(1,1,1)}&=&x^{\frac{3}{2}} \left(4+12x+30x^2+52x^3+52x^4+98x^5+170x^6+130x^7+106x^8+\mathcal{O}(x^9) \ \right)\nonumber\end{aligned}$$ and $$\begin{aligned} \label{3-unequal-distrib} I_{\Yboxdim4pt\yng(2,1)~\yng(1,1,1)}=I_{\Yboxdim4pt\yng(1,1,1)~\yng(2,1)} &=&x^{\frac{3}{2}}\left(0x^4+6x^5-10x^6-22x^7+88x^8+\mathcal{O}(x^9) \ \right)\nonumber\\ I_{\Yboxdim4pt\yng(3)~\yng(2,1)}= I_{\Yboxdim4pt\yng(2,1)~\yng(3)}&=&x^{\frac{3}{2}} \left(2x^6-8x^7+16x^8+\mathcal{O}(x^9)\ \right)\\ I_{\Yboxdim4pt\yng(3)~\yng(1,1,1)}=I_{\Yboxdim4pt\yng(1,1,1)~\yng(3)} &=&x^{\frac{3}{2}}\left(\ \mathcal{O}(x^{12})\ \right)\ .\nonumber\end{aligned}$$ The last three pairs of saddle points with unequal distributions are expected to enter from $(2k\!+\!2)$’th, $(4k\!+\!2)$’th and $(6k\!+\!6)$’th level above the lowest level for general $k$, respectively. Coefficients $0$ written above could be accidental.[^6] From gravity, one can construct states carrying three units of momenta in the following three ways: one graviton carrying three momenta, one graviton with one momentum and another graviton with two momenta, three identical particles where each carries one momentum. Three contributions are $$\begin{aligned} \hspace*{-1cm}I_3(x)\!&\!=\!&\!x^{\frac{3}{2}}\left(4+4x+2x^2+4x^3+2x^4+ 4x^5+2x^6+4x^7+2x^8\!+\!\mathcal{O}(x^9)\right)\nonumber\\ \hspace*{-1cm}I_1(x)I_2(x)\!&\!=\!&\!x^{\frac{3}{2}}\left(6\!+\!20x\!+\!26x^2\!+\! 36x^3\!+\!46x^4\!+\!52x^5\!+\!66x^6\!+\!68x^7\!+\!86x^8\!+\! \mathcal{O}(x^9)\right) \nonumber\\ \hspace*{-1cm} \frac{1}{3}I_1(x^3)\!+\!\frac{1}{2}I_1(x)I_1(x^2)\!+\!\frac{1}{6}I_1(x)^3 \!&\!=\!&\! x^{\frac{3}{2}}\left(4\!+\!12x\!+\!26x^2\!+\!48x^3\!+\!64x^4\!+\! 96x^5\!+\!122x^6\!+\!168x^7\!+\!194x^8\!+\!\mathcal{O}(x^9)\right)\nonumber\end{aligned}$$ From this we find $$\begin{aligned} &&\hspace{-1cm} I_{\Yboxdim4pt\yng(3)~\yng(3)}+I_{\Yboxdim4pt\yng(2,1)~\yng(2,1)} +I_{\Yboxdim4pt\yng(1,1,1)~\yng(1,1,1)}+ 2I_{\Yboxdim4pt\yng(2,1)~\yng(1,1,1)}+ 2I_{\Yboxdim4pt\yng(3)~\yng(2,1)}+ 2I_{\Yboxdim4pt\yng(3)~\yng(1,1,1)}\nonumber\\ &&=I_3(x)+I_1(x)I_2(x)+ \frac{1}{3}I_1(x^3)+\frac{1}{2}I_1(x)I_1(x^2)+\frac{1}{6}I_1(x)^3 +\mathcal{O}(x^{\frac{3}{2}+9})\ ,\end{aligned}$$ which is again a perfect agreement. Conclusion ========== In this paper we calculated the complete superconformal index for $\mathcal{N}\!=\!6$ Chern-Simons-matter theory with gauge group $U(N)_k\times U(N)_{-k}$ at level $k$. The low energy and large $N$ limit shows a perfect agreement with the index over supersymmetric gravitons in M-theory on $AdS_4\times S^7/\mathbb{Z}_k$ in all sample calculations we did. Though we strongly suspect that the two large $N$ indices will completely agree, it would definitely be desirable to develop a general proof of this claim. Since the two indices assume very different forms apparently, this would be a nontrivial check of the AdS$_4$/CFT$_3$ proposal based on superconformal Chern-Simons theories. Perhaps identities of unitary matrix integrals similar to those explored in [@Dolan:2008qi] could be found to show this. We expect our result to provide hints towards a better understanding of ’t Hooft operators and their roles in AdS$_4$/CFT$_3$. In particular we found that monopole operators in nontrivial representations, beyond those considered in [@Aharony:2008ug; @Berenstein:2008dc; @Klebanov:2008vq], played important roles for the agreement of gauge/gravity indices. In our calculation in the deformed theory, the degrees of freedom in the vector multiplets turned out to be important, starting from nontrivial contribution to the determinant and Casimir energy. It would be interesting to understand it directly in the physical Chern-Simons-matter theory, in which there are no propagating degrees for the fields in vector multiplets. In a preliminary study, we find that interaction between gauge fields and matters in background fluxes transmutes some of the matter scalar degrees into vector-like ones [@Kim:2009ia]. We have kept the chemical potentials to be finite and order $1$ as we take the large $N$ limit to obtain the low energy index. This setting is also partly motivated from the physics that one expects from the *partition function*, namely a deconfinement phase transition at order $1$ temperature. In the context of 4-dimensional Yang-Mills theory, a first order deconfinement transition is found in [@Sundborg:1999ue; @Aharony:2003sx; @Aharony:2005bq]. Although the index does not see this either in the $d\!=\!4$ $\mathcal{N}\!=\!4$ Yang-Mills theory or here, possibly due to cancelations from $(-1)^F$, it is not clear to us whether this means that the trace of (supersymmetric) black holes and deconfined phase is completely absent. It is a famous fact that the situation in $d\!=\!3$ is more mysterious due to the replacement of $N^2$ by $N^{\frac{3}{2}}$, or $N^{\frac{3}{2}}\sqrt{k}=\frac{N^2}{\sqrt{\lambda}}$ in our case, in the ‘deconfined’ degrees of freedom. Another new aspect in $d\!=\!3$ compared to $d\!=\!4$ is the presence of sectors with topological charges. It might be interesting to systematically investigate finite $N$ effects in the flux distributions. We think the finite $N$ and $k$ index that we obtained can be straightforwardly extended to other superconformal Chern-Simons theories. This can be used to solidly test many interesting ideas in these theories. For example, a non-perturbative ‘parity duality’ and its generalization were proposed by [@Aharony:2008gk] for Chern-Simons theories with gauge groups $O(M)\times Sp(N)$ and $U(M)\times U(N)$, respectively, based on the study of their gravity duals. The details of duality transformation involves changes of parameters $M$, $N$ and the Chern-Simons level $k$. The information on these parameters is of course wiped out in the index in ’t Hooft limit [@Choi:2008za], and perhaps also in the large $M$, $N$ limit keeping $k$ finite. The index with finite $M$, $N$, $k$ should have a delicate structure for the duality to hold. It would be interesting to explore this. An analysis of finite $N$ indices for a class of 4 dimensional SCFT was reported in [@Romelsberger:2007ec; @Dolan:2008qi]. For a class of superconformal Chern-Simons theories, the superconformal index exhibits an interesting large $N$ phase transition. For instance, it was explained that the index for $\mathcal{N}\!=\!2,3$ $U(N)_k$ Chern-Simons theories with $N_f$ flavors (presented in [@Gaiotto:2007qi]) can undergo third order phase transitions [@Bhattacharya:2008zy] in the so-called Veneziano limit, which is very similar to that in the lattice gauge theory explored by Gross, Witten and Wadia [@Gross:1980he; @Wadia:1979vk]. An interesting related issue is a proposal by Giveon and Kutasov on Seiberg duality in these theories [@Giveon:2008zn], based on the study of brane constructions. See also [@Niarchos:2008jb]. The $\mathcal{N}\!=\!3$ models were suggested to be self strong-weak dual in that $\frac{N}{k}$ cannot be small for both of the Seiberg-dual pair theories. In some cases, it seems that the calculation of [@Bhattacharya:2008zy] applies to only one of the two theories in the pair. We think that it could be important to consider the sectors with magnetic fluxes to see the dual phase transition. We hope to come back to this problem in a near future. Finally, there have been active studies on superconformal Chern-Simons theories which could be relevant to condensed matter systems. For example, nonrelativistic versions of Chern-Simons-matter theories were studied and some properties of the superconformal indices were pointed out [@Nakayama:2008qm]. The study of the last references in [@Nakayama:2008qm] suggests that monopole operators are expected to play important roles. More recent works include Chern-Simons-matter theories with fundamental matters and their gravity duals [@Hohenegger:2009as; @Gaiotto:2009tk]. It should be interesting to apply our methods to these examples. 0.5cm 0.2cm I am grateful to Shiraz Minwalla for his early collaboration and also for many helpful discussions and suggestions throughout this work. I also thank Sayantani Bhattacharyya, Kallingalthodi Madhu for collaborations, and Ofer Aharony, James Bedford, Amihay Hanany, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, Noppadol Mekareeya, Kyriakos Papadodimas, Mark van Raamsdonk, Soo-Jong Rey, Riccardo Ricci, Ashoke Sen, Ho-Ung Yee, Piljin Yi and Shuichi Yokoyama for helpful discussions. I acknowledge the support and hospitality of Tata Institute of Fundamental Research and Korea Institute for Advanced Study, respectively, where part of this work was done. Notes on radial quantization ============================ In this appendix we summarize the conversion between conformal field theories on $\mathbb{R}^{2+1}$ and $S^2\times\mathbb{R}$ via radial quantization. In particular we would like to obtain the action on the latter space starting from that on the former. The first procedure is analytic continuation $x^0=-ix^3_E$ to $\mathbb{R}^3$ and obtaining the action in Euclidean $S^2\times\mathbb{R}$ by setting $r=e^{\tau}$, where $\tau$ is the time of the latter space. One may then continue back to Lorentzian $S^2\times\mathbb{R}$ with Lorentzian time $t=-i\tau$, though in this paper we mainly consider the Euclidean theory. We first consider kinetic terms, involving derivatives acting on fields. It suffices to consider free fields with ordinary derivatives: covariantizing with gauge fields would take obvious forms once the gauge fields are appropriately transformed into fields in $S^2\times\mathbb{R}$ [@Grant:2008sk]. Let us start from a (real) scalar field $\Phi$ on $\mathbb{R}^3$ with scale dimension $\frac{1}{2}$. The field $\Phi_S$ on $S^2\times\mathbb{R}$ is related to $\Phi$ as $$\Phi=r^{-\frac{1}{2}}\Phi_S\ .$$ The kinetic term on $\mathbb{R}^3$ can be rewritten as $$\begin{aligned} \int d^3x\left(\nabla_{\mathbb{R}^3}\Phi\right)^2&=& \int {\rm vol}_{S^2}\ r^2dr\left[ \partial_r\left(r^{-\frac{1}{2}}\Phi_S\right)^2+ \frac{1}{r^2}\nabla_{S^2}\left(r^{-\frac{1}{2}}\Phi_S\right)^2\right] \nonumber\\ &=&\int{\rm vol}_{S^2}\ d\tau\left[ \left(\partial_\tau\Phi_S\right)^2+ \left(\nabla_{S^2}\Phi_S\right)^2+\frac{1}{4}\left(\Phi_S\right)^2\right]\end{aligned}$$ after eliminating surface terms. This is nothing but the action for a scalar conformally coupled to the curvature. This result applies to all eight real scalars $A_a,B_{\dot{a}}$. The gauge fields $A_\mu,\tilde{A}_\mu$ and scalars $\sigma,\tilde\sigma$ with dimensions $1$ can also be transformed appropriately. We do not present the result here since we will mostly work directly in $\mathbb{R}^3$ when we consider these fields. One may see [@Grant:2008sk] for the details on $\mathbb{R}^4$, which is essentially the same as our case. In particular, $\sigma=r^{-1}\sigma_S$, $A_r=r^{-1}(A_S)_\tau$ and $A_a=(A_S)_a$ for $a=\theta,\phi$. We note that the Chern-Simons term takes the same form on $S^2\times\mathbb{R}$. We also consider complex matter fermions $\Psi_\alpha$ with dimension $1$ whose (Euclidean) kinetic term is given by $$-i\bar\Psi^\alpha(\gamma^m)_{\alpha\beta}\nabla_m\Psi^\beta =\bar\Psi_\alpha(\bar\sigma^m)^{\alpha\beta} \nabla_m\Psi_\beta\ \ \ (m=1,2,3\ {\rm for\ Cartesian\ coordi.})\ .$$ In the representation with $\gamma^m_E=(-\sigma^2,\sigma^1,\sigma^3)$, one obtains $\bar\sigma^m=(1,i\sigma^3,-i\sigma^1)$ or an $SO(3)$ rotated one as explained in section 2.1. Since we are considering fermions, we should first change our dreibein frame on $\mathbb{R}^3$ from Cartesian to spherical curvilinear one: $$e^r=dr\ ,\ \ e^\theta=rd\theta\ ,\ \ e^\phi=r\sin\theta d\phi\ .$$ This frame is related to the Cartesian frame by a local $SO(3)$ transformation which we call $\Lambda$. One obtains $$\bar\Psi^\alpha(\gamma^m_E)_\alpha^{\ \beta}\nabla_m\Psi_\beta =\bar\Psi_{\rm cur}^\alpha \left[U(\Lambda^\dag)\gamma^m_EU(\Lambda)\right]_\alpha^{\ \ \beta} \Lambda_m^{\ \ n} E_n^\mu\nabla_\mu\Psi_{{\rm cur}\beta}=\bar\Psi_{\rm cur}^\alpha (\gamma^n_E)_\alpha^{\ \beta}E_n^\mu\nabla_\mu\Psi_{{\rm cur}\beta}\ ,$$ where $U(\Lambda)$ is the spinor representation of $\Lambda$, $\Psi=U(\Lambda)\Psi_{\rm cur}$, and $E_m$ is the inverse of the above frame. At the last step we used the fact that action of any local $SO(3)$ leaves $(\gamma^m)_\alpha^{\ \beta}$ invariant. Here the final derivative $\nabla_\mu$ is covariantized with the following spin connection (yet with zero curvature): $$\omega^{\theta r}=\frac{1}{r}e^\theta=d\theta\ ,\ \ \omega^{\phi r}=\frac{1}{r}e^{\phi}=\sin\theta d\phi \ ,\ \ \omega^{\phi\theta}=\frac{\cot\theta}{r}e^\phi= \cos\theta d\phi\ :\ \ \nabla_\mu=\partial_\mu+\frac{1}{4}\omega_\mu^{mn}\gamma_{mn}\ .$$ Finally we change the field $\Psi_{\rm cur}$ to $\Psi_S$ living on $S^2\times\mathbb{R}$, according to its dimension $1$, as $$\Psi_{\rm cur}=r^{-1}\Psi_S\ .$$ This yields $$\label{fermion-redefine} \int {\rm vol}_{S^2}\ r^2 dr\ \bar\Psi^\alpha(\gamma^m_E)_\alpha^{\ \beta} \nabla_m\Psi_\beta= \int {\rm vol}_{S^2}\ dr\left[\bar\Psi_S^\alpha (\gamma^n_E)_\alpha^{\ \beta}E_n^\mu\nabla_\mu\Psi_{S\beta} +\frac{1}{r}\bar\Psi_S^\alpha(\sigma^2)_\alpha^{\ \beta} \Psi_{S\beta}\right]\ .$$ The covariant derivative $\nabla_\mu$ is still that on $\mathbb{R}^3$. Since we are trying to obtain an action on $S^2\times\mathbb{R}$, with metric changing from $ds^2_{\mathbb{R}^3}=dr^2+r^2ds^2_{S^2}=r^2ds^2_{S^2\times\mathbb{R}}$ to $ds^2_{S^2\times\mathbb{R}}$, we rewrite the spin connection of $\mathbb{R}^3$ in terms of that of $S^2\times\mathbb{R}$. The covariant derivatives are related as $$\nabla=\nabla_S+\frac{1}{2}\left(\frac{1}{r}e^\theta\gamma_{\theta r}+ \frac{1}{r}e^\phi\gamma_{\phi r}\right)= \nabla_S-\frac{i}{2r}\left(e^\theta\sigma^3-e^\phi\sigma^1\right)\ .$$ Thus one finds that $$\label{connection-redefine} \gamma^n_EE_n^\mu\nabla_\mu=\frac{1}{r}\gamma^n_E(E_S)_n^\mu (\nabla_S)_\mu-\frac{1}{r}\sigma^2\ ,$$ where $E_S$ denotes the inverse frame for the metric $ds^2_{S^2\times\mathbb{R}}$. The final Lagrangian for $\Psi_S$ is $$\int{\rm vol}_{S^2}d\tau\ \bar\Psi_S\gamma^m_E(E_S)_m^\mu(\nabla_S)_\mu\Psi_S\ ,$$ where the second term in (\[fermion-redefine\]) is canceled by that in (\[connection-redefine\]). Note that there is no analogue of conformal mass terms for scalars, which is well-known from literatures on CFT in curved spaces [@Nicolai:1988ek]. Finally we consider the kinetic term for the gaugino fields $\lambda_\alpha$, $\tilde\lambda_\alpha$ in the $Q$-exact deformation $$\int d^3x\ r\lambda\gamma^m_E\nabla_m\bar\lambda\ ,$$ and a similar term for $\tilde\lambda$. These fields have dimensions $\frac{3}{2}$. The analysis is almost the same as the previous paragraph, apart from the fact that the fields on $S^2\times\mathbb{R}$ are given by $$\lambda_{\rm cur}=r^{-\frac{3}{2}}\lambda_S\ ,\ \ \tilde\lambda_{\rm cur}=r^{-\frac{3}{2}}\tilde\lambda_S\ .$$ Therefore the step analogous to (\[fermion-redefine\]) yields $$\label{gaugino-redefine} \int d^3x\ r\lambda^\alpha(\gamma^m_E)_\alpha^{\ \beta}\nabla_m \bar\lambda_\beta=\int{\rm vol}_{S^2}\ dr\left[\lambda_S^\alpha (\gamma^m_E)_\alpha^{\ \beta}E_m^\mu\nabla_\mu\bar\lambda_{S\beta} +\frac{3}{2r}\lambda_S^\alpha(\sigma^2)_\alpha^{\ \beta} \bar\lambda_{S\beta}\right]\ .$$ As in the previous paragraph, rewriting the covariant derivative on $\mathbb{R}^3$ in terms of that on $S^2\times\mathbb{R}$ provides a term $-\frac{1}{r}\lambda_S^\alpha(\sigma^2)_\alpha^{\ \beta}\bar\lambda_{S\beta}$, which in this case *does not* completely cancel the second term in (\[gaugino-redefine\]). The final kinetic term for $\lambda_S$ is, in terms of $(\sigma^\mu)_{\alpha\beta}=(1,i\vec\sigma)$, $$\int{\rm vol}_{S^2}\ d\tau\ \lambda_S^\alpha\left[ (\sigma^m)_{\alpha\beta}(E_S)_m^\mu (\nabla_S)_\mu -\frac{1}{2}\delta_{\alpha\beta}\right]\bar\lambda_S^\beta\ ,$$ where $\delta_{\alpha\beta}$ comes from $(\bar\sigma^0)_{\alpha\beta}$. Action for $\tilde\lambda_S$ is similar. The terms in the action which do not involve derivatives, such as potential, Yukawa interaction, etc., transform rather obviously. Starting from $\int d^3x\mathcal{L}(\Phi)$ and transforming the fields according to their dimensions $\Delta$ as $\Phi=r^{-\Delta}\Phi_S$, one obtains $\int_{S^2\times\mathbb{R}}\mathcal{L}(\Phi_S)$. Details of 1-loop calculation ============================= The 1-loop determinant comes from two contributions: firstly from the quadratic fluctuations of the matter fields and secondly from the fields in vector multiplets. We explain them in turn. Determinant from matter fields ------------------------------ We first consider the matter scalar fields. The quadratic action for the scalars (on $S^2\times S^1$) in the presence of nonzero $A_\mu,\tilde{A}_\mu,\sigma,\tilde\sigma$ background takes the following form: $$\begin{aligned} \label{quad-scalar} \mathcal{L}_{s2}&=&{\rm tr}\left[\frac{}{}\right.\! -\bar{A}^aD^\mu D_\mu A_a -\bar{B}^{\dot{a}}D^\mu D_\mu B_{\dot{a}} +\frac{1}{4}\left(A_a\bar{A}^a+B_{\dot{a}} \bar{B}^{\dot{a}}\right)\nonumber\\ &&\hspace{1cm}\left.+(\sigma A_a\!-\!A_a\tilde\sigma) (\bar{A}^a\sigma\!-\!\tilde\sigma\bar{A}^a)+ (\tilde\sigma B_{\dot{a}}\!-\!\sigma B_{\dot{a}}) (\bar{B}^{\dot{a}}\tilde\sigma\!-\!\bar{B}^{\dot{a}}\sigma)\frac{}{}\right]\end{aligned}$$ where derivatives $D_\mu$ are covariantized with the background gauge fields, external gauge field corresponding to the twisting and $S^2$ spatial connection. The fields are regarded as those on $S^2\times\mathbb{R}$, $A_{aS}$, etc. $\sigma,\tilde\sigma$ assume their background values $(\sigma_S)_{ij}=-\frac{n_i}{2}\delta_{ij}$ and $(\tilde\sigma_S)_{ij}=-\frac{\tilde{n}_i}{2}\delta_{ij}$ where $i,j=1,2,\cdots, N$. The third term in the trace is the conformal mass term. Since there are $U(1)^N\times U(1)^N$ background magnetic fields on $S^2$, each component of the scalars has to be spanned by appropriate monopole spherical harmonics on $S^2$ [@Wu:1976ge]. Scalars are either in $(N,\bar{N})$ or $(\bar{N},N)$ representation of $U(N)\times U(N)$. We consider the $ij$’th component of the scalar, where $i$ ($j$) runs over $1,2,\cdots,N$ and labels the components in the first (second) gauge group. A scalar $\Phi_{ij}$ couples to the background magnetic field whose strength is $$\frac{s(n_i\!-\!\tilde{n}_j)}{2} \sin\theta d\theta\wedge d\phi\ \ \ \ \ (s=\pm 1\ {\rm for}\ (N,\bar{N}),\ (\bar{N},N))\ ,$$ corresponding to a Dirac monopole carrying $s(n_i\!-\!\tilde{n}_j)$ units of minimal charge. The monopole spherical harmonics $Y_{jm}$ in this background, with angular momentum quantum numbers $j,m$ given by $$j=\frac{|n_i\!-\!\tilde{n}_j|}{2},\ \frac{|n_i\!-\!\tilde{n}_j|}{2}+1,\ \cdots\ {\rm and}\ \ m=-j,-(j\!-\!1)\cdots,j\!-\!1,j\ ,$$ diagonalize the spatial Laplacian on $S^2$ as $$\label{monopole-eigen} -D^aD_aY_{jm}=\left(j(j+1)-\frac{(n_i\!-\!\tilde{n}_j)^2}{4}\right) Y_{jm}$$ where $a=1,2$ labels the coordinates of $S^2$. Plugging this mode expansion into the quadratic action, one can easily see that the second term on the right hand side of (\[monopole-eigen\]) is canceled by the second line of (\[quad-scalar\]). Collecting all, the $Y_{jm}$ mode $\Phi_{ij}^{jm}$ has a quadratic term $$\bar\Phi_{ij}^{jm}\left[-(D_\tau)^2+\left(j+\frac{1}{2}\right)^2 \right]\Phi_{ij}^{jm}\ .$$ We hope our bad notation of using two kinds of $j$ is not too confusing. The time derivative is $$D_\tau=\partial_\tau-is\frac{\alpha_i\!-\!\tilde\alpha_j} {\beta\!+\!\beta^\prime} -\frac{\beta-\beta^\prime}{\beta+\beta^\prime}m +\frac{\beta^\prime}{\beta+\beta^\prime}h_3-\frac{\gamma_1} {\beta+\beta^\prime}h_1-\frac{\gamma_2}{\beta+\beta^\prime}h_2\ ,$$ where $h_{1,2,3}$ are eigenvalues of $SO(6)_R$ Cartans for the given field $\Phi$. The determinant is evaluated for each conjugate pair of scalar fields $(\Phi,\bar\Phi)$, where $\Phi$ may run over four complex scalars, say, $A_a,B_{\dot{a}}$. The determinant from the pair $\Phi,\bar\Phi$ is given by $$\begin{aligned} &&\hspace{-0.5cm} \prod_{j=\frac{|n_i\!-\!\tilde{n}_j|}{2}}^\infty\prod_{j_3=-j}^j \det\left[-\left(\partial_\tau-is\frac{\alpha_i\!-\!\tilde\alpha_j} {\beta\!+\!\beta^\prime}-\frac{\beta-\beta^\prime}{\beta+\beta^\prime}j_3 +\frac{\beta^\prime}{\beta+\beta^\prime}h_3- \frac{\gamma_1h_1+\gamma_2h_2}{\beta+\beta^\prime} \right)^2+\left(j+\frac{1}{2}\right)^2\right]\nonumber\\ &&\hspace{-0.3cm}=\prod_{n=-\infty}^\infty\prod_{j,j_3} \left[\left(\frac{2\pi n}{\beta+\beta^\prime} +s\frac{\alpha_i\!-\!\tilde\alpha_j}{\beta\!+\!\beta^\prime} -i\frac{\beta-\beta^\prime}{\beta+\beta^\prime}j_3 +i\frac{\beta^\prime}{\beta+\beta^\prime}h_3- i\frac{\gamma_1h_1+\gamma_2h_2}{\beta+\beta^\prime} \right)^2+\left(j+\frac{1}{2}\right)^2\right]\ .\nonumber\end{aligned}$$ Following the prescription in [@Aharony:2003sx][^7], we factor out a divergent constant, set it to unity, and obtain $$\begin{aligned} \label{scalar-pair-det} &&\hspace{-0.5cm}\prod_{j,j_3}(-2i) \sin\left[\frac{1}{2}\left(\frac{}{}\!s (\tilde\alpha_j\!-\!\alpha_i)+i\beta(\epsilon_j\!+\!j_3)+ i\beta^\prime(\epsilon_j\!-\!h_3\!-\!j_3)+i(\gamma_1h_1\!+\!\gamma_2h_2) \right)\right]\nonumber\\ &&\times(-2i) \sin\left[\frac{1}{2}\left(\!\!\frac{}{}\!-\!s (\tilde\alpha_j\!-\!\alpha_i)+i\beta(\epsilon_j\!-\!j_3)+ i\beta^\prime(\epsilon_j\!+\!h_3\!+\!j_3)-i(\gamma_1h_1\!+\!\gamma_2h_2) \right)\right],\end{aligned}$$ where $\epsilon_j\equiv j\!+\!\frac{1}{2}$. Generalizing [@Aharony:2003sx], the two sine factors in the final form has an obvious interpretation as contributions from a pair of particle and anti-particle modes, since all charges except ‘energy’ $\epsilon_j$ have different signs. Therefore, the determinant from the scalars admits a simple form $$\begin{aligned} \label{scalar-det} \hspace*{-0.7cm}\det\!{}_{\rm scalar}&=&\prod_{i,j} \prod_{8\ {\rm scalars}}\prod_{j,j_3} \sin\left[\frac{1}{2}\left(\frac{}{}\!s (\tilde\alpha_j\!-\!\alpha_i)+i\beta(\epsilon_j\!+\!j_3)+ i\beta^\prime(\epsilon_j\!-\!h_3\!-\!j_3)+i(\gamma_1h_1\!+\!\gamma_2h_2) \right)\right]\\ &=&\prod_{i,j}\prod_{8\ {\rm scalars}}\prod_{j,j_3} e^{\frac{is}{2}(\alpha_i\!-\!\tilde\alpha_j)+\frac{\beta}{2}(\epsilon_j\!+\!j_3)+ \frac{\beta^\prime}{2}(\epsilon_j\!-\!h_3\!-\!j_3)+\gamma_1h_2+\gamma_2h_2} \left(1-e^{is(\tilde\alpha_j\!-\!\alpha_i)} x^{\epsilon_j\!+\!j_3} (x^\prime)^{\epsilon_j\!-\!h_3\!-\!j_3}y_1^{h_1}y_2^{h_2}\right)\nonumber\end{aligned}$$ where $x^\prime\equiv e^{-\beta^\prime}$. The product is over $8$ scalars regarding conjugate pairs as independent fields. We would like to write $(\det_{\rm scalar})^{-1}$, appearing in the index, in terms of functions which we call the indices over ‘letters,’ or modes. We find that $$\begin{aligned} \label{scalar-det-log} \hspace*{-0cm}\log(\det\!{}_{\rm scalar})^{-1} &\equiv&-\sum_{i,j}\sum_{{\rm scalars}}\sum_{j,j_3} \left[\frac{is}{2}(\alpha_i\!-\!\tilde\alpha_j)+\frac{\beta}{2}(\epsilon_j\!+\!j_3)+ \frac{\beta^\prime}{2}(\epsilon_j\!-\!h_3\!-\!j_3)+\gamma_1h_2+\gamma_2h_2 \right]\\ &&+\sum_{i,j=1}^N\sum_{n=1}^\infty\frac{1}{n}\left[ f^{+B}_{ij}(x^n,(x^\prime)^n,y_1^n,y_2^n) e^{in(\tilde\alpha_j-\alpha_i)}+f^{-B}_{ij} (x^n,(x^\prime)^n,y_1^n,y_2^n)e^{in(\alpha_i-\tilde\alpha_j)} \!\frac{}{}\right]\ .\nonumber\end{aligned}$$ The first line provides a quantity analogous to the Casimir energy, which will be computed in appendix B.3. The contribution from scalars to the letter index is given by $$f^{\pm B}_{ij}(x,x^\prime,y_1,y_2)\equiv \sum_{\substack{4\ {\rm scalars}\\s=\pm 1}} \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|}{2}}^\infty\sum_{j_3=-j}^j \left(\frac{}{}\!x^{\epsilon_j+j_3}(x^\prime)^{\epsilon_j-h_3-j_3} y_1^{h_2} y_2^{h_2}\right)$$ where the first summation is restricted to fields with one of $s=\pm 1$. Explicitly summing over the scalar modes, one obtains $$\begin{aligned} \hspace*{-1cm}f^{+B}_{ij}(x,x^\prime,y_1,y_2)&=& \left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}\right) \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|}{2}}^\infty x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\\ &&+\left(\!\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}\right) \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|}{2}}^\infty x^\prime x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\nonumber\end{aligned}$$ where the two lines come from $\bar{B}^{\dot{a}}$ and $A_a$, respectively, and $$\begin{aligned} \hspace*{-1cm}f^{-B}_{ij}(x,x^\prime,y_1,y_2)&=& \left(\!\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}\right) \sum_{j=\frac{|\tilde{n}_i\!-\!n_j|}{2}}^\infty x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\\ &&+\left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}\right) \sum_{j=\frac{|\tilde{n}_i\!-\!n_j|}{2}}^\infty x^\prime x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\ .\nonumber\end{aligned}$$ where the two lines come from $\bar{A}^a$ and $B_{\dot{a}}$, respectively. The dependence on $x^\prime$ is to be canceled against the contribution from fermions. We also consider the determinant from fermions. Fermionic quadratic action is given by $$\mathcal{L}_{f2}=\bar\psi^a_\alpha(\bar\sigma^\mu)^{\alpha\beta}D_\mu \psi_a^\beta+\bar\psi^{\dot{a}}_\alpha(\bar\sigma^\mu)^{\alpha\beta}D_\mu \chi_{\dot{a}\beta}\ ,$$ where $D_3\psi_a=-i\sigma\psi_a+i\psi_a\tilde\sigma$ and $D_3\chi_{\dot{a}}=i\chi_{\dot{a}}\sigma-i\tilde\sigma\chi_{\dot{a}}$. As explained in section 2.1, $\bar\sigma^\mu=(1,i\sigma^3,-i\sigma^1,i\sigma^2)$ is changed to $\bar\sigma^\mu=(1,-i\sigma^1,-i\sigma^2,-i\sigma^3)$ by an $SO(3)$ frame rotation. Since the latter basis is more convenient in that spin operator on $S^2$ is diagonalized, we do our computation in this basis. Let us denote by $\Psi_{ij}$ the $i$’th and $j$’th component of fermions $\psi_a$ or $\chi_{\dot{a}}$ in the first and second gauge group, respectively. We want to compute the determinant of the matrix differential operator $$\label{spinor-operator} \bar\sigma^\mu D_\mu= D_\tau-i\sigma^aD_a+s\sigma^3\frac{n_i\!-\!\tilde{n}_j}{2}\ ,$$ where $a=1,2$, and the last term comes from the coupling with background $\sigma,\tilde\sigma$. We would first like to obtain the complete basis of spinor spherical harmonics diagonalizing $$\label{spinor-eigen} \left(i\sigma^aD_a-s\sigma^3\frac{n_i\!-\!\tilde{n}_j}{2}\right) \Psi=\lambda\Psi$$ with eigenvalue $\lambda$. Acting the same operator again on the above equation, one obtains $$\label{spinor-square} \left(-D^aD_a+\frac{1-s(n_i\!-\!\tilde{n}_j)\sigma^3}{2}+ \frac{|n_i\!-\!\tilde{n}_j|^2}{4}\right)\Psi=\lambda^2\Psi$$ where the second term comes from the commutator of two covariant derivatives and is the sum of the spatial curvature and the field strength. The first operator $-D^aD_a$ is a $2\times 2$ diagonal matrix since the derivative involves $+\frac{i}{2}\omega^{\theta\phi}\sigma^3$. The spectrum of this operator is known and may be found, for instance, in [@Kim:2001tw]. For the spinor component $\alpha=\pm$, its eigenvalue is given by $$l_\pm\left(l_\pm+|s(n_i\!-\!\tilde{n}_j)\mp 1|+1\frac{}{}\!\right)+ \frac{|s(n_i\!-\!\tilde{n}_j)\mp 1|}{2}$$ where $l_\pm=0,1,2,\cdots$, and $\Psi_\pm$ is given by scalar monopole harmonics with $j_\pm=l_\pm+\frac{|s(n_i\!-\!\tilde{n}_j)\mp 1|}{2}$, coupled to $s(n_i\!-\!\tilde{n}_j)\mp 1$ units of minimal Dirac monopoles. Plugging this in (\[spinor-square\]) and studying the upper/lower components, one obtains $$\begin{aligned} \label{spinor-eigenvalue} \lambda^2&=&\left(l_+ +\frac{|n_i\!-\!\tilde{n}_j|}{2}\right)^2= \left(l_- +\frac{|n_i\!-\!\tilde{n}_j|}{2}+1\right)^2\ \ \ (l_+\!=\!l_-\!+\!1=1,2,3,\cdots) \ \ \ {\rm if}\ s(n_i\!-\!\tilde{n}_j)>0\nonumber\\ \lambda^2&=&\left(l_+ +\frac{|n_i\!-\!\tilde{n}_j|}{2}+1\right)^2= \left(l_- +\frac{|n_i\!-\!\tilde{n}_j|}{2}\right)^2\ \ \ (l_-\!=\!l_+\!+\!1=1,2,3,\cdots) \ \ \ {\rm if}\ s(n_i\!-\!\tilde{n}_j)<0\nonumber\\ \lambda^2&=&(l_\pm+1)^2\ \ \ (l_+=l_-=0,1,2,\cdots)\ \ \ {\rm if}\ n_i\!=\!\tilde{n}_j\ .\end{aligned}$$ The eigenspinors are given as follows. In all three cases, one finds a pair of eigenspinors corresponding to $\lambda\gtrless 0$, $$\label{eigen-paired} \left(\begin{array}{c}\Psi_+\\ \Psi_-\end{array}\right)= \left(\begin{array}{c}Y_{jm}\\ \pm Y_{jm}\end{array}\right)$$ with $j\equiv j_+=j_-$, where the latter two are equal if one relates $l_+$ and $l_-$ as explained in (\[spinor-eigenvalue\]). $j\geq\frac{|n_i\!-\!\tilde{n}_j|+1}{2}$ is the total angular momentum of the mode. Apart from the above modes, there is a set of exceptional modes in the complete set if $n_i\neq\tilde{n}_j$. For the first and second cases in (\[spinor-eigenvalue\]), there exist nonzero modes $$\begin{aligned} \label{eigen-unpaired} &&\left(\begin{array}{c}Y_{\frac{|n_i\!-\!\tilde{n}_j|-1}{2},m}\\0 \end{array}\right)\ \ \ \ {\rm if}\ \ \ s(n_i\!-\!\tilde{n}_j)>0\nonumber\\ &&\left(\begin{array}{c}0\\Y_{\frac{|n_i\!-\!\tilde{n}_j|-1}{2},m} \end{array}\right)\ \ \ \ {\rm if}\ \ \ s(n_i\!-\!\tilde{n}_j)<0\ .\end{aligned}$$ These modes corresponds to $l_\pm=0$ on the first/second line of (\[spinor-eigenvalue\]), respectively. By directly studying (\[spinor-eigen\]), one finds that the eigenvalue is always negative for both cases, i.e. $\lambda=-\frac{|n_i\!-\!\tilde{n}_j|}{2}=-(j\!+\!\frac{1}{2})$. Expanding the operator (\[spinor-operator\]) in the above basis, and following steps similar to those for the scalar determinant, one obtains $$\begin{aligned} \hspace*{-1cm}\det\!{}_{\rm f}\!&\!=\!&\!\prod_{i,j} \prod_{8\ \rm fermions} \prod_{j\geq\frac{|n_i\!-\!\tilde{n}_j|\!+\!1}{2}}\prod_{j_3} (-2i)\sin\left[\frac{1}{2}\left(\frac{}{}\!s (\tilde\alpha_j\!-\!\alpha_i)+i\beta(\epsilon_j\!+\!j_3)+ i\beta^\prime(\epsilon_j\!-\!h_3\!-\!j_3)+i(\gamma_1h_1\!+\!\gamma_2h_2) \right)\right]\nonumber\\ &&\times\prod_{i,j}\prod_{\bar\psi^a,\bar\chi^{\dot{a}}} \prod_{j_3=-\frac{|n_i\!-\!\tilde{n}_j|\!-\!1}{2}}^{ \frac{|n_i\!-\!\tilde{n}_j|\!-\!1}{2}}(-2i)\sin\left[\frac{1}{2}\left(\frac{}{}\!s (\tilde\alpha_j\!-\!\alpha_i)+i\beta(\epsilon_j\!+\!j_3)+ i\beta^\prime(\epsilon_j\!-\!h_3\!-\!j_3)+i(\gamma_1h_1\!+\!\gamma_2h_2) \right)\right]\nonumber\end{aligned}$$ where $\epsilon_j=j+\frac{1}{2}$ for fermions as well. Let us explain how each term is derived. In the first line, 8 fermions in the product denote $\psi_{a},\bar\psi^a,\chi_{\dot{a}},\bar\chi^{\dot{a}}$. This comes from the paired eigenmodes (\[eigen-paired\]) as one evaluates the determinant of the operator (\[spinor-operator\]). The second line is multiplied over four fields only since the modes in (\[eigen-unpaired\]) do not appear in a paired form. From the fact that $\lambda$ is negative when the differential operator acts on the chiral spinors $\psi_a,\chi_{\dot{a}}$, one can easily check that only the charges of $\bar\psi^a,\bar\chi^{\dot{a}}$ have to be inserted on the second line. One can also write this determinant in terms of indices over letters as follows: $$\begin{aligned} \label{ferm-det-log} \hspace*{-0.3cm}\log(\det\!{}_{\rm fermion})\!&\!=\!&\! +\sum_{i,j}\sum_{{\rm fermions}}\sum_{j,j_3} \left[\frac{is}{2}(\alpha_i\!-\!\tilde\alpha_j)+ \frac{\beta}{2}(\epsilon_j\!+\!j_3)+\frac{\beta^\prime}{2} (\epsilon_j\!-\!h_3\!-\!j_3)+\gamma_1h_2+\gamma_2h_2\right]\\ \hspace*{-0.3cm}&&+\sum_{i,j=1}^N\sum_{n=1}^\infty\frac{1}{n}\left[ f^{+B}_{ij}(x^n,(x^\prime)^n,y_1^n,y_2^n) e^{in(\tilde\alpha_j-\alpha_i)}+f^{-B}_{ij} (x^n,(x^\prime)^n,y_1^n,y_2^n)e^{in(\alpha_i-\tilde\alpha_j)} \!\frac{}{}\right]\ ,\nonumber\end{aligned}$$ where $$\begin{aligned} \hspace*{-1.5cm}f^{+F}_{ij}(x,x^\prime,y_1,y_2)\!&=&\! -\left(\!\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}\right) \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|+1}{2}}^\infty x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\nonumber\\ &&\!-\left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}\right) \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|-1}{2}}^\infty x^\prime x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\nonumber\end{aligned}$$ from $\psi_{a\alpha}$ and $\bar\chi^{\dot{a}}_\alpha$, and $$\begin{aligned} \hspace*{-1.5cm}f^{-F}_{ij}(x,x^\prime,y_1,y_2)\!&=&\! -\left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}\right) \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|+1}{2}}^\infty x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\nonumber\\ &&\!-\left(\!\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}\right) \sum_{j=\frac{|n_i\!-\!\tilde{n}_j|-1}{2}}^\infty x^\prime x^{\frac{1}{2}} \left[(x^\prime)^{2j}+(x^\prime)^{2j-1}x^{}+\cdots+ x^\prime x^{2j\!-\!1}+x^{2j}\right]\nonumber\end{aligned}$$ from $\chi_{\dot{a}\alpha}$ and $\bar\psi^a_\alpha$. We combine the bosonic and fermionic determinants and obtain $$\begin{aligned} \label{matter-det-log} \hspace*{-0cm}\log\left(\frac{\det\!{}_{\rm fermion}} {\det\!{}_{\rm scalar}}\right)&=& -\sum_{i,j}\sum_{{\rm matter}}\sum_{j,j_3}(-1)^F \left[\frac{is}{2}(\alpha_i\!-\!\tilde\alpha_j)+ \frac{\beta}{2}(\epsilon_j\!+\!j_3)+\frac{\beta^\prime}{2} (\epsilon_j\!-\!h_3\!-\!j_3)+\gamma_1h_2+\gamma_2h_2\right]\nonumber\\ &&+\sum_{i,j=1}^N\sum_{n=1}^\infty\frac{1}{n}\left[ f^{+}_{ij}(x^n,(x^\prime)^n,y_1^n,y_2^n) e^{in(\tilde\alpha_j-\alpha_i)}+f^{-}_{ij} (x^n,(x^\prime)^n,y_1^n,y_2^n)e^{in(\alpha_i-\tilde\alpha_j)} \!\frac{}{}\right]\ ,\nonumber\end{aligned}$$ where $$f^+_{ij}(x,y_1,y_2)=f^{+B}_{ij}+f^{+F}_{ij}= x^{|n_i\!-\!\tilde{n}_j|}\left[\frac{x^{\frac{1}{2}}}{1-x^2} \left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}\right) -\frac{x^{\frac{3}{2}}}{1-x^2} \left(\!\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}\right)\right]$$ and similarly $$f^-_{ij}(x,y_1,y_2)= x^{|n_i\!-\!\tilde{n}_j|}\left[\frac{x^{\frac{1}{2}}}{1-x^2} \left(\!\sqrt{y_1y_2}\!+\!\frac{1}{\sqrt{y_1y_2}}\right) -\frac{x^{\frac{3}{2}}}{1-x^2} \left(\!\sqrt{\frac{y_1}{y_2}}\!+\!\sqrt{\frac{y_2}{y_1}}\right) \right]\ .$$ This proves the assertion in section 2.3 on determinant from matter fields. Determinant from fields in vector multiplets -------------------------------------------- We also consider the 1-loop determinant from fields in vector multiplets. We consider the multiplet $A_\mu,\sigma,\lambda_\alpha$: the other vector multiplet can be treated in a completely same way. We start from the bosonic part. We expand the quadratic fluctuation in the $Q$-exact deformation, which is dominant in the limit $g\rightarrow 0$. Denoting the fluctuation by $\delta A_\mu,\delta\sigma$, one finds the following quadratic term: $$\label{vec-bos-quadrataic} \left|\vec{D}\times\delta\vec{A}-\vec{D}\delta\sigma -i[\sigma,\delta\vec{A}]\right|^2\ .$$ We are directly working in $\mathbb{R}^3$ with $1\leq r\leq e^\beta$ rather than going to $S^2\times S^1$. The boundary conditions are $$\delta\vec{A}(r\!=\!e^{\beta})=e^{-\beta}\delta\vec{A}(r\!=\!1)\ ,\ \ \delta\sigma(r\!=\!e^\beta)=e^{-\beta}\delta\sigma(r\!=\!1)\ ,$$ associated with their scale dimensions $1$. $\delta\sigma_{ij}$ is expanded with monopole spherical harmonics with $n_i\!-\!n_j$ units of magnetic charge. We can also expand $\delta\vec{A}_{ij}$ using monopole vector spherical harmonics, which is nicely presented in [@Weinberg:1993sg]. For $j\geq q+1$ where $q\equiv\frac{n_i\!-\!n_j}{2}\geq 0$, it has three components $\vec{C}^{\lambda}_{qjm}$ (with $\lambda=+1,0,-1$) and are related to the scalar harmonics as $$\begin{aligned} \vec{C}^{+1}_{qjm}&=&\frac{1}{\sqrt{2(\mathcal{J}^2+q)}} \left(\vec{D}+i\hat{r}\times\vec{D}\right)Y_{qjm}\ \ \ \ \ (j\geq q>0)\\ \vec{C}^0_{qjm}&=&\frac{\hat{r}}{r}Y_{qjm}\ \ \ \ \ (j\geq q\geq 0)\\ \vec{C}^{-1}_{qjm}&=&\frac{1}{\sqrt{2(\mathcal{J}^2-q)}} \left(\vec{D}-i\hat{r}\times\vec{D}\right)Y_{qjm}\ \ \ \ \ (j>q\geq 0)\end{aligned}$$ where $\mathcal{J}^2\equiv j(j+1)-q^2$. Knowledge on vector spherical harmonics for $q\geq 0$ would turn out to be enough to calculate the 1-loop determinant. For $j=q$, $\vec{C}^{-1}_{qjm}$ is absent instead of the above. For $j=q-1$, both $\vec{C}^{-1}$ and $\vec{C}^0$ are absent. We expand the fields as $$\delta\vec{A}=\sum_{n=-\infty}^\infty\sum_{j,m}\sum_{\lambda=0,\pm 1}a^\lambda_{njm}r^{-i\frac{2\pi n}{\beta+\beta^\prime}} \vec{C}^\lambda_{jm}\ ,\ \ \delta\sigma=\sum_{n,j,m}b_{njm}r^{-i\frac{2\pi n}{\beta+\beta^\prime}} \frac{Y_{jm}}{r}\ .$$ One can expand (\[vec-bos-quadrataic\]) by inserting these expansions, using the following properties of vector harmonics[^8], $$\begin{aligned} &&\vec{D}\cdot\vec{C}^0_{qjm}=\frac{1}{r^2}Y_{qjm}\ ,\ \ \vec{D}\cdot\vec{C}^{\pm 1}_{qjm}=-\frac{1}{r^2} \sqrt{\frac{\mathcal{J}^2\pm q}{2}}Y_{qjm}\\ &&\vec{D}\times\vec{C}^0_{qjm}=\frac{i}{r}\left( \sqrt{\frac{\mathcal{J}^2+q}{2}}\ \vec{C}^{+1}_{qjm}- \sqrt{\frac{\mathcal{J}^2-q}{2}}\ \vec{C}^{-1}_{qjm}\right)\\ &&\vec{D}\times\vec{C}^{\pm 1}_{qjm}= \frac{i}{r}\left(\mp\sqrt{\frac{\mathcal{J}^2\pm q}{2}} \ \vec{C}^0_{qjm}\right)\end{aligned}$$ and $$\vec{D}\left(\frac{1}{r}Y_{qjm}\right)= \frac{1}{r}\left(-\vec{C}^0_{qjm}+\sqrt{\frac{\mathcal{J}^2+q}{2}} \ \vec{C}^{+1}_{qjm}+\sqrt{\frac{\mathcal{J}^2-q}{2}}\ \vec{C}^{-1}_{qjm}\right)\ .$$ From (\[vec-bos-quadrataic\]) one finds $$\label{quad-generic} \sum_{i,j=1}^N\sum_{n=-\infty}^\infty\sum_{j,m} v_{-n,j,-m}^T\left(\mathcal{M}^{\frac{n_j\!-\!n_i}{2}}_{-n,j,-m}\right)^T \mathcal{M}^{\frac{n_i\!-\!n_j}{2}}_{njm}v_{njm}$$ where $$\label{matrix-generic} \mathcal{M}^q_n=\left(\begin{array}{cccc} -\lambda+iq&0&is_+&-s_+\\0&\lambda+iq&-is_-&-s_-\\ -is_+&is_-&iq&i\lambda+1 \end{array}\right)\ ,\ \ v_n=\left(\begin{array}{c}a_+\\a_-\\a_0\\b\end{array}\right)$$ for the modes with $j\geq q+1$, $\lambda=\frac{2\pi n}{\beta+\beta^\prime}-i\frac{\beta-\beta^\prime}{\beta+\beta^\prime}m +(\alpha_i\!-\!\alpha_j)$, and this result is for $q=\frac{n_i\!-\!n_j}{2}\geq 0$.[^9] We also defined $s_\pm\equiv\sqrt{\frac{\mathcal{J}^2\pm q}{2}}$. For $j=q$, there is no $a_-$ modes and one finds $$\mathcal{M}^q_n=\left(\begin{array}{ccc} -\lambda+iq&is_+&-s_+\\-is_+&iq&i\lambda+1 \end{array}\right)\ ,\ \ v_n=\left(\begin{array}{c}a_+\\a_0\\b\end{array}\right)$$ where $s_+=\sqrt{q}$. Finally for $j=q-1$ (possible only when $q\geq 1$), both $a_-$ and $a_0$ modes are absent. The scalar monopole harmonics mode $b$ is also absent. One simply finds $$\mathcal{M}^q_n=\left(\begin{array}{c} -\lambda+iq\end{array}\right)\ ,\ \ v_n=\left(\begin{array}{c}a_+\end{array}\right)$$ where we used $s_+=0$ in this case. Before evaluating the determinant we fix the gauge for these fluctuations. For the $ij$’th mode for which $n_i\!=\!n_j$, we choose the Coulomb gauge $s_+a_++s_-a_-=0$ following [@Aharony:2003sx]. Since they are not coupled to magnetic fields, the corresponding infinitesimal gauge transformation, call it $\epsilon$, is expanded by ordinary spherical harmonics. The Coulomb gauge condition requires $\partial^a\partial_a\epsilon=0$ on $S^2$, which leaves the s-wave component of $\epsilon$ unfixed. We impose a residual gauge condition $\frac{d}{d\tau}\int_{S^2}A_\tau=0$ to fix this. The corresponding Faddeev-Popov determinant can be calculated following [@Aharony:2003sx]. For the Coulomb gauge, The Faddeev-Popov determinant is that of the operator $D^a\partial_a\approx \partial^a\partial_a$ over nonzero modes. For the residual gauge, the determinant is given by $$\label{residual-fp} \prod_{\substack{i<j;\\n_i\!=\!n_j}}\left[2\sin \frac{\alpha_i\!-\!\alpha_j}{2}\right]^2\ .$$ We also fix the gauge for the modes for which $n_i\neq n_j$. Analogous to the previous case, we choose the ‘background Coulomb gauge’ $s_+a_++s_-a_-=0$. The infinitesimal gauge transformation $\epsilon$ acquires the condition $D^aD_a\epsilon=0$. The corresponding Faddeev-Popov determinant is $\det D^aD_a$. This, and $\det\partial^a\partial_a$ above, will be canceled by a factor in the 1-loop determinant to be calculated below. See [@Aharony:2003sx] for the similar results. Contrary to the operator $\partial^a\partial_a$, $D^2$ has no zero modes due to the absence of s-waves in monopole spherical harmonics. So we do not have a residual gauge fixing or a corresponding measure like (\[residual-fp\]). For $j=q$, our gauge implies $a_+\!=\!0$. For $j=q\!-\!1$, there is no need to fix the gauge. In the Coulomb gauge, we may write $a_+=s_- a$ and $a_-=-s_+ a$. Now the quadratic terms for modes with $j\geq q\!+\!1$ takes the form (\[quad-generic\]) with $\mathcal{M}_n$ and $v_n$ given by $$\mathcal{M}^q_n=\left(\begin{array}{ccc}-s_-(\lambda-iq)&is_+&-s_+\\ -s_+(\lambda+iq)&-is_-&-s_-\\-2is_+s_-&iq&i(\lambda-i) \end{array}\right)\ ,\ \ v_n=\left(\begin{array}{c}a\\a_0\\b\end{array}\right)\ .$$ The determinant of this matrix is $\det(\mathcal{M}^q_n)=-\mathcal{J}^2\left[\left(\lambda-\frac{i}{2}\right)^2\!+\! \left(j+\frac{1}{2}\right)^2\right]$. $-\mathcal{J}^2$ is nothing but the eigenvalue of $D^aD_a$, whose determinant partly cancels with the Faddeev-Popov measure as claimed. The remaining determinant of bosonic fields with $j\geq q\!+\!1$ is $$\prod_{i,j=1}^N\prod_{n=-\infty}^\infty\prod_{j,j_3}\det\left( \mathcal{M}_{njj_3}^{\frac{|n_i\!-\!n_j|}{2}}\right) =\prod_{i,j}\prod_{n=-\infty}^\infty \prod_{j,j_3}\left[\left(j+\frac{1}{2}\right)^2+\left( \lambda-\frac{i}{2}\right)^2\right]\ .$$ We can arrange the product over $n$ to sine functions: $$\hspace*{-1cm}\prod_{i,j}\prod_{j=\frac{|n_i\!-\!n_j|}{2}+1}^\infty \prod_{j_3}\sin\left[\frac{1}{2}\left(\beta(j\!-\!j_3)+ \beta^\prime(j\!+\!j_3) -i(\alpha_i\!-\!\alpha_j)\!\frac{}{}\right)\right] \sin\left[\frac{1}{2}\left(\beta(j\!+1\!+\!j_3)+\beta^\prime (j\!+1\!-\!j_3)+i(\alpha_i\!-\!\alpha_j)\!\frac{}{}\right)\right]\ .$$ Note that in each of the two sine factors, the role of energy is played by the quantities $j$ and $j+1$, respectively. Signs of $j_3$ are not important since the product is symmetric under $j_3\rightarrow-j_3$. We also consider the modes with $j=q$ and $j=q\!-\!1$ (for $q\geq 1$). For $j=q$, with the gauge choice $a_+=0$, one finds $$\mathcal{M}^q_n=\left(\begin{array}{cc}i\sqrt{q}&-\sqrt{q}\\ iq&i\lambda+1\end{array}\right)\ ,\ \ v_n=\left(\begin{array}{c}a_0\\b \end{array}\right)\ .$$ From $\det(\mathcal{M}^q_n)=-\sqrt{q}(\lambda-i-iq)=-\frac{\sqrt{q}} {\beta+\beta^\prime}\left[2\pi n\!-\!i\beta(q\!+\!1\!+\!j_3)-i\beta^\prime(q\!+\!1\!-\!j_3)+ (\alpha_i\!-\!\alpha_j)\right]$, one obtains $$\prod_{n=-\infty}^\infty\prod_{j_3=-q}^q\sin\left[ \frac{1}{2}\left(\frac{}{}\!\beta(q\!+\!1\!+\!j_3)+\beta^\prime(q\!+\!1\!-\! j_3)+i(\alpha_i\!-\!\alpha_j)\right)\right]\ .$$ Note that this is a contribution from modes with energy $q\!+\!1(=j\!+\!1)$. For $j=q\!-\!1$, one similarly finds $$\prod_{n=-\infty}^\infty\prod_{j_3=-q}^q\sin\left[ \frac{1}{2}\left(\frac{}{}\!\beta(q\!+\!j_3)+\beta^\prime(q\!-\! j_3)+i(\alpha_i\!-\!\alpha_j)\right)\right]\ .$$ This is again a contribution from modes with energy $q$($=j\!+\!1$). Collecting all, the determinant of bosonic modes can be casted in the following form (after relabeling $j_3\rightarrow-j_3$ for some terms) $$\begin{aligned} \log(\det\!{}_{\rm boson})^{-1}&=& -\frac{1}{2}{\rm tr}_B\left[ \left(\frac{}{}\!\beta(j\!+\!j_3)+\beta^\prime(j\!-\!j_3)\right)+ \left(\frac{}{}\!\beta(j\!+1\!+\!j_3)+\beta^\prime(j+\!1\!-\!j_3)\right) \right]\nonumber\\ &&+\sum_{i,j=1}^\infty\sum_{n=1}^\infty\frac{1}{n}f^{\rm adj,B}_{ij}(x)e^{-in(\alpha_i\!-\!\alpha_j)}\end{aligned}$$ where ${\rm tr}_B$ is trace over all bosonic modes explained above. Contribution to the adjoint letter index from modes with $j\geq\frac{|n_i\!-\!n_j|}{2}\!+\!1$ is $$\begin{aligned} \hspace*{-0.5cm}f^{\rm adj,B}_{ij}&\leftarrow& \sum_{j=\frac{|n_i\!-\!n_j|}{2}\!+\!1}^\infty \left[(x^\prime)^{2j}+(x^\prime)^{2j\!-\!1}x+\cdots x^{2j}\right]+ \left[(x^\prime)^{2j\!+\!1}x+(x^\prime)^{2j}x^2+\cdots x^\prime x^{2j\!+\!1}\right].\end{aligned}$$ Additional contribution from modes with $j=\frac{|n_i\!-\!n_j|}{2}$ is given by $$f^{\rm adj,B}_{ij}\leftarrow (x^\prime)^{|n_i\!-\!n_j|\!+\!1}x+(x^\prime)^{|n_1\!-\!n_j|}x^2+ \cdots+x^\prime x^{|n_i\!-\!n_j|\!+\!1}\ .$$ Finally, when $\frac{|n_i\!-\!n_j|}{2}\geq 1$, $$f^{\rm adj,B}_{ij}\leftarrow (x^\prime)^{|n_i\!-\!n_j|\!-\!1}x+(x^\prime)^{|n_1\!-\!n_j|\!-\!2}x^2+ \cdots+x^\prime x^{|n_i\!-\!n_j|\!-\!1}$$ from modes with $j=\frac{|n_i\!-\!n_j|}{2}\!-\!1$. A similar determinant from $\tilde{A}_\mu,\tilde\sigma$ is obtained with $\alpha_i,n_i$ replaced by $\tilde\alpha_i,\tilde{n}_i$. To complete the computation we consider contribution from the fermion $\lambda_\alpha$. The Lagrangian on $S^2\times S^1$ is given in appendix A, with a novel mass-like term. The calculation is similar to that in appendix B.1 for matter fermions except for the addition of this term. The operator acting on $(\bar\lambda^\alpha)_{ij}$ is $$D_\tau+i\sigma^a D_a-\frac{n_i\!-\!n_j}{2}\sigma^3-\frac{1}{2}\ .$$ The eigenvalue problem for the operator consisting of second and third terms is solved, replacing $s(n_i\!-\!\tilde{n}_j)$ by $n_i\!-\!n_j$ here. Again there appears eigenspinors (\[eigen-paired\]) with eigenvalues $\lambda=\pm\frac{j+1}{2}$ for $j\geq\frac{|n_i\!-\!n_j|\!+\!1}{2}$, as well as additional modes only if $n_i\!\neq\!n_j$ with $j\!=\!\frac{|n_i\!-\!n_j|\!-\!1}{2}$ and $\lambda\!=\!-\frac{|n_i\!-\!n_j|}{2}$. The combination appearing in the determinant gets shifted by $-\frac{1}{2}$: $$D_\tau+\lambda-\frac{1}{2}\rightarrow-\frac{i}{\beta+\beta^\prime} \left[2\pi n+i\beta\left(\lambda-j_3-\frac{1}{2}\right)+ i\beta^\prime\left(\lambda-\frac{1}{2}+h_3+j_3\right)+ (\alpha_i\!-\!\alpha_j)\right]\ .$$ For the modes with $j\geq\frac{|n_i\!-\!n_j|\!+\!1}{2}$, since $\lambda$ appears in both signs, the ‘energy’ $|\lambda|$ appearing in the determinant gets shifted in two ways $|\lambda|\rightarrow|\lambda|\mp\frac{1}{2}$ where upper (lower) sign is for the positive (negative) $\lambda$. However, since $\lambda$ is always negative for modes with $j\!=\!\frac{|n_i\!-\!n_j|\!-\!1}{2}$, the shifted energy is always $|\lambda|+\frac{1}{2}$ in this case. For $\bar\lambda_\alpha$, one inserts $h_3=1$. Collecting all, the fermionic determinant is ($j_3\rightarrow-j_3$ relabeled for some terms) $$\begin{aligned} \hspace*{-1cm}\log(\det\!{}_{\rm fermion})&=& \frac{1}{2}{\rm tr}_F\left[ \left(\frac{}{}\!\beta(j\!+\!j_3)+\beta^\prime(j\!+\!1\!-\!j_3)\right)+ \left(\frac{}{}\!\beta(j\!+1\!+\!j_3)+\beta^\prime(j\!-\!j_3)\right) \right]\nonumber\\ &&+\sum_{i,j=1}^\infty\sum_{n=1}^\infty\frac{1}{n}f^{\rm adj,F}_{ij}(x)e^{-in(\alpha_i\!-\!\alpha_j)}\ .\end{aligned}$$ The modes with $j\geq\frac{|n_i\!-\!n_j|\!+\!1}{2}$ contribute to the letter index as $$f^{\rm adj,F}_{ij}\leftarrow -\!\!\!\!\sum_{j=\frac{|n_i\!-\!n_j|\!+\!1}{2}}^\infty\!\!\!\! \left[(x^\prime)^{2j\!+\!1}+(x^\prime)^{2j\!-\!1}x+\cdots+ x^\prime x^{2j}\right]+ \left[(x^\prime)^{2j}x+(x^\prime)^{2j\!-\!1}x^2+\cdots+ x^{2j\!+\!1}\right]\ .$$ Additional contribution from modes with $j\!=\!\frac{|n_i\!-\!n_j|\!-\!1}{2}$ is given by $$f^{\rm adj,F}_{ij}\leftarrow- \left[(x^\prime)^{|n_i\!-\!n_j|\!-\!1}x+(x^\prime)^{|n_1\!-\!n_j|\!-\!2}x^2+ \cdots+x^{|n_i\!-\!n_j|}\right]$$ if $n_i\!\neq\!n_j$. Comparing the determinants from bosons and fermions, one can immediately find a vast cancelation. In fact, contribution from bosonic modes with $j\geq\frac{|n_i\!-\!n_j|}{2}$ completely cancels with that from fermionic modes with $j\geq\frac{|n_i\!-\!n_j|\!+\!1}{2}$. In particular, this means that there is no net contribution from modes which do not feel the flux, i.e. $q=0$. This is of course consistent with the result of [@Bhattacharya:2008bja], in which the authors use combinatoric methods in the free theory where the gauge fields play no role. In our case, there are exceptional modes when $n_i\neq n_j$. Contributions from fermion modes with $j=q\!-\!\frac{1}{2}$ and bosonic modes with $j=q\!-\!1$ (if they exist) do not perfectly cancel and yield $$f^{\rm adj}_{ij}=-x^{|n_i\!-\!n_j|}\ \ \ \ ({\rm if}\ n_i\neq n_j)\ .$$ Generally one can write $f^{\rm adj}_{ij}(x)=-(1-\delta_{n_in_j}) x^{|n_i\!-\!n_j|}$. The final result is simply $$\begin{aligned} \frac{\det\!{}_{\rm fermion}}{\det\!{}_{\rm boson}} =\prod_{i,j=1}^N\exp\left[\sum_{n=1}^\infty\frac{1}{n} \left(f^{\rm adj}_{ij}(x^n)e^{-in(\alpha_i\!-\!\alpha_j)}+ \tilde{f}^{\rm adj}_{ij}(x^n)e^{-in(\tilde\alpha_i\!-\!\tilde\alpha_j)} \!\frac{}{}\right)\right]\ ,\end{aligned}$$ with similarly defined $\tilde{f}^{\rm adj}_{ij}(x)$. The evaluation of the Casimir-like energy is relegated to appendix B.3 below. Casimir energy -------------- We finally turn to the Casimir-energy like shift in the effective action $$\beta\epsilon_0\equiv \frac{1}{2}{\rm tr}\left[(-1)^F\left(\frac{}{}\!\beta(\epsilon\!+\!j_3)+ \beta^\prime(\epsilon\!-\!h_3\!-\!j_3) +\gamma_1h_1+\gamma_2h_2\right)\right]\ ,$$ where we have dropped the holonomy variables inside the trace, $is(\alpha_i\!-\!\tilde\alpha_j)$ for matters and $i(\alpha_i\!-\!\alpha_j)$ etc., for adjoints, since their traces are trivially zero. To compute this formally divergent quantity, one has to correctly regularize it. A constraint is that it has to be compatible with our special supersymmetry. The most general regularization would be insertion of $$x^{\epsilon+j_3}(x^\prime)^{\epsilon-h_3-j_3}y_1^{h_1}y_2^{h_2}\ .$$ inside the trace. The parameters $x,x^\prime,y_1,y_2$ are not to be confused with the chemical potentials in the rest of this paper: they are regulators and should be taken to $x,x^\prime\rightarrow 1^-$, $y_1,y_2\rightarrow 1$ after computing the trace. Anyway, the trace is formally very similar to the total summation of all letter indices we computed in the previous subsections. Actually the above trace, regularized as above, is $$\beta\epsilon_0=\frac{1}{2}\lim_{x,x^\prime,y_1,y_2\rightarrow 1} \left(\beta\partial_x\!+\! \beta^\prime\partial_{x^\prime}\!+\!\gamma_1\partial_{y_1}\!+\! \gamma_2\partial_{y_2}\right) \sum_{i,j=1}^N \left[f^{+}_{ij}(x,y_1,y_2)+f^-_{ij}(x,y_1,y_2)+f^{\rm adj}_{ij}(x) +\tilde{f}^{\rm adj}_{ij}(x)\right]\ .$$ Since $x^\prime$ disappears in the letter indices, $\partial_{x^\prime}$ is zero. Also, it is easy to see from the $y_1,y_2$ dependence of $f^{\pm}_{ij}$ that $\partial_{y_1}$ and $\partial_{y_2}$ are zero at $y_1,y_2=1$. Thus we only need to compute $\beta\partial_x$ acting on various functions. At $y_1=y_2=1$, they are given by $$\begin{aligned} &&\lim_{x\rightarrow 1}\partial_xf^+_{ij}= +|n_i\!-\!\tilde{n}_j|\ \ ,\ \ \lim_{x\rightarrow 1}\partial_xf^-_{ij}= +|n_i\!-\!\tilde{n}_j|\nonumber\\ &&\lim_{x\rightarrow 1}\partial_xf^{\rm adj}_{ij}(x)=-|n_i\!-\!n_j| \ \ ,\ \ \lim_{x\rightarrow 1}\partial_x\tilde{f}^{\rm adj}_{ij}(x) =-|\tilde{n}_i\!-\!\tilde{n}_j|\ .\end{aligned}$$ Therefore one finds $$\epsilon_0=\sum_{i,j=1}^N|n_i\!-\!\tilde{n}_j|-\sum_{i<j}|n_i\!-\!n_j| -\sum_{i<j}|\tilde{n}_i\!-\!\tilde{n}_j|\ .$$ We list a few nonzero values of $\epsilon_0$ for some positive flux distributions in Table 2. $$\begin{array}{c|c|c|c|c|c} \hline {\rm flux}&~\Yboxdim7pt\yng(2)~~\yng(1,1)~& ~\Yboxdim7pt\yng(2,1)~~\yng(1,1,1)~&\Yboxdim7pt\yng(3)~\yng(2,1)& \Yboxdim7pt\yng(3)~~\yng(1,1,1)~&\Yboxdim5pt\yng(6,4,3,2)~\yng(5,5,2,2,1)\\ \hline \epsilon_0&2&2&2&6&39\!-\!13\!-\!22\!=\!4\\ \hline \end{array}$$ We explain a few useful properties of $\epsilon_0$. The fluxes may involve positive, negative integers and zero. We first show that contributions to $\epsilon_0$ from modes carrying $U(1)$ indices with zero fluxes cancel to zero. Then we show that contributions from modes ending on one $U(1)$ with positive flux and another $U(1)$ with negative flux also cancel. To show the first, since modes ending on two $U(1)$’s both with zero flux is trivial, we restrict to the modes whose one end has zero flux and another nonzero. Then contribution of these modes to $\epsilon_0$ is $$\left(\!N_2\!\sum_{n_i\neq 0}|n_i|\!+\!N_1\! \sum_{\tilde{n}_i\neq 0}|\tilde{n}_i|\!\right)- N_1\sum_{n_i\neq 0}|n_i|-N_2\sum_{\tilde{n}_i\neq 0}|\tilde{n}_i|= (N_1\!-\!N_2)\left(\!\sum_{\tilde{n}_i\neq 0}|\tilde{n}_i|\!-\! \sum_{n_i\neq 0}|n_i|\!\right)\ .$$ Here we use the fact that, for the index to be nonzero, total sum of positive (negative) fluxes on both gauge groups should be equal. This implies that expression in the second parenthesis is zero, proving our claim. This result implies that, to compute $\epsilon_0$, one only has to consider contribution from modes connecting nonzero fluxes. To show the second, note that for such modes $|n_i\!-\!\tilde{n}_j|=|n_i|+|\tilde{n}_j|$, $|n_i\!-\!n_j|=|n_i|+|n_j|$ and $|\tilde{n}_i\!-\!\tilde{n}_j|=|\tilde{n}_i|+|\tilde{n}_j|$ due to the opposite sign of the two fluxes. After an analysis similar to the previous parenthesis, their contribution to $\epsilon_0$ is $$(M^-_1\!-\!M^-_2)\left(\sum|\tilde{n}^+_i|\!-\! \sum|n^+_i|\right)+(M^+_1\!-\!M^+_2)\left(\sum|\tilde{n}^-_i|\!-\! \sum|n^-_i|\right)\ .$$ Again from the equality of total positive/negative fluxes on two gauge groups, this quantity is zero. This result implies that one can separate $\epsilon_0=\epsilon_0^++\epsilon_0^-$, first one coming from modes connecting positive fluxes only and second from modes connecting negative fluxes only. This property will be important when we discuss the large $N$ factorization in section 3. Finally, we show that $\epsilon_0$ is always non-negative, and becomes zero if and only if the two sets $\{n_i\}$ and $\{\tilde{n}_i\}$ are the same. We shall actually prove a slightly more general claim. Suppose we have two decreasing functions $f(x)$ and $g(x)$ defined on $0\leq x\leq\ell$. Then we claim that the functional $\mathcal{E}[f,g]$ defined by $$\mathcal{E}[f,g]\equiv\int dxdy\left(|f(x)-g(y)|-\frac{1}{2}|f(x)-f(y)| -\frac{1}{2}|g(x)-g(y)|\right)$$ is always non-negative, and assumes its minimum at $0$ if and only if $f(x)=g(x)$ everywhere.[^10] To prove our claim, we vary the functional by $\delta f(x)$. Note that $$\begin{aligned} &&\delta|f(x)-g(y)|=\delta f(x)\left[2\theta(f(x)\!-\!g(y))-1\right] =\delta f(x)\left[2\theta(y\!-\!g^{-1}\!f(x))-1\right]\ ,\nonumber\\ &&\delta|f(x)\!-\!f(y)|=\delta f(x)\left[2\theta(y\!-\!x)-1\right]+ \delta f(y)\left[2\theta(x\!-\!y)-1\right]\end{aligned}$$ under this variation, where $\theta(x)$ is the step function (assuming $1$ for $x>0$, and $0$ for $x<0$). From these one finds $$\label{variation} \delta\mathcal{E}[f,g]=\int dx\ 2\delta f(x) \left[x\!-\!g^{-1}\!f(x)\right]\ .$$ The condition for the extremal points is $x=g^{-1}\!f(x)$, or $f(x)=g(x)$. Same result is obtained by the variation $\delta g(x)$. To show this is a minima, we compute the Hessian. From (\[variation\]) and analogous result for $\delta g(x)$, one finds $$\left(\begin{array}{cc}\frac{\delta^2 \mathcal{E}}{\delta f(x)\delta f(y)}& \frac{\delta^2 \mathcal{E}}{\delta f(x)\delta g(y)}\\ \frac{\delta^2 \mathcal{E}}{\delta g(x)\delta f(y)}& \frac{\delta^2 \mathcal{E}}{\delta g(x)\delta g(y)}\end{array}\right)=2 \left(\begin{array}{cc}-\frac{\delta(x\!-\!y)}{g^\prime(g^{-1}\!f(x))}& \frac{\delta(g^{-1}\!f(x)\!-\!y)}{g^\prime(g^{-1}\!f(x))} \\\frac{\delta(f^{-1}\!g(x)\!-\!y)}{f^\prime(f^{-1}\!g(x))}& -\frac{\delta(x\!-\!y)}{f^\prime(f^{-1}\!g(x))}\end{array}\right)$$ where we used $\delta f^{-1}(x)=-\frac{\delta f(f^{-1}(x))}{f^\prime(f^{-1}(x))}$ and similar formula for $\delta g^{-1}$. At the extrema $f=g$, the last matrix becomes $$2\delta(x\!-\!y)\left(\begin{array}{cc}-\frac{1}{f^\prime(x)}& \frac{1}{f^\prime(x)}\\\frac{1}{f^\prime(x)}& -\frac{1}{f^\prime(x)}\end{array}\right)= -\frac{2\delta(x\!-\!y)}{f^\prime(x)} \left(\begin{array}{cc}1&-1\\-1&1\end{array}\right)\ .$$ Since $f$ is decreasing, the coefficient in front of the matrix is positive. The last $2\times 2$ matrix has eigenvalue $0$ for $\delta f(x)=\delta g(x)$ and $+2$ for $\delta f(x)=-\delta g(x)$. The first is the expected zero direction since the variation leaves the relation $f=g$ unchanged. The second shows that the extrema $f=g$ is actually a minima, proving our claim. Index over gravitons in $AdS_4\times S^7/\mathbb{Z}_k$ ====================================================== Index of many gravitons in $AdS_4\times S^7/\mathbb{Z}_k$ can be obtained from the index of single graviton in $AdS_4\times S^7$, as explained in [@Bhattacharya:2008bja]. The index of single graviton in $AdS_4\times S^7$ is given by $$I^{\rm sp}(x,y_1,y_2,y_3)= \frac{\rm (numerator)}{\rm (denominator)}$$ where $$\begin{aligned} {\rm numerator}&=&\sqrt{y_1y_2y_3}\left(1+y_1y_2+y_2y_3+y_3y_1\right) x^{\frac{1}{2}}-\sqrt{y_1y_2y_3} (y_1\!+\!y_2\!+\!y_3\!+\!y_1y_2y_3)x^{\frac{7}{2}} \nonumber\\ &&+(y_1y_2\!+\!y_2y_3\!+\!y_3y_1\!+\!y_1y_2y_3(y_1\!+\!y_2\!+\!y_2)) (x^3-x)\\ {\rm denominator}&=&(1-x^2)(\sqrt{y_3}-\sqrt{xy_1y_2}) (\sqrt{y_1}-\sqrt{xy_2y_3})(\sqrt{y_2}-\sqrt{xy_3y_1}) (\sqrt{y_1y_2y_3}-\sqrt{x})\ .\nonumber\end{aligned}$$ A very useful property of this function is $$\begin{aligned} \label{relation} I^{\rm sp}&=&\frac{\left(1-x\sqrt{xy_1y_2y_3}\right) \left(1-x\sqrt{\frac{xy_3}{y_1y_2}}\right) \left(1-x\sqrt{\frac{xy_1}{y_2y_3}}\right) \left(1-x\sqrt{\frac{xy_2}{y_1y_3}}\right)} {\left(1-\sqrt{\frac{xy_1y_3}{y_2}}\right) \left(1-\sqrt{\frac{xy_2y_3}{y_1}}\right) \left(1-\sqrt{\frac{xy_1y_2}{y_3}}\right)\left(1-\sqrt{\frac{x}{y_1y_2y_3}}\right) (1-x^2)^2}-\frac{1-x^2+x^4}{(1-x^2)^2}\nonumber\\ &\equiv&\frac{F(x,y_1,y_2,y_3)}{(1-x^2)^2} -\frac{1-x^2+x^4}{(1-x^2)^2}\end{aligned}$$ where the function $F(x,y_i)$ is defined in section 3.2. The index of single gravitons in $AdS_4\times S^7/\mathbb{Z}_k$ is obtained by expanding $I^{\rm sp}$ in $y_3$ as $$I^{\rm sp}=\sum_{n=-\infty}^\infty y_3^{\frac{n}{2}}I^{\rm sp}_n(x,y_1,y_2)$$ and keeping terms in which $n$ is a multiplet of $k$: $$I^{\rm sp}_{\mathbb{Z}_k}(x,y_1,y_2,y_3)\equiv\sum_{n=-\infty}^\infty y_3^{\frac{kn}{2}}I^{\rm sp}_{kn}(x,y_1,y_2)\ .$$ Each term $I_{kn}$ represents a single particle index of gravitons carrying Kaluza-Klein momentum $kn$. Finally, the index of multiplet gravitons in $AdS_4\times S^7/\mathbb{Z}_k$ is given by $$I_{\rm mp}(x,y_1,y_2,y_3)=\exp\left[\sum_{n=1}^\infty\frac{1}{n} I^{\rm sp}_{\mathbb{Z}_k}(x^n,y_1^n,y_2^n,y_3^n)\right]\ .$$ One can decompose this index into three factors, each coming from gravitons with positive/negative/zero KK-momenta, respectively, as $$I_{\rm mp}(x,y_1,y_2,y_3)=I^{(0)}_{\rm mp}I^{(+)}_{\rm mp} I^{(-)}_{\rm mp}\ ,$$ where $$I^{(0)}_{\rm mp}=\exp\left[\sum_{n=1}^\infty\frac{1}{n} I^{\rm sp}_0(\cdot^n)\right]\ ,\ \ I^{(\pm)}_{\rm mp}=\exp\left[\sum_{n=1}^\infty \frac{1}{n}I^{\rm sp(\pm)}_{\mathbb{Z}_k}(\cdot^n)\right]$$ and $$I^{\rm sp(\pm)}_{\mathbb{Z}_k}=\sum_{n=1}^{\infty} y_3^{\pm\frac{kn}{2}}I^{\rm sp}_{\pm kn}(x,y_1,y_2)\ .$$ $I^{(\pm)}_{\rm mp}$ satisfies the property $I^{(-)}_{\rm mp}(x,y_1,y_2,y_3)= I^{(+)}_{\rm mp}(x,y_1,1/y_2,1/y_3)$, similar to the relation between $I^{(-)}$ and $I^{(+)}$ for the gauge theory index defined in section 3. 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See [@Witten:1992xu] for more explanation. Fortunately, this problem appears to be absent with the saddle points we find below. [^4]: More generally, one can consider fractional fluxes $n_i=m_i+\frac{B}{k}$, $\tilde{n}_i=\tilde{m}_i+\frac{B}{k}$ where $m_i,\tilde{m}_i,B$ are integers. See [@Klebanov:2008vq; @Berenstein:2009sa] for details. Since fields couple to differences of fluxes, $B$ appears only through the phase factor (\[phase\]) in (\[exact-index\]), and in particular disappears in our large $N$ calculation in section 3. [^5]: Letters are defined by single basic fields with many derivatives acting on them [@Aharony:2003sx; @Kinney:2005ej; @Bhattacharya:2008bja]. In path integral calculation, they are simply (monopole) spherical harmonics modes of the fields. [^6]: Unfortunately, we could not go to $\mathcal{O}(x^{12})$ where one can start testing the last line of (\[3-unequal-distrib\]), due to the long execution time. The bottleneck was at the third line of (\[3-equal-distribute\]), in which we had to integrate over the $U(3)\times U(3)$ holonomy with nine factors of $F$ functions, etc., in the integrand. We hope we can improve our calculation in the near future. We thank Sehun Chun for his advice. [^7]: See eqns. (4.23), (4.24) there and surrounding arguments. [^8]: We correct sign typos of [@Weinberg:1993sg] in some of these formulae. [^9]: The matrix $\mathcal{M}$ with $n_i<n_j$ can be easily obtained as follows. In (\[quad-generic\]), one of $\mathcal{M}^T$ and $\mathcal{M}$ satisfy $n_i\geq n_j$. Suppose $\mathcal{M}$ satisfies this condition. Firstly complex conjugate all $i$ which are explicit in (\[matrix-generic\]). Then change the sign of $\lambda$, which is due to the sign changes of $n,j_3$ and $\alpha_i\!-\!\alpha_j$. Transposing it gives the pair $\mathcal{M}^T$. [^10]: Requiring these functions to assume integral values, admitting decreasing step function like singularities, brings us back to our original problem. We think discontinuity would not cause any problem, but if one prefers, one may slightly regularize them to smooth decreasing functions while staying arbitrarily close to our problem.

--- abstract: | In this paper, the problem of tracking desired longitudinal and lateral motions for a vehicle is addressed. Let us point out that a “good” modeling is often quite difficult or even impossible to obtain. It is due for example to parametric uncertainties, for the vehicle mass, inertia or for the interaction forces between the wheels and the road pavement. To overcome this type of difficulties, we consider a model-free control approach leading to “intelligent” controllers. The longitudinal and the lateral motions, on one hand, and the driving/braking torques and the steering wheel angle, on the other hand, are respectively the output and the input variables. An important part of this work is dedicated to present simulation results with actual data. Actual data, used in Matlab as reference trajectories, have been previously recorded with an instrumented Peugeot 406 experimental car. The simulation results show the efficiency of our approach. Some comparisons with a nonlinear flatness-based control in one hand, and with a classical PID control in another hand confirm this analysis. Other virtual data have been generated through the interconnected platform SiVIC/RTMaps, which is a virtual simulation platform for prototyping and validation of advanced driving assistance systems. **Keywords** Longitudinal and lateral vehicle control, model-free control, intelligent controller, algebraic estimation, flatness property, classical PID controllers, ADAS (Advanced Driving Assistance Systems) author: - | Lghani MENHOUR, Brigitte d’ANDRÉA-NOVEL, Michel FLIESS, Dominique GRUYER\ and Hugues MOUNIER [^1] [^2] [^3] [^4] [^5] title: 'An efficient model-free setting for longitudinal and lateral vehicle control. Validation through the interconnected pro-SiVIC/RTMaps prototyping platform' --- [Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{} Introduction {#introd} ============ The vehicle longitudinal and lateral control problem has been widely investigated in the literature via model-based techniques (see, *e.g.*, [@Ackermann95; @Novel01; @Choi09; @Cerone09; @Chou05; @Fernandez11; @formentin; @Fuchsumer2005; @Hatipoglu03; @Khodayari10; @Manceur13; @Menhour13a; @Ge15; @Wei15; @Nobe01; @Odenthal99; @Poussot11a; @Tagne16; @Villagra09], and the references therein). Their performances are guaranteed in the vicinity of the model used for their implementation. In this paper, a description of the evolution of our developments on longitudinal and lateral vehicle control problem is given. Thereafter, the poor knowledge of vehicle mathematical models has led us to develop a vehicle controller based on *model-free* setting [@ijc13]. In fact, obtaining of a “good” mathematical modeling is a difficult task, if not an impossible one, since uncertainties and disturbances, like frictions and tire nonlinear behaviors, should be taken into account. The vehicle behavior is highly dependent on tire road forces which are usually modeled by linear tire models. Fig. \[switching\_rule\] illustrates an example of the tire characteristic obtained during an experimental braking maneuver on a real race track. This maneuver has highlighted the different nonlinear dynamics of the tire forces which cannot be characterized by a linear tire model. Unfortunately, there are other emergency driving situations for which simplified vehicle models cannot provide a realistic behavior of the actual car, such as the rollover phenomenon due to high values of Load Transfer Ratio (LTR), and the under-steering and over-steering behaviors due to high values of front or rear sideslip angles. ![Experimental lateral tire force characteristic and its nonlinear behavior[]{data-label="switching_rule"}](Switching_rule.pdf) The parametric dependency and the tedious features vanish here thanks to a new model-free approach where: - the need to exploit the flatness property of a simplified model disappears, - the flat output depending on uncertain parameters is replaced by a simpler and perhaps more natural one, which is the lateral deviation. Our paper is organized as follows. A brief description of the required vehicle models used for simulation or control purposes is given in Section \[Section\_1\]. A short summary on the model-free control approach with application to the design of a longitudinal/lateral vehicle control and a lateral deviation computation algorithm are presented in Section \[Section\_3\]. Section \[Section\_4\] displays several numerical simulations established using actual data as reference trajectories and a full nonlinear 10DoF vehicle model of an instrumented Peugeot 406 experimental car. The obtained results show the relevance and efficiency of our approach. Some comparisons are provided with a nonlinear flatness-based controller and with classical PID controllers, briefly recalled in Appendix \[appendix1\], in order to highlight the advantages of model-free control. Secondly, a validation of model-free control is conducted under an interconnected platform SiVIC/RTMaps (See [@Gruyer10a]). This platform provides an efficient simulation tool in order to replace an actual system (sensors, actuators, embedded algorithms) by a virtual one in a physically realistic environment. It allows advanced prototyping and validation of perception and control algorithms including quite realistic models of vehicles, sensors and environment. Some concluding remarks are given in Section \[Section\_5\]. Description of vehicle models for simulation or control purposes {#Section_1} ================================================================ The following section details some vehicle dynamics elements related to the longitudinal/lateral vehicle control problem. Three control approaches are developed: the first one is based on an algebraic nonlinear estimation and differential algebraic flatness of a 3DoF nonlinear two wheels vehicle model , the second one is based on simple PID controllers, in order to overcome the modeling problems, the last one is based on the so-called model-free control approach developped in [@ijc13], since, under high dynamic loads, such models may become not sufficient to handle the critical maneuvers. For this purpose, the last model-free control law is developed to overcome the modeling problems. For vehicle simulation and control design problems, a large set of vehicle models exist. Usually, the linear bicycle model is widely used for control and estimation algorithms implementation. Furthermore, more complex models exist which are, however, not appropriated for such applications. For the flat nonlinear control issue, the following nonlinear coupled two wheels vehicle model is used (see e.g. [@Menhour13a]): $$\label{Non_linear_bicycle_model} \left \{ \begin{array}{l} m a_x=m(\dot{V}_x-\dot{\psi}V_y)= (F_{xf}+F_{xr}) \\ m a_y=m(\dot{V}_y+\dot{\psi}V_x)=(F_{yf}+F_{yr}) \\ I_z\ddot{\psi}= L_fF_{yf} -L_rF_{yr}%{M_{zf}+M_{zr}} \end{array} \right.$$ Notations are defined in Table \[notations\_vehicle\]. Notice that the the front and rear longitudinal forces in model are expressed using the following dynamical model of the tire forces: $$\label{Front_rear_wheel} \left \{ \begin{array}{lcl} F_{xf} &= &(1/R)(- I_r\dot{\omega}_f + T_{m}-T_{bf})\\ F_{xr} &= &-(1/R)(T_{br} + I_r\dot{\omega}_r) \end{array} \right.$$ The linear tire model of lateral forces in tha case of small slip angles is defined as follows: $$\label{lin_lat_for} \left \{ \begin{array}{l} F_{yf}=C_f \left(\delta-\frac{V_y+\dot{\psi}L_f}{V_x}\right)\\[2mm] F_{yr}=-C_r \left(\frac{V_y-\dot{\psi}L_r}{V_x}\right) \end{array} \right.$$ For preliminary simulation results, a 10-DoF nonlinear vehicle model and real data recorded previously with a laboratory vehicle are used. The vehicle simulator considered here is a full nonlinear four wheels vehicle model [@Menhour13a] of a Peugeot 406 vehicle. Such a model is composed of longitudinal $V_x$, lateral $V_y$ and vertical $V_z$ translational motions, roll $\phi$, pitch $\theta$ and yaw $\psi$ rotational motions and dynamical models of the four wheels (see [@Menhour13a] for more details on the 10DoF nonlinear simulation vehicle model). All steps of design and simulation are summarized in Fig. \[diag\_valid\_control\_NB\_V0\].  In this document, the variables and notations in Table \[notations\_vehicle\], with their meaning are used. Symbol Variable name ------------------------ --------------------------------------------------------------- $V_x$ longitudinal speed \[$km.h$\] $V_y$ lateral speed \[$km.h$\] $\dot{\psi}$ yaw rate $[rad/s]$ $\psi$ yaw angle $[rad]$ $y$ lateral deviation $[m]$ $\dot{y}$ derivative of lateral deviation $[m/s]$ $\beta$ sideslip angle at the CoG $[rad]$ $\delta$ wheel steering angle $[rad]$ $T_{\omega}$ acceleration/braking torque $[Nm]$ $F_{yf}$, $F_{yr}$ front and rear lateral forces in the vehicle coordinate $[N]$ $\alpha_f$, $\alpha_r$ front and rear tire slip angles $[rad]$ $\rho_x$ aerodynamics drag coefficient $C_f$, $C_r$ front and rear cornering tire stiffnesses $[N/rad]$ $L_f$, $L_r$ distances from the CoG to the front and rear axles $[m]$ $I_z$ yaw moment of inertia $[kg.m^{-2}]$ $I_x$ moment of inertia about $x$ axis $[kg.m^{2}]$ $m$ vehicle mass $[kg m^2]$ $g$ acceleration due to gravity $[m/s^2]$ $\mu$ adhesion coefficient : notations[]{data-label="notations_vehicle"} Model-free control {#Section_3} ================== A background of the model-free control approach ----------------------------------------------- Model-free control was already applied and used quite successfully in a lot of various concrete examples (see the references in [@ijc13; @ecc]). For obvious reasons let us insist here on its applications to *intelligent transportation systems*: see [@Abouaissa12; @Choi09; @Villagra09; @vil2], and [@Menhour13b]. This last reference was briefly discussed in Section \[introd\]. ### The ultra-local model {#A} Replace the unknown SISO system by the *ultra-local model*: $$z^{(\nu)} = F + \alpha u \label{ultralocal}$$ where - $u$ is the control input, - $z$ is the controlled output, - $F$ is estimated via the measurements of the control input $u$ and the controlled output $z$. It does not distinguish between the unknown model of the system, the perturbations and uncertainties. - $\nu \geq 1$ is the derivation order, - $\alpha \in \mathbb{R}$ is chosen such that $\alpha u$ and $z^{(\nu)}$ are of the same order of magnitude, - $\nu$ and $\alpha$ are chosen by the practitioner. In all the existing concrete examples $$\nu = \ 1 \ \text{or} \ 2$$ Until now from our knowledge, in the context of model-free control, the example of magnetic bearings [@Miras13] with their low friction, provides the only instance where the order $\nu = 2$ is necessary. ### Intelligent controllers Set $\nu = 2$ in Equation : $$\label{MFC_4_n_2} \ddot{z} = F+ \alpha u$$ The corresponding *intelligent Proportional-Integral-Derivative controller*, or *iPID*, reads $$\label{iPID_c} u = - \frac{\left( F - \ddot{z}^d + K_P e + K_I\int e dt + K_D \dot{e}\right)}{\alpha}$$ where - $z^d$ is the desired reference trajectory, - $e = z-z^d$ is the tracking error and $z^d$ is a desired signal, - $K_P$, $K_I$, $K_D \in \mathbb{R}$ are the usual gains. Combining Equations and yields $$\ddot{e} + K_P e + K_I \int e dt + K_D \dot{e}= 0$$ where $F$ does not appear anymore. Gain tuning becomes therefore quite straightforward. This is a major benefit when compared to “classic” PIDs. If $K_I = 0$ we obtain the *intelligent Proportional-Derivative controller*, or *iPD*, $$\label{iPD_c} u = - \frac{\left( F - \ddot{z}^d + K_P e + K_D \dot{e}\right)}{\alpha}$$ Set $\nu = 1$ in Equation : $$\label{MFC_4_n_1} \dot{z} = F+ \alpha u$$ The corresponding *intelligent Proportional-Integral controller*, or *iPI*, reads: $$\label{iPI_c} u = - \frac{\left(F - \dot{z}^d + K_P e + K_I\int e dt \right)}{\alpha}$$ If $K_I = 0$ in Equation , we obtain the *intelligent proportional controller*, or *iP*, which turns out until now to be the most useful intelligent controller: $$\label{ip} {u = - \frac{F - \dot{z}^d + K_P e}{\alpha}}$$ ### Algebraic estimation of $F$ {#F} $F$ in Equation is assumed to be “well” approximated by a piecewise constant function $F_{\text{est}} $. According to the algebraic parameter identification developed in [@sira1; @sira2], where the probabilistic properties of the corrupting noises may be ignored, if $\nu = 1$, Equation rewrites in the operational domain (see, *e.g.*, [@Yosida84]) $$s Z = \frac{\Phi}{s}+\alpha U +z(0)$$ where $\Phi$ is a constant. We get rid of the initial condition $z(0)$ by multiplying the both sides on the left by $\frac{d}{ds}$: $$Z + s\frac{dZ}{ds}=-\frac{\Phi}{s^2}+\alpha \frac{dU}{ds}$$ Noise attenuation is achieved by multiplying both sides on the left by $s^{-2}$. It yields in the time domain the realtime estimation $$\label{integral} F_{\text{est}}(t) = % \left[ %\begin{array}{c} -\frac{6}{\tau^3}\int_{t-\tau}^t \left\lbrack \begin{array}{c} (\tau -2\sigma)z(\sigma)\\ +\alpha\sigma(\tau -\sigma)u(\sigma) \end{array} \right\rbrack d\sigma %\end{array} %\right]$$ Notice that the extension to the case $\nu = 2$ is straightforward. For this case, the estimation of $F$ is performed by the following estimator: $$\label{integral_2} F_{\text{est}}(t) = \left[ \begin{array}{c} -\frac{60}{\tau^5}\int_{t-\tau}^t(\tau^2 +6\sigma^2 - 6\tau \sigma)z(\sigma)d\sigma \\[2mm] -\frac{30\alpha}{\tau^5} \int_{t-\tau}^t(\tau - \sigma)^2\sigma^2 u(\sigma)d\sigma \end{array} \right]$$ where $\tau > 0$ might be quite small. This integral may, of course, be replaced in practice by a classic digital filter (for more details see e.g. [@ijc13]). See [@nice] for a cheap and small hardware implementation of our controller. It should be emphasized that the above-mentioned estimation methods are not of asymptotic type. Application to a vehicle control -------------------------------- As previously mentioned, to take advantage of model-free approach, an appropriate choice of the outputs and the corresponding inputs control is required. Since to avoid any vehicle modeling problem and ensure a desired tracking longitudinal and lateral motions, the following input and output variables are selected: 1. the acceleration/braking torque $u_1 = T_\omega$ and the longitudinal speed $z_1$, 2. the steering wheel angle $u_2 = \delta$ and the lateral deviation $z_2$. It is obvious that the second output, which is the lateral deviation output, gives a kinematic relationship between the vehicle motions like longitudinal, lateral and yaw motions. This allow us to include some coupling effects between these motions. According to the background on the model-free setting, the above inputs/outputs, and the Newton’s second law, the following two local models are deduced: $$\begin{aligned} \label{MFC_LLVC2_1} \text{longitudinal local model:} & \dot{z}_1 = F_1+ \alpha_1 u_1\\ \label{MFC_LLVC2_2} \text{lateral local model:} & \ddot{z}_2 = F_2+ \alpha_2 u_2 \end{aligned}$$ Note the following properties: - Equations - seem decoupled, but the coupling effects are included in the terms $F_1$ and $F_2$. - Equation is an order $2$ formula with respect to the derivative of $z_2$. For Equation (resp. ), the loop is closed by an iP (resp. iPD ) as follows: $$\begin{aligned} \label{MFC_Control_1} u_1 & = & -\frac{1}{\alpha_1} \left( F_1 - {\dot{z}}_1^{d} + K_P^{z_1} e_{z_1} \right)\\ \label{MFC_Control_2} u_2 & = & -\frac{1}{\alpha_2}\left( F_2 - {\ddot{z}}_2^{d} + K_P^{z_2}e_{z_2} + K_D^{z_2}\dot{e}_{z_2} \right)\end{aligned}$$ Moreover, the estimators and are powerful tools to estimate respectively $F_1$ and $F_2$ as follows: - For $\nu = 1$ $$\label{integral_app} F_{1_\text{est}}(t) = -\frac{6}{\tau^3}\int_{t-\tau}^t \left\lbrack \begin{array}{c} (\tau -2\sigma)z_1(\sigma)\\ +\alpha\sigma(\tau -\sigma)u_1(\sigma) \end{array} \right\rbrack d\sigma$$ - For $\nu = 2$ $$\label{integral_2_app} F_{2_\text{est}}(t) = \left[ \begin{array}{c} -\frac{60}{\tau^5}\int_{t-\tau}^t(\tau^2 +6\sigma^2 - 6\tau \sigma)z_2(\sigma)d\sigma\\ -\frac{30\alpha}{\tau^5} \int_{t-\tau}^t(\tau - \sigma)^2\sigma^2 u_2(\sigma)d\sigma \end{array} \right]$$ Table \[Gains\_MFC\] gives the gains of the model-free control used in Section \[Section\_4\]. $\alpha$ $K_P$ $K_D$ $\tau\, [ms]$ f\[Hz\] ---------------------------- ---------- ------- ------- --------------- --------- Longitudinal control $u_1$ 1.5 2 0 0.25 200 Lateral control $u_2$ 1.95 1.9 0.5 0.25 200 : Model-Free Control parameters[]{data-label="Gains_MFC"} Lateral deviation calculation {#Section_42} ----------------------------- The parameters that easily characterize a desired reference trajectory are the road curvature $\rho_d$, the yaw angle $\psi_d$, the path length coordinate $s_d$, the coordinates $x_d$ and $y_d$. The road curvature is computed by longitudinal speed and lateral acceleration using the following formula: $$\label{corbure} \rho_d(s_d)=\frac{1}{R(s_d)}=\frac{a_y(s_d)}{V_x^2(s_d)}$$ This method computes the yaw angle from the curvature , then the coordinates $\psi_d$, $x_d$ and $y_d$ are deduced as follows by projection of the path length coordinate $s_d$ in the vehicle frame ($s_0$ being the initial condition): $$\label{Model_trajectory} \left\{ \begin{array}{l} \psi_{d}(s_{d}) = \int_{s_{0}}^{s}{\rho_{d}(s_{d})ds} \\ x_{d}(s_{d})= \int_{s_{0}}^{s}{\cos(\psi_d(s_{d}))ds}\\ y_{d}(s_{d})= \int_{s_{0}}^{s}{\sin(\psi_d(s_{d}))ds}%\\ %x_{d}(s_{d}) = \int_{s_{0}}^{s}{V_x\cos{(\psi_d(s_{d}))}-V_y\sin{(\psi_d(s_{d}))}}ds\\ %y_{d}(s_{d}) = \int_{s_{0}}^{s}{V_x\sin{(\psi_d(s_{d}))}+V_y\cos{(\psi_d(s_{d}))}}ds \end{array}\right.$$ ![GPS trajectory and trajectory of the algorithm []{data-label="Trajectories_Model_GPS"}](GPS_algorithm.pdf) Fig. \[Trajectories\_Model\_GPS\] shows an experimental validation of algorithm . The actual GPS data (grey curve) are quite close to the computed trajectory (dash blue curve). Moreover, this algorithm only depends on measured data and it is independent of any vehicle parameters. Unfortunately, without accurate ground truth requiring very expensive embedded means and sensors, it is very difficult to quantify precisely the impact and the quality of the obtained result. In the real experiments, we have observed that the results were interesting and relevant. In order to obtain an objective view of the level of quality of the proposed model-free approach comparatively to classical one, the use of simulation platforms is performed. Simulation results {#Section_4} ================== As we mentioned previously in this paper, actual condition tests often are expensive and require specially equipped vehicles in order to study the interactions between embedded applications and events encountered in the road environment. Moreover in actual conditions, tens of thousands of kilometers are required in order to obtain good enough estimation of an application quality and robustness. Furthermore, with the development of autonomous, cooperative and connecting systems, applications become increasingly complex to implement and validate. Then the Simulation allows to approach the reality of a situation and allows the control of specific events and their reproducibility. The simulation allows too to play with weather factors and to study the influence and the impact of physical parameters on the robustness of the functions used in embedded applications, especially connected and active ones (cooperative and automated driving assistance systems). Difficult or dangerous tests are easily accessible to simulation as well as the analysis of the functions performance in a singular event (emergency braking or road departure warning due to obstacle detection or road friction problem), whereas in real condition, waiting hours could be required before the generation of such an event. Waiting for this real event would be inefficient and unreasonable. In this study, the simulation stage is carried out according to the diagram block of Fig. \[diag\_valid\_control\_NB\_V1\] and uses two levels of simulation and three different platforms: firstly a Matlab platform is used with a full nonlinear model of the instrumented Peugeot 406 car as in [@Menhour13b], and secondly thanks to the pro-SiVIC[^6] platform [@Gruyer10b; @Gruyer10a; @Vanholme10; @Gruyer09] interconnected with RTMaps platform[^7], we will simulate complex vehicle dynamical modeling [@IJACSA13], environment, infrastructure, and realistic embedded sensors. Moreover pro-SiVIC will provide mechanisms allowing to generate very accurate ground truth for the evaluation and validation stages. RTMaps is a platform which allows to record, replay, manage and process multiple data flows in real-time.  Simulation under Matlab ----------------------- The first simulation is conducted according to the diagram block of Fig. \[diag\_valid\_control\_NB\_V2\] using a 10DoF nonlinear vehicle model[^8] and real data recorded previously with a prototype vehicle from CAOR laboratory (Mines Paristech). Several experiments have been realized with an instrumented car to compute the coordinates of the race track depicted in Figure \[Trajectories\_Model\_GPS\]. For each trial, several dynamical variables have been recorded; among them: lateral and longitudinal accelerations, longitudinal and lateral speeds, yaw and roll rates, wheel rotation speeds, moments on the four wheels, longitudinal, lateral and vertical forces on the four wheels, steering angle, etc. Moreover, the road geometry (road bank and slope angles) is considered in the closed-loop system. In these simulations, the efficiency and relevance of the model-free control as well as its performances are compared to those of nonlinear flat control and PID controllers. The Table \[Inputs\_outputs\] summarizes the control inputs and the controlled outputs used for each control law. Two simulation scenarios are conducted for two values of the road friction coefficient: the first one for dry asphalt $\mu = 1$ and the second one for wet asphalt $\mu = 0.7$. These scenarios mean that the adhesion capability of the ground is reduced, thus, the vehicle becomes unstable, i.e that the vehicle maneuverability and controllability become more difficult.  -- ----------------------------------- ----------- ---------- ---------- -- Driving/braking torque $T_\omega$ $\times$ $\times$ $\times$ Steering angle $\delta$ $\times $ $\times$ $\times$ Longitudinal speed $V_x$ $\times$ $\times$ $\times$ Second flat output $z_2$ NA $\times$ NA Lateral deviation $y$ $\times$ NA $\times$ -- ----------------------------------- ----------- ---------- ---------- -- For a dry road ($\mu = 1$), results are shown in Figs. \[Vx\_comp\], \[Ey\_Epsi\] and \[Twheel\_Delta\]. These simulations demonstrate that the model-free control (MFC) gives quite satisfying results, better than those obtained with PID control and nonlinear flat control. It should be noticed that the test track which has been considered implies strong lateral and longitudinal dynamical requests. This track involves different types of curvatures linked to straight parts, and all these configurations represent a large set of driving situations. Fig. \[Vx\_comp\] shows that all controllers produce accurate behavior for autonomous driving applications. It is obvious that the results displayed on Fig. \[Ey\_Epsi\], show that the tracking errors on the lateral deviation and on the yaw angle outputs produced by the model-free control (MFC) are better compared to those produced by the other controllers. These errors are less than $10\, cm$ and $0.5\, deg$ in the case of model-free control (MFC). Finally, Fig. \[Twheel\_Delta\] shows that the control signals computed by all control strategies are quite close to the actual ones provided by the driver along the track race. ![The actual longitudinal speed versus the closed-loop simulated longitudinal velocities[]{data-label="Vx_comp"}](Vx_VV2.pdf) ![Tracking trajectory errors on lateral deviation and yaw angle[]{data-label="Ey_Epsi"}](ey_epsi_V2.pdf) ![Wheel torques and steering angles control signals: actual and simulated[]{data-label="Twheel_Delta"}](Tw_Delta_V2.pdf) Moreover, the effectiveness of the model-free control (MFC) results is also evaluated through the following normalized error: $$\label{norma_errors} e_z(i)={100 \left|z_{s}(i)-z_{act}(i) \right|}/{ \max \left| z_{act} \right|}$$ -- ---------------------------------- ------- ------ ------- -- Longitudinal speed $e_{V_x}$ (%) 0.93 0.45 0.186 Yaw angle $e_{\psi}$ (%) 1.76 1.21 0.45 Lateral deviation $e_{y}$ (%) 2.8 1.4 0.35 Longitudinal speed $e_{V_x}$ (%) 5.54 4.23 2.31 Yaw angle $e_{\psi}$ (%) 13.54 7.67 2.7 Lateral deviation $e_{y}$ (%) 16.64 9.37 3.49 -- ---------------------------------- ------- ------ ------- -- Table \[errors\_dyn\_para\] summarizes a comparison between the closed-loop simulated results $z_{s}$ and the actual data $z_{act}$ using the normalized error . The normalized norm is computed for dry and wet road scenarios. In the two cases, the model-free control (MFC) provides better results than PID control and nonlinear flat control. Using the model-free control (MFC), the maximum normalized error values are less than $0.5 \,\%$ for dry road test and less than $3.5 \,\%$ for wet road test. However, all normalized errors obtained with other controllers are deteriorated mainly with wet road test. These results confirm that the model-free control (MFC) with natural outputs produces a satisfactory behavior. Simulation under pro-SiVIC interconnected with RTMaps ----------------------------------------------------- After the Matlab stage for the model-free approach validation, a second simulation level is addressed. This second stage proposes to implement the model-free control algorithms in a more realistic way. In this case, very realistic and complex vehicle modeling is used. This vehicle level of complexity is presented in [@IJACSA13]. Attached to this vehicle model, a set of virtual sensors are simulated. All this functionalities are available in pro-SiVIC platform. The data generated by pro-SiVIC and its sensors are sent to RTMaps which is the real software environment used in a real prototype vehicle in order to implement ADAS (Advanced Driving Assistance Systems) applications. Using an efficient way to interconnect these 2 platforms, we obtain an efficient Software In the Loop (SIL) solution and a full real-time simulation architecture presented in Fig. \[SiVIC\_RTMaps\_architecture\].  ### The pro-SiVIC platform From the past decade, a great set of researches aimed to improve the road safety through the implementation of driving assistance systems. These researches generally take into account a local perception (from the ego-vehicle point of view) and the ego-vehicle maneuvers (e.g. braking and acceleration). Nevertheless, in many actual road configurations, a local perception is no longer sufficient. Additional information is needed to minimize risk and maximize the driving security level. This additional information requires additional resources which are both time-consuming and expensive. It therefore becomes essential to have a simulation environment allowing to prototype and to evaluate extended, enriched and cooperative driving assistance systems in the early design stages. An efficient and functional simulation platform has to integrate different important capabilities: Road environments, virtual embedded sensors (proprioceptive, exteroceptive), infrastructure sensors and cooperative devices (transponders, communication means, ...), according to the physical laws. In the same way, a physics-based model for vehicle dynamics (potentially with steering wheel column and powertrain models) coupled with actuators (steering wheel angle, torques on each wheel) is necessary. pro-SiVIC takes into account these remarks and these requirements, and is therefore a very efficient and dedicated platform to develop and prototype a high level autonomous driving system with cooperative and extended environment perception. This platform is currently shared in two version: pro-SiVIC and pro-SiVIC research. The first version is the industrial one. The second version, which we have used, is the research version developed in IFSTTAR institute. The research version, in its current state, includes a set of different exteroceptive and proprioceptive sensors, thus communication means. The exteroceptive sensors set involved several types of cameras (classical, omnidirectional, and fisheye), several models of laser scanner, and some levels of automotive RADAR modeling. The proprioceptive sensors model odometers (curvilinear distance, speed, impulses) and Inertial Navigation systems (accelerometers, gyrometers). Then communication means for cooperative systems include both 802.11p communication media and beacon (transponder). For all these sensors and medias, it is possible, in real-time and during the simulation stages, to tune and to change the sensor’s sampling time, operating modes, and intrinsic/ extrinsic parameters. Some examples of the rendering of these sensors are shown in Fig. \[SiVIC\_Sensors\].  ### RTMaps platform RTMaps is a software platform originally developed at Mines ParisTech by B. Steux [@Steux01].[^9] for the needs of the European Carsens project. The main objective of this platform was to provide an efficient way to collect, to replay and to process, in real real-time, a great set of heterogeneous and asynchronous sensors. So this platform can provide a relevant solution for the embedded and multi-sensors applications prototyping, test and evaluation. In the processing stages, RTMaps allows to manage and to process in same time raw data flows coming from images, laser scanner, GPS, odometric, and INS sensors. The algorithms, which can be applied to the sensor data, are provided in several libraries (RTMaps packages) dedicated to specific processing (image processing and multi-sensors fusion). In its current user configuration an efficient new package and new module development procedure is available. This capability has been used in order to implement our different perception and control modules. This type of architecture gives a powerful tool in order to prototype embedded ADAS (Advanced Driving Assistance Systems) with either informative outputs or orders to control vehicle dynamics. At each stage, the sensor data and module outputs are time-stamped for an accurate and a reliable time management. ### pro-SiVIC/RTMaps: an interconnected platform for efficient Advanced Driving Assistance Systems prototyping The interconnection between pro-SiVIC and RTMaps platforms brings to RTMaps the ability to replace real raw data coming from embedded sensors by simulated ones. This platforms coupling provides an efficient and a solid framework for the prototyping and the evaluation of control/command and perception algorithms dedicated to autonomous driving and SIL applications. In this multiple platform, the raw data coming from virtual sensors, vehicles, and ground truth are sent to RTMaps with, for each one, a time stamp. These data are used as inputs in the applications developed in a specific package involving a perception/control/command module. Then the outputs of the control/command module provide the orders which are sent back to the interface module which allows, in pro-SiVIC, to control the longitudinal and lateral maneuvers of the virtual ego-vehicle. In our case, we do not use the powertrain and steering wheel models. This implies that the orders act directly on the dynamics vehicle model (front wheel angle and wheel torques). This chain of design is very efficient because the algorithms developed in RTMaps can then be directly transferred either as a micro-software on real hardware devices, or in an embedded version of RTMaps in an actual prototype vehicle. In this way and with the high level of vehicle and sensors modeling, the design process can be considered very close to reality (real vehicles, real sensors). The different modes, flow of data, types of data, type of peripherals handled by this interconnection mechanism are shown in Fig. \[SiVIC\_Interconnections\].  Several mechanisms have been implemented and tested. The best solution is clearly the optimized FIFO (First In, First Out) method which allows the transfer of a great number of data in a short time. It is a very critical functionality in order to guarantee a real-time link between pro-SiVIC and the perception/data processing/control algorithms. In order to correctly manage time, a synchronization module is available. This synchronization allows providing a time reference from pro-SiVIC to RTMaps. Then RTMaps is fully synchronized with pro-SiVIC components (vehicle, pedestrian and sensors). The pro-SiVIC/RTMaps simulation platform also enables to build reference scenarios and allows evaluating and testing of control/command and perception algorithms. In fact, the pro-SiVIC/RTMaps platform constitutes a full simulation environment because it provides the same types of interactivity found on actual vehicles: steering wheel angle, acceleration/braking torques, etc. From the different modules and functionalities available in both pro-SiVIC and RTMaps platforms, and following the diagram block of Fig. \[diag\_valid\_control\_NB\_V1\], we have implemented a complete operational architecture in order to test and to evaluate the model-free controller with a real-time generated reference. In this architecture, shown in Fig. \[Inter\_Sivic\_RTMAPS\] and Fig. \[Inter\_Sivic\], pro-SiVIC provides the virtual environment, the complex vehicle dynamics modeling, and the sensors simulation. Then data coming from a vehicle observer sensor are sent towards RTMaps platform by using a dedicated interface library. This “observer” sensor, which is a state vector with 40 parameters, includes all informations concerning the ego-vehicle state and dynamics (positioning, angles, speeds, angle speeds, accelerations, wheel speeds, ...). In RTMaps, we take into account these vehicle observer data in order to calculate the longitudinal and lateral control outputs. Then, these outputs are sent back to pro-SiVIC’s vehicle and more specifically to the virtual actuators. In addition, an events mechanism has been implemented in order to apply some dynamic constraints in the simulation (obstacle appearance, vehicle parameters modification, ...). In order to provide speed limit constraints in different areas, infrastructure sensors like road side beacons have been put on the virtual environment. The real-time software in the loop implementation with some data viewers is presented in Fig. \[Inter\_Sivic\_RTMAPS\].   ### Simulation results with the interconnected platforms pro-SiVIC and RTMaps In order to test the model-free control in realistic scenario, the Satory’s test tracks have been used. The modeling of these tracks in the pro-SiVIC Platform was made from surveyor’s data with a centimeter accuracy including road coordinates and road geometry. In addition, the entire virtual infrastructure corresponds to the real infrastructure present on the real tracks. These tracks are shared into three different areas. The first is the “routiere” track with a range of 3.4 km corresponding to a national road with very strong curvatures. The second track is a speedway (2 km) similar to a highway scene. The third track is the "\`val d’or" area corresponding to a rural area. In our tests, we only used the first track. The level of reality between the track modeling in the simulation platform and the actual track can be seen in Fig. \[Satory\_test\_track\]. Moreover, the inputs/outputs used in this second test are the same as those used in the first one (see Table \[Inputs\_outputs\]).  Using this track, the Figs. \[X\_Y\_Sivic\], \[Vx\_SiVIC\], \[Tw\_Delta\_SiVIC\] and \[EVx\_Ey\_SiVIC\] give the simulation results obtained by implementing the control law under the interconnected pro-SiVIC/RTMaps platforms. These results confirm the efficiency of the proposed control law even under a complex and a full simulation environment very close to real conditions. The tracking performances, in terms of longitudinal speed and lateral deviation tracking errors, are also depicted in Fig. \[EVx\_Ey\_SiVIC\]. With this scenario included very strong curvatures, we can observe a little degradation of the lateral deviation accuracy. Nevertheless the results stay in an acceptable domain allowing to control position of the vehicle in the current lane. About the speed profile, the vehicle follows closely the reference with an absolute error lower than 0.2  km/h. ![Reference trajectory versus the simulated closed-loop trajectory[]{data-label="X_Y_Sivic"}](X_Y_Sivic_V1.pdf) ![Longitudinal speed: actual and simulated[]{data-label="Vx_SiVIC"}](Vx_SiVIC_V1.pdf) ![Control inputs: Wheel torques and steering angles control signals[]{data-label="Tw_Delta_SiVIC"}](Tw_Delta_SiVIC_V1.pdf) ![Tracking errors on longitudinal speed and lateral deviation[]{data-label="EVx_Ey_SiVIC"}](EVx_Ey_SiVIC_V1.pdf) Moreover, Figs. \[Vx\_mu1\_05\_03\], \[Cw\_delat\_mu\_1\_05\_03\] and \[EVx\_Ey\_mu1\_05\_03\] show three simulation tests with three values of the road friction coefficient. We can observe that these tests highlight the performance of the model-free control approch under low road friction coefficient and high longitudinal speed. ![Longitudinal speed: actual and simulated for $\mu = 1$, $\mu = 0.5$ and $\mu = 0.3$[]{data-label="Vx_mu1_05_03"}](Vx_mu1_05_03.pdf) ![Control inputs: Wheel torques and steering angles control signals for $\mu = 1$, $\mu = 0.5$ and $\mu = 0.3$[]{data-label="Cw_delat_mu_1_05_03"}](Cw_delat_mu_1_05_03.pdf) ![Tracking errors on longitudinal speed and lateral deviation for $\mu = 1$, $\mu = 0.5$ and $\mu = 0.3$[]{data-label="EVx_Ey_mu1_05_03"}](EVx_Ey_mu1_05_03.pdf) Conclusion {#Section_5} ========== In recent years, many researches and many studies are carried out to allow the development of co-pilot functions for autonomous driving. In order to converge to a reliable, robust and functional solution in driving limit conditions, it is essential to be able to find answers to many research problems affecting both the perception of the environment, interpretation of road scenes, the safe path planning, the decision making and finally the control/command of the vehicle. In our work, we mainly address the final stage to produce control input for the vehicle control. Generally, in this stage, solutions require knowledge of vehicle dynamics and a accurate estimation of both the parameters and the variables of these dynamics models (often complex). Moreover, in real conditions, obtaining knowledge on these complex dynamic models is, in general, difficult and needs to implement expensive and complex means (sensors, hardware architecture). It is the same problem for the ground truth generation. Nevertheless, in case of poor estimation of the attributes of these dynamic models, the control algorithms could produce orders that will lead the system to diverge and by extension to dangerous maneuvers for the vehicle. In this paper, we propose a new approach of vehicle control using no vehicle evolution model. To demonstrate and to prove the relevance of such an approach, we have compared it with the results of several other more traditional approaches such as a PID controller and a flatness based controller. In order to test, to evaluate, and to validate this new approach, two simulation stages with different levels of dynamics vehicle modeling are used. The first level uses Matlab with a dynamic model of a Peugeot 406 (with 10 DoF). In this first stage of evaluation, the reference is provided by a trajectory obtained from a real Peugeot 406. The results clearly show a significant gain by using the model-free setting comparatively to the other. The accuracy obtained from the model-free controller is, in this first case of simulation, good enough in order to use it in an autonomous driving application. Another conclusion can also be inferred from the first results. Indeed, the use of the model-free controller also shows that the question of vehicle control can be effectively addressed and processed without need to a complex dynamics vehicle modeling. Only simple methods and functions are needed in order to solve this problem. Secondly, to validate this approach in a simulation environment very close to the prototyping conditions and real condition use, we have implemented the model-free approach in a complex platform made up of pro-SiVIC (simulation of vehicles, infrastructure and sensors) and RTMaps (management and processing of data flows. Also used in the real vehicle prototypes of LIVIC and CAOR laboratories). The interconnection of these two platforms has enabled to integrate the model-free controller in a SIL architecture. In addition to generating information from embedded sensors, pro-SiVIC has made it possible to generate a very accurate ground truth. Again, the results were very successful and have validated the relevance of this new model-free controller approach even in extreme driving conditions with very low curvature radii. For this second level of simulation, the vehicle was evolving on a very realistic modeling of Satory’s test tracks. Currently, the tests were carried out using sensor data without noise or malfunctions. In future works, we will discuss the impact of noise and failures of the embedded sensors on the robustness of this model-free approach (see [@ijc13]). With these new studies, we hope that we will prove that the model-free controller can remain sufficiently robust and sensors failure tolerant to ensure safe control stage enough longer in time to give back the vehicle control to the driver. This step is crucial to enable the deployment of safe driving automation applications. Flatness-based and PID controllers {#appendix1} ================================== A flatness-based controller --------------------------- After some straightforward computations, model can be rewritten in the following standard form (see [@Menhour13a] for details): $$\label{affine_NL_modele} \dot{x}=f(x,t)+g(x,t)u$$ Moreover the flatness property [@Fliess95; @levine; @hsr] of system is established according to the following proposition. The following outputs: $$\label{flatness_outputs} \left \{\begin{array}{l} z_1=V_x \\ [1mm] z_2= L_f m V_y -I_z \dot{\psi} \end{array} \right.$$ are flat outputs for system . Some algebraic manipulations (see [@Menhour13a] for more details) yield: $$\label{x_A_y1_y2_y2p} \begin{array}{c} x= \left[ \begin{array}{ccc} V_x & V_y & \dot{\psi} \end{array} \right]^T \begin{array}{c} =\end{array}\\ \left[ \begin{array}{c} z_1\\ [2mm] \frac{z_2}{L_f m} - \frac{I_z }{L_f m} \left(\frac{L_f m z_1\dot{z}_2 + C_r(L_f+L_r)z_2 }{C_r(L_f+L_r)(I_z -L_rL_fm)+ (L_f mz_1)^2 }\right) \\ [2mm] %%%% -\left(\frac{L_f m y_1\dot{z}_2 + C_r(L_f+L_r)z_2 }{C_r(L_f+L_r)(I_z -L_rL_fm)+ (L_f mz_1)^2 }\right) %%%%%%% \end{array} \right] \end{array}$$ and $$\label{U_B_y1_y2_y2p} \begin{array}{c} \left[ \begin{array}{c} \dot{z}_1\\ \ddot{z}_2 \end{array} \right] = \Delta(z_1,z_2,\dot{z}_2) \left( \begin{array}{c} u_1\\ u_2 \end{array} \right)+\Phi(z_1,z_2,\dot{z}_2) %\\[4mm] \end{array}$$ The terms $\Delta_{11}$, $\Delta_{12}$, $\Delta_{21}$, $\Delta_{22}$, $\Phi_1$ and $\Phi_2$ of the matrices $\Delta$ and $\Phi$ are in detailed in [@Menhour13a]. Thus $$\label{u_B_y1_y1p_y2_y2p_y2pp} \begin{array}{c} u=\left[ \begin{array}{c} T_\omega\\ \delta \end{array} \right]= \Delta^{-1}(z_1,z_2,\dot{z}_2) \left( \left[ \begin{array}{c} \dot{z}_1\\ \ddot{z}_2 \end{array} \right] - \Phi(z_1,z_2,\dot{z}_2) \right) \end{array}$$ with $r_x=1$ and $r_u=2$. Consequently, the system is flat with flat outputs . Then, in order to track the desired output $z_1^{ref}$ and $z_2^{ref}$, set $$\label{lin_contro} \left [ \begin{array}{c} \dot{z}_1 \\ \ddot{z}_2 \end{array} \right]= \left [ \begin{array}{c} \dot{z}_1^{ref}+K_1^1e_{z_1}+K_1^2\int{e_{z_1} dt}\\[2mm] \ddot{z}_2^{ref}+K_2^1\dot{e}_{z_2}+K_2^2e_{z_2}+K_2^3\int{e_{z_2} dt} \end{array} \right]$$ where, $e_{z_1}=z_1^{ref}-z_1=V_x^{ref}-V_x$ and $e_{z_2}=z_2^{ref}-z_2$. Choosing the gains $K_1^1$, $K_1^2$, $K_2^1$, $K_2^2$ and $K_2^3$ is straightforward. The control law contains derivatives of measured signals, which are of course noisy. This estimation is performed using the recent advances algebraic estimation techniques in [@Fliess08; @Mboup09; @sira1; @sira2]. \[Remark\_0\] Let us notice that the second flat output $z_2$ is the angular momentum of a point located on the axis between the centers of the rear and front axles. PID controllers {#Remark_1} --------------- It seems interesting, in order to test the validity and the limits of the proposed control methods, to compare them with the classical well known PID controllers. Numerous methods to design PID controllers exist in the literature, see e.g. [@Astrom88; @Ho98; @Saeki06; @Wang07]. For our case study, the gains $(K_p^{V_x},\, K_d^{V_x},\,K_i^{V_x})$ (gains of longitudinal PID$_{V_x}$ control) and $(K_p^{y},\, K_d^{y},\,K_i^{y})$ (gains of lateral deviation PID$_{y}$ control) are computed in vicinity of $V= 20 m/s$. Then, using the oscillation method [@Astrom88], the chosen numerical values of the gains are $(K_p^{V_x} = 1.51,\, K_d^{V_x} = 0.52,\,K_i^{V_x} = 0.75)$ for PID$_{V_x}$ and $(K_p^{y} = 0.95,\, K_d^{y}=0.36,\,K_i^{y} = 46)$ for PID$_{y}$. Moreover, in order to avoid the impact of the noisy measurements on the longitudinal speed and the lateral deviation, a first order filter is associated to the derivative actions of these controllers (see [@Astrom88] for more details). The following PID controllers are used: $$\label{PID_controllers} \left \{ \begin{array}{lcl} u_1(t) = K_p^{V_x}e_{V_x}+ K_d^{V_x}\dot{e}_{V_x} + K_i^{V_x} \int e_{V_x} dt\\ u_2(t) =K_p^{y}e_{y}+ K_d^{y}\dot{e}_{y} + K_i^{y} \int e_{y} dt \end{array} \right.$$ [00]{} H. Abouaïssa, M. Fliess, C. Join, “On ramp metering: towards a better understanding of ALINEA via model-free control”, [*Int. J. Control*]{}, Vol. 90(5), 2017, pp. 1018-1026. J. Ackermann, J. Guldner, W. Sienel, R. Steinhauser, V. I. 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Wang, “An overview of recent developments in automated lateral and longitudinal vehicle controls”, in [*IEEE Int. Conf. Syst. Man Cyber.*]{}, Tucson, 2001. D. Odenthal, T. Bunte, J.Ackerman, “Nonlinear steering and breaking control for vehicle rollover avoidance”, in [*Europ. Control Conf.*]{}, Karlsruhe, 1999. D. Gruyer, S. Demmel, B. d’Andrea-Novel, G. Larue, A. Rakotonirainy, “Simulating Cooperative Systems Applications: a New Complete Architecture”, in [*International Journal of Advanced Computer Science and Applications*]{} (IJACSA), Volume 4, 2013. C. Poussot-Vassal, O. Sename, L. Dugard, P. Gaspar, Z. Szaboc, J. Bokor, “Attitude and handling improvements through gain-scheduled suspensions and brakes control”, [*Control Engin. Practice*]{}, vol. 19, 252-263, 2011. H. Sira-Ramírez, S. Agrawal, “Differentially Flat Systems”, [*Marcel Dekker*]{}, 2004. B. Steux, “RTMaps, un environnement logiciel dédié à la conception d’applications embarquées temps-réel. Utilisation pour la détection automatique de véhicules par fusion radar/vision”, [*Thèse*]{}, Ecole des Mines de Paris, 2001. G. Tagne and R. Talj and A. Charara, “Design and comparison of Robust Nonlinear Controllers for the Lateral Dynamics of Intelligent Vehicles”, [*IEEE Transactions on Intelligent Transportation Systems*]{}, vol. 17(3), 796-809, 2016. B. Vanholme, D. Gruyer, S. Glaser, S. Mammar, “Fast prototyping of highly autonomous cooperative driving system for public roads”, [*in proc. IV2010*]{}, SanDiego, June 2010. J. Villagra, B. d’Andréa-Novel, S. Choi, M. Fliess, H. Mounier, “Robust stop-and-go control strategy: an algebraic approach for nonlinear estimation and control”, [*Int. J. Vehicle Autonomous Systems*]{}, 7, 270-291, 2009. Preprint available at J. Villagra, D. Herrero-Pérez, “A comparison of control techniques for robust docking maneuvers for an AVG”, *IEEE Trans. Control Systems Techno.*, 20, 1116-1123, 2012. Wei Yue, Ge Guo, Wang liyuan, Wei Wang, “Nonlinear Vehicle Platoon Allowing Range-limited Sensors: Application to Arduino Cars”, in **International Journal of Control**, 1037-1050, 8(5), 2015. K. Yosida, *Operational Calculus* (translated from the Japanese), Springer, 1984. [Lghani MENHOUR]{} obtained his Ph.D. in automatic control in 2010 from the Laboratoire Heuristique et diagnostic des systèmes complexes of the Université de Technologie de Compiègne, Compiègne, France. Since 2011, he is an Assistant Professor at the Laboratoire CReSTIC of the Université de Reims-Champagne-Ardenne, Reims, France. His research interests include linear and nonlinear systems, control and estimation, switched systems, Model-free control, algebraic approaches, intelligent transportation systems. [Brigitte d’ANDRÉA-NOVEL]{} graduated from École Supérieure d’Informatique, Électronique, Automatique in 1984. She received the Ph. D. degree from École Nationale Supérieure des Mines de Paris in 1987 and the Habilitation degree from Université Paris-Sud in 1995. She is currently a Professor of Systems Control Theory and responsible for the research group in Advanced Control Systems at the Center for Robotics from MINES ParisTech. Her current research interests include nonlinear control theory and applications to underactuated mechanical systems, control of wheeled vehicles with applications to automated highways. In that context she has developed regular collaborations with PSA-Peugeot-Citroën and Valeo on “Global Chassis Control”, “Stop and Go algorithms” and “Automatic parking”. Moreover, she has also been interested in modeling and boundary control of dynamical systems coupling ODEs and PDEs, with applications to the control of irrigation canals and wind musical instruments. [Michel FLIESS]{} works at the École polytechnique (Palaiseau, France). In 1991 he invented with J. Lévine, P. Martin, and P. Rouchon (Ecole des Mines de Paris, France), the notion of differentially flat systems which is playing a crucial role in many concrete situations all over the world. In 2002 he introduced with H. Sira-Ramírez (CINVESTAV, Mexico City) algebraic estimation and identification techniques, which are most useful in control and signal processing. Ten years ago he started with C. Join (Université de Lorraine, Nancy, France) a model-free control setting which is now being used in a number of concrete industrial applications. He got several prizes, including three ones from the French Academy of Sciences. [Dominique GRUYER]{} was born in France, in 1969. He received the M.S. and Ph.D. degree respectively in 1995 and 1999 from the University of Technology of Compiègne. Since 2001, he is a researcher at INRETS, into the perception team of the LIVIC department (Laboratory on interactions between vehicles, Infrastructure and drivers) and he works on the study and the development of multi-sensors/sources association, combination and fusion. His works enter into the conception of on-board driving assistance systems and more precisely on the carry out of multi-obstacle detection and tracking, extended perception, accurate ego-localization. He is involved for multi-sensor fusion tasks and sensors modeling and simulation in several European and French projects dealing with intelligent vehicles (HAVEit, Isi-PADAS, CVIS, CARSENSE, eMOTIVE, LOVe, ARCOS, MICADO, ABV, eFuture, CooPerCom, SINETIC, ...). He is responsible and the main inventor of the SiVIC platform (Simulation for Vehicle, Infrastructure and sensors). Since 2010, He is the team leader of the LIVIC’s Perception team. For 5 years, he is a network researcher in AUTO21 (Canada). He was the co-founder and technical Director of the CIVITEC Company until 2015. Since April 2015, CIVITEC is a subsidiary of the ESI group company. He is now the head of the LIVIC laboratory (since January 2015), a Research Director (since 2014) in IFSTTAR, and Perception system and data fusion Scientific Director (since April 2015) for ESI group (CIVITEC). [Hugues MOUNIER]{} obtained his Ph.D. in automatic control in 1995 from the Laboratoire des Signaux et Systèmes of the Université Paris Sud, Orsay, France. From 1998 to 2010 he has been with the Institut d’Electronique Fondamentale of the same University. He is currently Professor at the Laboratoire des Signaux et Systèmes. His research interests include automotive and real-time control, neuroscience, delay systems and systems modelled by partial differential equations [^1]: L. Menhour is with Université de Reims, 9, Rue du Québec, 10000 Troyes, France [lghani.menhour@univ-reims.fr]{} [^2]: B. d’Andréa-Novel is with Centre de Robotique, Mines ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272 Paris cedex 06, France. [brigitte.dandrea-novel@mines-paristech.fr]{} [^3]: M. Fliess is with LIX (CNRS, UMR 7161), École polytechnique, 91128 Palaiseau, France. [Michel.Fliess@polytechnique.edu]{} and AL.I.E.N. (ALgèbre pour Identification et Estimation Numériques), 24-30 rue Lionnois, BP 60120, 54003 Nancy, France. [michel.fliess@alien-sas.com]{} [^4]: D. Gruyer is with IFSTTAR-CoSys-LIVIC, 77 rue des Chantiers, 78000 Versailles, France. [dominique.gruyer@ifsttar.fr]{} [^5]: H. Mounier is with L2S (UMR 8506), CNRS – Supélec – Université Paris-Sud, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France. [hugues.mounier@lss.supelec.fr]{} [^6]: pro-SiVIC is a professional software of CIVITEC ([http://www.civitec.com]{}). [^7]: RT-Maps is developed by Intempora ([http://www.intempora.com]{}). [^8]: See [@Menhour13a] and Section \[Section\_1\] for more details on 10DoF nonlinear vehicle model [^9]: It is now marketed by the company Intempora.

= 6.6truein = 9truein = 0.9 in = -1truein = -.75truein plus 0.2pt minus 0.1pt = 44by .5cm .5cm plus 1pt UH-IfA-94/35\ SU-ITP-94-13\ YITP/U-94-15\ hep-th/9405187\ 2 truecm [**REHEATING AFTER INFLATION**]{} 1.6cm [**Lev Kofman**]{}\ .1cm [Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr., Honolulu, HI 96822, USA]{}[^1] .2cm [**Andrei Linde**]{}\ .1cm [Department of Physics, Stanford University, Stanford, CA 94305, USA]{}[^2]\ .3cm [and]{}\ .3cm [** Alexei A. Starobinsky**]{}\ .1cm [Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan\ .1cm and Landau Institute for Theoretical Physics, Kosygina St. 2, Moscow 117334, Russia\ ]{} 1.5cm > **ABSTRACT** > > .2cm > > The theory of reheating of the Universe after inflation is developed. We have found that typically at the first stage of reheating the classical inflaton field $\phi$ rapidly decays into $\phi$-particles or into other bosons due to a broad parametric resonance. Then these bosons decay into other particles, which eventually become thermalized. Complete reheating is possible only in those theories where a single particle $\phi$ can decay into other particles. This imposes strong constraints on the structure of inflationary models, and implies that the inflaton field can be a dark matter candidate. > > 1.3cm > > PACS numbers:  98.80.Cq,  04.62.+v 1\. The theory of reheating of the Universe after inflation is the most important application of the quantum theory of particle creation, since almost all matter constituting the Universe at the subsequent radiation-dominated stage was created during this process [@MyBook]. At the stage of inflation all energy was concentrated in a classical slowly moving inflaton field $\phi$. Soon after the end of inflation this field began to oscillate near the minimum of its effective potential. Gradually it produced many elementary particles, they interacted with each other and came to a state of thermal equilibrium with some temperature $T_r$, which was called the reheating temperature. An elementary theory of reheating was first developed in [@1] for the new inflationary scenario. Independently a theory of reheating in the $R^2$ inflation was constructed in [@st81]. Various aspects of this theory were further elaborated by many authors, see e.g. [@Dolg]. Still, a general scenario of reheating was absent. In particular, reheating in the chaotic inflation theory remained almost unexplored. The present paper is a short account of our investigation of this question [@REH]. We have found that the process of reheating typically consists of three different stages. At the first stage, which cannot be described by the elementary theory of reheating, the classical coherently oscillating inflaton field $\phi$ decays into massive bosons (in particular, into $\phi$-particles) due to parametric resonance. In many models the resonance is very broad, and the process occurs extremely rapidly (explosively). Because of the Pauli exclusion principle, there is no explosive creation of fermions. To distinguish this stage from the stage of particle decay and thermalization, we will call it [*pre-heating*]{}. Bosons produced at that stage are far away from thermal equilibrium and typically have enormously large occupation numbers. The second stage is the decay of previously produced particles. This stage typically can be described by methods developed in [@1]. However, these methods should be applied not to the decay of the original homogeneous inflaton field, but to the decay of particles and fields produced at the stage of explosive reheating. This considerably changes many features of the process, including the final value of the reheating temperature. The third stage is the stage of thermalization, which can be described by standard methods, see e.g. [@MyBook; @1]; we will not consider it here. Sometimes this stage may occur simultaneously with the second one. In our investigation we have used the formalism of the time-dependent Bogoliubov transformations to find the density of created particles, $n_{\vec k}(t)$. A detailed description of this theory will be given in [@REH]; here we will outline our main conclusions using a simple semiclassical approach. 2\. We will consider a simple chaotic inflation scenario describing the classical inflaton scalar field $\phi$ with the effective potential $V(\phi) = \pm {1\over2} m_\phi^2 \phi^2+{\lambda\over 4}\phi^4$. Minus sign corresponds to spontaneous symmetry breaking $\phi \to \phi +\sigma$ with generation of a classical scalar field $\sigma = {m_\phi \over\sqrt\lambda}$. The field $\phi$ after inflation may decay into bosons $\chi$ and fermions $\psi$ due to the interaction terms $- { 1\over2} g^2 \phi^2 \chi^2$ and $- h \bar \psi \psi \phi$. Here $\lambda$, $ g$ and $h$ are small coupling constants. In case of spontaneous symmetry breaking, the term $- { 1\over2} g^2 \phi^2 \chi^2$ gives rise to the term $- g^2 \sigma\phi \chi^2$. We will assume for simplicity that the bare masses of the fields $\chi$ and $\psi$ are very small, so that one can write $ m_\chi (\phi) = g \phi$, $m_{\psi}(\phi) = |h\phi|$. Let us briefly recall the elementary theory of reheating [@MyBook]. At $\phi > M_p$, we have a stage of inflation. This stage is supported by the friction-like term $3H\dot\phi$ in the equation of motion for the scalar field. Here $H\equiv \dot a/a$ is the Hubble parameter, $a(t)$ is the scale factor of the Universe. However, with a decrease of the field $\phi$ this term becomes less and less important, and inflation ends at $\phi {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }M_p/2$. After that the field $\phi$ begins oscillating near the minimum of $V(\phi)$ [@f2]. The amplitude of the oscillations gradually decreases because of expansion of the universe, and also because of the energy transfer to particles created by the oscillating field. Elementary theory of reheating is based on the assumption that the classical oscillating scalar field $\phi (t)$ can be represented as a collection of scalar particles at rest. Then the rate of decrease of the energy of oscillations coincides with the decay rate of $\phi$-particles. The rates of the processes $\phi \to \chi\chi$ and $\phi \to \psi\psi$ (for $m_\phi \gg 2m_\chi, 2m_\psi$) are given by $$\label{7} \Gamma ( \phi \to \chi \chi) = { g^4 \sigma^2\over 8 \pi m_{\phi}}\ , \ \ \ \ \ \Gamma( \phi \to \psi \psi ) = { h^2 m_{\phi}\over 8 \pi}\ .$$ Reheating completes when the rate of expansion of the universe given by the Hubble constant $H=\sqrt{8\pi \rho\over 3 M^2_p} \sim t^{-1}$ becomes smaller than the total decay rate $\Gamma = \Gamma (\phi \to \chi \chi) + \Gamma (\phi \to \psi \psi )$. The reheating temperature can be estimated by $T_r \simeq 0.1\, \sqrt{\Gamma M_p}$. As we already mentioned, this theory can provide a qualitatively correct description of particle decay at the last stages of reheating. Moreover, this theory is always applicable if the inflaton field can decay into fermions only, with a small coupling constant $h^2 \ll m_{\phi}/M_p$. However, typically this theory is inapplicable to the description of the first stages of reheating, which makes the whole process quite different. In what follows we will develop the theory of the first stages of reheating. We will begin with the theory of a massive scalar field $\phi$ decaying into particles $\chi$, then we consider the theory ${\lambda\over 4} \phi^4$, and finally we will discuss reheating in the theories with spontaneous symmetry breaking. 3\. We begin with the investigation of the simplest inflationary model with the effective potential ${m^2_\phi\over 2}\phi^2$. Suppose that this field only interacts with a light scalar field $\chi$ ($m_{\chi} \ll m_{\phi}$) due to the term $-{ 1\over2} g^2 \phi^2 \chi^2$. The equation for quantum fluctuations of the field $\chi$ with the physical momentum $\vec k/a(t)$ has the following form: $$\label{M} \ddot \chi_k + 3H \dot \chi_k + \left({k^2\over a^2(t)} + g^2 \Phi^2\, \sin^2(m_{\phi}t) \right) \chi_k = 0 \ ,$$ where $k = \sqrt {\vec k^2}$, and $\Phi$ stands for the amplitude of oscillations of the field $\phi$. As we shall see, the main contribution to $\chi$-particle production is given by excitations of the field $\chi$ with $k/a \gg m_\phi$, which is much greater than $H$ at the stage of oscillations. Therefore, in the first approximation we may neglect the expansion of the Universe, taking $a(t)$ as a constant and omitting the term $3H \dot \chi_k$ in (\[M\]). Then the equation (\[M\]) describes an oscillator with a variable frequency $\Omega_k^2(t)= k^2a^{-2} + g^2\Phi^2\, \sin^2(m_{\phi}t) $. Particle production occurs due to a nonadiabatic change of this frequency. Equation (\[M\]) can be reduced to the well-known Mathieu equation: $$\label{M1} \chi_k'' + \left(A(k) - 2q \cos 2z \right) \chi_k = 0 \ ,$$ where $A(k) = {k^2 \over m_\phi^2 a^2}+2q$, $q = {g^2\Phi^2\over 4m_\phi^2} $, $z = m_{\phi}t$, prime denotes differentiation with respect to $z$. An important property of solutions of the equation (\[M1\]) is the existence of an exponential instability $\chi_k \propto \exp (\mu_k^{(n)}z)$ within the set of resonance bands of frequencies $\Delta k^{(n)}$ labeled by an integer index $n$. This instability corresponds to exponential growth of occupation numbers of quantum fluctuations $n_{\vec k}(t) \propto \exp (2\mu_k^{(n)} m_{\phi} t)$ that may be interpreted as particle production. The simplest way to analyse this effect is to study the stability/instability chart of the Mathieu equation, which is sketched in Fig. 1. White bands on this chart correspond to the regions of instability, the grey bands correspond to regions of stability. The curved lines inside white bands show the values of the instability parameter $\mu_k$. The line $A = 2q$ shows the values of $A$ and $q$ for $k = 0$. All points in the white regions above this line correspond to instability for any given $q$. As one can see, near the line $A = 2q$ there are regions in the first, the second and the higher instability bands where the unstable modes grow extremely rapidly, with $\mu_k \sim 0.2$. We will show analytically in [@REH] that for $q \gg 1$ typically $\mu_k \sim {\ln 3\over 2\pi} \approx 0.175$ in the instability bands along the line $A = 2q$, but its maximal value is ${\ln(1+\sqrt{2}) \over \pi} \approx 0.28$. Creation of particles in the regime of a broad resonance ($q > 1$) with $2\pi \mu_k = O(1)$ is very different from that in the usually considered case of a narrow resonance ($ q \ll 1$), where $2\pi \mu_k \ll 1$. In particular, it proceeds during a tiny part of each oscillation of the field $\phi$ when $1-\cos z \sim q^{-1}$ and the induced effective mass of the field $\chi$ (which is determined by the condition $m^2_{\chi}= g^2\Phi^2/2$) is less than $m_{\phi}$. As a result, the number of particles grows exponentially within just a few oscillations of the field $\phi$. This leads to an extremely rapid (explosive) decay of the classical scalar field $\phi$. This regime occurs only if $q {\ \lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } \pi^{-1}$, i.e. for $g\Phi {\ \lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } m_\phi$, so that $m_\phi \ll gM_p$ is the necessary condition for it. One can show that a typical energy $E$ of a particle produced at this stage is determined by equation $A-2q \sim \sqrt{q}$, and is given by $E \sim \sqrt{g m_\phi M_p}$ [@REH]. Creation of $\chi$-particles leads to the two main effects: transfer of the energy from the homogeneous field $\phi (t)$ to these particles and generation of the contribution to the effective mass of the $\phi$ field: $m^2_{\phi ,eff}=m^2_{\phi}+g^2\langle\chi^2 \rangle_{ren}$. The last term in the latter expression quickly becomes larger than $m^2_{\phi}$. One should take both these effects into account when calculating backreaction of created particles on the process. As a result, the stage of the broad resonance creation ends up within the short time $t\sim m_{\phi}^{-1} \ln (m_{\phi}/g^5M_p)$, when $\Phi^2 \sim \langle\chi^2\rangle$ and $q = {g^2\Phi^2\over 4m_{\phi ,eff}^2}$ becomes smaller than $1$. At this time the energy density of produced particles $\sim E^2 \langle\chi^2\rangle \sim g m_\phi M_p \Phi^2$ is of the same order as the original energy density $\sim {m_\phi^2} M_p^2$ of the scalar field $\phi$ at the end of inflation. This gives the amplitude of oscillations at the end of the stage of the broad resonance particle creation: $\Phi^2 \sim \langle\chi^2\rangle \sim g^{-1} m_\phi M_p \ll M_p^2$. Since $E\gg m_{\phi}$, the effective equation of state of the whole system becomes $p\approx \varepsilon /3$. Thus, explosive creation practically eliminates a prolonged intermediate matter-dominated stage after the end of inflation which was thought to be characteristic to many inflationary models. However, this does not mean that the process of reheating has been completed. Instead of $\chi$-particles in the thermal equilibrium with a typical energy $E \sim T \sim (mM_p)^{1/2}$, one has particles with a much smaller energy $\sim (g m_\phi M_p)^{1/2}$, but with extremely large mean occupation numbers $n_k \sim g^{-2} \gg 1$. After that the Universe expands as $a(t)\propto \sqrt t$, and the scalar field $\phi$ continues its decay in the regime of the narrow resonance creation $q\approx {\Phi^2\over 4 \langle\chi^2\rangle} \ll 1$. As a result, $\phi$ decreases rather slowly, $\phi \propto t^{-3/4}$. This regime is very important because it makes the energy of the $\phi$ field much smaller than that of the $\chi$-particles. One can show that the decay finally stops when the amplitude of oscillations $\Phi$ becomes smaller than $g^{-1} m_\phi$ [@REH]. This happens at the moment $t\sim m_{\phi}^{-1} (gM_p/m_{\phi})^{1/3}$ (in the case $m < g^7 M_p$ decay ends somewhat later, in the perturbative regime). The physical reason why the decay stops is rather general: decay of the particles $\phi$ in our model occurs due to its interaction with another $\phi$-particle (interaction term is quadratic in $\phi$ and in $\chi$). When the field $\phi$ (or the number of $\phi$-particles) becomes small, this process is inefficient. The scalar field can decay completely only if a single scalar $\phi$-particle can decay into other particles, due to the processes $\phi \to \chi \chi$ or $\phi \to \psi \psi$, see eq. (\[7\]). If there is no spontaneous symmetry breaking and no interactions with fermions in our model, such processes are impossible. At later stages the energy of oscillations of the inflaton field decreases as $a^{-3}(t)$, i.e. more slowly than the decrease of energy of hot ultrarelativistic matter $\propto a^{-4}(t)$. Therefore, the relative contribution of the field $\phi(t)$ to the total energy density of the Universe rapidly grows. This gives rise to an unexpected possibility that the inflaton field by itself, or other scalar fields can be cold dark matter candidates, [*even if they strongly interact with each other*]{}. However, this possibility requires a certain degree of fine tuning; a more immediate application of our result is that it allows one to rule out a wide class of inflationary models which do not contain interaction terms of the type of $g^2\sigma\phi\chi^2$ or $h\phi\bar\psi\psi$. 4\. So far we have not considered the term ${\lambda \over 4} \phi^4$ in the effective potential. Meanwhile this term leads to production of $\phi$-particles, which in some cases appears to be the leading effect. Let us study the $\phi$-particle production in the theory with $V(\phi) = {m^2_{\phi}\over 2} \phi^2 + {\lambda\over 4}\phi^4$ with $m^2_{\phi} \ll \lambda M_p^2$. In this case the effective potential of the field $\phi$ soon after the end of inflation at $\phi \sim M_p$ is dominated by the term ${\lambda\over 4} \phi^4$. Oscillations of the field $\phi$ in this theory are not sinusoidal, they are given by elliptic functions, but with a good accuracy one can write $\phi(t) \sim \Phi \sin (c\sqrt \lambda \int \Phi dt)$, where $c={\Gamma^2(3/4)\over \sqrt \pi} \approx 0.85$. The Universe at that time expands as at the radiation-dominated stage: $a(t)\propto \sqrt t$. If one neglects the feedback of created $\phi$-particles on the homogeneous field $\phi (t)$, then its amplitude $\Phi (t) \propto a^{-1}(t)$, so that $a\Phi =const$. Using a conformal time $\eta$, exact equation for quantum fluctuations $\delta \phi$ of the field $\phi$ can be reduced to the Lame equation. The results remain essentially the same if we use an approximate equation $$\label{lam1} {d^2(\delta\phi_k)\over d\eta^2} + {\Bigl[{k^2} + 3\lambda a^2\Phi^2\, \sin^2 (c\sqrt\lambda a\Phi \eta)\Bigr]} \delta\phi_k = 0 \ ,~~~\eta =\int {dt\over a(t)}={2t\over a(t)}\, ,$$ which leads to the Mathieu equation with $A = {k^2\over c^2\lambda a^2\Phi^2} + {3\over 2c^2} \approx {k^2\over c^2\lambda a^2\Phi^2} + 2.08$, and $q = {3\over 4c^2} \approx 1.04$. Looking at the instability chart, we see that the resonance occurs in the second band, for $k^2 \sim 3\lambda a^2\Phi^2$. The maximal value of the coefficient $\mu_k$ in this band for $q \sim 1$ approximately equals to $0.07$. As long as the backreaction of created particles is small, expansion of the Universe does not shift fluctuations away from the resonance band, and the number of produced particles grows as $\exp (2c\mu_k\sqrt\lambda a\Phi \eta) \sim \exp ({\sqrt\lambda\Phi t\over 5})$. After the time interval $\sim M_p^{-1}\lambda^{-1/2}|\ln \lambda|$, backreaction of created particles becomes significant. The growth of the fluctuations $\langle\phi^2\rangle$ gives rise to a contribution $3\lambda \langle\phi^2\rangle$ to the effective mass squared of the field $\phi$, both in the equation for $\phi (t)$ and in Eq. (\[lam1\]) for inhomogeneous modes. The stage of explosive reheating ends when $\langle\phi^2\rangle$ becomes greater than $\Phi^2$. After that, $\Phi^2 \ll \langle\phi^2\rangle$ and the effective frequency of oscillations is determined by the term $\sqrt{3\lambda \langle\phi^2\rangle}$. The corresponding process is described by Eq. (4) with $A(k) = 1 + 2q + {k^2 \over 3\lambda a^2\langle\phi^2\rangle}$, $q = {\Phi^2\over 4 \langle\phi^2\rangle}$. In this regime $q \ll 1$, and particle creation occurs in the narrow resonance regime in the second band with $A \approx 4$. Decay of the field in this regime is extremely slow: the amplitude $\Phi$ decreases only by a factor $t^{1/12}$ faster that it would decrease without any decay, due to the expansion of the Universe only, i.e., $\Phi \propto t^{-7/12}$ [@REH]. Reheating stops altogether when the presence of non-zero mass $m_{\phi}$ though still small as compared to $\sqrt{3\lambda \langle\phi^2\rangle}$ appears enough for the expansion of the Universe to drive a mode away from the narrow resonance. It happens when the amplitude $\Phi$ drops up to a value $\sim m_{\phi}/\sqrt \lambda$. In addition to this process, the field $\phi$ may decay to $\chi$-particles. This is the leading process for $g^2\gg \lambda$. The equation for $\chi_k$ quanta has the same form as eq. (\[lam1\]) with the obvious change $\lambda \to g^2/3$. Initially parametric resonance is broad. The values of the parameter $\mu_k$ along the line $A = 2q$ do not change monotonically, but typically for $q \gg 1$ they are 3 to 4 times greater than the parameter $\mu_k$ for the decay of the field $\phi$ into its own quanta. Therefore, this pre-heating process is very efficient. It ends at the moment $t\sim M_p^{-1}\lambda^{-1/2} \ln (\lambda /g^{10})$ when $\Phi^2 \sim \langle \chi^2 \rangle \sim g^{-1}\sqrt \lambda M_p^2$. The typical energy of created $\chi$-particles is $E \sim (g^2\lambda)^{1/4}M_p$. The following evolution is essentially the same as that described in Sec. 3. 5\. Finally, let us consider the case with symmetry breaking. In the beginning, when the amplitude of oscillations is much greater than $\sigma$, the theory of decay of the inflaton field is the same as in the case considered above. The most important part of pre-heating occurs at this stage. When the amplitude of the oscillations becomes smaller than $m_\phi/\sqrt\lambda$ and the field begins oscillating near the minimum of the effective potential at $\phi = \sigma$, particle production due to the narrow parametric resonance typically becomes very weak. The main reason for this is related to the backreaction of particles created at the preceding stage of pre-heating on the rate of expansion of the universe and on the shape of the effective potential [@REH]. However, importance of spontaneous symmetry breaking for the theory of reheating should not be underestimated, since it gives rise to the interaction term $g^2\sigma\phi\chi^2$ which is linear in $\phi$. Such terms are necessary for a complete decay of the inflaton field in accordance with the perturbation theory (\[7\]). 6\. In this paper we discussed the process of reheating of the universe in various inflationary models. We have found that decay of the inflaton field typically begins with a stage of explosive production of particles at a stage of a broad parametric resonance. Later the resonance becomes narrow, and finally this stage of decay finishes altogether. Interactions of particles produced at this stage, their decay into other particles and subsequent thermalization typically require much more time that the stage of pre-heating, since these processes are suppressed by the small values of coupling constants. The corresponding processes in many cases can be described by the elementary theory of reheating. However, this theory should be applied not to the decay of the original large and homogeneous oscillating inflaton field, but to the decay of particles produced at the stage of pre-heating, as well as to the decay of small remnants of the classical inflaton field. This makes a lot of difference, since typically coupling constants of interaction of the inflaton field with matter are extremely small, whereas coupling constants involved in the decay of other bosons can be much greater. As a result, the reheating temperature can be much higher than the typical temperature $T_r {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } 10^9$ GeV which could be obtained neglecting the stage of parametric resonance [@REH]. On the other hand, such processes as baryon creation after inflation occur best of all outside the state of thermal equilibrium. Therefore, the stage of pre-heating may play an extremely important role in our cosmological scenario. Another consequence of the resonance effects is an almost instantaneous change of equation of state from the vacuum-like one to the equation of state of relativistic matter. This leads to suppression of the number of primordial black holes which could be produced after inflation. L.K. was supported in part by the Canadian Institute for Advance Research cosmology program, and CITA. A.L. was supported in part by NSF grant PHY-8612280. A.S. is grateful to Profs. Y. Nagaoka and J. Yokoyama for their hospitality at the Yukawa Institute for Theoretical Physics, Kyoto University. A.S. was supported in part by the Russian Foundation for Basic Research, Project Code 93-02-3631, and by Russian Research Project “Cosmomicrophysics”. [999]{} A.D. Linde, [*Particle Physics and Inflationary Cosmology*]{} (Harwood, Chur, Switzerland, 1990). A.D. Dolgov and A.D. Linde, Phys. Lett. [**116B**]{}, 329 (1982); L.F. Abbott, E. Fahri and M. Wise, Phys. Lett. [**117B**]{}, 29 (1982). A.A. Starobinsky, in: [*Quantum Gravity, Proc. of the Second Seminar “Quantum Theory of Gravity” (Moscow, 13-15 Oct. 1981)*]{}, eds. M.A. Markov and P.C. West (Plenum, New York, 1984), p. 103. A.D. Dolgov and D.P. Kirilova, Sov. Nucl. Phys., [**51**]{}, 273 (1990); J. Traschen and R. Brandenberger, Phys. Rev. D [**42**]{}, 2491 (1990). L.A. Kofman, A.D. Linde and A.A. Starobinsky, in preparation. [ **Fig. 1.**]{}  The sketch of the stability/instability chart of the canonical Mathieu equation (3). [^1]: On leave of absence from Institute of Astrophysics and Atmospheric Physics, Tartu EE-2444, Estonia [^2]: On leave of absence from Lebedev Physical Institute, Moscow 117924, Russia

--- abstract: 'We propose to study the accelerating expansion of the universe in the double complex symmetric gravitational theory (DCSGT). The universe we live in is taken as the real part of the whole spacetime ${\cal M}^4_C(J)$ which is double complex. By introducing the spatially flat FRW metric, not only the double Friedmann Equations but also the two constraint conditions $p_J=0$ and $J^2=1$ are obtained. Furthermore, using parametric $D_L(z)$ ansatz, we reconstruct the $\omega^{''}(z)$ and $V(\phi)$ for dark energy from real observational data. We find that in the two cases of $J=i,p_J=0$ and $J=\varepsilon,p_J\neq 0$, the corresponding equations of state $\omega^{''}(z)$ remain close to -1 at present ($z=0$) and change from below -1 to above -1. The results illustrate that the whole spacetime, i.e. the double complex spacetime ${\cal M}^4_C(J)$, may be either ordinary complex ($J=i,p_J=0$) or hyperbolic complex ($J=\varepsilon,p_J\neq 0$). And the fate of the universe would be Big Rip in the future.' author: - 'Shao Ying,[^1] Gui Yuan-Xing [^2], Wang Wei' title: Reconstructing the Equation of State for Dark Energy In the Double Complex Symmetric Gravitational Theory --- [^3] INTRODUCTION ============ Current astrophysical observations have indicated that the universe undergoes accelerated expansion during recent redshift times$^{[1,2]}$. The accelerating expansion has been attributed to the existence of mysterious dark energy$^{[3]}$ with negative pressure which can induce repulsive gravity and thus cause accelerated expansion. The cosmological constant $\Lambda$ with equation of state $\omega=\frac{p}{\rho}=-1$$^{[4]}$ is the simplest and most obvious candidate for dark energy. However this model raises theoretical problems related to the fine tunned value. These difficulties have led to a variety of alternative models where the dark energy component varies with time (eg. quintessence, phantom etc.)$^{[5]}$. Other physically motivated models predicting late accelerated expansion include modified gravity$^{[6]}$, chaplygin gas$^{[7]}$, braneworld$^{[8]}$, quintom$^{[9,10]}$ etc. It is interesting that the equation of state $\omega$ for the quintom model crosses -1 in the near past. In addition, some researches have presented to reconstruct the properties of dark energy from observations$^{[11]}$.\ On the other hand, differing from these models, Moffat proposed the nonsymmetric gravitational theory (NGT)$^{[12]}$ as a possible alternative of dark energy$^{[13]}$. Meanwhile, a so-called the double complex symmetric gravitational theory (DCSGT) have been established$^{[14]}$. Moreover, the corresponding double complex Einstein’s field equations have been obtained. NGT and DCSGT are also the generalized gravitational theory presented as the unified field theory of gravity and electromagnetism. Thus, the questions arise whether we can obtain the corresponding equations of state for dark energy in the DCSGT or whether the accelerating expansion of the universe can be studied in the DCSGT. In this paper, we will answer these questions. For instance, the universe we live in has been taken as the real part of the whole spacetime ${\cal M}^4_C(J)$ which is double complex. This paper is organized as follows. We obtain in Sec two, by introducing spatially flat FRW metric, the double Friedmann Equations in the DCSGT and the two constraint conditions $p_J=0$ and $J^2=1$. In Sec three, the equation of state $\omega^{'}(z)$, potential $V(\phi)$ and scalar field $\phi$ for dark energy are reconstructed from real observational data. Sec four is a conclusion. THE FRIEDMANN EQUATIONS IN THE DCSGT ==================================== In the double complex symmetric gravitational theory (DCSGT), metric tensor is a double complex symmetric tensor. Correspondingly, connection and curvature are forced to be double complex. The real diffeomorphism symmetry of standard Riemannian geometry is extended to complex diffeomorphism symmetry. In the double complex manifold of coordinates ${\cal M}^4_C(J)$, the double complex symmetric metric $g_{\mu\nu}(J)$ is defined by $${g_{\mu\nu}}(J)={s_{\mu\nu}}+J{a_{\mu\nu}},\eqno(1)$$ where ${s_{\mu\nu}}$ and ${a_{\mu\nu}}$ are the real symmetric tensors, and the double imaginary unit $J=i,\varepsilon$ ($J=i, J^2=-1; J=\varepsilon, J^2=1, J\neq 1$)$^{[15,16]}$. The real contravariant tensor $s^{\mu\nu}$ is associated with $s_{\mu\nu}$ by the relation $$s^{\mu\nu}s_{\mu\sigma}=\delta^\nu_\sigma,\eqno(2)$$ and also $$g^{\mu\nu}g_{\mu\sigma}=\delta^\nu_\sigma.\eqno(3)$$ The double complex symmetric connection $ {\Gamma}_{\mu\nu}^{\lambda}(J)$ and curvature $R_{\mu\nu}(J)$ are determined by the equations $${g_{\mu\nu;\lambda}(J)}={\partial_\lambda}{g_{\mu\nu}(J)}-{g_{\rho\nu}(J)}{g_{\mu\lambda}^\rho}(J)-{g_{\mu\rho}(J)}{\Gamma_{\nu\lambda}^\rho(J)}=0,\eqno(4)$$ $$R_{\mu\nu\sigma}^{\lambda}(J)=-{\partial_\sigma}{{\Gamma}_{\mu\nu}^{\lambda}(J)}+ {\partial_\nu}{{\Gamma}_{\mu\sigma}^{\lambda}(J)}+{{\Gamma}_{\rho\nu}^{\lambda}(J)}{{\Gamma}_{\mu\sigma}^{\rho}(J)} -{{\Gamma}_{\rho\sigma}^{\lambda}(J)}{{\Gamma}_{\mu\nu}^{\rho}(J)}.\eqno(5)$$ From curvature tensor, we can obtain the double complex Bianchi identities $${{\bigg(}{R^{\mu\nu}}(J)-\frac{1}{2}g^{\mu\nu}(J)R(J){\bigg)}_{;\nu}}=0.\eqno(6)$$ The action is denoted by $S={S_{grav}}+{S_M}$, where $S_{grav}$ and $S_{M}$ are respectively gravity action and matter action $${S_{grav}} =\frac{1}{2}\int{d^4}x{\bigg[}{{\cal G}^{\mu\nu}(J)}{R_{\mu\nu}(J)}+{{\big(}{{\cal G}^{\mu\nu}(J)}{R_{\mu\nu}(J)}{\big)}^\dag}{\bigg]},\eqno(7)$$ $$\frac{1}{\sqrt{-g(J)}}(\frac{\delta S_M}{\delta g^{\mu\nu}(J)})=8\pi G{T_{\mu\nu}}(J),\eqno(8)$$ where ${\cal G}^{\mu\nu}(J):=\sqrt{-g(J)}g^{\mu\nu}(J)={\cal S}^{\mu\nu}+J{\cal A}^{\mu \nu}$, “$\dag$”denotes complex conjugation and $T_{\mu\nu}(J)=\tau_{\mu\nu}+J{\tau_{\mu\nu}^{'}}$ is a double complex symmetric source tensor. The variation with respect to $g^{\mu\nu}(J)$ and dividing the equation by $\sqrt{-g(J)}$ can lead to $$R_{\mu\nu}(J)-\frac{1}{2}{g_{\mu\nu}(J)}R(J)=-8\pi G{T_{\mu\nu}(J)}.\eqno(9)$$ Eq.(9) is the double complex Einstein’s field equation in the DCSGT$^{14}$. Eq.(9) is written as $$R_{\mu\nu}(J)=-8\pi G\big{(}T_{\mu\nu}(J)-\frac{1}{2}g_{\mu\nu}(J)T(J)\big{)}.$$ Let us consider the spatially flat FRW metric $${ds^2}=-{dt^2}+{a^2}(t)[{dr^2}+{r^2}({d\theta^2}+{\sin^2}\theta{d\phi^2})].\eqno(10)$$ The double complex symmetric metric tensor $g_{\mu\nu}(J)$ is determined by $${g_{00}(J)}=-(1+J),$$ $${g_{11}(J)}=a^2(t)+Ja^2(t),$$ $${g_{22}(J)}=a^2(t)r^2+Ja^2(t)r^2,$$ $${g_{33}(J)}=a^2(t)r^2\sin^2\theta+Ja^2(t)r^2\sin^2\theta.\eqno(11)$$ The energy-momentum tensor $T_{\mu\nu}(J)$ in the double complex symmetric spacetime is defined $$T^{\mu\nu}(J)=[({\rho_C}+{p_C}){U^\mu}{U^\nu}+{p_C}{g^{\mu\nu}}]+J[({\rho_J}+{p_J}){U^{'\mu}}{U^{'\nu}}+{p_J}{g^{\mu\nu}}],\eqno(12)$$ where $\rho_C,p_C$ and $\rho_J,p_J$ are energy density and pressure respectively in real and imaginary spacetime respectively. We define $$s_{\mu\nu}{U^\mu}{U^\nu}=-1,~~~~~a_{\mu\nu}{U^{'\mu}}{U^{'\nu}}=-1,\eqno(13)$$ $$T_{\mu\nu}(J)={g_{\mu\alpha}}{g_{\nu\beta}}{T^{\alpha\beta}(J)},\eqno(14)$$ Substituting Eqs.(11) and (12) into Eq.(9), the double evolutive equations of the universe are expressed $$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}[(3{p_C}+{\rho_C})+{J^2}({p_J}-{\rho_J})],\eqno(15)$$ $${p_C}-{\rho_C}+(2-{J^2}){\rho_J}+(4-{J^2}){p_J}=0,\eqno(16)$$ $$2\frac{\dot{a}^2}{a^2}+\frac{\ddot{a}}{a}=4\pi G[({\rho_C}-{p_C})+{J^2}({\rho_J}-{p_J})],\eqno(17)$$ $$2\frac{\dot{a}^2}{a^2}+\frac{\ddot{a}}{a}=4\pi G[({\rho_C}-{p_C})+{J^2}{\rho_J}+({J^2}-2){p_J}],\eqno(18)$$ where dot means derivative with respect to time.\ From Eqs.(15)-(18), we can obtain two constraint conditions $$p_J=0~~~~or~~~~J^2=1,\eqno(19)$$ and Hubble parameter $$H^2=\frac{8\pi G}{3}[\rho_C+\frac{1}{2}J^2(\rho_J-p_J)],\eqno(20)$$ $$\dot{H}=-4\pi G(\rho_C+p_C).\eqno(21)$$ Eqs.(20) and (21) are the Friedmann Equations in the DCSGT.\ In the following section, we will study the reconstructions of the equations of state for dark energy in the two constraints $p_J=0$ and $J^2=1$, respectively. THE RECONSTRUCTIONS OF EQUATIONS OF STATE $\omega_\phi$ AND POTENTIAL $V(\phi)$ FOR DARK ENERGY =============================================================================================== If the density and pressure of the matter of the universe are respectively $$\rho_C=\rho_m+{\rho^{'}},p_C=p_m+p^{'},\eqno(22)$$ where $\rho_m=\frac{3H^2_0}{8\pi G}{\Omega_{0m}}(1+z)^3$ and $p_m=0$ are energy density and pressure of nonrelativistic matter, $\rho^{'}$ and $p^{'}$ are energy density and pressure for scalar field $\phi$ $$\rho^{'}=\frac{\lambda}{2}\dot{\phi}^2+V(\phi), p^{'}=\frac{\lambda}{2}\dot{\phi}^2-V(\phi),\eqno(23)$$ where $\lambda=\pm1$ corresponding to ordinary scalar field and phantom scalar field. Substituting Eqs.(22) and (23) into Eqs.(20) and (21), then we can obtain $$H^2=\frac{8\pi G}{3}[\rho_m+\frac{\lambda}{2}\dot{\phi}^2+V(\phi)+\frac{1}{2}J^2(\rho_J-p_J)],\eqno(24)$$ $$\dot{H}=-4\pi G(\rho_m+\lambda\dot{\phi}^2).\eqno(25)$$ the case of $p_J=0$ ------------------- If the first constraint condition $p_J=0$, Eq.(24) is rewritten as $$H^2=\frac{8\pi G}{3}{\bigg[}\frac{4-J^2}{2(2-J^2)}{\rho_m}+\frac{\lambda}{2}\dot{\phi}^2+\frac{2}{2-J^2}V(\phi){\bigg]}.\eqno(26)$$ From Eqs.(25) and (26), the double potential $V(z)$ and scalar field $\phi$ can be obtained $$V(z)=\frac{3{H_0}^2}{8\pi G}{\bigg[}\frac{(2-J^2)H^2}{2{H_0}^2}-\frac{1}{2}{\Omega_{0m}}(1+z)^3-\frac{(2-J^2)(1+z)H}{6{H_0}^2}\frac{dH}{dz}{\bigg]}, \eqno(27)$$ $$(\frac{d\phi}{dz})^2=\frac{1}{\lambda}\frac{3{H_0}^2}{8\pi G}{\bigg[}\frac{2}{3{H_0}^2(1+z)H}\frac{dH}{dz}-\frac{{\Omega_{0m}}(1+z)}{H^2}{\bigg]},\eqno(28)$$ where $\frac{dz}{dt}=-(1+z)H(z)$. And the equation of state for dark energy is $$\omega^{'}(z)=\frac{p^{'}}{\rho^{'}}=\frac{\frac{(4-J^2)(1+z)}{3(2-J^2)H}\frac{dH}{dz}-1}{1-\frac{2H_0^2}{(2-J^2)H^2}{\Omega_{0m}}(1+z)^3+\frac{J^2(1+z)}{3(2-J^2)H}\frac{dH}{dz}}.\eqno(29)$$ Eqs.(27)-(29) show that potential $V(z)$ and equation of state $\omega^{'}(z)$ are only dependent of double imaginary unit $J$, but field function $\phi$ is dependent of $\lambda$, i.e. $V(z)$ and $\omega^{'}(z)$ are model-independent. Therefore, the following quantities are shown for $\lambda=1$ (ordinary scalar field). When $J=i$ (corresponding to the ordinary complex 4D spacetime), Eqs.(27)-(29) are potential $V(z)$, field $\phi$ and equation of state $\omega^{'}(z)$ in the ordinary complex symmetric gravitational theory (OCSGT). When $J=\varepsilon$ (corresponding to the hyperbolic complex 4D spacetime), $V(z)$, $\phi$ and $\omega^{'}(z)$ are obtained in the hyperbolic complex symmetric gravitational theory (HCSGT)$^{[14]}$.\ In the following we will concretely study the state parameters $\omega^{'}(z)$ corresponding to $J=i,\varepsilon$ from the observational data. Hence we will deny the equation of state $\omega^{'}(z)$ in the HCSGT ($J=\varepsilon$) for the constraint condition $p_J=0$. The observational data we will adopt are the Full Gold dataset (FG) (157 data points $0<z<1.7$) compiled by Riess et al$^{[17]}$, which is one of the most reliable and robust SnIa datasets existing. And all reconstructed quantities are shown for $\Omega_{m}=0.3$.\ In spatially flat cosmology, the luminosity distance $D_L(z)$ and Hubble parameter $H(z)$ are simply related as $(c=1)$ $$H(z)\equiv\frac{\dot{a}}{a}={\bigg[}\frac{d}{dz}{\bigg(}\frac{D_L(z)}{1+z}{\bigg)}{\bigg]}^{-1}.\eqno(30)$$ We use a rational ansatz for the luminosity distance $D_L(z)$$^{[11]}$ $$\frac{D_L}{1+z}\equiv\frac{2}{H_0}{\bigg[}\frac{z-\alpha\sqrt{1+z}+\alpha}{\beta z+\gamma\sqrt{1+z}+2-\alpha-\gamma}{\bigg]},\eqno(31)$$ where $\alpha,\beta$ and $\gamma$ are fitting parameters, which are determined by minimizing the function $$\chi^2(a_1...a_n)=\sum_{i=1}^{N}\frac{(\mu_{obs}(z_i)-\mu_{th}(z_i))^2}{\sigma_i^2},\eqno(32)$$ where the total error published for the FG dataset $\sigma_i^2=\sigma_{\mu i}^2+\sigma_{int}^2+\sigma_{\nu i}^2$. The observational and theoretical distance modulus are defined as $\mu_{obs}(z_i)=m_{obs}(z_i)-M$ and $\mu_{th}(z_i)=5\log_{10}(D_L(z))+\mu_0$, respectively. The luminosity distance $D_L$ is related to the measured quantity, the corrected apparent peak $B$ magnitude $m_B$ as $m_B=M+25+5\log_{10}D_L$, where $M$ is the absolute peak luminosity. The minimization of Eq.(32) is made using the FindMinimum command of Mathematica. Moreover, the following constraints$^{[11]}$ are applied to minimize $\chi^2$, $$\Omega_m=(\frac{\beta^2}{\alpha\beta+\gamma})^2,\eqno(33)$$ and $$\frac{4\beta+2\gamma-\alpha}{2-\alpha}\geq 3\Omega_m.\eqno(34)$$ Our reconstructions for $\omega^{'}(z)$ are shown in fig1.(The curves plotted are for the best-fit values of the parameters $\alpha=1.492,\beta=0.583,\gamma=-0.190$) Notice that the divergence of equation of state $\omega^{'}(z) (J=\varepsilon)$ in the hyperbolic complex spacetime does not accord with observation facts. So we deny this situation. According to the evolution of $\omega^{'}(z) (J=i)$ in the OCSGT, we find that the $\omega^{'}(z)$ remains close to -1 at present ($z=0$) and the fate of the universe would be Big Rip in the future. It is interesting that $\omega^{'}(z)$ changes from below -1 to above -1, which agrees with the Quintom model$^{[9,10]}$. In fig 2 and 3, the potential $V(z)$ reconstructed and the age of the universe obtained using $ t(z)=H_0^{-1}\int^\infty_z\frac{H_0dz}{(1+z)H}$, are respectively shown. And we see that the age of universe is about 14 Gyr, which is consistent with the observations. ![The plot shows the evolution of $\omega^{'}(z)$ for the $\Omega_m=0.3$ in the DCSGT. The solid line and the dashed line respectively denote the $\omega^{'}(z)$ in the OCSGT $(J=i)$ and HCSGT $(J=\varepsilon)$. Notice that the divergence of $\omega^{'}(z)$ in the HCSGT is not consistent with the observations while the $\omega^{'}(z)$ accords with the observations in the OCSGT and changes from below $-1$ to above $-1$.[]{data-label="fig1"}](fig1.eps) ![The effective potential $V(z)$ is shown in units of $3H_0^2/8\pi G$.[]{data-label="fig2"}](fig2.eps) ![The age of the Universe at a redshift $z$, given in Gyr, for the value of $H_0=70km s^{-1} Mpc^{-1}$.[]{data-label="fig3"}](fig3.eps) the case of $J^2=1$ ------------------- We have discussed the case of $p_J=0, J=\varepsilon$ above. Below we will study the case of $J^2=1 (J=\varepsilon)$, but $p_J\neq 0$. Eq.(24) turns into $$H^2=\frac{8\pi G}{3}{\bigg[}\frac{5}{6}{\rho_m}+\frac{\lambda}{2}\dot{\phi}^2+\frac{2}{3}V(\phi)+\frac{2}{3}{\rho_J}{\bigg]}.\eqno(35)$$ If the density parameter $\Omega_J$ in the imaginary part of the whole spacetime ${\cal M}^4_C(J)$ is $ \Omega_J\equiv\frac{\rho_J}{\rho_0}=\rho_J/(\frac{3H_0^2}{8\pi G})=1, $ then we can obtain $$V(z)=\frac{3H_0^2}{8\pi G}{\bigg[}\frac{3H^2}{2H_0^2}-\frac{1}{2}\Omega_{0m}(1+z)^3-\frac{(1+z)H}{2H_0^2}\frac{dH}{dz}-1{\bigg]},\eqno(36)$$ $$(\frac{d\phi}{dz})^2=\frac{1}{\lambda}\frac{3H_0^2}{8\pi G}{\bigg[}\frac{2}{3H_0^2(1+z)H}\frac{dH}{dz}-\frac{1}{H^2}\Omega_{0m}(1+z){\bigg]}.\eqno(37)$$ Furthermore, the equation of state for dark energy $$\omega^{'}(z)=\frac{p^{'}}{\rho^{'}}=\frac{-1+\frac{5(1+z)}{9H}\frac{dH}{dz}+\frac{2H_0^2}{3H^2}}{1-\frac{2H_0^2}{3H^2}{\Omega_{0m}}(1+z)^3-\frac{1+z}{9H}\frac{dH}{dz}-\frac{2H_0^2 }{3H^2}}.\eqno(38)$$ Eqs. (36)-(38) are potential $V(\phi)$, field $\phi$ and equation of state $\omega^{'}(z)$ in the hyperbolic complex symmetric gravitational theory (HCSGT). We still apply the $D_L(z)$ ansatz (31) and the two constraints (33) and (34) to minimize $\chi^2$. Hence the equations of state $\omega^{'}(z)$ and potential $V(z)$ for dark energy are shown in fig4 and 5, respectively. Note that the $\omega^{'}(z)$ remains close to -1 at present and changes from below -1 to above -1 for the whole redshift range $0<z<1.7$. The fate of the universe would be Big Rip in the future. ![The equation of state $\omega^{'}(z)=p^{'}/\rho^{'}$ as a function of redshift $z$ for $p_J=0$ in the HCSGT ($J=\varepsilon$). The curve plotted is for $\Omega_m=0.3$. The $\omega^{'}(z)$ changes from below -1 to above -1 and remains close to -1 at present.[]{data-label="fig4"}](fig4.eps) ![The effective potential $V(z)$ is shown in units of $3H_0^2/8\pi G$.[]{data-label="fig5"}](fig5.eps) CONCLUSIONS =========== In this paper, we have proposed to study the accelerating expansion of the universe in the double complex symmetric gravitational theory (DCSGT). The universe we live in is taken as the real part of the whole spacetime ${\cal M}^4_C(J)$ which is double complex. By introducing the spatially flat FRW metric, the double Friedmann Equations and two constraint conditions $p_J=0$ and $J^2=1$ have been obtained. Furthermore, using parametric $D_L(z)$ ansatz and real observational data, we have reconstructed the equation of state $\omega^{'}(z)$ and potential $V(\phi)$ for the two constraints, respectively. The results have indicated that the corresponding state parameters $\omega^{'}(z)$ are consistent with the observations for the two cases of $J=i,p_J=0$ and $J^2=1 (J=\varepsilon),p_J\neq 0$. So we have concluded that the whole spacetime ${\cal M}^4_C(J)$, may be either ordinary complex ($J=i$) for $p_J=0$ or hyperbolic complex ($J=\varepsilon (J^2=1)$) for $ p_J\neq 0$. Moreover, we find (see fig1 and 4) that the $\omega^{'}(z)$ remains close to the $-1$ at present ($z=0$) and changes from below -1 to above -1, which is consistent with the Quintom model$^{[9,10]}$. And fig1 and 4 show that when $z<0, \omega^{'}(z)<-1$ which tell us that the fate of the universe would be Big Rip in the future. Since we have only studied the corresponding properties of the dark energy in the DCSGT, some questions are not deeply discussed (eg. the relation (16) of matter between real and imaginary space and the properties of the whole spacetime). Hence, we will investigate these issues in detail in the forthcoming work. ACKNOWLEDGEMENTS ================ This work was supported by National Science Foundation of China under Grant NO.10573004. [99]{} A.G.Riess *et al.*, Astron.J. **116**(1998)1009. S.Perlmutter *et al.*, Astrophys.J. **517**(1999)565. D.Huterer and M.S.Turner, Phys.Rev.**D64** (2001)123527. V.Sahni and A.A.Starobinsky, Int.J.Mod.Phys. **D9**, 373 (2000); T.Padmanabhan, Phys.Rept. **380** (2003)235. P.G.Ferreira and M.Joyce, Phys.Rev.**D58** (1998)023503; B.Ratra and P.J.E.Peebles, Rev.Mod.Phys. **75**(2003)559. K.Freese and M.Lewis, Phys.Lett. **B540** (2002)1. A.Y.Kamenshchik, U.Moschella and V.Pasquier, Phys.Lett. **B511**(2001)265. V.Sahni and Y.Shtanov, JCAP **0311**(2003)014. B.Feng, X.L.Wang and X.M.Zhang, Phys.Lett. **B607**(2005)35. Z.K.Guo, Y.S.Piao, X.M.Zhang and Y.Z.Zhang, Phys.Lett. **B608**(2005)177. T.D.Saini *et al.*, Phys.Rev.Lett. **85** (2000)6. J.W.Moffat, J.Math.Phys. **21**(1980)1978; J.W.Moffat, J.Math.Phys. **29**(1988)1655. J.W.Moffat, astro-ph/0403266. Y.B.Wu, Y.Shao and P.Dong, Acta.Phys.Sin. **53** (2004)2846; J.W.Moffat, Phys.Lett. **B491**(2000)345; J.Moffat, Proc.Camb.Phil.Soc. **53**(1957)473. Z.Z.Zhong, J.Math.Phys. **26**(1985)2589. J.Huck, J.Math.Phys. **34**(1993)12. A.G.Riess *et al.*, Astrophys.J. **607** (2004)665. S.Nesseris and L.Perivolaropoulos, astro-ph/0511040. [^1]: Email: sybb37@student.dlut.edu.cn; Photo code: 0411-84707869(office) [^2]: Email: thphys@dlut.edu.cn; Photo code: 0411-84706203(office) [^3]: Supported by National Science Foundation of China under Grant No 10573004

--- abstract: 'Being a complex subject of major importance in AI Safety research, value alignment has been studied from various perspectives in the last years. However, no final consensus on the design of ethical utility functions facilitating AI value alignment has been achieved yet. Given the urgency to identify systematic solutions, we postulate that it might be useful to start with the simple fact that for the utility function of an AI not to violate human ethical intuitions, it trivially has to be *a model* of these intuitions and reflect their *variety* – whereby the most accurate models pertaining to human entities being biological organisms equipped with a brain constructing concepts like moral judgements, are *scientific* models. Thus, in order to better assess the variety of human morality, we perform a transdisciplinary analysis applying a security mindset to the issue and summarizing variety-relevant background knowledge from neuroscience and psychology. We complement this information by linking it to *augmented utilitarianism* as a suitable ethical framework. Based on that, we propose first practical guidelines for the design of approximate *ethical goal functions* that might better capture the variety of human moral judgements. Finally, we conclude and address future possible challenges.' author: - 'Nadisha-Marie Aliman $^1$' - | Leon Kester $^2$ $^1$ Utrecht University, Utrecht, Netherlands\ $^2$ TNO Netherlands, The Hague, Netherlands nadishamarie.aliman@gmail.com bibliography: - 'ijcai19.bib' title: 'Requisite Variety in *Ethical* Utility Functions for AI Value Alignment' --- Introduction ============ AI value alignment, the attempt to implement systems adhering to human ethical values has been recognized as highly relevant subtask in AI Safety at an international level and studied by multiple AI and AI Safety researchers across diverse research subareas [@hadfield2016cooperative; @soares2017agent; @yudkowsky2016ai] (a review is provided in [@taylor2016alignment]). Moreover, the need to investigate value alignment has been included in the Asilomar AI Principles  with a worldwide support of researchers from the field. While value alignment has often been tackled using reinforcement learning [@abel2016reinforcement] (and also reward modeling [@leike2018scalable]) or inverse reinforcement learning [@abbeel2004apprenticeship] methods, we focus on the approach to explicitly formulate cardinal ethical utility functions crafted by (a representation of) society and assisted by science and technology which has been termed *ethical goal functions* [@Delphi; @werkhoven2018telling]. In order to be able to formulate utility functions that do not violate the ethical intuitions of most entities in a society, these ethical goal functions will have to be a model of human ethical intuitions. This simple but important insight can be derived from the good regulator theorem in cybernetics [@conant1970every] stating that *“every good regulator of a system must be a model of that system"*. We believe that instead of learning models of human intuitions in their apparent complexity and ambiguity, AI Safety research could also make use of the already available scientific knowledge on the nature of human moral judgements and ethical conceptions as made available e.g. by neuroscience and psychology. The human brain did not evolve to facilitate rational decision-making or the experience of emotions, but instead to fulfill the core task of allostasis (anticipating the needs of the body in an environment before they arise in order to ensure growth, survival and reproduction) [@barrett2017emotions; @kleckner2017evidence]. Thereby, psychological functions such as cognition, emotion or moral judgements are closely linked to the predictive regulation of physiological needs of the body [@kleckner2017evidence] making it indispensable to consider the embodied nature of morality when aspiring to model it for AI value alignment. For the purpose of facilitating the injection of requisite knowledge reflecting the variety of human morality in ethical goal functions, Section \[var\] provides information on the following variety-relevant aspects: 1) the essential role of affect and emotion in moral judgements from a modern constructionist neuroscience and cognitive science perspective followed by 2) dyadic morality as a recent psychological theory on the nature of cognitive templates for moral judgements. In Section \[aug\], we propose first guidelines on how to approximately formulate ethical goal functions using a recently proposed non-normative socio-technological ethical framework grounded in science called *augmented utilitarianism* [@aliman2019augmented] that might be useful to better incorporate the requisite variety of human ethical intuitions (especially in comparison to classical utilitarianism). Thereafter, we propose how to possibly validate these functions within a socio-technological feedback-loop [@Delphi]. Finally, in Section \[conc\], we conclude and specify open challenges providing incentives for future work. Variety in Embodied Morality {#var} ============================ ![Intuitive illustration for the Law of Requisite Variety. Taken from [@norman2018special]. []{data-label="pic"}](Variety){height="0.3\textheight"} While value alignment is often seen as a safety problem, it is possible to interpret and reformulate it as a related security problem which might offer a helpful different perspective on the subject emphasizing the need to capture the variety of embodied morality. One possible way to look at AI value alignment is to consider it as being an attempt to achieve advanced AI systems exhibiting adversarial robustness against malicious adversaries attempting to lead the system to action(s) or output(s) that are perceived as violating human ethical intuitions. From an abstract point of view, one could distinguish different means by which an adversary might achieve successful attacks: e.g. 1) by fooling the AI at the perception-level (in analogy to classical adversarial examples [@goodfellow2018defense], this variant has been denoted *ethical adversarial examples* [@aliman2019augmented]) which could lead to an unethical behavior even if the utility function would have been aligned with human ethical intuitions or 2) simply by disclosing dangerous (certainly unintended from the designer) unethical implications encoded in its utility function by targeting specific mappings from perception to output or action (this could be understood as ethical adversarial examples on the utility function itself). While the existence of point 1) yields one more argument for the importance of research on adversarial robustness at the perception-level for AI Safety reasons [@goodf] and a sophisticated combination of 1) and 2) might be thinkable, our exemplification focuses on adversarial attacks of the type 2). One could consider the explicitly formulated utility function $ U $ as representing a separate model[^1] that given a sample, outputs a value determining the perceived ethical desirability of that sample which should ideally be in line with the society that crafted this utility function. The attacker which has at his disposal the knowledge on human ethical intuitions, can attempt targeted misclassifications at the level of a single sample or at the level of an ordering of multiple samples whereby the ground-truth are the ethical intuitions of most people in a society. The Law of Requisite Variety from cybernetics [@ashby1961introduction] states that *“only variety can destroy variety"*, with other words in order to cope with a certain variety of problems or environmental variety, a system needs to exhibit a suitable and sufficient variety of responses. Figure \[pic\] offers an intuitive explanation of this law. Transferring it to the mentioned utility function $ U $, it is for instance conceivable that if $ U $ does not encode affective information that might lead to a difference in ethical evaluations, an attacker can easily craft a sample which $ U $ might misclassify as ethical or unethical or cause $ U $ to generate a total ordering of samples that might appear unethical from the perspective of most people. Given that $ U $ does not have an influence on the variety of human morality, the only way to respond to the disturbances of the attacker and reduce the variety of possible undesirable outcomes, is by increasing the own variety – which can be achieved by encoding more relevant knowledge. Role of Emotion and Affect in Morality --------------------------------------- One fundamental and persistent misconception about human biology (which does not only affect the understanding of the nature of moral judgements) is the assumption that the brain incorporates a layered architecture in which a battle between emotion and cognition is given through the very anatomy of the *“triune brain"* [@maclean1990triune] exhibiting three hierarchical layers: a reptilian brain on top of which an emotional animalistic paleomammalian limbic system is located and a final rational neomammalian cognition layer implemented in the neocortex. This flawed view is not in accordance with neuroscientific evidence and understanding [@barrett2017emotions; @miller2018happily]. In fact, the assumed reactive and animalistic limbic regions in the brain are predictive (e.g. they send top-down predictions to more granular cortical regions), control the body as well as attention mechanisms while being the source of the brain’s internal model of the body [@barrett2015interoceptive; @barrett2017theory]. Emotion and cognition do not represent a dichotomy leading to a conflict in moral judgements [@helion2015beyond]. Instead, the distinction between the experience of an instance of a concept as belonging to the category of emotions versus the category of cognition is grounded in the focus of attention of the brain [@barrett2015conceptual] whereby *“the experience of cognition occurs when the brain foregrounds mental contents and processes"* and *“the experience of emotion occurs when, in relation to the current situation, the brain foregrounds bodily changes"*  [@hoe]. The mental phenomenon of actively dynamically simulating different alternative scenarios (including anticipatory emotions) has also been termed conceptual consumption [@gilbert2007prospection] and plays a role in decision-making and moral reasoning. While emotions are discrete constructions of the human brain, core affect allows a low-dimensional experience of interoceptive sensations (sensory array from within the body) and is a continuous property of conciousness with the dimensions of valence (pleasantness/unpleasantness) and arousal (activation/deactivation) [@kleckner2017evidence]. It has been argued that core affect provides a basis for moral judgements in which different events are qualitatively compared to each other [@cabanac2002emotion]. Like other constructed mental states, moral judgements involve domain-general brain processes which simply put combine 1) the interoceptive sensory array, 2) the exteroceptive sensory inputs from the environment and 3) past experience/ knowledge for a goal-oriented situated conceptualization (as tool for allostasis) [@oosterwijk2012states]. From these key constituents of mental constructions one can extract the following: concepts (including morality) are *perceiver-dependent* and *time-dependent*. Thereby, affect, (but not emotion [@cameron2015constructionist]) is a necessary ingredient of every moral judgement. More fundamentally, *“the human brain is anatomically structured so that no decision or action can be free of interoception and affect"* [@barrett2017emotions] – this includes any type of thoughts that seem to correspond to the folk terms of “rational" and “cold". Therefore, a utility function without affect-related parameters might not exhibit a sufficient variety and might lead to the violation of human ethical intuitions. Morality cannot be separated from a model of the body, since the brain constructs the human perception of reality based on what seems of importance to the brain for the purpose of allostasis which is inherently strongly linked to interoception [@barrett2017emotions]. Interestingly, even the imagination of future not yet experienced events is facilitated through situated recombinations of sensory-motor and affective nature in a similar way as the simulation of actually experienced events [@addis2018episodic]. To sum up, there is no battle between emotion and cognition in moral judgements. Moreover, there is also no specific moral faculty in the brain, since moral judgements are based on domain-general processes within which affect is always involved to a certain degree. One could obtain insufficient variety in dealing with an adversary crafting ethical adversarial examples on a utility model $ U $ if one ignores affective parameters. Further crucial parameters for ethical utility functions could be e.g. of cultural, social and socio-geographical nature. Variety through “Dyadicness" ---------------------------- The psychological theory of dyadic morality [@schein2018theory] posits that moral judgements are based on a fuzzy cognitive template and related to the perception of an intentional agent ($ iA $) causing damage ($ d $) to a vulnerable patient ($ vP $) denoted $ iA\xrightarrow{\text{d}}vP $. More precisely, the theory postulates that the perceived immorality of an act is related to the following three elements: norm violations, negative affect and importantly perceived harm. According to a study, the reaction times in describing an act as immoral predict the reaction times in categorizing the same act as harmful [@schein2015unifying]. The combination of these basic constituents is suggested to lead to the emergence of a rich diversity of moral judgements [@gray2017think]. *Dyadicness* is understood as a continuum predicting the condemnation of moral acts. The more a human entity perceives an intentional agent inflicting damage to a vulnerable patient, the more immoral this human perceives the act. As stated by Schein and Gray, the dyadic harm-based cognitive template *“is rooted in innate and evolved processes of the human mind; it is also shaped by cultural learning, therefore allowing cultural pluralism"*. Importantly, the nature of this cognitive template reveals that moral judgements besides being perceiver-dependent, might vary across diverse parameters such as especially e.g. in relation to the perception of agent, act and patient in the outcome of the action. Further, the theory also foresees a possible time-dependency of moral judgements by introducing the concept of a *dyadic loop*, a feedback cycle resulting in an iterative polarization of moral judgements through social discussion modulating the perception of harm as time goes by. Overall, moral judgements are understood as constructions in the same way visual perception, cognition or emotion are constructed by the human mind. Similarly to the existence of variability in visual perception, variability in morality is the norm which often leads to moral conflicts [@schein2016visual]. However, the understanding that humans share the same harm-based cognitive template for morality has been described as reflecting *“cognitive unity in the variety of perceived harm"* [@schein2018theory]. Analyzing the cognitive template of dyadic morality, one can deduce that human moral judgements do not only consider the outcome of an action as prioritized by consequentialist frameworks like classical utilitarianism, nor do they only consider the state of the agent which is in the focus of virtue ethics. Furthermore, as opposed to deontological ethics, the focus is not only on the nature of the performed action. The main implications for the design of utility functions that should ideally be aligned with human ethical values, is that they might need to encode information on agent, action, patient as well as on the perceivers – especially with regard to the cultural background. This observation is fundamental as it indicates that one might have to depart from classical utilitarian utility functions $ U(s') $ which are formulated as total orders at the abstraction level of outcomes i.e. states (of affairs) $ s' $. In line with this insight, is the context-sensitive and perceiver-dependent type of utility functions considering agent, action and outcome which has been recently proposed within a novel ethical framework denoted *augmented utilitarianism* [@aliman2019augmented] (abbreviated with AU in the following). Reconsidering the dyadic morality template $ iA\xrightarrow{\text{d}}vP $, it seems that in order to better capture the variety of human morality, utility functions – now transferring it to the perspective of AI systems – would need to be at least formulated at the abstraction level of a *perceiver-dependent* evaluation of a transition $ s\xrightarrow{\text{a}}s'$ leading from a state $ s $ to a state $ s' $ via an action $ a $. We encode the required novel type of utility function with $ U_x(s,a,s') $ with $ x $ denoting a specific perceiver. This formulation could enable an AI system implemented as utility maximizer to jointly consider parameters specified by a perceiver which are related to its perception of agent, the action and the consequences of this action on a patient. Since the need to consider time-dependency has been formulated, one would consequently also require to add the time dimension to the arguments of the utility function leading to $ U_x((s,a,s'),t) $. Approximating Ethical Goal Functions {#aug} ==================================== While the psychological theory of dyadic morality was useful to estimate the abstraction level at which one would at least have to specify utility functions, the closer analysis on the nature of the construction of mental states performed in Section \[var\], abstractly provides a superset of primitive relevant parameters that might be critical elements of every moral judgement (being a mental state). Given a perceiver $ x $, the components of this set are the following subsets: 1) parameters encoding the interoceptive sensory array $ B_x $ (from within the body) which are accessible to the human consciousness via the low-dimensional core affect, 2) the exteroceptive sensory array $ E_x $ encoding information from the environment and 3) the prior experience $ P_x $ encoding memories. Moreover, these set of parameters obviously vary in time. However, to simplify, it has been suggested within the mentioned AU framework, that ethical goal functions will have to be updated regularly (leading to a so-called socio-technological feedback-loop [@Delphi]) in the same way as votes take place at regular intervals in a democracy. One could similarly assume that this regular update will be sufficient to reflect a relevant change in moral opinion and perception. Injecting Requisite Variety in Utility -------------------------------------- For simplicity, we assume that the set of parameters $ B_x $, $ P_x $ and $ E_x $ are invariant during the utility assignment process in which a perceiver $ x $ has to specify the ethical desirability of a transition $ s\xrightarrow{\text{a}}s'$ by mapping it to a cardinal value $U_x(s,a,s')$ obtained by applying a not-nearer defined type of scientifically determined transformation $ v_x $ (chosen by $ x $) on the mental state of $ x $. This results in the following naive and simplified mapping however adequately reflecting the property of *mental-state-dependency* formulated in the AU framework (the required dependency of ethical utility functions on parameters of the own mental state function $ m_x $ in order to avoid perverse instantiation scenarios [@aliman2019augmented]): $$\begin{aligned} U_x(s,a,s')= v_x(m_x((s,a,s'), B_x, P_x, E_x))\end{aligned}$$ Conversely, the utility function of classical utilitarianism is only defined at the impersonal and context-independent abstraction level of $ U(s') $ which has been argued to lead to both *perverse* instantiation problem but also to the *repugnant* conclusion and related impossibility theorems in population ethics for consequentialist frameworks which do not apply to mental-state-dependent utility functions [@aliman2019augmented]. The idea to restrict human ethical utility functions to the considerations of outcomes of actions alone – ignoring affective parameters of the own current self – as practiced in classical utilitarianism while later referring to the resulting total orders with emotionally connoted adjectives such as “repugnant" or “perverse" has been termed the *perspectival fallacy of utility assignment* [@Delphi]. The use of consequentialist utility functions affected by the impossibility theorems of Arrhenius  has been justifiably identified by Eckersley  as a safety risk if used in AI systems without more ado. It seems that the isolated consideration of outcomes of actions (for consequentialism) or actions (for deontological ethics) or the involved agents (for virtue ethics) does not represent a good model of human ethical intuitions. It is conceivable, that if a utility model $ U $ is defined as utility function $ U(s') $, the model cannot possibly exhibit a sufficient variety and might more likely violate human ethical intuitions than if it would be implemented as a context-sensitive utility function $ U_x(s,a,s') $. (Beyond that, it has been argued that consequentialism implies the rejection of *“dispositions and emotions, such as regret, disappointment, guilt and resentment"* from “rational" deliberation  [@verbeek2001consequentialism] and should i.a. for this reason be disentangled from the notion of rationality for which it cannot represent a plausible requirement.) It is noteworthy that in the context of reinforcement learning (e.g. in robotics) different types of reward functions are usually formulated ranging from $ R(s') $ to $ R(s,a,s')$. For the purpose of ethical utility functions for advanced AI systems in critical application fields, we postulate that one does not have the choice to specify the abstraction level of the utility function, since for instance $ U(s') $ might lead to safety risks. Christiano et al.   considered the elicitation of human preferences on trajectory (state-action pairs) segments of a reinforcement learning agent i.a. realized by human feedback on short movies. For the purpose of utility elicitation in an AU framework exemplarily using a naive model as specified in equation (1), people will similarly have to assign utility to a movie representing a transition in the future (either in a mental mode or augmented by technology such as VR or AR [@Delphi]). However, it is obvious that this naive utility assignment would not scale in practice. Moreover, it has not yet been specified how to aggregate ethical goal functions at a societal level. In the following Subsection \[approx\], we will address these issues by proposing a practicable approximation of the utility function in (1) and a possible societal aggregation of this approximate solution. Approximation, Aggregation and Validation {#approx} ----------------------------------------- So far, it has been stated throughout the paper that one has to adequately increase the variety of a utility function meant to be ethical in order to avoid violations of human ethical intuitions and vulnerability to attackers crafting ethical adversarial examples against the model. However, it is important to note that despite the negatively formulated motivation of the approach, the aim is to craft a utility model $ U $ which represents a better model of human ethical intuitions in general, thus ranging from samples that are perceived as highly unethical to those that are assigned a high ethical desirability. In order to craft practical solutions that lead to optimal results, it might be advantageous to perform a thought experiment imagining a utopia and from that impose practical constraints on its viability. It might not seem realistic to deliberate a future *utopia 1* as a sustainable society which is stable across a very large time interval in which every human being acts according to the ethical intuitions of all humans including the own and every artificial intelligent system fulfills the ethical intuitions of all humans. However, it seems more likely that within a *utopia 2* being a stable society in which every human achieves a high level of a scientific definition of well-being (such as e.g. PERMA [@seligman2012flourish]) with artificial agents acting as to maximize context-sensitive utility according to which (human or artificial) agents promoting the (measurable) well-being of human patients is regarded as the most utile type of events, the ethical intuitions of humans might tend to get closer to each other. The reason being that the variety of human moral judgements might interestingly *decrease* since it is conceivable that they will tend to exhibit more similar prior experiences (all imprinted by well-being) and have more similar environments (full of stable people with a high level of well-being). The main factor drawing differences could be the body – especially biological factors. However, the parameters related to interoception might be closer to each other, since all humans exhibit a high level of well-being which classically includes frequent positive affect. It is conceivable that with time, such a society could converge towards the utopia 1. In the following, we will denote the mentioned utopia-related ideal cognitive template of a (human or artificial) agent $ A $ performing an act $ w $ that contributes to the well-being of a human patient $ P $ with $ A\xrightarrow{\text{w}}P $ in analogy to the cognitive template of dyadic morality. (Thereby, $ A\xrightarrow{\text{w}}P $ is perceiver-dependent i.a. because psychological measures of well-being include subjective and self-reported elements such as e.g. life satisfaction or furthermore positive emotions [@seligman2012flourish].) Augmented utilitarianism foresees the need to at least depict a final goal at the abstraction level of a perceiver-dependent function on a transition as reflected in $ U_x(s,a,s') $. The ideal cognitive template $ A\xrightarrow{\text{w}}P $ formulated for utopia 2 by which it has been argued that a decrease in the variety of human morality might be achievable in the long-term exhibits an abstraction level that is compatible with $ U_x(s,a,s') $. A thinkable strategy for the design of a utility model $ U $ that is robust against ethical adversarial examples and a model of human ethical intuitions is to try to adequately increase its variety using relevant scientific knowledge and to complementarily attempt to decrease the variety of human moral judgements for instance by considering $ A\xrightarrow{\text{w}}P $ as high-level final goal such that the described utopia 2 ideally becomes a self-fulfilling prophecy. For it to be realizable in practice, we suggest that the appropriateness of a given aggregated societal ethical goal function could be approximately validated against its quantifiable impact on well-being for society across the time dimension. Since it seems however unfeasible to directly map all important transitions of a domain to their effect on the well-being of human entities, we propose to consider perceiver-specific and domain-specific utility functions indicating combined preferences that each perceiver $ x $ considers to be relevant for well-being from the viewpoint of $ x $ himself in that specific domain. For these combined utility functions to be grounded in science, they will have to be based on scientifically measurable parameters. We postulate that a possible aggregation at a societal level could be performed by the following steps: 1) agreement on a common validation measure of an ethical goal function (for instance the temporal development of societal satisfaction with AI systems in a certain domain or with future AGI systems, their aptitude to contribute to sustainable well-being), 2) agreement on *superset* $ O $ of scientifically measurable and relevant parameters (encoding e.g. affective, dyadic, cultural, social, political, socio-geographical but importantly also law-relevant information) that are considered as important across the whole society, 3) specification of personal utility functions for each member $ n $ of a society of $ N $ members allowing personalized and tailored combinations of a subset of $ O $, 4) aggregation to a societal ethical goal function $ U_{Total}(s,a,s') $. Taken together, these considerations lead us to the following possible approximation for an aggregated societal ethical goal function given a domain: $$\begin{aligned} U_{Total}(s,a,s')= \sum_{n=1}^{N} \sum_{i=1}^{j} w_{i}^nf_{fi}^n(C_i)\end{aligned}$$ with $ N $ standing for the number of participating entities in society, $ C_i= (p_{i1}, p_{i2},...,p_{im}) $ being a cluster of $ m\geq 1 $ correlated parameters (whereby independent factors are assigned an own cluster each) and $ f $ representing a set of preference functions (*form functions*). For instance $ f=\{f_1,f_2,...,f_f\} $ where $ f_1 $ could be a linear transformation, $ f_2 $ a concave, $ f_3 $ a convex preference function and so on. Each entity $ n $ assigns a weight $ w_{i}^n $ to a form function $ f_{fi}^n $ applied to a cluster of parameters $ C_i $ whereby $\sum_{i=1}^{j} w_{i}^n=1 $. We define $ O=\{C_1, C_2,...\} $ as the superset of all parameters considered in the overall aggregated utility function. Moreover, $ a \in A $ with $ A $ representing the foreseen discrete action space at the disposal of the AI. (It is important to note that while the AI could directly perform actions in the environment, it could also be used for policy-making and provide plans for human agents.) Further, we consider a continuous state space with the states $ s$ and $s' \in S = \mathbb{R}^{|O|} $. Other aspects including e.g. legal rules and norms on the action space can be imposed as constraints on the utility function. In a nutshell, the utility aggregation process can be understood as a voting process in which each participating individual $ n $ distributes his vote across scientifically measurable clusters of parameters $ C_i $ on which he applies a preference function $ f_{fi}^n $ to which weights $ w_{i}^n $ are assigned as identified as relevant by $ n $ given a to be approximated high-level societal goal (such as $ A\xrightarrow{\text{w}}P $). In short, people do not have to agree on personal preferences and weightings, but only on a superset of acceptable parameters, an aggregation method and an overall validation measure. (Note that instead of involving society as a whole for each domain, the utility elicitation procedure can as well be approximated by a transdisciplinary set of representative experts (e.g. from the legislative) crafting *expert ethical goal functions* that attempt to ideally emulate $ U_{Total}(s,a,s') $). Finally, it is important to note that the societal ethical goal function specified in (2) will need to be updated (and evalutated) at regular intervals due to the mental-state-dependency of utility entailing time-dependency [@aliman2019augmented]. This leads to the necessity of a socio-technological feedback-loop which might concurrently offer the possibility of a *dynamical ethical enhancement* [@Delphi; @werkhoven2018telling]. Pre-deployment, one could in the future attempt a validation via selected preemptive simulations [@Delphi] in which (a representation of) society experiences simulations of future events $ (s,a,s') $ as movies, immersive audio-stories or later in VR and AR environments. During these experiences, one could approximately measure the temporal profile of the so-called *artificially simulated future instant utility* [@Delphi] denoted $ U_{TotalAS} $ being a potential constituent of future well-being. Thereby, $ U_{TotalAS} $ refers to the instant utility [@kahneman1997back] experienced during a technology-aided simulation of a future event whereby instant utility refers to the affective dimension of valence at a certain time $ t $. The temporal integral that a measure of $ U_{TotalAS} $ could approximate is specified as: $$\begin{aligned} U_{TotalAS}(s,a,s')\approx \sum_{n=1}^{N}\int_{t_0}^{T} I_n(t) dt \end{aligned}$$ with $ t_0 $referring to the starting point of experiencing the simulation of the event $ (s,a,s') $ augmented by technology (movie, audio-story, AR, VR) and $ T $ the end of this experience. $ I_n(t) $ represents the valence dimension of core affect experienced by $ n $ at time $ t $. Finally, post-deployment, the ethical goal function of an AI system can be validated using the validation measure agreed upon before utility aggregation (such as the temporal development of societal-level satisfaction with an AI system, well-being or even the perception of dyadicness) that has to be a priori determined. Conclusion and Future Work {#conc} ========================== In this paper, we motivated the need in AI value alignment to attempt to model utility functions capturing the variety of human moral judgements through the integration of relevant scientific knowledge – especially from neuroscience and psychology – (instead of learning) in order to avoid violations of human ethical intuitions. We reformulated value alignment as a security task and introduced the requirement to increase the variety within classical utility functions positing that a utility function which does not integrate affective and perceiver-dependent dyadic information does not exhibit sufficient variety and might not exhibit robustness against corresponding adversaries. Using augmented utilitarianism as a suitable non-normative ethical framework, we proposed a methodology to implement and possibly validate societal perceiver-dependent ethical goal functions with the goal to better incorporate the requisite variety for AI value alignment. In future work, one could extend and refine the discussed methodology, study a more systematic validation approach for ethical goal functions and perform first experimental studies. Moreover, the *“security of the utility function itself is essential, due to the possibility of its modification by malevolent actors during the deployment phase"* [@aliman2019augmented]. For this purpose, a blockchain-based solution might be advantageous. In addition, it is important to note that even with utility functions exhibiting a sufficient variety for AI value alignment, it might still be possible for a malicious attacker to craft adversarial examples against a utility maximizer at the perception-level which might lead to unethical behavior. Besides that, one might first need to perform policy-by-simulation [@werkhoven2018telling] prior to the deployment of advanced AI systems equipped with ethical goal functions for safety reasons. Last but not least, the usage of ethical goal functions might represent an interesting approach to the AI coordination subtask in AI Safety, since an international use of this method might contribute to reduce the AI race to the problem-solving ability dimension [@Delphi]. [^1]: a conceptually similar separation of objective function model and optimizing agent has been recently performed for reward modeling [@leike2018scalable]

--- abstract: 'We study the rate of decay of the probability of error for distinguishing between a sparse signal with noise, modeled as a sparse mixture, from pure noise. This problem has many applications in signal processing, evolutionary biology, bioinformatics, astrophysics and feature selection for machine learning. We let the mixture probability tend to zero as the number of observations tends to infinity and derive oracle rates at which the error probability can be driven to zero for a general class of signal and noise distributions via the likelihood ratio test. In contrast to the problem of detection of non-sparse signals, we see the log-probability of error decays sublinearly rather than linearly and is characterized through the $\chi^2$-divergence rather than the Kullback-Leibler divergence for “weak” signals and can be independent of divergence for “strong” signals. Our contribution is the first characterization of the rate of decay of the error probability for this problem for both the false alarm and miss probabilities.' address: - | Coordinated Science Laboratory\ and\ Department of Electrical and\ Computer Engineering\ University of Illinois at\ Urbana-Champaign\ Urbana, IL 61801, USA\ \ - | Department of Electrical and\ Computer Engineering\ University of Patras\ 26500 Rio, Greece\ and\ Department of Computer Science\ Rutgers University\ New Brunswick, NJ 08854, USA\ author: - - - title: 'Detecting Sparse Mixtures: Rate of Decay of Error Probability ' --- , Introduction ============ We consider the problem of detecting a sparse signal in noise, modeled as a mixture, where the unknown sparsity level decreases as the number of samples collected increases. Of particular interest is the case where the unknown signal strength relative to the noise power is very small. This problem has many natural applications. In signal processing, applications include detecting a signal in a multi-channel system [@dobrushin; @ingster] and detecting covert communications [@donoho]. In evolutionary biology, the problem manifests in the reconstruction of phylogenetic trees in the multi-species coalescent model [@mosselroch]. In bioinformatics, the problem arises in the context of determining gene expression from gene ontology datasets [@goeman]. In astrophysics, detection of sparse mixtures is used to compare models of the cosmic microwave background to observed data [@cayon]. Also, statistics developed from the study of this problem have been applied in machine learning to anomaly detection on graphs [@saligrama] and high-dimensional feature selection when useful features are rare and weak [@feature]. Prior work on detecting a sparse signal in noise has been primarily focused on Gaussian signal and noise models, with the goal of determining the trade-off in signal strength with sparsity required for detection with vanishing probability of error. In contrast, this work considers a fairly general class of signal and noise models. Moreover, in this general class of sparse signal and noise models, we provide the first analysis of the rate at which the false alarm [(Type-I)]{} and miss detection [(Type-II)]{} error probabilities vanish with sample size. We also provide simple to verify conditions for detectability, [ which are derived using simpler tools than previously used]{}. In the problem of testing between $n$ i.i.d. samples from two known distributions, it is well known that the rate at which the error probability decays is $e^{-c n}$ for some constant $c>0$ bounded by the Kullback-Leibler divergence between the two distributions [@cover; @dz]. In this work, we show for the problem of detecting a sparse signal in noise that the error probability for an oracle detector decays at a slower rate determined by the sparsity level and the $\chi^2$-divergence between the signal and noise distributions, with different behaviors possible depending on the signal strength. In addition to determining the optimal trade-off between signal strength and sparsity for consistent detection, an important contribution in prior work has been the construction of adaptive (and, to some extent, distribution-free) tests that achieve the optimal trade-off without knowing the model parameters [@castro; @caijengjin; @caiwu; @donoho; @ingster; @jager; @walther]. We discuss prior work in more detail in Sec. \[sec:rw\]. However, the adaptive tests that have been proposed in these papers are not amenable to an analysis of the [*rate*]{} at which the error probability goes to zero. We show that in a Gaussian signal and noise model that an adaptive test based on the sample maximum has miss detection probability that vanishes at the optimal rate when the sparse signal is sufficiently strong. Problem Setup ============= Let $\{\f_{0,n}(x)\}, \{\f_{1,n}(x)\}$ be sequences of probability density functions (PDFs) for real valued random-variables. We consider the following sequence of composite hypothesis testing problems with sample size $n$, called the (sparse) *mixture detection problem*: $$\begin{aligned} \Hyp_{0,n}:&~~ X_1, \ldots, X_n \sim \f_{0,n}(x) \text{ i.i.d. (null)}\\ \Hyp_{1,n}:&~~ X_1, \ldots, X_n \sim (1-\epsilon_n) \f_{0,n}(x) + \epsilon_n \f_{1,n}(x) \text{ i.i.d. (alternative)} \label{eq:altdef}\end{aligned}$$ where $\{\f_{0,n}(x)\}$ is known, $\{\f_{1,n}(x)\}$ is from some known family $\mathcal{F}$ of sequences of PDFs, and $\{\epsilon_n\}$ is a sequence of positive numbers such that $\epsilon_n \to 0$. We will also assume $n \epsilon_n \to \infty$ so that a typical realization of the alternative is distinguishable from the null. Let $\Pro_{0,n},\Pro_{1,n}$ denote the probability measure under $\Hyp_{0,n},\Hyp_{1,n}$ respectively, and let $\Exp_{0,n},\Exp_{1,n}$ be the corresponding expectations, with respect to the particular $\{\f_{0,n}(x)\}$, $\{\f_{1,n}(x)\}$ and $\{\epsilon_n\}$. When convenient, we will drop the subscript $n$. Let $\LR_n \triangleq \frac{\f_{1,n}(x)}{\f_{0,n}(x)}$. When $\f_{0,n}(x) = \f_0(x)$ and $\f_{1,n}(x) = \f_0(x-\mu_n)$, we say that the model is a *location model*. [For the purposes of presentation, we will assume that $\{\mu_n\}$ is a positive and monotone sequence.]{} When $\f_{0}(x)$ is a standard normal PDF, we call the location model a *Gaussian location model*. The distributions of the alternative in a location model are described by the set of sequences $\{ (\epsilon_n,\mu_n)\}$. The location model can be considered as one where the null corresponds to pure noise, while the alternative corresponds to a sparse signal (controlled by $\epsilon_n$), with signal strength $\mu_n$ contaminated by additive noise. The relationship between $\epsilon_n$ and $\mu_n$ determines the signal-to-noise ratio (SNR), and characterizes when the hypotheses can be distinguished with vanishing probability of error. In the general case, $\f_{0,n}(x)$ can be thought of as the noise and $\f_{1,n}(x)$ as the signal distribution. We define the *probability of false alarm* for a hypothesis test $\delta_n$ between $\Hyp_{0,n}$ and $\Hyp_{1,n}$ as $$\Pro_{\rm FA}(n) \triangleq \Pro_{0,n} [ \delta_n = 1 ]$$ and the *probability of missed detection* as $$\Pro_{\rm MD}(n) \triangleq \Pro_{1,n} [ \delta_n = 0 ].$$ A sequence of hypothesis tests $\{\delta_n\}$ is *consistent* if $\Pro_{\rm FA}(n),\Pro_{\rm MD}(n) \to 0$ as $n \to \infty$. We say we have a *rate characterization* for a sequence of consistent hypothesis tests $\{\delta_n\}$ if we can write $$\lim_{n \to \infty} \frac{\log \pfa(n)}{g_0(n)} = -c,~~~ \lim_{n \to \infty} \frac{\log \pmd(n)}{g_1(n)} = -d,$$ where $g_0(n),g_1(n) \to \infty$ as $n \to \infty$ and $0 < c,d<\infty$. The rate characterization describes decay of the error probabilities for large sample sizes. All logarithms are natural. For the problem of testing between i.i.d. samples from two fixed distributions, $g_0(n)=g_1(n)=n$, and $c,d$ are called the *error exponents* [@cover]. In the mixture detection problem, $g_0(n)$ and $g_1(n)$ will be sublinear functions of $n$. The log-likelihood ratio between the corresponding probability measures of $\Hyp_{1,n}$ and $\Hyp_{0,n}$ is $$\text{LLR}(n)=\sum_{i=1}^n \log\big(1-\epsilon_n + \epsilon_n \LR_n(X_i)\big). \label{eq:llr}$$ In order to perform an *oracle rate* characterization for the mixture detection problem, we consider the sequence of oracle likelihood ratio tests (LRTs) between $\Hyp_{0,n}$ and $\Hyp_{1,n}$ (i.e. with $\epsilon_n, \f_{0,n}, \f_{1,n}$ known): $$\delta_n (X_1,\ldots,X_n) \triangleq \begin{cases} 1 & \text{LLR}(n) \geq 0 \\0 & \text{otherwise} \end{cases}. \label{eq:lrt}$$ It is well known that is optimal for testing between $\Hyp_{0,n}$ and $\Hyp_{1,n}$ in the sense of minimizing $\frac{\pfa(n)+\pmd(n)}{2}$, which is the average probability of error when the null and alternative are assumed to be equally likely [@lehmann; @poor]. It is valuable to analyze $\pfa(n)$ and $\pmd(n)$ separately since many applications incur different costs associated with false alarms and missed detections. [**Location Model:**]{} The detectable region for a location model is the set of sequences $\{(\epsilon_n, \mu_n)\}$ such that a sequence of consistent oracle tests $\{\delta_n\}$ exist. For convenience of analysis, we introduce the parameterization $$\epsilon_n = n^{-\beta}$$ where $\beta \in (0,1)$ as necessary. Following the terminology of [@castro], when $\beta \in (0,\frac{1}{2})$, the mixture is said to be a “dense mixture”. If $\beta \in (\frac{1}{2},1)$, the mixture is said to be a “sparse mixture”. Related Work {#sec:rw} ------------ Prior work on mixture detection has been focused primarily on the Gaussian location model. The main goals in these works have been to determine the detectable region and construct *optimally adaptive* tests (i.e. those which are consistent independent of knowledge of $\{(\epsilon_n$, $\mu_n)\}$, whenever possible). The study of detection of mixtures where the mixture probability tends to zero was initiated by Ingster for the Gaussian location model [@ingster]. Ingster characterized the detectable region, and showed that outside the detectable region the sum of the probabilities of false alarm and missed detection is bounded away from zero for any test. Since the generalized likelihood statistic tends to infinity under the null, Ingster developed an increasing sequence of simple hypothesis tests that are optimally adaptive. Donoho and Jin introduced the Higher Criticism test, which is optimally adaptive and is computationally efficient relative to Ingster’s sequence of hypothesis tests, and also discussed some extensions to Subbotin distributions and $\chi^2$-distributions [@donoho]. Cai et al. extended these results to the case where $\f_{0,n}(x)$ is standard normal and $\f_{1,n}(x)$ is a normal distribution with positive variance, derived limiting expressions for the distribution of $\text{LLR}(n)$ under both hypotheses, and showed that the Higher Criticism test is optimally adaptive in this case [@caijengjin]. Jager and Wellner proposed a family of tests based on $\phi$-divergences and showed that they attain the full detectable region in the Gaussian location model [@jager]. Arias-Castro and Wang studied a location model where $\f_{0,n}(x)$ is some fixed but unknown symmetric distribution, and constructed an optimally adaptive test that relies only on the symmetry of the distribution when $\mu_n >0$ [@castro]. In a separate paper, Arias-Castro and Wang also considered mixtures of Poisson distributions and showed the problem had similar detectability behavior to the Gaussian location model [@castropoisson]. Cai and Wu gave an information-theoretic characterization of the detectable region via an analysis of the sharp asymptotics of the Hellinger distance for a wide variety of distributions, and established a strong converse result showing that reliable detection is impossible outside the detectable region in many cases if is not consistent [@caiwu]. This work also gave general conditions for the Higher Criticism test to be consistent. [Our work complements [@caiwu] by providing conditions for consistency (as well as asymptotic estimates of error probabilities) for optimal tests, with simple to verify conditions for a fairly general class of models. ]{} While the Hellinger distance used in [@caiwu] provides bounds on $\Pro_{\rm FA}(n)+ \Pro_{\rm MD}(n)$ for the test specified in , our analysis treats $\Pro_{\rm FA}(n),\Pro_{\rm MD}(n)$ separately as they may have different rates at which they tend to zero and different acceptable tolerances in applications. As we will show in Sec. \[sec:glma\] and Sec. \[sec:rtadap\], there are cases where $\Pro_{\rm FA}(n) \gg \Pro_{\rm MD}(n)$ for adaptive tests and $\Pro_{\rm FA}(n) \ll \Pro_{\rm MD}(n)$ for an oracle test. Walther numerically showed that while the popular Higher Criticism statistic is consistent, there exist optimally adaptive tests with significantly higher power for a given sample size at different sparsity levels [@walther]. Our work complements [@walther] by providing a benchmark to meaningfully compare the sample size and sparsity trade-offs of different tests with an oracle test. It should be noted that all of the work except [@castro; @caijengjin] has focused on the case where $\beta > \frac{1}{2}$, and no prior work has provided an analysis of the *rate* at which $\Pro_{\rm FA}(n),\Pro_{\rm MD}(n)$ can be driven to zero with sample size. Main Results for Rate Analysis ============================== General Case ------------ Our main result is a characterization of the oracle rate via the test given in . The sufficient conditions required for the rate characterization are applicable to a broad range of parameters in the Gaussian location model (Sec. \[sec:glma\]). We first look at the behavior of “weak signals”, where $\LR_n$ has suitably controlled tails under the null hypothesis. In the Gaussian location model in Sec. \[sec:glma\], this theorem is applicable to small detectable $\mu_n$. \[mthm\_main\] Let $\gamma_0 \in (0,1)$ and assume that for all $\gamma\in(0,\gamma_0)$ the following conditions are satisfied: $$\begin{gathered} \lim_{n \to \infty} \Exp_0 \left[ \frac{(\LR_n -1)^2}{D_n^2}\ind{\LR_n \geq 1 + \frac{\gamma}{\epsilon_n}}\right] =0 \label{eq:c1}\\ \epsilon_n D_n \to 0 \label{eq:c2}\\ \sqrt{n} \epsilon_n D_n \to \infty \label{eq:c3}\end{gathered}$$ where $$D_n^2 = \Exp_0 [(\LR_n-1)^2]< \infty.$$ Then for the test specified by , $$\lim_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 D_n^2} = -\frac{1}{8}. \label{eq:mainthm}$$ Moreover, holds if we replace $\Pro_{\rm FA}(n)$ with $\Pro_{\rm MD}(n)$. The quantity $D_n^2$ is known as the $\chi^2$-divergence between $\f_{0,n}(x)$ and $\f_{1,n}(x)$ [@gibbssu]. In contrast to the problem of testing between i.i.d. samples from two fixed distributions [@dz], the rate is not characterized by the Kullback-Leibler divergence for the mixture detection problem. We provide a sketch of the proof for $\Pro_{\rm FA}(n)$, and leave the details to Supplemental Material. We first establish that $$\limsup_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 D_n^2} \leq -\frac{1}{8} \label{eq:cub0}$$ By the Chernoff bound applied to $\Pro_{\rm FA}(n)$ and noting $X_1,\ldots,X_n$ are i.i.d., $$\begin{aligned} \Pro_{\rm FA}(n) &= \Pro_0\big[ \text{LLR}(n) \geq 0\big] \leq \left( \min_{0\leq s \leq 1} \Exp_0 \big[ \big(1- \epsilon_n + \epsilon_n \LR_n(X_1) \big)^s\big] \right)^n \\ &\leq \left(\Exp_0 \left[ \sqrt{1- \epsilon_n + \epsilon_n \LR_n(X_1) }\right] \right)^n \numberthis \label{eq:cub}\end{aligned}$$ By direct computation, we see $\Exp_0[\LR_n(X_1) -1] = 0$, and the following sequence of inequalities hold: $$\begin{gathered} \Exp_0 \left[ \sqrt{1- \epsilon_n + \epsilon_n \LR_n(X_1) }\right] = 1 - \frac{1}{2} \Exp_0\left[ \frac{\epsilon_n^2 (\LR_n(X_1) - 1)^2}{\big(1+\sqrt{1+\epsilon_n (\LR_n(X_1) -1)}\big)^2}\right] \\ \leq 1 - \frac{\epsilon_n^2}{2} \Exp_0\left[ \frac{(\LR_n(X_1) - 1)^2}{(1+\sqrt{1+\epsilon_n (\LR_n(X_1) -1)})^2}\ind{\epsilon_n (\LR_n(X_1) - 1) \leq \gamma} \right] \\ \leq 1 - \frac{\epsilon_n^2 D_n^2}{2(1+\sqrt{1+\gamma})^2} \Exp_0\left[ \frac{(\LR_n(X_1) - 1)^2}{D_n^2}\ind{\LR_n(X_1)\leq1+ \frac{\gamma}{\epsilon_n}} \right]\\ = 1 - \frac{\epsilon_n^2 D_n^2}{2(1+\sqrt{1+\gamma})^2} \left(1 - \Exp_0\left[ \frac{(\LR_n(X_1) - 1)^2}{D_n^2}\ind{\LR_n(X_1)\geq1+ \frac{\gamma}{\epsilon_n}} \right]\right) \end{gathered}$$ Since the expectation in the previous line tends to zero by , for sufficiently large $n$ it will become smaller than $\gamma$. Therefore we have by $$\frac{\log \Pro_{\rm FA}(n)}{n} \leq \log \left( 1 - \frac{1}{2} \frac{\epsilon_n^2 D_n^2}{(1+ \sqrt{1+\gamma})^2} (1-\gamma) \right).$$ Dividing both sides by $\epsilon_n^2 D_n^2$ and taking the $\limsup$ using , establishes $\limsup_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 D_n^2} \leq -\frac{1}{2} \frac{1- \gamma}{(1+\sqrt{1+\gamma})^2}$. Since $\gamma$ can be arbitrarily small, is established. We now establish that $$\liminf_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 D_n^2} \geq -\frac{1}{8}. \label{eq:clb0}$$ The proof of is similar to that of Cramer’s theorem (Theorem I.4, [@denhollander]). The key difference from Cramer’s theorem is that $\text{LLR}(n)$ is the sum of i.i.d. random variables for each $n$, but the distributions of the summands defining $\text{LLR}(n)$ in change for each $n$ under either hypothesis. Thus, we modify the proof of Cramer’s theorem by introducing a $n$-dependent tilted distribution, and replacing the standard central limit theorem (CLT) with the Lindeberg-Feller CLT for triangular arrays (Theorem 3.4.5, [@durrett]). We introduce the tilted distribution $\tilde{\f}_n(x)$ corresponding to $\f_{0,n}(x)$ by $$\tilde{\f}_n(x) = \frac{\big(1-\epsilon_n + \epsilon_n \LR_n(x)\big)^{s_n}}{\Lambda_n(s_n)} \f_{0,n}(x) \label{eq:tilt}$$ where $\Lambda_n(s) = \Exp_0\big[\big(1-\epsilon_n+\epsilon_n \LR_n(X_1)\big)^s\big]$, which is convex with $\Lambda_n(0)=\Lambda_n(1)=1$, and $s_n = \argmin_{0\leq s \leq 1} \Lambda_n(s)$. Let $\tilde{\Pro},\tilde{\Exp}$ denote the tilted measure and expectation, respectively (where we suppress the $n$ for clarity). A standard dominated convergence argument (Lemma 2.2.5, [@dz]) shows that $$\tilde{\Exp}\big[ \log\big(1-\epsilon_n + \epsilon_n \LR_n(X_1)\big) \big] =0.$$ Define the variance of the log-likelihood ratio for one sample as $$\sigma_n^2 = \tilde{\Exp} \left[ \big(\log\big(1+ \epsilon_n ( \LR_n(X_1) -1 ) \big) \big)^2\right]. \label{eq:sigma2}$$ For sufficiently large $n$ such that Lemma \[suplem1\] (proved in Supplementary Material) holds, namely that $C_1 \epsilon_n^2 D_n^2 \geq \sigma_n^2 \geq C_2 \epsilon_n^2 D_n^2$, we have: $$\begin{aligned} \pfa(n)&= \Pro_0 \left[\LLR(n) \geq 0\right]= \Exp_0 \left[\ind{ \LLR(n) \geq 0 }\right]\nonumber \\ &= \left(\Lambda_n(s_n)\right)^n \tilde{\Exp} \left[ e^{- \LLR(n)} \ind{ \LLR(n) \geq 0 }\right]\nonumber \\ &= \left(\Lambda_n(s_n)\right)^n \tilde{\Exp} \left[ e^{- \LLR(n)} | \LLR(n) \geq 0\right] \tilde{\Pro}\left[\LLR(n) \geq 0\right] \nonumber\\ &\geq \left(\Lambda_n(s_n)\right)^n e^{-\tilde{\Exp} \left[ \LLR(n) | \LLR(n) \geq 0\right]} \tilde{\Pro}\left[\LLR(n) \geq 0\right]\label{eq:b0}\\ &= \left(\Lambda_n(s_n)\right)^n e^{-\frac{\tilde{\Exp} \left[ \LLR(n) \ind{ \LLR(n) \geq 0 }\right]}{\tilde{\Pro}\left[\LLR(n)\geq 0\right]}} \tilde{\Pro}\left[\LLR(n) \geq 0\right]\nonumber\\ &\geq \left(\Lambda_n(s_n)\right)^n e^{-\frac{\tilde{\Exp} \left[ |\LLR(n)|\right]}{\tilde{\Pro}\left[\LLR(n)\geq 0\right]}} \tilde{\Pro}\left[\LLR(n) \geq0\right] \label{eq:b1}\\ &\geq \left(\Lambda_n(s_n)\right)^n e^{-\frac{\sqrt{\tilde{\Exp} \left[ \left(\LLR(n)\right)^2\right]}}{\tilde{\Pro}\left[\LLR(n)\geq 0\right]}} \tilde{\Pro}\left[\LLR(n) \geq 0\right]\label{eq:b2} \\ &= \left(\Lambda_n(s_n)\right)^n e^{-\frac{\sqrt{n \sigma_n^2}}{\tilde{\Pro}\left[\LLR(n)\geq0\right]}}\tilde{\Pro}\left[\LLR(n) \geq 0\right]\nonumber\\ &\geq \left(\Lambda_n(s_n)\right)^n e^{-\frac{\sqrt{n C_1 \epsilon_n^2 D_n^2}}{\tilde{\Pro}\left[\LLR(n)\geq0\right]}} \tilde{\Pro}\left[\LLR(n) \geq 0\right]\label{eq:b3}\end{aligned}$$ where follows from Jensen’s inequality, by $\LLR(n)\ind{\LLR(n)>0}$ $\leq|\LLR(n)|$, by Jensen’s inequality, and by Lemma \[suplem1\] proved in the Supplementary Material. Taking logarithms and dividing through by $n \epsilon_n^2 D_n^2$ gives $$\frac{\log \pfa(n)}{n \epsilon_n^2 D_n^2} \geq \frac{\log \Lambda_n(s_n)}{\epsilon_n^2 D_n^2} - \frac{\sqrt{C_1}}{\tilde{\Pro}\left[\LLR(n) \geq 0\right]} \frac{1}{\sqrt{n} \epsilon_n D_n} + \frac{\log \tilde{\Pro}\left[\LLR(n) \geq 0\right]}{n \epsilon_n^2 D_n^2}.$$ Taking $\liminf$ and applying Lemma \[suplem2\], in which it is established that $\tilde{\Pro}[\LLR(n) \geq 0] \to \frac{1}{2}$, and Lemma \[suplem3\] in which it is established that\ $\liminf_{n \to \infty} \frac{\log \Lambda_n(s_n)}{\epsilon_n^2 D_n^2} \geq - \frac{1}{8} $, (see Supplementary Material), along with the assumption $n \epsilon_n^2 D_n^2 \to \infty$ establishes that $\liminf_{n \to \infty} \frac{\log \pfa(n)}{n \epsilon_n^2 D_n^2} \geq -\frac{1}{8}$. The analysis under $\Hyp_{1,n}$ for $\Pro_{\rm MD}(n)$ relies on the fact that the $X_i$ are i.i.d. with pdf $(1-\epsilon_n + \epsilon_n \LR_n) \f_{0,n}(x)$, which allows the use of $1-\epsilon_n + \epsilon_n \LR_n$ to change the measure from the alternative to the null. The upper bound is established identically, by noting that the Chernoff bound furnishes $$\begin{aligned} \Pro_{\rm MD}(n) &= \Pro_{1,n}\big[ -\text{LLR}(n) >0 \big] \leq \left(\Exp_{1} \left[ \frac{1}{\sqrt{1- \epsilon_n + \epsilon_n \LR_n(X_1) }} \right] \right)^n \\ & = \left(\Exp_{0} \left[ \sqrt{1- \epsilon_n + \epsilon_n \LR_n(X_1) }\right] \right)^n \end{aligned}$$ Similarly, the previous analysis can be applied to show that holds with $\Pro_{\rm FA}(n)$ replaced with $\Pro_{\rm MD}(n)$. In order to study the behavior of tests when Thm \[mthm\_main\] does not hold, we rely on the following bounds for $\Pro_{\rm MD}(n),\Pro_{\rm FA}(n)$: \[lbthm\_main\](a) Let $\{\delta_n\}$ be any sequence of tests such that $$\limsup_{n \to \infty} \Pro_{\rm FA}(n) < 1,$$ then, $$\liminf_{n \to \infty} \frac{ \log \Pro_{\rm MD} (n)} {n \epsilon_n} \geq -1. \label{eq:unilb}$$ (b) The following upper and lower bounds for $\Pro_{\rm FA}(n)$ hold for the test specified by : $$\begin{gathered} \Pro_{\rm FA}(n) \leq 1 - (\Pro_0 [ \LR_n \leq 1 ])^n \label{eq:univfaub}\\ \Pro_{\rm FA}(n) \geq \Pro_0 \left[ \sum_{i=1}^n \log \max \big(1 - \epsilon_n, \epsilon_n \LR_n(X_i) \big) \geq 0 \right]. \label{eq:univfalb}\end{gathered}$$ These bounds are easily proved by noting if all observations under $\Hyp_{1,n}$ come from $\f_{0,n}$, then a miss detection occurs (a), and at least one sample must have $\LR_n\geq 1$ in order to raise a false alarm (b). Note that these are universal bounds in the sense that they impose no conditions on $\f_{1,n}(x), \f_{0,n}(x)$ and $\epsilon_n$. Also note that the bound of Thm \[lbthm\_main\](a) is independent of any divergences between $\f_{0,n}(x)$ and $\f_{1,n}(x)$, and it holds for any consistent sequence of tests because $\Pro_{\rm FA}(n) \to 0$. This is in contrast to the problem of testing between i.i.d. samples from fixed distributions, where the rate is a function of divergence [@dz]. When the conditions of Thm \[mthm\_main\] do not hold, we have the following rate characterization for “strong signals”, where $\LR_n$ is under the $\f_{1,n}(x)$ distribution in an appropriate sense. In the Gaussian location model in Sec. \[sec:glma\], this theorem is applicable to large detectable $\mu_n$. \[ubthm\] Let $M_0>1$, and assume that for all $M>M_0$, the following condition is satisified: $$\Exp_0 \left[ \LR_n \ind{ \LR_n > 1 + \frac{M}{\epsilon_n}} \right] \to 1 \label{eq:ubcond}.$$ Then for the test specified by , $$\begin{gathered} \limsup_{n \to \infty} \frac{\log \Pro_{\rm FA} (n)}{n \epsilon_n} \leq -1 \label{eq:gub}\\ \lim_{n \to \infty} \frac{\log \Pro_{\rm MD} (n)}{n \epsilon_n}=-1. \label{eq:ssmd}\end{gathered}$$ We first prove . Let $$\phi(x) = 1+ s x -(1+x)^s.$$ By Taylor’s theorem, we see for $s \in (0,1)$ and $x \geq -1$ that $\phi(x) \geq 0$. Since $\Exp_0[\LR_n-1]=0$, $$\Exp_0 [ (1- \epsilon_n + \epsilon_n \LR_n(X_1) )^s] = 1 - \Exp_0[\phi(\epsilon_n (\LR_n (X_1) -1))].$$ Note this implies $\Exp_0[\phi(\epsilon_n (\LR_n (X_1) -1))] \in [0,1]$ since $ \Exp_0 [ (1- \epsilon_n + \epsilon_n \LR_n(X_1) )^s]$ is convex in $s$ and is $1$ for $s=0,1$. As in the proof of Thm \[mthm\_main\], by the Chernoff bound, $$\Pro_{\rm FA}(n) \leq \left(\Exp_0 [ (1- \epsilon_n + \epsilon_n \LR_n(X_1) )^s] \right)^n$$ for any $s \in (0,1)$. Thus, supressing the dependence on $X_1$, and assuming $M>M_0$, we have $$\begin{aligned} \frac{\log \Pro_{\rm FA}(n)}{n} &\leq \log \Exp_0 \left[ (1- \epsilon_n + \epsilon_n \LR_n(X_1) )^s\right] \\ &= \log (1-\Exp_0\left[\phi(\epsilon_n (\LR_n -1))\right])\\ &\leq - \Exp_0\left[\phi(\epsilon_n (\LR_n -1))\right] \numberthis \label{eq:ubp1}\\ &\leq - \Exp_0\left[\phi(\epsilon_n (\LR_n -1)) \ind{ \epsilon_n (\LR_n-1) \geq M }\right] \numberthis \label{eq:ubp2} \\ &= - \Exp_0 \left[ \left( 1 + s \epsilon_n (\LR_n -1) - (1+ \epsilon_n (\LR_n -1))^s \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right] \\ &\leq - \Exp_0 \left[ \left( s \epsilon_n (\LR_n -1) - (1+ \epsilon_n (\LR_n -1))^s \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right] \\ &\leq - \Exp_0 \left[ \left( s \epsilon_n (\LR_n -1) - 2^s \epsilon_n^s (\LR_n -1)^s \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right] \numberthis \label{eq:ubp3}\\ &= - \Exp_0 \left[ \epsilon_n (\LR_n -1) \left( s - \frac{2^s} { (\epsilon_n (\LR_n -1))^{1-s}} \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right]\\ &\leq - \Exp_0 \left[ \epsilon_n (\LR_n -1) \left( s - \frac{2} { M^{1-s}} \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right] \numberthis \label{eq:ubp4}\\ &= - \epsilon_n \left( s - \frac{2}{M^{1-s}} \right) \Exp_0\left[ \LR_n \left(1 - \frac{1}{\LR_n} \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right] \\ &\leq - \epsilon_n \left( s - \frac{2}{M^{1-s}} \right) \Exp_0\left[ \LR_n \left(1 - \frac{1}{1+ \frac{M}{\epsilon_n}} \right) \ind{ \epsilon_n (\LR_n-1) \geq M } \right] \\ &= - \epsilon_n \left( s - \frac{2}{M^{1-s}} \right) \left(1 - \frac{1}{1+ \frac{M}{\epsilon_n}} \right) \Exp_0\left[ \LR_n \ind{ \epsilon_n (\LR_n-1) \geq M } \right]\end{aligned}$$ where follows from $\log(1-x) \leq -x$ for $x\leq 1$, follows from $\phi(x) \geq 0$, follows from $(1+x)^s \leq 2^s x^s$ for $x \geq 1$ and taking $M>M_0$, follows from $s\in (0,1)$. Dividing both sides of the inequality by $\epsilon_n$ and taking a $\limsup_{n \to \infty}$ establishes $$\limsup_{n \to \infty} \frac{\log \Pro_{\rm FA}}{n \epsilon_n} \leq -s + \frac{2}{M^{1-s}}.$$ Letting $M \to \infty$ and optimizing over $s \in (0,1)$ establishes the . By a change of measure between the alternative and null hypotheses, we see that also holds with $\Pro_{\rm FA}(n)$ replaced with $\Pro_{\rm MD}(n)$. Combining this with Thm \[lbthm\_main\] establishes . Theorem \[ubthm\] shows that the rate of miss detection is controlled by the average number of observations drawn from $\f_{1,n}(x)$ under $\Hyp_{1,n}$, independent of any divergence between $\f_{1,n}(x)$ and $\f_{0,n}(x)$ when holds. Interestingly, so long as the condition of Thm \[ubthm\] holds, by Thm \[lbthm\_main\](a), no non-trivial sequence of tests (i.e. $\limsup_{n \to \infty} \Pro_{\rm FA}(n), \Pro_{\rm MD}(n) < 1$) can achieve a better rate than under $\Hyp_{1,n}$. This is different from the case of testing i.i.d. observations from two fixed distributions, where allowing for a slower rate of decay for $\Pro_{\rm FA}(n)$ can allow for a faster rate of decay for $\Pro_{\rm MD}(n)$ (Sec. 3.4, [@dz]). In Sec. \[sec:glma\], we will show that Thm \[ubthm\] is not always tight under $\Hyp_{0,n}$, and the true behavior can depend on divergence between $\f_{0,n}(x)$ and $\f_{1,n}(x)$, using the upper and lower bounds of Thm \[lbthm\_main\](b). ### Comparison to Related Work Cai and Wu [@caiwu] consider a model which is essentially as general as ours, and characterize the detection boundary for many cases of interest, but do not perform a rate analysis. Note that our rate characterization depends on $D_n$, the $\chi^2$-divergence between $\f_{0,n}$ and $\f_{1,n}$. While the Hellinger distance used in [@caiwu] can be upper bounded in terms of the $\chi^2$-divergence, a corresponding lower bound does not exist in general [@gibbssu], and so our results cannot be derived using the methods of [@caiwu]. In fact, our results complement [@caiwu] in giving precise bounds on the error decay for this problem once the detectable region boundary has been established. Furthermore, as we will show in Thm \[maxtest\], there are cases where the rates derived by analyzing the likelihood ratio test are essentially achievable. Gaussian Location Model {#sec:glma} ----------------------- In this section, we specialize Thm \[mthm\_main\] and \[ubthm\] to the Gaussian location model. The rate characterization proved is summarized in Fig. \[fig:glmsummary\]. We first recall some results from the literature for the detectable region for this model. [0.48]{} ![Detectable regions for the Gaussian location model. Unshaded regions have $\Pro_{\rm MD}(n)+\Pro_{\rm FA}(n) \to 1$ for any test (i.e. reliable detection is impossible). Green regions are where corollaries \[densecase\] and \[ms\] provide an exact rate characterization. The red region is where Thm \[gausub\_main\] provides an upper bound on the rate, but no lower bound. The blue region is where Cor. \[ssig\] holds, and provides an upper bound on the rate for $\Pro_{\rm FA}(n)$ and an exact rate characterization for $\Pro_{\rm MD}(n)$.[]{data-label="fig:glmsummary"}](state1.pdf "fig:"){width="\textwidth"} [0.48]{} ![Detectable regions for the Gaussian location model. Unshaded regions have $\Pro_{\rm MD}(n)+\Pro_{\rm FA}(n) \to 1$ for any test (i.e. reliable detection is impossible). Green regions are where corollaries \[densecase\] and \[ms\] provide an exact rate characterization. The red region is where Thm \[gausub\_main\] provides an upper bound on the rate, but no lower bound. The blue region is where Cor. \[ssig\] holds, and provides an upper bound on the rate for $\Pro_{\rm FA}(n)$ and an exact rate characterization for $\Pro_{\rm MD}(n)$.[]{data-label="fig:glmsummary"}](state2.pdf "fig:"){width="\textwidth"} \[dc\] The boundary of the detectable region (in $\{(\epsilon_n,\mu_n)\}$ space) is given by (with $\epsilon_n = n^{-\beta})$: 1. If $0 < \beta \leq \frac{1}{2}$, then $\mu_{crit,n} = n^{\beta-\frac{1}{2}}$. (Dense) 2. If $\frac{1}{2} < \beta < \frac{3}{4}$, then $\mu_{crit,n} = \sqrt{2 (\beta - \frac{1}{2}) \log n}$. (Moderately Sparse) 3. If $\frac{3}{4} \leq \beta < 1 $, then $\mu_{crit,n} = \sqrt{2 (1- \sqrt{1- \beta})^2 \log n}$. (Very Sparse) If in the dense case $\mu_n = n^{r}$, then the LRT is consistent if $r> \beta -\frac{1}{2}$. Moreover, if $r< \beta-\frac{1}{2}$, then $\Pro_{\rm FA}(n)+ \Pro_{\rm MD}(n) \to 1$ for any sequence of tests as $n\to\infty$. If in the sparse cases, $\mu_n = \sqrt{2 r \log n}$, then the LRT is consistent if $\mu_n > \mu_{crit,n}$. Moreover, if $\mu_n < \mu_{crit,n}$, then $\Pro_{\rm FA}(n)+ \Pro_{\rm MD} (n)\to 1$ for any sequence of tests as $n\to\infty$. For the proof see [@castro; @caijengjin; @donoho]. We call the set of $\{(\epsilon_n,\mu_n)\}$ sequences where is consistent the *interior of the detectable region*. We now begin proving a rate characterization for the Gaussian location model by specializing Thm \[mthm\_main\]. Note that $\LR_n(x) = e^{\mu_n x-\frac{1}{2}\mu_n^2}$ and $D_n^2 = e^{\mu_n^2}-1$. A simple computation shows that the conditions in the theorem can be re-written as: For all $\gamma>0$ sufficiently small: $$\begin{gathered} \textstyle Q\left(-{\frac{3}{2}} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right)+\frac{1}{e^{\mu_n^2}-1}\Big\{Q\left(-\frac{3}{2} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right)\label{eq:c1g}\\ \textstyle - 2 Q\left(-\frac{1}{2} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right) + Q\left(\frac{1}{2} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right)\Big\} \to 0 \nonumber\\ \epsilon_n^2 (e^{\mu_n^2}-1) \to 0 \label{eq:c2g} \\ \textstyle n \epsilon_n^2 (e^{\mu_n^2}-1) \to \infty \label{eq:c3g}\end{gathered}$$ where $Q(x) = \int_x^\infty \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}x^2} dx$. \[densecase\](Dense case) If $\epsilon_n = n^{-\beta}$ for $\beta \in (0,\frac{1}{2})$ and $\mu_n = \frac{h(n)}{n^{\frac{1}{2}-\beta}}$ where $h(n)\to \infty$ and $\limsup_{n \to \infty} \frac{\mu_n}{\sqrt{\frac{2 }{3}\beta \log n}} <1$, then $$\lim_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 (e^{\mu_n^2}-1)} = -\frac{1}{8}. \label{eq:seventeen}$$ If $\mu_n \to 0$, can be rewritten as $$\lim_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 \mu_n^2} = -\frac{1}{8}.$$ This result holds when replacing $\Pro_{\rm FA}(n)$ with $\Pro_{\rm MD}(n)$. It is easy to verify and directly, and if $\mu_n$ does not tend to zero. To verify it suffices to show: If $\mu_n \to 0$, for any $\alpha \in \mathbb{R}$, then $\frac{Q(\alpha \mu_n + \frac{1}{\mu_n}\log(1+\frac{\gamma}{\epsilon_n}))}{e^{\mu_n^2}-1} \to 0$. Since $e^x -1 \geq x$, it suffices to show that $\frac{Q(\alpha \mu_n + \frac{1}{\mu_n}\log(1+\frac{\gamma}{\epsilon_n}))}{\mu_n^2} \to 0$. This can be verified by the standard bound $Q(x) \leq e^{-\frac{1}{2} x^2}$ for $x>0$, and noting that $\alpha \mu_n + \frac{1}{\mu_n}\log(1+\frac{\gamma}{\epsilon_n})>0$ for sufficiently large $n$ and that $\frac{x}{e^{x}-1} \to 1$ as $x \to 0$. The implication of this corollary is that our rate characterization of the probabilities of error holds for a large portion of the detectable region up to the detection boundary, as $h(n)$ can be taken such that $\frac{h(n)}{n^{\xi}} \to 0$ for any $\xi>0$, making it negligible with respect to $\mu_{crit,n}$ in Thm \[dc\]. \[ms\] (Moderately sparse case) If $\epsilon_n = n^{-\beta}$ for $\beta \in (\frac{1}{2},\frac{3}{4})$ and $\mu_n = \sqrt{ 2(\beta+\frac{1}{2}+\xi) \log n}$ for any $0<\xi < \frac{3-4 \beta}{6}$ then $$\lim_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 (e^{\mu_n^2}-1)} = -\frac{1}{8}$$ and the same result holds replacing $\Pro_{\rm FA}(n)$ with $\Pro_{\rm MD}(n)$. It is easy to verify and directly. To verify , note since $Q(\cdot) \leq 1$ and $\mu_n \to \infty$, we need $$\begin{gathered} \textstyle \frac{1}{e^{\mu_n^2}-1}\Big\{Q\left(-\frac{3}{2} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right)-\\ \textstyle 2 Q\left(-\frac{1}{2} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right) + Q\left(\frac{1}{2} \mu_n + \frac{1}{\mu_n}\log \left(1 + \frac{\gamma}{\epsilon_n}\right)\right) \Big\} \to 0.\end{gathered}$$ Thus, it suffices to show that$Q\big(\!-\frac{3}{2} \mu_n + \frac{1}{\mu_n}\log (1 + \frac{\gamma}{\epsilon_n})\big) \to 0 $, or equivalently, that $-\frac{3}{2} \mu_n + \frac{1}{\mu_n}\log (1 + \frac{\gamma}{\epsilon_n}) \to \infty$ for any fixed $\gamma>0$. Applying $\log (1 + \frac{\gamma}{\epsilon_n}) \geq \log(\frac{\gamma}{\epsilon_n}) = \beta \log n + \log \gamma$ shows that $$\begin{gathered} \textstyle -\frac{3}{2} \mu_n + \frac{1}{\mu_n}\log \big(1 + \frac{\gamma}{\epsilon_n}\big) \geq\\ -\frac{3}{2} \sqrt{ 2 (\beta -{\textstyle \frac{1}{2}} +\xi) \log n } + \frac{\beta \log n + \log \gamma }{\sqrt{2 (\beta - \frac{1}{2} +\xi) \log n}} \\ = \Big( -{\textstyle\frac{3}{2} }\sqrt{ 2 (\beta - {\textstyle \frac{1}{2}} +\xi) } + \frac{\beta}{\sqrt{2 (\beta - \frac{1}{2} +\xi) }}\Big) \sqrt{\log n} + \frac{ \log \gamma }{\sqrt{2 (\beta - \frac{1}{2} +\xi) \log n}} \numberthis \label{eq:vsp}\end{gathered}$$ where the last term tends to 0 with $n$. Thus, tends to infinity if the coefficient of $\sqrt{\log n}$ is positive, i.e. if $\frac{1}{2} (1- 2 \xi) < \beta <\frac{1}{4} (3-6 \xi)$ , which holds by the definition of $\xi$. Thus, tends to infinity and is proved. Note that $\xi$ can be replaced with an appropriately chosen sequence tending to $0$ such that and hold. For $\mu_n> \sqrt{\frac{2}{3}\beta\log n}$, does not hold. However, Thm \[ubthm\] and Thm \[lbthm\_main\] provide a partial rate characterization for the case where $\mu_n$ grows faster than $\sqrt{2 \beta \log n}$ which we present in the following corollary. \[ssig\]If $\epsilon_n = n^{-\beta}$ for $\beta \in (0,1)$ and $\liminf_{n\to \infty} \frac{\mu_n}{ \sqrt{2 \beta \log n}} > 1$, then $$\lim_{n\to \infty} \frac{ \log \Pro_{\rm MD}(n)}{n \epsilon_n} = -1.$$ If $\frac{n \epsilon_n}{\mu_n^2} \to \infty$, then $$\limsup_{n\to \infty} \frac{ \log \Pro_{\rm FA}(n)}{n \epsilon_n} = -1$$ Otherwise, if $\frac{n \epsilon_n}{\mu_n^2} \to 0$, then $$\limsup_{n\to \infty} \frac{ \log \Pro_{\rm FA}(n)}{\mu_n^2} \leq -\frac{1}{8}. \label{eq:degeneratebehavior}$$ The condition for Thm \[ubthm\] given by is $$\textstyle Q\left( \frac{1}{\mu_n}\log\big(1+\frac{M}{\epsilon_n}\big) - \frac{1}{2}\mu_n \right) \to 1.$$ This holds if $ \frac{1}{\mu_n}\log(1+\frac{M}{\epsilon_n}) - \frac{1}{2}\mu_n \to - \infty$, which is true if $r > \beta$. To show that $\liminf_{n\to \infty} \frac{ \log \Pro_{\rm FA}(n)}{n \epsilon_n} \geq -1$ if $\frac{n \epsilon_n}{\mu_n^2} \to \infty$, we can apply a similar argument to the lower bound for Thm \[mthm\_main\] to the lower bound given by and is thus omitted. Instead, we show a short proof of $\liminf_{n\to \infty} \frac{ \log \Pro_{\rm FA}(n)}{n \epsilon_n} \geq -C$ for $C \geq 1$ using . Note that we can loosen to $$\Pro_{\rm FA}(n) \geq \Pro_0 \left[ \sum_{i=1}^k \log \left(1 - \epsilon_n \right) + \sum_{i=k+1}^n \log \big( \epsilon_n \LR_n(X_i) \big) \geq 0 \right]$$ for any $k$ and explicitly compute a lower bound to $\Pro_{\rm FA}(n)$ in terms of the standard normal cumulative distribution function. Optimizing this bound over the choice of $k$ establishes that $\liminf_{n\to \infty} \frac{ \log \Pro_{\rm FA}(n)}{n \epsilon_n} \geq -C$ for some constant $C \geq 1$ (with $C=1$ if $\frac{\mu_n}{\sqrt{\log n}} \to \infty$). The lower bounding of in a manner similar to recovers the correct constant when $\mu_n$ scales as $\sqrt{2 r \log n}$. To see that the log-false alarm probability scales faster than $n \epsilon_n$ when $\frac{n \epsilon_n}{\mu_n^2} \to 0 $, one can apply . In this case, $$\log \Pro_{\rm FA} (n) \leq \log\left(1 - \big(1- Q({\textstyle\frac{1}{2}}\mu_n)\big)^n\right).$$ Applying the standard approximation $$\frac{x e^{-\frac{1}{2}x^2}}{\sqrt{2 \pi}(1+x^2)} \leq Q(x) \leq \frac{e^{-\frac{1}{2}x^2}}{x \sqrt{2 \pi}} \text{ for } x>0 \label{eq:qapprox},$$ we see $\limsup_{n\to \infty} \frac{ \log \Pro_{\rm FA}(n)}{\mu_n^2} \leq - \frac{1}{8}$. Note that shows an asymmetry between the rates for the miss detection and false alarm probabilities, since there is a fundamental lower bound due to the sparsity under the alternative for the miss probability, but not under the null. Theorems \[mthm\_main\] and \[ubthm\] do not hold when $\epsilon_n = n^{-\beta}$ and $\mu_n = \sqrt{2 r \log n}$ where $r \in ( \frac{\beta}{3}, \beta )$ for $\beta \in (0, \frac{3}{4})$ or $r \in ((1-\sqrt{1-\beta})^2,\beta)$ for $\beta \in (\frac{3}{4},1)$. For the remainder of the detectable region, we have an upper bound on the rate derived specifically for the Gaussian location setting. One can think of this as a case of “moderate signals”. \[gausub\_main\] Let $\epsilon_n = n^{-\beta}$ and $\mu_n = \sqrt{2 r \log n}$ where $r \in \left( \frac{\beta}{3}, \beta \right)$ for $\beta \in (0, \frac{3}{4})$ or $r \in ((1-\sqrt{1-\beta})^2,\beta)$ for $\beta \in (\frac{3}{4},1)$. Then, $$\limsup_{n \to \infty} \frac{ \log \Pro_{\rm FA} (n)}{n \epsilon_n^2 e^{\mu_n^2} \Phi\left( \big( \frac{\beta}{2r} - \frac{3}{2} \big) \mu_n \right)} \leq -\frac{1}{16}. \label{eq:gausubbd_main}$$ where $\Phi(x) = 1-Q(x) = \int_{-\infty}^x \frac{1}{\sqrt{2 \pi}} e^{-x^2/2} dx $ denotes the standard normal cumulative distribution function. Moreover, holds replacing $\Pro_{\rm FA}$ with $\Pro_{\rm MD}$. The proof is based on a Chernoff bound with $s=\frac{1}{2}$. Details are given in the Supplemental Material. It is useful to note that $n \epsilon_n^2 e^{\mu_n^2} \Phi\big( ( \frac{\beta}{2r} - \frac{3}{2} ) \mu_n \big)$ behaves on the order of $\frac{n^{1-2 \beta + 2r - r( 1.5 - \beta/2r)^2}}{\sqrt{2 r \log n}}$ for large $n$ in Thm \[gausub\_main\]. Rates and Adaptive Testing in the Gaussian Location Model {#sec:rtadap} ========================================================= No adaptive tests prior to this work have had precise rate characterization. Moreover, optimally adaptive tests for $0<\beta<1$ such as the Higher Criticism (HC) [@donoho] test or the sign test of Arias-Castro and Wang (ACW) ([@castro], Sec. 1.4)[^1]are not amenable to rate analysis based on current analysis techniques. This is due to the fact that the consistency proofs of these tests follow from constructing functions of order statistics that grow slowly under the null and slightly quicker under the alternative via a result of Darling and Erdös [@darlingerdos]. We therefore analyze the *max test*: $$\delta_{\max}(X_1,\ldots,X_n) \triangleq \begin{cases} 1 & \max_{i=1,\ldots,n} X_i \geq \tau_n \\0 & \text{otherwise} \end{cases} \label{eq:maxtest}$$ where $\tau_n$ is a sequence of test thresholds. While the max test is not consistent everywhere is [@castro; @donoho], it has a few advantages over other tests that are adaptive to all $\{(\epsilon_n,\mu_n)\}$ possible (i.e. optimally adaptive). The first advantage is a practical perspective; the max test requires a linear search and trivial storage complexity to find the largest element in a sample, whereas computing the HC or ACW test requires on the order of $n \log n$ operations to compute the order statistics of a sample of size $n$ (which may lead to non-trivial auxiliary storage requirements), along with computations depending on $Q$-functions or partial sums of the signs of the data. Moreover, the max test has been shown to work in applications such as astrophysics [@cayon]. It does not require specifying the null distribution, which allows it to be applied to the Subbotin location models as in [@castro]. The second advantage is analytical, as the cumulative distribution function of the maximum of an i.i.d. sample of size $n$ with cumulative distribution function $F(x)$ has the simple form of $F(x)^n$. This also provides a simple way to set the test threshold to meet a pre-specified false alarm probability for a given sample size $n$. As most applications focus on the regime where $\epsilon_n = n^{-\beta}$ for $\beta > \frac{1}{2}$, the following theorem shows the max test provides a simple test with rate guarantees for almost the entire detectable region in this case. \[maxtest\] For the max test given by with threshold $\tau_n = \sqrt{2 \log n}$: The rate under the null is given by $$\lim_{n \to \infty} \frac{\log \Pro_{\rm FA}(n)}{\log \log n} = - \frac{1}{2}. \label{eq:ratemaxfa}$$ Under the alternative, if $\liminf_{n \to \infty} \frac{\mu_n}{\sqrt{2 (1-\sqrt{1-\beta})^2 \log n}} >1$ with $\epsilon_n = n^{-\beta}$, $$\lim_{n \to \infty} \frac{\log \Pro_{\rm MD}(n)}{n \epsilon_n Q(\sqrt{2 \log n} - \mu_n)} = -1. \label{eq:ratemax1md}$$ In particular, if $\liminf_{n \to \infty} \frac{\mu_n}{\sqrt{2 \log n}} >1$, the max test achieves the optimal rate under the alternative $$\lim_{n \to \infty} \frac{\log \Pro_{\rm MD}(n)}{n \epsilon_n} = -1. \label{eq:ratemax2md}$$ Otherwise, the max test is not consistent. ![Detectable region of the Max test. White denotes where detection is impossible for any test. Black denotes where the max test is inconsistent. Green denotes where the max test is consistent, but has suboptimal rate under the alternative compared to . Blue denotes where the max test achieves the optimal rate under the alternative. Compare to Fig. \[fig:glmsummarya\].[]{data-label="fig:maxtest"}](maxtest.pdf){width="50.00000%"} The error probabilities for the max test given by with threshold $\tau_n$ $$\begin{gathered} \Pro_{\rm FA} (n) = 1-\Phi(\tau_n)^n \label{eq:famax}\\ \Pro_{\rm MD} (n) = \big( (1- \epsilon_n) \Phi(\tau_n) + \epsilon_n \Phi(\tau_n - \mu_n) \big)^n \label{eq:mdmax}\end{gathered}$$ follow from the cumulative distribution function of the maximum of an i.i.d. sample. The rates ,, as well as the condition for inconsistency are derived by applying the approximation to and . The results of Thm. \[maxtest\] are summarized in Fig. \[fig:maxtest\]. In particular, if we take $\mu_n = \sqrt{2 r \log n}$ with $r \in \big((1-\sqrt{1-\beta})^2 , 1\big)$, we see $\log \Pro_{\rm MD}(n)$ scales on the order of $\frac{n^{1-\beta-(1-\sqrt{r})^2}}{(1- \sqrt{r}) \sqrt{2 \log n}}$. This is suboptimal compared to the rates achieved by the (non-adaptive) likelihood ratio test , but is of polynomial order (up to a sub-logarithmic factor). Note that the rate of decay of the sum error probability can be slower than that of the miss detection probability, since the false alarm probability is fixed by the choice of threshold, independent of the true $\{(\epsilon_n,\mu_n)\}$ for adaptivity. Numerical Experiments ===================== In this section, we provide numerical simulations to verify the rate characterization developed for the Gaussian location model as well as some results comparing the performance of adaptive tests. Rates for the Likelihood Ratio Test ----------------------------------- [0.48]{} ![Simulation results for Cor. \[densecase\] and \[ms\]](beta04.pdf "fig:"){width="\textwidth"} [0.48]{} ![Simulation results for Cor. \[densecase\] and \[ms\]](beta06.pdf "fig:"){width="\textwidth"} We first consider the dense case, with $\epsilon_n = n^{-0.4}$ and $\mu_n=1$. The conditions of Cor. \[densecase\] apply here, and we expect $\frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 (e^{\mu_n^2}-1)} \to -\frac{1}{8}$. Simulations were done using direct Monte Carlo simulation with $10000$ trials for the errors for $n \leq 10^6$. Importance sampling via the hypothesis alternate to the true hypothesis (i.e. $\Hyp_{0,n}$ for simulating $\Pro_{\rm MD}(n)$, $\Hyp_{1,n}$ for simulations $\Pro_{\rm FA}(n)$) was used for $10^6 < n \leq 2 \times 10^7$ with between $10000-15000$ data points. The performance of the test given is shown in Fig. \[fig:b04m1\]. The dashed lines are the best fit lines between the log-error probabilities and $n \epsilon_n^2 (e^{\mu_n^2}-1)$ using data for $n \geq 350000$. By Cor. \[densecase\], we expect the slope of the best fit lines to be approximately $-\frac{1}{8}$. This is the case, as the line corresponding to missed detection has slope $-0.13$ and the line corresponding to false alarm has slope $-0.12$. The moderately sparse case with $\epsilon_n = n^{-0.6}$ and $\mu_n = \sqrt{2 (0.19) \log n}$ is shown in Fig. \[fig:b06\]. The conditions of Cor. \[ms\] apply here, and we expect $\frac{\log \Pro_{\rm FA}(n)}{n \epsilon_n^2 (e^{\mu_n^2}-1)} \to -\frac{1}{8}$. Simulations were performed identically to the dense case. The dashed lines are the best fit lines between the log-error probabilities and $n \epsilon_n^2 (e^{\mu_n^2}-1)$ using data for $n \geq 100000$. By Cor. \[ms\], we expect the slope of the best fit lines to be approximately $-\frac{1}{8}$. Both best fit lines have slope of $-0.11$. It is important to note that $\Pro_{\rm FA}(n),\Pro_{\rm MD}(n)$ are both large even at $n=2 \times 10^{7}$ and simulation to larger sample sizes should show better agreement with Cor. \[ms\]. Adaptive Testing ---------------- In order to implement an adaptive test, the threshold for the test statistic must be chosen in order to achieve a target false alarm probability. This can be done analytically for the max test by inverting . For other tests, which do not have tractable expressions for the false alarm probability, we set the threshold by simulating the test statistic under the null. The threshold is chosen such that the empirical fraction of exceedances of the threshold matches the desired false alarm. As expected, the adaptive tests cannot match the rate under the null with non-trivial behavior under the alternative, and therefore we report the results for adaptive tests at the standard $0.05$ and $0.10$ levels. The miss detection probabilities reported for the max test were computed analytically via . Note that the likelihood ratio test with threshold set to meet a given false alarm level is the oracle test which minimizes the miss detection probability [@dz]. As multiple definitions of the Higher Criticism test exist in literature, we use the following version from [@caijengjin]: Given a sample $X_1, \ldots, X_n$, let $p_i = Q(X_i)$ for $1\leq i \leq n$. Let $\{p_{(i)}\}$ denote $\{p_i\}$ sorted in ascending order. Then, the higher criticism statistic is given by $$\text{HC}^*_n = \max_{1 \leq i \leq n} \text{HC}_{n,i} \text{ where } \text{HC}_{n,i} = \frac{ \frac{i}{n} - p_{(i)}}{\sqrt{p_{(i)} (1- p_{(i)})}} \sqrt{n}$$ and the null hypothesis is rejected when $\text{HC}^*_n$ is large. The HC test is optimally adaptive, i.e. is consistent whenever is. The ACW test [@castro] is implemented as follows: Given the samples $X_1,\ldots,X_n$, let $X_{[i]}$ denote the $i$-th largest sample by absolute value. Then, $$S^*= \max_{1\leq k \leq n} \frac{\sum_{i=1}^k \text{sgn}(X_{[i]})}{\sqrt{k}}$$ and the null hypothesis is rejected when $S^*$ is large. The ACW test is adaptive for $\beta>\frac{1}{2}$. It is unknown how the ACW test behaves for $\beta\leq \frac{1}{2}$. Note that like the Max test (and unlike the HC test), the ACW test does not exploit exact knowledge of the null distribution (but assumes continuity and symmetry about zero). [|c|c|c|]{}\ $n$ & $\pfa(n)$ & $\pmd(n)$\ $10$ & 0.307 & 0.388\ $10^2$ & 0.258 & 0.320\ $10^3$ & 0.213 & 0.256\ $10^4$ & 0.166 & 0.193\ $10^5$ & 0.119 & 0.134\ $10^6$ & 0.074 & 0.084\ 0.5cm -------- ------- ------- ------- ------- ------- ------- ------- ------- $n$ LRT Max HC ACW LRT Max HC ACW $10$ 0.776 0.845 0.790 0.807 0.665 0.744 0.666 0.706 $10^2$ 0.667 0.814 0.775 0.816 0.542 0.704 0.630 0.722 $10^3$ 0.548 0.789 0.728 0.792 0.417 0.672 0.561 0.653 $10^4$ 0.403 0.762 0.688 0.751 0.281 0.639 0.491 0.617 $10^5$ 0.252 0.733 0.623 0.685 0.158 0.603 0.396 0.539 $10^6$ 0.119 0.699 0.546 0.602 0.064 0.562 0.295 0.446 -------- ------- ------- ------- ------- ------- ------- ------- ------- : Miss Detection probabilities for $\mu_n=\sqrt{2(0.19)\log n}$, $\epsilon_n=n^{-0.6}$, for False Alarm probability 0.05 and 0.10.[]{data-label="table:spweakfa"} The performance of test is summarized in Table \[table:spweak\] with a comparison of adaptive tests in the moderately sparse example from the previous section is given in Table \[table:spweakfa\]. We used $115000$ realizations of the null and alternative. The sample sizes illustrated were chosen to be comparable with applications of sparse mixture detection, such as the WMAP data in [@cayon] which has $n \approx 7 \times 10^4$. Thus, our simulations provide evidence for both larger and smaller sample sizes than used in practice. We see there is a large gap in performance between the likelihood ratio test and the adaptive tests, but the Higher Criticism test performs significantly better than the Max or ACW tests. [|c|c|c|]{}\ $n$ & $\pfa(n)$ & $\pmd(n)$\ $10$ & 1.62e-1 & 2.75e-1\ $10^2$ & 6.31e-2 & 1.12e-1\ $10^3$ & 7.63e-3 & 1.36e-2\ $10^4$ & 5.38e-5 & 8.83e-5\ 0.5cm -------- --------- --------- ------------- --------- --------- --------- ------------- --------- $n$ LRT Max HC ACW LRT Max HC ACW $10$ 4.66e-1 5.66e-1 7.18e-1 5.88e-1 3.59e-1 4.36e-1 3.38e-1 5.88e-1 $10^2$ 1.28e-1 2.56e-1 6.24e-1 4.80e-1 8.45e-2 1.61e-1 1.07e-1 4.80e-1 $10^3$ 3.69e-3 4.40e-2 2.48e-2 1.33e-1 1.89e-3 1.80e-2 4.20e-3 1.33e-1 $10^4$ 2.12e-7 8.08e-4 $\leq$ 1e-5 4.43e-3 7.10e-8 1.32e-4 $\leq$ 1e-5 1.25e-3 -------- --------- --------- ------------- --------- --------- --------- ------------- --------- : Miss Detection probabilities for $\mu_n=\sqrt{2(0.66)\log n}$, $\epsilon_n=n^{-0.6}$ for False Alarm probability 0.05 and 0.10. []{data-label="table:spstrongfa"} For the case of strong signals, we calibrate as $\mu_n = \sqrt{2 (0.66) \log n}$ for $\epsilon_n = n^{-0.6}$. This corresponds to the rates given by Thm. \[ubthm\]. The performance of test is summarized in Table \[table:spstrong\] with a comparison of adaptive tests in the moderately sparse example from the previous section is given in Table \[table:spstrongfa\]. Here we used $180000$ realizations of the null and alternative. As even the max test has error probabilities sufficiently small for many applications in this regime at moderate sample sizes (which are still on the order used in applications [@cayon]), we only consider sample sizes up to $n=10^4$. We see that in the strong signal case, the likelihood ratio test performs better than the adaptive tests, but all tests produce sufficiently small error probabilities for most applications. Conclusions and Future Work =========================== In this paper, we have presented an rate characterization for the error probability decay with sample size in a general mixture detection problem for the likelihood ratio test. In the Gaussian location model, we explicitly showed that the rate characterization holds for most of the detectable region. A partial rate characterization (an upper bound on the rate under both hypotheses and universal lower bound on the rate under $\Hyp_{1,n}$) was provided for the remainder of the detectable region. In contrast to usual large deviations results [@cover; @dz] for the decay of error probabilities, our results show that the log-probability of error decays sublinearly with sample size. There are several possible extensions of this work. One is to provide corresponding lower bounds for the rate in cases not covered by Thm \[mthm\_main\]. Another is to provide a general analysis of the behavior that is not covered by Thm \[mthm\_main\] and \[ubthm\], present in Thm \[gausub\_main\] in the Gaussian location model. As noted in [@caijengjin], in some applications it is natural to require $\pfa(n) \leq \alpha$ for some fixed $\alpha>0$, rather than requiring $\pfa(n) \to 0$. While Thm \[dc\] shows the detectable region is not enlarged under in the Gaussian location model (and similarly for some general models [@caiwu]), it is conceivable that the oracle optimal test which fixes $\pfa(n)$ (i.e. one which compares $\text{LLR}(n)$ to a non-zero threshold) can achieve a better rate for $\pmd(n)$. It is expected that the techniques developed in this paper extend to the case where $\pfa(n)$ is constrained to a level $\alpha$. In the Gaussian location model, the analysis of constrained to level $\alpha$ problem has been studied in [@ingster] via contiguity arguments. Finally, it is important to develop tests that are amenable to a rate analysis and are computationally simple to implement over $0<\beta<1$. In the case of weak signals in the Gaussian location model, we see that the error probabilities for the likelihood ratio test, which establish the fundamental limit on error probabilities, decay quite slowly even with large sample sizes. In this case, closing the gap between the likelihood ratio test and adaptive tests is important for applications where it is desirable to have high power tests. In the case of strong signals, we see the miss detection probability for even the simplest adaptive test, the max test, are very small for moderate sample sizes at standard false alarm levels so the rate of decay is not as important as the weak signal case for applications. [24]{} (). . . (). . . , (). . . (). . . , (). . . (). . (). . . (). . . (). . (). . . (). . . (). . . (). , ed. . (). . . (). . . (). . . (). . . (). . . , (). . In . (). . In . . . ,. (). , ed. , . (). In . (). . In . . . , . [^1]: We avoid the use of the acronym CUSUM since it is reserved for the most popular test for the quickest change detection problem in Sequential Analysis.

--- abstract: 'We study Online Convex Optimization in the unbounded setting where neither predictions nor gradient are constrained. The goal is to simultaneously adapt to both the sequence of gradients and the comparator. We first develop parameter-free and scale-free algorithms for a simplified setting with hints. We present two versions: the first adapts to the squared norms of both comparator and gradients separately using $O(d)$ time per round, the second adapts to their squared inner products (which measure variance only in the comparator direction) in time $O(d^3)$ per round. We then generalize two prior reductions to the unbounded setting; one to not need hints, and a second to deal with the range ratio problem (which already arises in prior work). We discuss their optimality in light of prior and new lower bounds. We apply our methods to obtain sharper regret bounds for scale-invariant online prediction with linear models.' bibliography: - 'biblio.bib' title: 'Lipschitz and Comparator-Norm Adaptivity in Online Learning' --- Online Convex Optimization, Parameter-Free Online Learning, Scale-Invariant Online Algorithms Introduction {#sec:intro} ============ We consider the setting of online convex optimization where the goal is to make sequential predictions to minimize a certain notion of *regret*. Specifically, at the beginning of each round $t\geq 1$, a *learner* predicts $\what\w_t$ in some convex set $\cW \subseteq \reals^d$ in dimension $d\in\mathbb{N}$. The *environment* then reveals a convex loss function $f_t\colon \cW \rightarrow \reals$, and the learner suffers loss $f_t(\what\w_t)$. The goal of the learner is to minimize the regret $\sum_{t=1}^T f_t(\what \w_t) - \sum_{t=1}^T f_t(\w)$ after $T\geq 1$ rounds against any “comparator” prediction $\w\in \cW$. Typically, an online learning algorithm outputs a vector $\what\w_t$, $t\geq 1$, based only on a sequence of observed sub-gradients $(\g_s)_{s< t}$, where $\g_s\in \partial f_s(\what\w_s), s<t$. In this paper, we are interested in online algorithms which can guarantee a good regret bound (by a measure which we will make precise below) against any comparator vector $\w\in \cW$, even when $\cW$ is unbounded, and without prior knowledge of the maximum norm $L\coloneqq\max_{t\leq T}\|\g_t\|$ of the observed sub-gradients. In what follows, we refer to $L$ as the *Lipschitz constant*. By assuming an upper-bound $D>0$ on the norm of the desired comparator vector $\w$ in hindsight, there exist *Lipschitz-adaptive* algorithms that can achieve a sub-linear regret of order $L D \sqrt{T}$, without knowing $L$ in advance. A Lipschitz-adaptive algorithm is also called *scale-free* (or scale-invariant) if its predictions do not change when the loss functions $(f_t)$ are multiplied by a factor $c>0$; in this case, its regret bound is expected to scale by the same factor $c$. When $L$ is known in advance and $\cW=\reals^d$, there exists another type of algorithms, so-called *parameter-free*, which can achieve a $\bigto(\|\w\| L \sqrt{T})$ regret bound, where $\w$ is the desired comparator vector in hindsight (the notation $\bigto$ hides log-factors). Up to an additive lower-order term, this type of regret bound is also achievable for bounded $\cW$ via the unconstrained-to-constrained reduction [@cutkosky2019artificial]. The question of whether an algorithm can simultaneously be *scale-free* and *parameter-free* was posed as an open problem by [@orabona2016open]. It was latter answered in the negative by [@cutkosky2017online]. Nevertheless, [@cutkosky2019artificial] recently presented algorithms which achieve a $\bigto(\|\w\| L\sqrt{T}+ L \|\w\|^3)$ regret bound, without knowing either $L$ or $\norm{\w}$. This does not violate the earlier lower bound of [@cutkosky2017online], which insists on norm dependence $\wtilde O(\norm{\w})$. Though [@cutkosky2019artificial] designed algorithms that can to some extent adapt to both $L$ and $\norm{\w}$, their algorithms are still *not* scale-free. Multiplying $(f_t)$, and as a result $(\g_t)$, by a positive factor $c>0$ changes the outputs $(\what\w_t)$ of their algorithms, and their regret bounds scale by a factor $c'$, not necessarily equal to $c$. Their algorithms depend on a parameter $\epsilon>0$ which has to be specified in advance. This parameter appears in their regret bounds as an additive term and also in a logarithmic term of the form $\log (L^{\alpha} /\epsilon)$, for some $\alpha>1$. As a result of this type of dependence on $\epsilon$ and the fact that $\alpha>1$, there is no prior choice of $\epsilon$ which can make their regret bounds scale-invariant. What is more, without knowing $L$, there is also no “safe” choice of $\epsilon$ which can prevent the $\log (L^{\alpha} /\epsilon)$ term from becoming arbitrarily large relative to $L$ (it suffices for $\epsilon$ to be small enough relative to the “unknown” $L$). #### Contributions. Our main contribution is a new scale-free, parameter-free learning algorithm for OCO with a regret at most $O(\|\w\| \sqrt{V_T\log (\|\w\|T)})$, for any comparator $\w \in \cW$ in a bounded set $\cW$, where $V_T \coloneq \sum_{t=1}^T \|\g_t\|^2$. When the set $\cW$ is unbounded, the algorithm achieves the same guarantee up to an additive $O(L \sqrt{\max_{t\leq T}B_t}+ L \|\w\|^3)$, where $B_t\coloneqq \sum_{s=1}^t\| \g_s \|/L_t$ and $L_t \df \max_{s \le t} \norm{\g_s}$, for all $t\in[T]$. In the latter case, we also show a matching lower bound; when $\cW$ is unbounded and without knowing $L$, any online learning algorithm which insists on a $\bigto(\sqrt{T})$ bound, has regret at least $\Omega (L \sqrt{B_T}+ L \|\w\|^3)$. We also provide a second scale-invariant algorithm which replaces the leading $\|\w\| \sqrt{V_T}$ term in the regret bound of our first algorithm by $\sqrt{\w^\top {\bm{V}}_T \w \ln \det \bm V_T}$, where ${\bm{V}}_T \coloneqq \sum_{t=1}^T\g_t \g_t^\top$. Our starting point for designing our algorithms is a known potential function which we show to be controlled for a unique choice of output sequence $(\what\w_t)$. As our main application, we show how our algorithms can be applied to learn linear models. The result is an online algorithm for learning linear models whose label predictions are invariant to coordinate-wise scaling of the input feature vectors. The regret bound of the algorithm is naturally also scale-invariant and improves on the bounds of existing state-of-the-art algorithms in this setting [@Kotlowski17; @KempkaKW19]. #### Related Work For an overview of Online Convex Optimization in the bounded setting, we refer to the textbook [@HazanOCOBook2016]. The unconstrained case was first studied by [@McMahanStreeter2010]. A powerful methodology for the unbounded case is Coin Betting by [@orabona2016coin]. Even though not always visible, our potential functions are inspired by this style of thinking. We build our unbounded OCO learner by targeting a specific other constrained problem. We further employ several general reductions from the literature, including gradient clipping [@cutkosky2019artificial], the constrained-to-unconstrained reduction [@cutkosky2018], and the restart wrapper to pacify the final-vs-initial scale ratio appearing inside logarithms by [@mhammedi19]. Our analysis is, at its core, proving a certain minimax result about sufficient-statistic-based potentials reminiscent of the Burkholder approach pioneered by [@FosterRakhlinSridharan2017; @Foster2018]. Applications for scale-invariant learning in linear models were studied by [@KempkaKW19]. For our multidimensional learner we took inspiration from the Gaussian Exp-concavity step in the analysis of the MetaGrad algorithm by [@Erven2016]. #### Outline In Section \[sec:prelim\], we present the setting and notation, and formulate our goal. In Section \[sec:mainalg\], we present our main algorithms. In Section \[sec:lower\], we present new lower-bounds for algorithms which adapt to both the Lipschitz constant and the norm of the comparator. In Section \[sec:linear\], we apply our algorithms to online prediction with linear models. Preliminaries {#sec:prelim} ============= Our goal is to design scale-free algorithms that adapt to the Lipschitz constant $L$ and comparator norm $\norm{\w}$. We will first introduce the setting, then discuss existing reductions, and finally state what needs to be done to achieve our goal. Setting and Notation -------------------- Let $\cW \subseteq \reals^d, d\in \mathbb{N},$ be a convex set, and assume without loss of generality that $\bm{0}\in \cW$. We allow the set $\cW$ to be unbounded, and we define its (possibly infinite) diameter $D\coloneqq\sup_{\w,\w' \in \cW} \|\w -\w'\|\in[0,+\infty]$. We consider the setting of Online Convex Optimization (OCO) where at the beginning of each round $t\geq 1$, the learner outputs a prediction $\what\w_t \in \cW$, before observing a convex loss function $f_t: \cW \rightarrow \reals$, or an element of its sub-gradient $\g_t \in \partial f_t(\what \w_t)$ at $\what\w_t$. The goal of the learner is to minimize the regret after $T\geq 0$ rounds $$\begin{aligned} \sum_{t=1}^T f_t(\what\w_t) - \sum_{t=1}^T f_t(\w) \quad\label{eq:linupper}\end{aligned}$$ for any comparator vector $\w\in \cW$. In this paper, we do not assume that $T$ is known to the learner, and so we are after algorithms with so called *any-time* guarantees. By convexity, we have $$\begin{aligned} \sum_{t=1}^T f_t(\what\w_t) - \sum_{t=1}^T f_t(\w) \leq \sum_{t=1}^T \inner{\g_t}{\what\w_t -\w}, \quad \text{for all $\w\in \cW$}, \label{eq:lin}\end{aligned}$$ and thus for the purpose of minimizing the regret, typical OCO algorithms minimize the RHS of , which is known as the *linearized regret*, by generating outputs $(\what\w_t)$ based on the sequence of observed sub-gradients $(\g_t)$. Likewise, we focus our attention exclusively on linear optimization. Given a sequence of sub-gradients $(\grad_t)$, it will be useful to define the running maximum gradient norm and the clipped sub-gradients $$\begin{aligned} \label{eq:Lip} L_t \coloneqq \max_{s\in [t]} \| \g_s\| \quad \text{and} \quad \bar\g_t\coloneq \g_t \cdot L_{t-1}/L_t, \end{aligned}$$ for $t\geq 1$, with the convention that $L_0=0$. We also drop the subscript $t$ from $L_t$ when $t=T$, *i.e.* we write $L$ for $L_T$. We denote by $\text{A}(\g_1,\dots, \g_{t-1};h_t)$ the output in round $t \ge 1$ of an algorithm <span style="font-variant:small-caps;">A</span>, which uses the observed sub-gradients so far and a *hint* $h_t \ge L_t$ on the upcoming sub-gradient $\g_t$. As per Section \[sec:intro\], we say that an algorithm is *scale-free* (or scale-invariant) if its predictions are invariant to any common positive scaling of the loss functions $(f_t)$ and, if applicable, the hints. #### Additional Notation. Given a closed convex set $\mathcal{X} \subseteq \reals^d$, we denote by $\Pi_{\mathcal{X}}(\bm x)$ the Euclidean projection of a point $\bm x \in \reals^d$ on the set $\mathcal X$; that is, $\Pi_{\mathcal{X}}(\bm x) \in \argmin_{\tilde{\bm{x}} \in \mathcal X} \| \bm{x} -\tilde{\bm{x}}\|$. Helpful Reductions {#sec:helpful} ------------------ The difficulty behind designing scale-free algorithms lies partially in the fact that $L_{t}$ is not-known at the start of round $t$; before outputting $\what\w_{t}$. The following result due to [@cutkosky2019artificial] quantifies the additional cost of proceeding with the plug-in estimate $L_{t-1}$ for $L_t$: \[lem:reduc1\] Let $\textsc{A}$ be an online algorithm which at the start of each round $t\geq 1$, has access to a hint $h_t\geq L_t$, and outputs $\textsc{A}(\g_1,\dots,\g_{t-1};h_t)\in \cW$, before observing $\g_t$. Suppose that $\textsc{A}$ guarantees an upper-bound $R^{\textsc{A}}_T(\w)$ on its linearized regret for the sequence $(\g_t)$ and for all $\w\in \cW, T\geq 1$. Then, algorithm $\textsc{B}$ which at the start of each round $t\geq 1$ outputs $\what\w_t = \textsc{A}(\bar{\g}_1,\dots, \bar{\g}_{t-1};L_{t-1})$, guarantees $$\begin{aligned} \sum_{t=1}^T \inner{\what\w_t-\w}{\g_t} \leq R^{\textsc{A}}_T(\w) + \max_{t\in[T]} \|\what\w_t\| L_t +\| \w \| L , \quad \text{$\forall \w\in \cW, T\geq 1.$} \label{eq:boundtransfer} \end{aligned}$$ First, we note that Lemma \[lem:reduc1\] is only really useful when $\cW$ is bounded; otherwise, depending on algorithm $\textsc{A}$, the term $\max_{t\in[T]}L_t \|\what\w_t\|$ on the RHS of  could in principle be arbitrarily large even for fixed $\w, L$, and $T$. The moral of Lemma \[lem:reduc1\] is that as long as the set $\cW$ is bounded, one does not really need to know $L_t$ before outputting $\what\w_t$ to guarantee a “good” regret bound against any $\w \in \cW$. For example, suppose that $\cW$ has a bounded diameter $D$ and algorithm <span style="font-variant:small-caps;">A</span> in Lemma \[lem:reduc1\] is such that ${R}^{\textsc{A}}_T(\w) = \wtilde{O}(\|\w\| L \sqrt{T} + D L)$, for all $\w\in \cW$. Then, from  and the fact that $\|\what\w_t\|\leq D$ (since $\what\w_t\in \cW$), it is clear that algorithm $\textsc{B}$ in Lemma \[lem:reduc1\] also guarantees the same regret bound $R_T^{\textsc{A}}(\w)$ up to an additive $2 DL$, despite not having had the hints $(h_t)$. It is possible to extend the result of Lemma \[lem:reduc1\] so that the regret bound of algorithm <span style="font-variant:small-caps;">B</span> remains useful even in the case where $\cW$ is unbounded. An approach suggested by [@cutkosky2019artificial] is to restrict the outputs $(\what\w_t)$ of algorithm <span style="font-variant:small-caps;">B</span> to be in a non-decreasing sequence $(\cW_t)$ of *bounded* convex subsets of $\cW$. In this case, the diameters $(D_t) \subset \reals$ of $(\cW_{t})$ need to be carefully chosen to achieve a desired regret bound. This approach, which essentially combines the idea of Lemma \[lem:reduc1\] and the unconstrained-to-constrained reduction due to [@cutkosky2018], is formalized in the next lemma (essentially due to [@cutkosky2019artificial]): \[lem:reduc2\] Let algorithm $\textsc{A}$ be as in Lemma \[lem:reduc1\], and let $(\cW_t)$ be a sequence of non-decreasing closed convex subsets of $\cW$ with diameters $(D_t)\subset \reals_{>0}$. Then, algorithm $\textsc{B}$ which at the start of round $t\geq 1$ outputs $\what\w_t = \Pi_{\cW_t}(\wtilde{\w}_t)$, where $$\begin{gathered} \wtilde{\w}_t\coloneqq \textsc{A}(\wtilde{\g}_1,\dots, \wtilde{\g}_{t-1};L_{t-1}) \ \ \ \text{and} \ \ \ \wtilde{\g}_s \coloneqq (\bar \g_s + \|\bar \g_s\| \cdot (\wtilde{\w}_s -\what{\w}_s) /\| \wtilde{\w}_s -\what{\w}_s\|)/2, \ \ s< t, \shortintertext{guarantees, for all $\w\in \cW$ and $T\geq 1,$} \sum_{t=1}^T \inner{\what\w_t-\w}{\g_t} \leq R^{\textsc{A}}_T(\w) + \sum_{t=1}^T \|\g_t\| \cdot \|\w -\Pi_{\cW_t}(\w)\|+ L D_T +L \| \w\|.\label{eq:boundtransfer2} \end{gathered}$$ We see that compared to Lemma \[lem:reduc1\], the additional penalty that algorithm <span style="font-variant:small-caps;">B</span> incurs for restricting its predictions to the sets $\cW_1,\dots, \cW_T \subseteq \cW$ is the sum $\sum_{t=1}^T \|\g_t\| \cdot \|\w -\Pi_{\cW_t}(\w)\|$. The challenge is now in choosing the diameters $(D_t)$ to control the trade-off between this sum and the term $L D_T$ on the RHS of . If $T$ is known in advance, one could set $D_1=\dots=D_T=\sqrt{T}$, in which case the RHS of  is at most $$\begin{aligned} R_T^{\textsc{A}}(\w) + L(\|\w\|^3+\|\w\|) + L\sqrt{T}. \label{eq:naive} \end{aligned}$$ We now instantiate the bound of Lemma \[lem:reduc2\] for another choice of $(D_t)$ when $T$ is unknown: \[cor:instan\] In the setting of Lemma \[lem:reduc2\], let $\cW_t$ be the ball of diameter $D_t \coloneqq \sqrt{\max_{s\leq t} B_s}$, $t\geq 1$, where $B_t\coloneqq \sum_{s=1}^t \|\g_s\|/L_t$, and let $\cW=\reals^d$. Then the RHS of  is bounded from above by $$\begin{aligned} R_T^{\textsc{A}}(\w)+ L \|\w\|^3 + L \sqrt{\max_{t\in[T]} B_t} +L \|\w\|, \quad \forall \w \in \cW =\reals^d, T\geq 1. \label{eq:neq}\end{aligned}$$ We see that by the more careful choice of $(D_t)$ in Corollary \[cor:instan\], one can replace the $L \sqrt{T}$ term in by the smaller quantity $L \sqrt{\max_{t\in[T]} B_t}$; whether this can be improved further to $\sqrt{V_T}$, where $V_T = \sum_{t=1}^T \|\g_t\|^{2}$, was raised as an open question by [@cutkosky2019artificial]. We will answer this in the negative in Theorem \[thm:lower1\]. We will also show in Theorem \[thm:lower2\] bellow that, if one insists on a regret of order $\wtilde{O}(\sqrt{T})$, it is essentially not possible to improve on the penalty $L \|\w\|^3$ in . Outlook ------- The conclusion that should be drawn from Lemmas \[lem:reduc1\] and \[lem:reduc2\] is the following; if one seeks an algorithm <span style="font-variant:small-caps;">B</span> with a regret bound of the form $\wtilde O(\|\w\| L \sqrt{T})$ up to some lower-order terms in $T$, without knowledge of $L$ and regardless of whether $\cW$ is bounded or not, it suffices to find an algorithm <span style="font-variant:small-caps;">A</span> which guarantees the sought type of regret whenever it has access to a sequence of hints $(h_t)$ satisfying (as in Lemmas \[lem:reduc1\] and \[lem:reduc2\]), $h_t \geq L_t$, for all $t\geq 1$. Thus, our first goal in the next section is to design a *scale-free* algorithm $\textsc{A}$ which accesses such a sequence of hints and ensures that its linearized regret is bounded from above by: $$\begin{aligned} O\left(\|\w\| \sqrt{V_T \ln (\|\w\| V_T)}\right), \ \ \ \text{where} \ \ V_T \coloneqq h_1^2+ \sum_{t=1}^T \|\g_t\|^2, \label{eq:freegrad0}\end{aligned}$$ for all $\w \in \reals^d, T\geq 0$, and $(\g_t)\subset\reals^d$. We show an analogous “full-matrix” upgrade of order $\sqrt{\w^\top \bm V \w \ln \del*{\w^\top \bm V \w \det \bm V}}$, with $\bm V = \sum_{t=1}^T \g_t \g_t^\top$. We note that if Algorithm <span style="font-variant:small-caps;">A</span> in Lemmas \[lem:reduc1\] and \[lem:reduc2\] is scale-free, then so is the corresponding Algorithm <span style="font-variant:small-caps;">B</span>. If the desired set $\cW$ has bounded diameter $D>0$, then using the unconstrained-to-constrained reduction due to [@cutkosky2018], it is straightforward to design a new algorithm based on $\textsc{A}$ with regret also bounded by up to an additive $L D$, for $\w\in \cW$ (this is useful for Lemma \[lem:reduc1\]). Finally, we also note that algorithms which can access hints $(h_t)$ such that $h_t \geq L_t$, for all $t\geq 1$, are of independent interest; in fact, it is the same algorithm $\textsc{A}$ that we will use in Section \[sec:linear\] as a scale-invariant algorithm for learning linear models. Scale-Free, Parameter-Free Algorithms for OCO {#sec:mainalg} ============================================= In light of the conclusions of Section \[sec:prelim\], we will design new unconstrained scale-free algorithms which can access a sequence of hints $(h_t)$ (as in Lemma \[lem:reduc1\]) and guarantee a regret bound of the form given in . In this section, we will make the following assumption on the hints $(h_t)$: \[assum:assum2\] We assume that (i) $(h_t)$ is a non-decreasing sequence; (ii) $h_t \geq L_t$, for all $t\geq 1$; and (iii) if the sub-gradients $(\g_s)$ are multiplied by a factor $c>0$, then the hints $(h_t)$ are multiplied by the same factor $c$. The third item of the assumption ensures that our algorithms are scale-free. We note that Assumption \[assum:assum2\] is satisfied by the sequence of hints that Algorithm <span style="font-variant:small-caps;">B</span> constructs when invoking Algorithm <span style="font-variant:small-caps;">A</span> in Lemmas \[lem:reduc1\] and \[lem:reduc2\]. For simplicity, we will also make the following assumption, which is without loss of generality, since the regret is zero while $\g_t = \vzero$. \[assum:assum1\] We assume that $L_1 = \norm{\g_1} > 0$. [$\textsc{FreeGrad}$]{}: An Adaptive Scale-Free Algorithm {#sec:commonVgrad} --------------------------------------------------------- In this subsection, we design a new algorithm based on a time-varying potential function, where the outputs of the algorithm are uniquely determined by the gradients of the potential function at its iterates—an approach used in the design of many existing algorithms [@CesaBianchiEtAl1997]. Let $t\geq 1$, $(\g_s)_{s\leq t}\subset\reals^d$ be a sequence of sub-gradients satisfying Assumption \[assum:assum1\], and $(h_t)$ be a sequence of hints satisfying Assumption \[assum:assum2\]. Consider the following potential function: $$\begin{aligned} &\Phi_t \coloneqq S_t + \frac{h_1^2}{\sqrt{V_t}} \cdot \exp\left( \frac{ \| \mathbf{G}_t \|^2}{2V_t + 2h_{t} \|\mathbf{G}_t \| } \right), \quad t\geq 0, \label{eq:potential} \\ \text{where }\ \ \ \ &S_t \coloneqq \sum_{s=1}^t \inner{\grad_s}{\what\w_s}, \quad \mathbf{G}_t \coloneqq \sum_{s=1}^t \grad_{s}, \quad V_t \coloneqq h_1^2 + \sum_{s=1}^t \|\g_s\|^2 . \label{eq:statistics} \end{aligned}$$ This potential function has appeared as a by-product in the analyses of previous algorithms such as the ones in [@cutkosky2018; @cutkosky2019artificial]. The expression of $\Phi_t$ in is interesting to us since it can be shown via the *regret-reward duality* [@mcmahan2014] (as we do in the proof of Theorem \[thm:firstbound\] below) that any algorithm which outputs vectors $(\what\w_t)$ such that $(\Phi_{t})$ is non-increasing for any sequence of sub-gradients $(\g_t)$, also guarantees a regret bound of the form . We will now design such an algorithm. Consider the unconstrained algorithm [[$\textsc{FreeGrad}$]{}]{} which at the beginning of round $t\geq 1$, uses the sequence of sub-gradients $(\g_s)_{s< t}$ seen so far and the available hint $h_{t} \ge L_{t}$ to output: $$\begin{aligned} \what \w_{t} \coloneqq -\G_{t-1} \cdot \frac{ (2V_{t-1} +h_{t} \|\G_{t-1}\|)\cdot h_1^2 }{2(V_{t-1}+h_{t} \|\G_{t-1} \|)^2 \ \sqrt{{}V_{t-1}}} \cdot \exp\left(\frac{\|\G_{t-1}\|^2}{2 V_{t-1} + 2h_{t} \|\G_{t-1}\|} \right). \label{eq:predictunbounded} \end{aligned}$$ where $(\G_t)$ and $(V_t)$ are as in . The output in is obtained by setting the gradient $\nabla_{\g_{t}}\Phi_{t}$ at $\g_{t}=\bm{0}$ to the zero vector, and solving the resulting equation for $\what\w_{t}\in \reals^d$. Thus, for any $\what\w_{t}$ other than the one in , one can find a vector $\g_{t}$ such that $\|\g_{t}\|\leq h_{t}$, and $\Phi_{t}>\Phi_{t-1}$. Therefore, given that our aim is to find a sequence $(\what\w_{t})$ which makes $(\Phi_{t})$ non-increasing, the outputs in are the *unique* candidates. Our main technical contribution in this subsection is to show that, in fact, with the choice of $(\what\w_{t})_{t\geq 1}$ as in , the potential functions $(\Phi_t)$ are non-increasing for any sequence of sub-gradients $(\g_t)$: \[thm:unconstlearner\] For $(\what\w_{t})$, and $(\Phi_t)$ as in , and , under Assumptions \[assum:assum2\] and \[assum:assum1\], we have: $$\begin{aligned} \Phi_T \leq \dots \leq \Phi_0 =h_1, \quad \text{for all $T\geq 1$.} \end{aligned}$$ The proof of the theorem is postponed to Appendix \[sec:potentialdiff\]. Theorem \[thm:unconstlearner\] and the regret-reward duality [@mcmahan2014] yield a regret bound for the algorithm that outputs the sequence $(\what \w_t)$. In fact, if $\Phi_T \leq \Phi_0$, then by the definition of $\Phi_T$ in , we have $$\begin{aligned} \hspace{-0.4cm} \sum_{t=1}^T \inner{\g_t}{\what\w_t} \leq \Phi_0 -\Psi_T(\G_T), \ \ \text{where} \ \ \Psi_T(\G)\coloneqq \frac{h_1^2}{\sqrt{{}V_T}} \exp\left( \frac{ \| \mathbf{G} \|^2}{2V_T + 2h_{T} \|\mathbf{G} \| } \right),\ \ \G \in \reals^d. \label{eq:dual} \end{aligned}$$Now by Fenchel’s inequality, we have $-\Psi_T(\G_T) \leq \inner{\w}{\G_T} +\Psi^\star_T(-\w)$, for all $\w\in \reals^d$, where $\Psi^{\star}_T(\w)\coloneqq \sup_{\z \in \reals^d}\{ \inner{\w}{\bm{z}} -\Psi_T(\z)\}$, $\w\in \reals^d$, is the Fenchel dual of $\Psi_T$ [@Hiriart-Urruty]. Combining this with , we obtain: $$\begin{aligned} \sum_{t=1}^T \inner{\g_t}{\what\w_t} \leq \inf_{\w\in \reals^d}\left\{ \sum_{t=1}^T \inner{\g_t}{\w} + \Psi^\star_T(-\w) +\Phi_0 \right\}, \label{eq:regret}\end{aligned}$$ Rearranging for a given $\w\in \reals^d$ leads to a regret bound of $\Psi^\star_T(-\w)+\Phi_0$. Further bounding this quantity using existing results due to [@cutkosky2018; @cutkosky2019artificial; @mcmahan2014], leads to the following regret bound (the proof is in Appendix \[sec:epsilon\]): \[thm:firstbound\] Under Assumptions \[assum:assum2\] and \[assum:assum1\], for $(\what\w_{t})$ as in , we have, with $\ln_+ (\cdot) \coloneqq 0\vee \ln (\cdot)$, $$\begin{aligned} \sum_{t=1}^T \inner{\g_t}{\what \w_t-\w} \leq \left[ 2 \|\w\| \sqrt{V_T\ln_+ \left(\frac{ 2\|\w\| V_T }{h_1^2} \right)} \right] \vee \left[ 4 h_T \|\w\| \ln \left( \frac{4 h_T \|\w\| \sqrt{V_T} }{ h_1^2} \right) \right]+h_1 , \end{aligned}$$ for all $\w \in \cW = \reals^d, T\geq 1$. #### Range-Ratio Problem. While the outputs $(\what\w_t)$ in of [[$\textsc{FreeGrad}$]{}]{} are scale-free for the sequence of hints $(h_t)$ satisfying Assumption , there remains one serious issue; the fractions $V_T/h_1^2$ and $h_T/h_1$ inside the log-terms in the regret bound of Theorem \[thm:firstbound\] could in principle be arbitrarily large if $h_1$ is small enough relative to $h_T$. Such a problematic ratio has appeared in the regret bounds of many previous algorithms which attempt to adapt to the Lipschitz constant $L$ [@ross2013normalized; @Wintenberger2017; @Kotlowski17; @mhammedi19; @KempkaKW19]. When the output set $\cW$ is bounded with diameter $D>0$, this ratio can be dispensed of using a recently proposed restart trick due to [@mhammedi19], which restarts the algorithm whenever $L_t/L_1> \sum_{s=1}^t \|\g_s\|/L_s$. The price to pay for this is merely an additive $O(LD)$ in the regret bound. However, this trick does not directly apply to our setting since in our case $\cW$ may be unbounded. Fortunately, we are able to extend the analysis of the restart trick to the unbounded setting where a sequence of hints $(h_t)$ satisfying Assumption \[assum:assum2\] is available; the cost we incur in the regret bound is an additive lower-order $\wtilde{O}(\|\w\| L)$ term. Algorithm \[alg:wrapper\] displays our restart “wrapper”, [$\textsc{FreeRange}$]{}, which uses the outputs of [[$\textsc{FreeGrad}$]{}]{} to guarantee the following regret bound (the proof is in Appendix \[sec:proofs\]): \[thm:freegrad1\] Let $(\what\w_t)$ be the outputs of [$\textsc{FreeRange}$]{} (Algorithm \[alg:wrapper\]). Then, $$\begin{gathered} \sum_{t=1}^T \inner{\g_t}{\what \w_t-\w} \leq 2 \|\w\| \sqrt{2V_T\ln_+ \left(\|\w\| b_T \right)} + h_T\cdot (16 \|\w\| \ln_+ (2 \|\w\| b_T)+2\|\w\| + 3),\end{gathered}$$ for all $\w \in \reals^d, T\geq 1$, and $(\g_t)\subset \reals^d$, where $b_T\coloneqq 2\sum_{t=1}^T\del[\big]{\sum_{s=1}^{t-1} \frac{\|\g_s\|}{h_s}+2}^2\leq (T+1)^3$. Hints $(h_t)$ satisfying Assumption \[assum:assum2\]. Set $\tau=1$; \[line:1\] Observe hint $h_{t}$; Set $\tau=t$; \[line:6\] Output $\what\w_{t}$ as in with $(h_1,V_{t-1},\G_{t-1})$ replaced by $(h_\tau$, $h_\tau^2 +\sum_{s=\tau}^{{t-1}}\|\g_s\|^2$,  $\sum_{s=\tau}^{{t-1}}\g_s$); \[line:7\] We next introduce our second algorithm, in which the variance is only measured in the comparator direction; the algorithm can be viewed as a “full-matrix” version of [$\textsc{FreeGrad}$]{}. [$\textsc{Matrix-FreeGrad}$]{}: Adapting to Directional Variance {#sec:multidim} ---------------------------------------------------------------- Reflecting on the previous subsection, we see that the potential function that we ideally would like to use is $S_t+ h_1 \exp \del*{\frac{1}{2} \G_t^\top \bm V_t^{-1} \G_t - \frac{1}{2} \ln \det \bm V_t}$, $t\geq 1$, where $\bm V_t =\sum_{s=1}^t \g_s \g_s^\top$. However, as we saw, this is a little too greedy even in one dimension, and we need to introduce some slack to make the potential controllable. In the previous subsection we did this by increasing the scalar denominator from $V$ to $V + \norm{\G}$, which acts as a barrier function restricting the norm of $\what\w_t$. In this section, we will instead employ a hard norm constraint. We will further need to include a fudge factor $\gamma > 1$ multiplying $\bm V$ to turn the above shape into a bona fide potential. To describe its effect, we define $$\begin{aligned} \rho(\gamma) \coloneqq \frac{1}{ 2 \gamma }\del[\Big]{ \sqrt{(\gamma +1)^2-4 e^{\frac{1}{2 \gamma } - \frac{1}{2}} \gamma ^{3/2}} + \gamma - 1 }, \quad \text{for $\gamma \ge 1$.} \label{def.xi} \end{aligned}$$ The increasing function $\rho$ satisfies $\lim_{\gamma \to 1} \rho(\gamma) = 0$, $\lim_{\gamma \to \infty} \rho(\gamma) = 1$, and $\rho(2) = 0.358649$. The potential function of this section is parameterized by a *prod factor* $\gamma > 1$ (which we will set to some universal constant). We define $$\begin{aligned} \Psi(\G, \bm V, h) \coloneqq \frac{h_1 \exp \del*{ \inf_{\lambda \ge 0} \set*{ \frac{1}{2} \G^\top \del*{\gamma h_1^2 \bm I + \gamma \bm V + \lambda \bm I}^{-1} \G + \frac{\lambda \rho(\gamma)^2}{2 h^2} } } }{ \sqrt{\det\del*{\bm I + \frac{1}{h_1^2} \bm V}} } , \label{eq:matrixpotential}\end{aligned}$$ where $\G \in \reals^d$, ${\bm{V}}\in \reals^{d\times d}$, and $h>0$. Given a sequence of sub-gradients $(\g_s)_{s< t}$, $t\geq 1$, and a hint $h_{t}\geq L_T$, we obtain the prediction at round $t$ from the gradient of $\Psi$ in the first argument $$\label{eq:FTLR.multid} \what\w_{t} ~\df~ - \nabla^{(1,0,0)} \Psi(\G_{t-1}, \bm V_{t-1}, h_{t}),$$ where $\G_{t-1} = \sum_{s=1}^{t-1} \g_s$ and $\bm V_{t-1} \coloneqq \sum_{s=1}^{t-1} \g_s \g_s^\top$. We can compute $\what\w_{t}$ in $O(d^3)$ time per round by first computing an eigendecomposition of $\bm V_{t-1}$, followed by a one-dimensional binary search for the $\lambda_\star$ which achieves the $\inf$ in with $(\G,{\bm{V}}, h) = (\G_{t-1},{\bm{V}}_{t-1},h_t)$. Then the output is given by $$\begin{aligned} \what\w_t = - \Psi(\G_{t-1}, \bm V_{t-1}, h_t) \cdot \del*{\gamma h_1^2 \bm I + \gamma \bm V_{t-1} + \lambda_\star \bm I}^{-1} \G_{t-1}.\end{aligned}$$ Our heavy-lifting step in the analysis is the following, which we prove in Appendix \[sec:pf.multidim\]: \[lemma:multidim.control\] For any vector $\g_t\in \reals^d$ and $h_t>0$ satisfying $\norm{\g_t} \leq h_t$, the vector $\what\w_t$ in ensures $$\g_t^\top \what\w_t ~\le~ \Psi(\G_{t-1}, \bm V_{t-1}, h_t) - \Psi(\G_t, \bm V_t, h_t).$$ From here, we obtain our main result using telescoping and regret-reward duality: \[thm:multidim\] Let $\bm \Sigma^{-1}_T \df \gamma h_1^2 \bm I + \gamma \bm V_T$. For $(\what\w_t)$ as in , we have $$\sum_{t=1}^T \tuple*{\what\w_t - \w, \g_t} ~\le~ h_1 + \sqrt{Q_T^\w \ln_+\del*{\frac{ \det\del*{\gamma h_1^2 \bm \Sigma_T}^{-1} }{ h_1^2 } Q_T^\w}},\quad \text{for all $\w\in \reals^d$, where}$$ $$Q_T^\w \coloneqq \max \set*{ \w^\top \bm \Sigma^{-1}_T \w , \frac{1}{2} \del*{\frac{h_T^2 \norm{\w}^2}{\rho(\gamma)^2} \ln \del*{ \frac{ \det\del*{\gamma h_1^2 \bm \Sigma_T}^{-1} }{ h_1^2 } \frac{h_T^2 \norm{\w}^2}{\rho(\gamma)^2} } + \w^\top \bm \Sigma^{-1}_T \w} } .$$ Note in particular that the result is scale-free. Expanding the main case of the theorem (modest $\norm{\w}$), we find regret bounded by $$\sum_{t=1}^T \tuple*{\what\w_t - \w, \g_t} ~\le~ h_1 + h_1 \sqrt{\gamma \w^\top \bm Q \w \ln_+\del*{\gamma \w^\top \bm Q \w \det \bm Q } } \quad \text{where} \quad \bm Q = \bm I + \bm V_T/h_1^2 .$$ This bound looks almost like an ideal upgrade of that in Theorem \[thm:firstbound\], though technically, the bounds are not really comparable since the $\ln \det \bm Q$ can be as large as $d \ln T$, potentially canceling the advantage of having $\w^\top \bm Q \w$ instead of $\|\w\|^2 \sum_{t=1}^T \|\g_t\|^2$ inside the square-root. The matrix $\bm Q$ and hence any directional variance $\w^\top \bm Q \bm w$ is scale-invariant. The only fudge factor in the answer is the $\gamma > 1$. We currently cannot tolerate $\gamma = 1$, for then $\rho(\gamma) = 0$ so the lower-order term would explode. We note that a bound of the form given in the previous display, with the $\ln \det \bm Q$ replaced by the larger term $d \ln \op{tr} \bm Q$, was achieved by a previous (not scale-free) algorithm due to [@cutkosky2018]. \[rem:restartremark\] As Theorem \[thm:freegrad1\] did in the previous subsection, our restarts method allows us to get rid of problematic scale ratios in the regret bound of Theorem \[thm:multidim\]; this can be achieved using [$\textsc{FreeRange}$]{} with $(\what\w_t)$ set to be as in instead of . The key idea behind the proof of Theorem \[thm:freegrad1\] is to show that the regrets from all but the last two epochs add up to a lower-order term in the final regret bound. This still holds when $(\what\w_t)$ are the outputs of [$\textsc{Matrix-FreeGrad}$]{} instead [$\textsc{FreeGrad}$]{}, since by Theorem \[thm:multidim\], the regret bound of [$\textsc{Matrix-FreeGrad}$]{} is of order at most $d$ times the regret of [$\textsc{FreeGrad}$]{} within any given epoch. As a final note about the algorithm, we may also develop a “one-dimensional” variant by replacing matrix inverse and determinant by their scalar analogues applied to $V_T = \sum_{t=1}^T \norm{\g_t}^2$. One effect of this is that the minimization in $\lambda$ can be computed in closed form. The resulting potential and corresponding algorithm and regret bound are very close to those of Section \[sec:commonVgrad\]. #### Conclusion The algorithms designed in this section can now be used in the role of algorithm <span style="font-variant:small-caps;">A</span> in the reductions presented in Section \[sec:helpful\]. This will yield algorithms which achieve our goal; they adapt to the norm of the comparator and the Lipschitz constant and are completely scale-free, for both bounded and unbounded sets, without requiring hints. We now show that the penalties incurred by these reductions are not improvable. Lower Bounds {#sec:lower} ============ As we saw in Corollary \[cor:instan\], given a base algorithm <span style="font-variant:small-caps;">A</span>, which takes a sequence of hints $(h_t)$ such that $h_t\geq L_t$, for all $t\geq 1$, and suffers regret $R^{\textsc{A}}_T(\w)$ against comparator $\w\in \cW$, there exists an algorithm <span style="font-variant:small-caps;">B</span> for the setting without hints which suffers the same regret against $\w$ up to an additive penalty $L_T \|\w\|^3+ L_T \sqrt{\max_{t\in[T]} B_t},$ where $B_t = \sum_{s=1}^t \|\g_s\|/L_t$. In this section, we show that the penalty $L_T \|\w\|^3$ is not improvable if one insists on a regret bound of order $\wtilde{O}(\sqrt{T})$. We also show that it is not possible to replace the penalty $L_T \sqrt{\max_{t\in[T]} B_t}$ by the typically smaller quantity $\sqrt{V_T}$, where $V_T=\sum_{t=1}^T \|\g_t\|^2$. Our starting point is the following lemma: \[lem:firstbound\] For all $t \ge 1$, past sub-gradients $(\g_s)_{s<t}$ and past and current outputs $(\what\w_s)_{s \le t} \in \reals^d$, $$\begin{aligned} \exists \g_t\in \reals^d, \quad \sum_{s=1}^{t} \inner{\g_s}{\what\w_s} \geq \|\what\w_t \| \cdot L_t/2, \quad \text{where $L_t=\max_{s\leq t}\|\g_s\|$.} \label{eq:baselower} \end{aligned}$$ We want to find $\g_t$ such that $\inner{\g_t}{\what\w_t}\geq \|\w_t\| L_{t}/2 - S_{t-1}$, where $S_{t-1}\coloneqq \sum_{s=1}^{t-1} \inner{\g_s}{\what\w_s}$. By restricting $\g_t$ to be aligned with $\what\w_t$, the problem reduces to finding $x = \|\g_t\|$ such that $$\begin{aligned} x \|\what\w_t\| - \left|\|\what\w_t\| \cdot (L_{t-1}\vee x)/2 - S_{t-1}\right|\geq 0. \label{eq:ineq} \end{aligned}$$ The LHS of is a piece-wise linear function in $x$ which goes to infinity as $x\to \infty$. Therefore, there exists a large enough $x\geq0$ which satisfies . Observe that if $\|\what\w_t\|\geq D_t>0$, for $t\geq 1$, then by Lemma \[lem:firstbound\], there exists a sub-gradient $\g_t$ which makes the regret against $\w=\bm{0}$ at round $t$ at least $D_t L/2$. This essentially means that if the sub-gradients $(\g_t)$ are unbounded, then the outputs $(\what\w_t)$ must be in a bounded set whose diameter will depend on the desired regret bound; if one insists on a regret of order $\wtilde{O}(\sqrt{T})$, then the norm of the outputs $\what\w_t, t\geq 1,$ must be in a ball of radius at most $\wtilde{O}(\sqrt{T})$. [@cutkosky2019artificial] posed the question of whether there exists an algorithm which can guarantee a regret bound of order $L\|\w\|^3 +(\|\w\| +1) \sqrt{V_T \ln T}$,with $V_T=\sum_{t=1}^T \|\g_t\|^2$, while adapting to both $L$ and $\|\w\|$. Here we ask the question whether $L\|\w\|^\nu +(\|\w\| +1) \sqrt{V_T \ln T}$ is possible for any $\nu \geq 1$. If such an algorithm exists, then by Lemma \[lem:firstbound\], there exists a constant $b>0$ such that its outputs $(\what \w_t)$ satisfy $\|\what\w_t\| \leq b \sqrt{V_t \ln t}/L_t$, for all $t\geq 1$. The next lemma, when instantiated with $\alpha=2$, gives us a regret lower-bound on such algorithms (the proof is in Appendix \[sec:lowerproof\]): \[lem:seconbound\] For all $b,c,\beta \geq 0$, $\nu\geq 1$, and $\alpha \in ]1,2]$, there exists $(\g_t)\in\reals^d$, $T\geq 1$, and $\w\in \reals^d$, such that for any sequence $(\what\w_t)$ satisfying $\|\what\w_t\| \leq b \cdot \sqrt{V_{\alpha, t} \ln(t)/L^\alpha_t}$, for all $t\in \mathbb[T]$, where $V_{\alpha,t} \coloneqq \sum_{s=1}^t \|\g_s\|^\alpha$, we have $$\begin{aligned} \sum_{t=1}^T \inner{\what\w_t-\w}{\g_t} \geq c \cdot \ln (1+\|\w\| T)^{\beta} \cdot (L_T \|\w\|^\nu + L_T^{1-\alpha/2} (\|\w\|+1) \sqrt{V_{\alpha,T} \ln T}). \end{aligned}$$ By combining the results of Lemma \[lem:firstbound\] and \[lem:seconbound\], we have the following regret lower bound for algorithms with can adapt to both $L$ and $\|\w\|$: \[thm:lower1\] For any $\alpha \in]1,2]$, $c>0$ and $\nu\geq 1$, there exists no algorithm that guarantees, up to log-factors in $\|\w\|$ and $T$, a regret bound of the form $c\cdot (L_T \|\w\|^\nu + L_T^{1-\alpha/2} (\|\w\|+1) \sqrt{V_{\alpha,T} \ln T})$, for all $T\geq 1$, $\w\in \reals^d$, and $(\g_t)\subset \reals^d$, where $V_{\alpha,T} \coloneqq \sum_{t=1}^T \|\g_t\|^\alpha$. By Lemma \[lem:firstbound\], the only candidate algorithms are those whose outputs $(\what \w_t)$ satisfy $\|\what\w_t\| \leq b \sqrt{V_{\alpha,t} \ln (t)/L^{\alpha}_t}$, for all $t\geq 1$, for some constant $b>0$. By Lemma \[lem:seconbound\], no such algorithms can achieve the desired regret bound. The regret lower bound in Theorem \[thm:lower1\] does not apply to the case where $\alpha =1$. In fact, thanks to Corollary \[cor:instan\] and our main algorithm in Section \[sec:mainalg\] (which can play the role of Algorithm <span style="font-variant:small-caps;">A</span> in Corollary \[cor:instan\]), we know that there exists an algorithm <span style="font-variant:small-caps;">B</span> which guarantees a regret bound of order $\wtilde{O}(L_T \|\w\|^3 + \|\w\| \sqrt{V_T} + L_T \sqrt{\max_{t\in[T]} B_t})$, where $B_t = \sum_{s=1}^t \|\g_s\|/L_t$. Next we show that if one insists on a regret bound of order $\sqrt{B_T}$, or even $\sqrt{T}$ (up to log-factors), the exponent in $\|\w\|^3$ is unimprovable (the proof of Theorem \[thm:lower2\] is in Appendix \[sec:proofoflower2\]). \[thm:lower2\] For any $\nu \in[1,3[$ and $c>0$, there exists no algorithm that guarantees, up to log-factors in $\|\w\|$ and $T$, a regret bound of the form $c\cdot( L_T \|\w\|^\nu + L_T (\|\w\|+1) \sqrt{T\ln T})$, for all $T\geq 1$, $\w\in \reals^d$, and $(\g_t)\subset \reals^d$. Application to Learning Linear Models with Online Algorithms {#sec:linear} ============================================================ In this section, we consider the setting of online learning of linear models which is a special case of OCO. At the start of each round $t\geq 1$, a learner receives a feature vector $\x_t \in \cW = \reals^d$, then issues a prediction $\what y_t\in \reals$ in the form of an inner product between $\x_t$ and a vector $\what\u_t\in \reals^d$, *i.e.* $\what y_t =\what\u_t^\top \x_t$. The environment then reveals a label $y_t \in \reals$ and the learner suffers loss $\ell(y_t,\what y_t)$, where $\ell\colon \reals^2 \rightarrow \reals$ is a fixed loss function which is convex and $1$-Lipschitz in its second argument; this covers popular losses such as the logistic, hinge, absolute and Huberized squared loss. (Technically, the machinery developed so far and the reductions in Section \[sec:helpful\] allow us to handle the non-Lipschitz case). In the current setting, the regret is measured against the best fixed “linear model” $\w\in \reals^d$ as $$\begin{aligned} \textsc{Regret}_T(\w) \coloneqq \sum_{t=1}^T \ell(y_t, \what y_t) - \sum_{t=1}^T \ell(y_t, \w^{\top} \x_t) \leq \sum_{t=1}^T \delta_t \inner{\x_t}{\what\u_t -\w}, \label{eq:regretlin}\end{aligned}$$ where the last inequality holds for any sub-gradients $\delta_t \in \partial^{(0,1)} \ell(y_t, \what y_t)$, $t\geq 1$, due to the convexity of $\ell$ in its second argument, which in turn makes the function $f_t(\w) \coloneqq \ell(y_t, \w^{\top} \x_t)$ convex for all $\w \in \cW = \reals^d$. Here, $\partial^{(0,1)} \ell$ denotes the sub-differential of $\ell$ with respect to its second argument. Thus, minimizing the regret in  fits into the OCO framework described in Section \[sec:prelim\]. In fact, we will show how our algorithms from Section \[sec:mainalg\] can be applied in this setting to yield scale-free, and even *rotation-free*, (all with respect to the feature vectors $(\x_t)$) algorithms for learning linear models. These algorithms can, without any prior knowledge on $\w$ or $(\w^{\top} \x_t)$, achieve regret bounds against any $\w\in \reals^d$ matching (up to log-factors) that of OGD with optimally tuned learning rate. As in Section \[sec:mainalg\], we focus on algorithms which make predictions based on observed sub-gradients ($\g_t$); in this case, $\g_t = \x_t \delta_t \in \x_t \cdot \partial^{(0,1)} \ell(y_t, \what y_t)= \partial f_t(\what \u_t)$, $t\geq 1$, where $f_t(\w)=\ell(y_t,\w^{\top} \x_t)$. Since the loss $\ell$ is $1$-Lipschitz, we have $|\delta_t|\leq 1$, for all $\delta_t \in \partial^{(0,1)}\ell(y_t, \what y_t)$ and $t\geq 1$, and so $\|\g_t\|\leq \|\x_t\|$. Since $\x_t$ is revealed at the beginning of round $t\geq 1$, the hint $$\begin{aligned} h_t = \max_{s \leq t }\|\x_s\| \geq L_T = \max_{s\leq t} \|\g_s\| \label{eq:lin2} \end{aligned}$$ is available ahead of outputting $\what \u_t$, and so our algorithms from Section \[sec:mainalg\] are well suited for this setting. #### Improvement over Current Algorithms. We improve on current state-of-the-art algorithms in two ways; First, we provide a (coordinate-wise) scale-invariant algorithm which guarantees regret bound, against any $\w\in \reals^d$, of order $$\begin{aligned} \sum_{i=1}^d |w_i| \sqrt{V_{T,i} \ln (|w_i| \sqrt{V_{T,i}}T)} + |w_i| \ln_+(|w_i|\sqrt{V_{T,i}}T), \label{eq:olbound}\end{aligned}$$ where $V_{T,i}\coloneqq |x_{1,i}|^2 + \sum_{t=1}^T \delta^2_t|x_{t,i}|^2, i\in[d]$, which improves the regret bound of the current state-of-the-art scale-invariant algorithm $\textsc{ScLnOL}_1$ [@KempkaKW19] by a $\sqrt{\ln (\|\w\| T)}$ factor. Second, we provide an algorithm that is both scale and rotation invariant with respect to the input feature vectors $(\x_t)$ with a state-of-the-art regret bound; by scale and rotation invariance we mean that, if the sequence of feature vectors $(\x_t)$ is multiplied by $c \bm{O}$, where $c >0$ and $\bm{O}$ is any special orthogonal matrix in $\reals^{d\times d}$, the outputs ($\what y_t$) of the algorithm remain unchanged. Arguably the closest algorithm to ours in the latter case is that of [@Kotlowski17] whose regret bound is essentially of order $\wtilde{O}(\sqrt{\w^{\top} \bm{S}_T \w})$ for any comparator $\w\in \reals^d$, where $\bm{S}_T = \sum_{t=1}^T \x_t \x_t^\top$. However, in our case, instead of the matrix $\bm{S}_T$, we have ${\bm{V}}_T \coloneqq \|\x_1\|^2 \bm I + \sum_{t=1}^T \x_t \x_t^\top \delta_t^2 $, where $\delta_t \in \partial^{(0,1)} \ell(y_t, \what y_t), t\geq 1$, which can yield a much smaller bound for small $(\delta_t)$ (this typically happens when the algorithm starts to “converge”). #### A Scale-Invariant Algorithm. To design our first scale-invariant algorithm, we will use the outputs ($\what\w_t$) of [$\textsc{FreeGrad}$]{} in with $(h_t)$ as in , and a slight modification of [$\textsc{FreeRange}$]{} (see Algorithm \[alg:newfreerange\]). This modification consists of first scaling the outputs $(\what \w_t)$ of [$\textsc{FreeGrad}$]{} by the initial hint of the current epoch to make the predictions $(\what y_t)$ scale-invariant. By Theorem \[thm:scaled\] below, the regret bound corresponding to such scaled outputs will have a lower-order term which, unlike in the regret bound of Theorem \[thm:firstbound\], does not depend on the initial hint. This breaks our current analysis of [$\textsc{FreeRange}$]{} in the proof of Theorem \[thm:freegrad1\] which we used to overcome the range-ratio problem. To solve this issue, we further scale the output $\what\w_t$ at round $t\geq 1$ by the sum $\sum_{s=1}^\tau \|\x_s\|/h_s$, where $\tau$ denotes the first index of the current epoch (see Algorithm \[alg:newfreerange\]). Due to this change, the proof of the next theorem differs slightly from that of Theorem \[thm:freegrad1\]. First, we study the regret bound of Algorithm \[alg:newfreerange\] in the case where $\cW=\reals$. \[thm:freegradol\] Let $d=1$ and $(h_t)$ be as in . If $(\what u_t)$ are the outputs of Algorithm \[alg:newfreerange\], then for all $w \in \reals; T\geq 1$; $(x_t,y_t)\subset \reals^2$, s.t. $h_1=|x_1|> 0$; and $\delta_t \in \partial^{(0,1)} \ell\left(y_t,x_t \what u_t \right)$, $t\in[T]$, $$\begin{aligned} \sum_{t=1}^T \delta_t x_t \cdot \left(\what u_t- w\right) & \leq 2 |w| \sqrt{V_{T}\ln_+ (2 |w|^2 V_{T} c_T )} \\ & \quad + h_{T} |w| ( 14 \ln _+( 2 |w| \sqrt{2V_{T} c_T } ) + 1 )+ {2+ \ln B_T}, \label{eq:regbound} \end{aligned}$$ where $V_T \coloneqq |x_1|^2 +\sum_{t=1}^T \delta_t^2 x_t^2$, $c_T\coloneqq 2 B_T^2 \sum_{t=1}^T\left(\sum_{s=1}^t \frac{|x_s |}{h_s}\right)^2 \leq T^5$, and $B_T =\sum_{s=1}^{T} \frac{|x_s |}{h_s}\leq T$. The proof of Theorem \[thm:freegradol\] is in Appendix \[sec:sec5proofs\]. If $(\what u_t)$ are the outputs of Algorithm \[alg:newfreerange\] in the one-dimensional case, then by Theorem \[thm:freegradol\] and , the algorithm which, at each round $t\geq 1$, predicts $\what y_t=x_t \what u_t$ has regret bounded from above by the RHS of . Note also that the outputs $(\what y_t)$ are scale-invariant. Now consider an algorithm $\textsc{A}$ which at round $t\geq 1$ predicts $\what y_t =\sum_{i=1}^d x_{t,i} \what u_{t,i}$, where $(\what u_{t,i}), i\in[d]$, are the outputs of Algorithm \[alg:newfreerange\] when applied to coordinate $i$; in this case, we will have a sequence of hints $(h_{t,i})$ for each coordinate $i$ satisfying $h_{t,i} = \max_{s\leq t}|x_{t,i}|$, for all $t\geq 1$. Algorithm $\textsc{A}$ is coordinate-wise scale-invariant, and due to and Theorem \[thm:freegradol\], it guarantees a regret bound of the form . We note, however, that a factor $d$ will appear multiplying the lower-order term $(2+\ln B_T)$ in (since the regret bounds for the different coordinates are added together). To avoid this, at the cost of a factor $d$ appearing inside the logarithms in , it suffices to divide the outputs of algorithm <span style="font-variant:small-caps;">A</span> by $d$. To see why this works, see Theorem \[thm:scaled\] in the appendix. The hints $(h_t)$ as in . Set $\tau=1$; Observe hint $h_{t}$; Set $\tau=t$; Output $\what\u_t= \what\w_{t} \cdot \left(h_\tau \cdot {\sum_{s=1}^{\tau } \frac{\|\x_s\|}{h_s}}\right)^{-1}$, where $\what \w_t$ is as in with $(h_1, V_{t-1}, \G_{t-1})$ replaced by $(h_{\tau}, h_\tau^2+ \sum_{s=\tau}^{t-1} \|\g_s\|^2,\ \sum_{s=\tau}^{t-1}\g_s)$; \[eq:lastline\] #### A Rotation-Invariant Algorithm. To obtain a rotation and scale-invariant online algorithm for learning linear models we will make use of the outputs of [$\textsc{Matrix-FreeGrad}$]{} instead of [$\textsc{FreeGrad}$]{}. Let $(\what y_t)$ be the sequence of predictions defined by $$\begin{aligned} \hspace{-0.2cm}\what y_t=\x_t^\top \what\w_t/h_{1}, \ t\geq 1, \label{eq:pred2}\end{aligned}$$ with $(h_t)$ as in and where $\what\w_t$ are the predictions of a variant of [$\textsc{Matrix-FreeGrad}$]{}, where the leading $h_1$ in the potential is replaced by $1$ (we analyze this variant in Appendix \[sec:multidim.control.real\]). \[thm:multidimlinear\] Let $\gamma>0$ and $(h_t)$ be as in . If $(\what y_t)$ are as in , then $$\forall \w \in \reals^d, \forall T\geq 1, \forall (\g_t)\subset \reals^d, \ \ \textsc{Regret}_T(\w) ~\le~ 1 + \sqrt{Q_T^\w \ln_+\del*{{ \det\del*{ \gamma h_1^2 \bm \Sigma_T}^{-1} } Q_T^\w}},\ \ \text{where}$$ $$Q_T^\w \coloneqq \max \set*{ \w^{\top} \bm \Sigma^{-1}_T \w , \frac{1}{2} \del*{\frac{h_T \norm{\w}^2}{\rho(\gamma)^{2}} \ln \del*{ \frac{h_T \norm{\w}^2}{\rho(\gamma)^{2}} \det \left(\gamma h^2_1 \bm \Sigma_T\right)^{-1} } + \w^{\top} \bm \Sigma^{-1}_T \w} } ,$$ and $\bm \Sigma^{-1}_T \df \gamma h_1^2 \bm I + \gamma \sum_{t=1}^T \g_t \g_t^\top$. It suffices to use and instantiate the regret bound in Theorem \[thm:rephrased\] with $(\epsilon,\sigma^{-2})= (1,\gamma h^2_1)$. The range-ratio problem manifests itself again in Theorem \[thm:multidimlinear\] through the term $ \det(\gamma h^2_1 \bm \Sigma_T)^{-1}$. This can be solved using the outputs of Algorithm \[alg:newfreerange\], where in Line \[eq:lastline\], $\what \w_t$ is taken to be as in (see Remark \[rem:restartremark\]). \[3\][\#3]{} \[3\][\#2]{}

--- abstract: 'It has long been accepted that the multiple-ion single-file transport model is appropriate for many kinds of ion channels. However, most of the purely theoretical works in this field did not capture all of the important features of the realistic systems. Nowadays, large-scale atomic-level simulations are more feasible. Discrepancy between theories, simulations and experiments are getting obvious, enabling people to carefully examine the missing parts of the theoretical models and methods. In this work, it is attempted to find out the essential features that such kind of models should possess, in order that the physical properties of an ion channel be adequately reflected.' author: - 'K.K. Liang' bibliography: - 'cooptran.bib' title: 'On the crucial features of a single-file transport model for ion channels' --- Introduction ============ The mechanism with which an ion channel conducts the ions through it had been the aim of a tremendous amount of researches since the time of Hodgkin and Keynes[@hodgkin1955a]. It will not be unfair to say that due to the importance of this problem, the subject of *single-file transport* made it to the center stage of theoretical biophysics. Besides the purely theoretical interests of it, understanding of the transport mechanism is also useful in many practical aspects. The complicated current-voltage (I-V) relations of different ion channels, the different manners in which they can be gated and regulated, together with the selectivity of them, imply that they are like contemporary semiconductor electronic devices that can be the logical operation components in complicated circuitries. They are even more complicated than semiconductor devices because there are more than two kinds of charge carriers in a biological ionic circuitry, and each of them are regulated by many others in many ways. Knowledge with which one can tune the electrophysiological characters of one of the devices may enable one to fix malfunctioning biological circuitry. Synthetic biologists may even design components with desired characters to implement artificial organic machines[@l:inpress; @acs:inpress]. Structural biological studies provide quantitative information based on which in depth studies can be performed. Combinations of biophysical experiments and mutagenic manipulations perturb the system in well-controlled manners to extend the dynamic range of investigations. It is hoped that theoretical studies can join the parade and provide solid understanding and models with high predicting power. The seminal works of Hille were perhaps the prototype of most, if not all, of the single-file transport models of ion channels[@hille1975b; @hille1978a]. The channel is modeled by a chain of saturable binding sites. It can be occupied by a number of ions. The ions move in single file, that is, they do not penetrate through one another. Each ion moves to one of its immediate neighboring vacant sites with absolute rates determined through Eyring equation by the barrier heights between sites, the membrane potential, and the ion-ion repulsion. Many works followed this idea and the early ones were summarized in several good reviews[@levitt1986; @nm5:1105; @hille2001]. Specifically for the stochastic models, there is an excellent recent review[@rmp85:135], although the subject of ion channels is only a small portion of that review. Recent works that are not categorized into the stochastic models and are not yet reviewed include, but not exclusively, the works from several groups that focused on continuum models and some others that transformed Hille’s picture into exclusion-process models. It is worthy of noting here that a major common technique in these theories is to use steady-state fluxes as the quantities to compare with the observed ion currents. However, some features of the above abstract models are not satisfactory. Most of these works ignored Coulomb interaction between the permeating particles. The particles are either non-interacting or the interaction is short-ranged. Interestingly, this approximation rarely introduced troubles because in most of these works only one particle was allowed to move in the channel pore each moment. On the other hand, very strong Coulomb repulsion between ions had indeed been considered, but that was done also by assuming that only one particle can be inside the channel pore at a time[@prl100:038104]. The interaction of the particles with the channel pore is an even more intriguing problem. In the abstract models the potential surface seen by the permeating particles should, of course, be artificially defined. In order to mimic the effect of binding sites, it was invariably assumed that local minima of the potential along the pore that are able to tentatively trap the particles were present. In exclusion-process models, these traps are further simplified into more abstract ‘cells’ that house the particles tentatively. It was rarely reflected if the structure and therefore the potential has anything to do with the presence (or absence) of the permeating particles. The potential of mean force (PMF) experienced by the permeating particles, by definition, is averaged over the degrees of freedom of the remaining parts of the system, including the protein, the membrane, the solvent, and other ions. In principle it can be a static function of the space. The problem, however, is whether the more relevant reaction path were chosen and how the total free energy of the system was calculated. This point will be further discussed later. Thanks to the progress in structural biology studies[@jgp115:269; @zhou2001b; @nature417:515; @nature417:523; @nature423:33; @science309:897; @nature450:376; @nature471:336], nowadays atomistic simulations can be performed to study the transport processes with increasingly realistic details included. Recently there was a review of the computational studies[@cr112:6250]. Many other works mentioned in that review, especially those that were not directly related to ion transport, were not cited in the above list. Besides, several of the works mentioned above were newer than that review. It seemed that atomistic simulations are now capable of replacing abstract theoretical models. However, for many practical applications, abstract theoretical models are still more convenient than simulations, when efficiency is considered. Moreover, even though atomic-level details are included as much as possible, the designs of these computer experiments were still not perfect in a few subtle yet crucial aspects. Those are the focus of this article. When the structures of KcsA at different potassium ion concentrations were solved, it was specially emphasized that[@zhou2001b] “in the selectivity filter a large number of negatively-charged carbonyl oxygen atoms point into the channel pore, making the pore very unlikely stable structure, unless several potassium ions are in the pore.” Indeed, it is natural to suppose that in order to form the native structure those few potassium ions observed should be in the channel at the right places at the beginning, and the binding would be rather strong. The entrance of a few ions into an empty channel pore seems to happen routinely during reactivation after deactivation in simulations on gating process[@science336:229]: When the voltage-gated potassium ion channel was reactivated, water molecules and potassium ions flew into the channel to re-establish the conducting structure, in early time in a molecular dynamics simulation, before permeation processes could have been observed. Conversely, the channel stabilized the configuration of several positive ions, otherwise the repulsion between ions would make it impossible for them to line up in single file to move across the membrane[@roux2001a]. In other words, the structural features seem to suggest that when the channel is conducting, a number of ions have to be in the channel to stabilize the structure, and the really meaningful potential surface for ion conduction should be established inside the channel together with these first few ions. The particles that enter the channel during conduction process will probably only perturb the overall channel-plus-ions configuration relatively weakly to generate the ion current through the so-called *knock-on* type of transfer[@hodgkin1955a]. If that were the case, question arises as what “the reaction path for single file transport” will look like. That is equivalent to asking whether we are looking at one ion moving between bulk solutions at both ends through the channel, or we are looking at the transformation of the whole system with one more ion in the bulk solution on one side as the initial state, and the state with one more ion in the bulk on the other side as the final state. In the latter case, obviously, the reaction path is generally different from the spatial coordinate along the channel pore. Indeed, this is why nowadays simulation scientists emphasize more and more on multiple-ion potential of mean force. In this work, it is intended to set up a model in which ion-ion repulsion is included. Besides, it is also intended that the role of the permeating ionic species in stabilizing the channel structure is explicitly considered. Moreover, it is hoped that the model can be as simple as possible. The purpose is to qualitatively demonstrate the impact of these effects on the current-voltage-concentration relation of an ion channel[@pnas107:5833; @bpj101:2671; @jgp141:619]. To that end, a chemical kinetics model is employed. The channel is thought of as an enzyme which catalyzes the uptake of one ion on one side of the membrane and the release of one ion to the other side of the membrane. Therefore, the model involves the binding of ions into the channel as well as the transport of the ions out of the channel. The major features are as following. First, the rate in which an ion exits the channel increases with the number of ions inside the pore. In other words, the exit of ions is a positively cooperative process. Second, the binding constants of the first few ions into the channel are high, but the binding constants decreases drastically when the number of ions further increases. However, a simple model of ion transport is still rather difficult to handle, not because the chemical kinetics model is difficult to set up, but because approximations are difficult to apply to get results that are simple to manipulate. Therefore, instead of directly studying the ion current of a channel system, the unblocking kinetics of a blocked channel is studied[@jgp92:549; @jgp92:569]. In a previous work[@chang2009], the kinetics of ion transport in the inner vestibule of inward-rectifying potassium ion channel Kir2.1 had been studied by blocking the selectivity filter part of the channels with barium ions and analyzing the exit rate of the barium ions. Compared to the apparent rate for a potassium ion to traverse the channel pore, the rate in which the blocking barium be “kicked out” under the action of both the membrane potential and the intra-channel potassium ions is much slower. The slowing down of potassium ion motion permitted higher-quality experimental observations. For setting up the theoretical model, there are also great advantages. If the potassium ions will exit the channel more rapidly when the number of intra-channel ions increases, so will the blocking barium ion exit more rapidly. Indeed this was observed in the previous work. More importantly, since the exit of the blocking barium ion is much slower than the transport through an unblocked channel of a potassium ion, the process of barium ion exit can be thought of as the rate-limiting step, and pre-equilibrium can be assumed for the processes of potassium ions binding into the channel. If the potassium ion transport process were directly studied, probably the pre-equilibrium approximation cannot be applied. If that is the case, instead of proposing several binding constants, a lot of rate constants have to be proposed, and the analysis will be distracted. Nevertheless, two major difference between the unblocking and transport processes persist. First, the unblocking process can only be better compared to the uni-directional transport. The block-free transport is approximately uni-directional only when the electrochemical gradient is steep. Second, the blocking barium ion is positively charged and will influence the channel structure non-trivially. It is impossible to assess the influence of this fact on the soundness of the following discussions in this work. These problems can only be answered by further studies that take the suggestions given in this work into account. In the next Section, the details of the model is presented. Most importantly, the relations between various binding constants and rate constants are discussed carefully. In Section \[s:res\], simulations of the I-V curve as well as the concentration effect of the model system in different conditions are presented and the implications are discussed. Then this work is concluded with some suggestions to both the simulation scientists and theoreticians. Theory {#s:theory} ====== Consider a channel with $N$ possible ion-binding sites in it. On one end of it, a barium ion can block the channel. Upon referring to the above $N$ binding sites, the barium ion blocking site is not counted. Potassium ions can enter from the other end of the channel pore. Up to $N$ ions can be filled into the channel, if the physical condition permits. In fact, it is not necessary to mention that there are $N$ binding sites. Even if there is not any binding site or if there are a lot more binding sites inside, the important feature is simply the maximum number of ions that can be in the channel simultaneously. In other words, for simplicity, the different binding states of a channel with $n$ ions inside are considered indifferent in their effects on the barium ion exit rate. In reality, the arrangement of the intra-channel potassium ions certainly have concrete effects on the transport properties. However, in the present abstract model, especially under pre-equilibrium approximation, it is assumed that these details can be grossly lumped together. Being thought about in this way, this model becomes exactly the same as the sequential-binding model introduced by Weiss[@weiss1997]. The sequential binding of potassium ions into the channel can be described by the following chemical equations: $$\left\{ \begin{aligned} \cee{K+(f) + Ba^{2+}(b) & <=>[K_1] K+(b) + Ba^{2+}(b)}\\ \cee{K+(f) + K+(b) + Ba^{2+}(b) & <=>[K_2] 2 K+(b) + Ba^{2+}(b)}\\ &\vdots \\ \cee{K+(f) + $\left(N-1\right)$ K+(b) + Ba^{2+}(b) & <=>[K_N] $N$ K+(b) + Ba^{2+}(b)} \end{aligned}\right. \label{e:rkbind}$$ where K$^+$(f) means a free potassium ion in the bulk, while K$^+$(b) and Ba$^{2+}$(b) means bound potassium and barium ions, respectively. The equilibrium constants labeled above the arrows are binding constants of these reactions. The subscript $j$ indicates that this is the process of the $j$-th potassium ion binding to the channel. Since in all cases one barium ion is in the channel, it can be neglected from the reaction quotient. Similarly, in each of the reactions, the numbers of bound potassium ions in the initial and final states of the reaction are fixed. They can also be removed from the reaction quotient. Eventually, the pre-equilibrium of this set of $N$ reactions is only determined by the values of the binding constants and the concentration of the free potassium ions in the bulk. The other set of reactions to be considered are the exit processes of the barium ion under different conditions. When there is not any potassium ions in the channel, according to the instantaneous membrane potential, there is a probability that the barium ion will exit. With the increase of the number of ions in the channel, primarily due to the Coulomb repulsion, it can be expected that the exit rate of the barium ion will increase. These processes can be described by the following chemical equations: $$\left\{ \begin{aligned} \cee{ Ba^{2+}(b) & ->[k_0] Ba^{2+}(f) }\\ \cee{ K+(b) + Ba^{2+}(b) & ->[k_1] K+(b) + Ba^{2+}(f) }\\ \cee{ 2 K+(b) + Ba^{2+}(b) & ->[k_2] 2 K+(b) + Ba^{2+}(f) }\\ & \vdots \\ \cee{ $N$ K+(b) + Ba^{2+}(b) & ->[k_N] N K+(b) + Ba^{2+}(f) } \end{aligned} %\begin{array}{rcl} %{\rm Ba^{2+}\!\left(b\right)} %&\stackrel{k_0}{\harrow{+}{8}}& %{\rm Ba^{2+}\!\left(f\right)}\\ %{\rm K^+\!\left(b\right)+Ba^{2+}\!\left(b\right)} %&\stackrel{k_1}{\harrow{+}{8}}& %{\rm K^+\!\left(b\right)+Ba^{2+}\!\left(f\right)}\\ %2\,{\rm K^+\!\left(b\right)+Ba^{2+}\!\left(b\right)} %&\stackrel{k_2}{\harrow{+}{8}}& %{\rm2\, K^+\!\left(b\right)+Ba^{2+}\!\left(f\right)}\\ %\vdots\\ %N\,{\rm K^+\!\left(b\right)+Ba^{2+}\!\left(b\right)} %&\stackrel{k_N}{\harrow{+}{8}}& %N{\rm\, K^+\!\left(b\right)+Ba^{2+}\!\left(f\right)} %\end{array} \right. \label{e:rbaexit}$$ where Ba$^{2+}$(f) means free barium ion that just exits into the bulk solution on the opposite side of the channel. The quantities $k_j$ over the arrow are the effective zeroth-order rate constants. They are zeroth-order because each of them corresponds to a specific number of intra-channel potassium ions, therefore this factor is already absorbed into the expression of the rate. They are also only meaningful when the barium ion still blocks the channel. Therefore effective zeroth-order rate constants are sufficient. The subscript $j$ indicates the barium ion exit rate under the influence of $j$ intra-channel potassium ions. Here it is emphasized again that since the barium ion exit processes, Reactions , are much slower than the potassium binding processes, pre-equilibrium approximation can be applied. That is why in Reactions  only the binding constants are introduced, while in Reactions  the reaction rate constants are used to describe the reactions. Furthermore, in real experiments, after a preparatory step in which most of the channels were found blocked with barium ion, the bulk solution was perfused so that the bulk barium concentration was zero, before the unblocking experiment began. Therefore, throughout the experiment, the bulk concentration of barium ions was so low that the re-blocking of the channel was negligibly possible. The rate constants of the reversed reactions do not have to be included in Reaction . With the pre-equilibrium approximation, under a given condition, the probability that there are $j$ ions in the channel can be easily derived[@weiss1997]: $$p_j=\frac{\left(\prod_{m=1}^jK_m\right)\left[{\rm K}^+\right]^j}{ \sum_{n=0}^N\left(\prod_{m=1}^{n}K_m\right)\left[{\rm K}^+\right]^n} \label{e:probj}$$ where \[K$^+$\] is the bulk potassium ion concentration on the open-end side of the channel. Besides the bulk concentration of potassium ion, this probability also depends on $N$, but for brevity it is not explicitly labeled. For example, if $N=3$ and $j=2$, $$p_2=\frac{K_1K_2\left[{\rm K}^+\right]^2}{ 1+K_1\left[{\rm K}^+\right]+K_1K_2\left[{\rm K}^+\right]^2+K_1K_2K_3\left[{\rm K}^+\right]^3}$$ The expectation value of the barium exit rate under the given binding constants, rate constants, and bulk potassium ion concentration, is $$\bar{v}=\sum_{n=0}^Np_nk_n$$ This is the major quantity of the theory. It is directly proportional to the steady-state outward current. Another quantity of interests is the mean number of potassium ions in the channel $\bar{n}$ given by $$\bar{n}=\sum_{n=0}^Np_nn$$ The explicit expressions of $\bar{v}$ and $\bar{n}$ are trivial but tedious, and are not shown here. It has to be noted that since the barium exit rate is supposed to be proportional to the potassium permeation rate and therefore it is used to represent the permeation rate in the following, this model cannot be used to simulate negative ion current (or inward current). Nevertheless, as long as the outward current is positive, the model may still work fine even if the membrane potential is negative. Therefore probably the reversal potential is still meaningful. To perform simulations with this model, the relative values of the parameters have to be determined. Since the I-V curve is to be simulated, the dependence of the parameters on the membrane potential $V$ also have to be determined. Since the conventional definition of membrane potential is the intracellular potential minus the extra-cellular potential, and since all of the ions explicitly considered are positive ions, the open-end of the channel is considered to be intracellular and the barium-ion-blocked end is extracellular. In the following, the rules used in the present work are discussed. When the membrane potential is zero, there is still a natural barium exit rate constant even without potassium ion in the channel. This rate constant is labeled as $k_0^*$ (the star-sign indicates the field-free condition). If one potassium ion entered the channel, the barium ion exit rate constant will increase, and this increase is considered to be related to the Coulomb interaction between the potassium ion and the barium ion. With $j$ potassium ions in the channel, there are totally $j$ units of positive charge repulsing the barium ion. It is assumed that the activation energy of barium ion exit decreases linearly with the number of potassium ions in the channel, therefore the barium ion exit rate constant increases exponentially with the number of potassium ions in the channel: $$k_j^*=k_0^*\exp\left(\beta j\right)$$ where $\beta$ is related to the sensitivity of the activation energy to the number of intra-channel potassium ions as well as temperature. In the present model, it was proposed that the positive ions participate in stabilizing the structure of the channel pore. Therefore, the change in the number of ions in the channel affects the structure or structural stability. The barium exit rate will not only be influenced by the Coulomb repulsion. Microscopic theories have to be used to find out the real form of its dependence on the value $j$. On the other hand, two properties assumed here are reasonable for the barium ion exit rate, but are not really appropriate for describing the ion transport rate. First, in the barium exit process, the extracellular concentrations of different ion species are not very important. Therefore the concept of reversal potential is not very relevant. When the membrane potential is zero or even negative, theoretically, there is always a finite barium exit rate. However, for the ion transport process, there is always a thermodynamically well-defined reversal potential, below which the macroscopic ion current turns zero and then negative. In contrast, the barium exit rate is always positive. Second, when the membrane potential is increased, the barium exit rate will increase, and it diverges at some point. This divergence is not strictly physical, but is understandable because the barium ions will unblock the channel in almost no time. However, typically, the ion transport rate will converge to a finite value[@bj41:119]. Two reasons may be the most critical. On one hand, due to the knock-on model of ion transport and the finite capacity of the channel pore, at high voltage the channel is probably fully occupied and is waiting for the bulk ions to knock on. On the other hand, the friction due to ion-pore interaction and ion-ion repulsion make the transport a drifting process. There will probably be a terminal velocity. Both ideas could lead to the same conclusion that the rate in which the intra-cellular ions hit the entrance of the channel determines the maximal ion transfer rate. Therefore it is reasonable that the observed asymptotic value of the I-V curve is closely related to the diffusion rate of the permeating ion. In this work, the discussions are focused on the change in the magnitude of the I-V curve. Therefore, the discrepancy between this model and the experiments at the negative- and positive-voltage extremes will not be seriously treated. In comparison, a more complicated dependence of the field-free binding constants on the potassium ion number is proposed. As mentioned in the Introduction, it is desired that the first few potassium ions bind with higher affinity, while the binding constant decreases drastically for excess potassium ions. In the following, the number of preferentially bound potassium ions is set to two. To realize such a model, a Fermi-distribution-like dependence of $K_j^*$ on $j$ with the “Fermi level” at $j=\mu$ is proposed: $$K_j^*=B/\left(1+e^{a\left(j-\mu\right)}\right)\label{e:Kjstar}$$ where $a$ and $B$ are constants. In the next Section, $B$ will be represented by another parameter $b$ such that $b=\ln B$ just for numerical convenience. Notice that when $B$ is changed, all of the binding constants are changed by the same multiplicative factor. The ratios between the magnitudes of the binding constants do not change with $B$. This is certainly not the perfect model. Nevertheless, depending on the emphasis of the study, it may be not a poor model, either. Although the ratios of the form $K_i/K_j$ is independent of $B$, when $j\lesssim\mu$, the terms of the form $\prod_{m=1}^jK_m$ in Eq.  are proportional to $B^j$. These quantities are of interests because $K_j=\exp\left(-\Delta G_{j,j-1}/RT\right)$ and $\Delta G_{j,j-1}=G\left(j\right)-G\left(j-1\right)$ where $G\left(j\right)$ is supposed to be the absolute free energy of the system with $j$ potassium ions inside the channel. Therefore $\prod_{m=1}^nK_m=\exp\left(-\left[G\left(j\right)-G\left(0\right)\right]/RT\right)$. Defining $\Delta G_j=G\left(j\right)-G\left(0\right)$ then $\prod_{m=1}^nK_m=\exp\left(-\Delta G_j/RT\right)$. When $j\lesssim\mu$, the denomiator part of Eq. is around 1, and $\prod_{m=1}^jK_m$ is dominated by $B^j$. Therefore the relative probability of having larger number of intra-channel ions increases significantly with $B$, until there are enough positive ions to stabilize the channel. The free energy of the channel, compared with its empty state, decreases with increasing number of ions inside. For $j>\mu$, the denominator part of Eq. increases rapidly with $j$, and the product turns decreasing with $j$. That is, the free energy of the system starts to increase with more ions inside. These are indeed the qualitative features desired by the present model, which are not trivial and difficult to embody quantitatively. The good point is that only a very limited number of arbitrary parameters are introduced. The parameter $a$ also controls how abruptly the binding affinity decreases when the intra-channel potassium ion number increases beyond $\mu$. Throughout the following text, its value will be fixed at $a=2.5$, while $\mu=3$. When the membrane potential $V$ is finite, the parameters have to be further adjusted. The simplest exponential-form I-V curve commonly saw in microelectronics textbook is used. Again, this is the proper model for barium exit and qualitatively related to the ion current. By representing the sensitivity of the exit rate constants to the membrane potential and temperature with a factor $\kappa$, the full form of the barium ion exit rate constants $k_j$ in this model is $$k_j=k_0^*\exp\left(\kappa V+\beta j\right) \label{e:baexitk}$$ The ion binding reactions are considered as barrier-crossing processes of charged particles. The dependence of the activation energies on membrane potential is assumed to be linear, from the most straight-forward and lowest-order energetic consideration. Therefore, under the action of membrane potential $V$, the potassium ion binding constants are $$K_j=e^{b+\delta V}/\left(1+e^{a\left(j-3\right)}\right) \label{e:eqkdef}$$ where $\delta$ represents the degree of sensitivity of the binding process to the membrane potential. Again, notice that when the membrane potential is raised, all binding constant $K_j$ increases with the same proportion. As discussed earlier, in the present model as long as the ion can enter the channel it can be considered as bound. Therefore in any case the increased membrane potential should shift the binding equilibrium to the forward direction. Thus this property is also qualitatively correct. A plot of $K_j$ for the $N=6$ case is shown in Fig. \[f:fig01\] for reference. In the next Section, both $N=6$ and $N=4$ cases will be simulated and discussed. If all other parameters are kept the same, the first four binding constants $K_1$ through $K_4$ will be the same in both cases. ![The values of $K_j$ for $N=6$, $\mu=3$ and $a=2.5$ are presented. The filled circles indicates the $K_j$ values, while the dotted lines shows the trend. The data shown, from the lowest to the highest curve, correspond to $b+\delta V=3$, 4, 5, 6 and 7, respectively. \[f:fig01\]](figure01.pdf){width="6cm"} In the following Section, selected results of numerical calculations based on this model are presented. Since $\bar{n}$ is obviously dimensionless, and $\bar{v}/k_0^*$ is also dimensionless, actually the time scale, voltage, current, and ion concentration are all relative quantities. Those results will only be discussed qualitatively. Results and Discussions {#s:res} ======================= First, consider a channel with six potassium binding sites. In the first set of results, the following values of the parameters are used. The sensitivity factors of the potassium ion binding constants and the barium ion exit rate constants on membrane potential are $\delta=4$ and $\kappa=1$, respectively. The bulk potassium ion concentration is $\rm\left[K^+\right]=0.1$. The sensitivity factor of the barium exit rate constant on intra-channel number of potassium ions is $\beta=1$. The cases corresponding to five different values of $b=\ln B$ from 0 to 6 in steps of 1.5 are presented in Fig. \[f:fig02\]. In either Panels, the cases of $b=0, 1.5, 3, 4.5$ and 6 are plotted in thin solid line, thin dashed line, dotted line, dashed line and solid line, respectively. In Panel (a), the I-V curves for positive membrane potential are presented, where the voltage range is from 0 to 1. The $y$-axis is the outward current, but actually what is plotted is the scaled expected barium exit rate $\bar{v}$. Also plotted are the data obtained from the literature. The filled circles are the experimentally (single-channel recording) measured I-V relation of gramicidin A channel by Anderson[@bj41:119]. The filled squares are the simulated ion current by Jensen *et al.*[@jgp141:619]. The comparison between the present model with these data will be discussed later. All of the theoretical curves look like exponential curves, but actually they are not. The important feature is that, with increasing $b$ or $B$, the whole I-V curve shifts in the direction of higher current. To explain the trend of the I-V curves, in Panel (b) the mean number of potassium ions in the channel, $\bar{n}$, is plotted with respect to membrane potential. For zero or small values of $b$, it is seen that at low membrane potential there is hardly any potassium ion inside the channel. However, $\bar{n}$ increases more rapidly after some turning point. When $V$ is increased to about 1.0, the channels are averagely half-filled. Therefore, the corresponding I-V curves have much smaller slopes at lower voltage. For higher $b$ value, of course the number of ions in the channel is much larger even at very small membrane potential, because the first few binding constants are significantly higher. However, since $K_5$ and $K_6$ are still very small, it is very difficult to saturate the channel, and $\bar{n}$ only increases slowly. But this is already sufficient to explain why the magnitude of the current is higher with higher $b$ value. Besides, if the current axis is logarithmic instead, the I-V curves have, respectively, very similar shapes as their corresponding $\bar{n}$-V curves. In other words, the upper-most curve $\left(b=6\mbox{ or }B=400\right)$ is close to exponential, but the lowest curve $\left(b=0\mbox{ or }B=1\right)$ is highly non-exponential. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (a) (b) ![The properties of a system of channels with six binding sites inside. The I-V curves of the system, (a), and the mean number of potassium ions in the channel, (b), are plotted as functions of membrane potential. The unit of the membrane potential is arbitrary, but the parameters were chosen so that it seems the unit of the potential is Volt. The five curves in each panel correspond to $b=0$ (thin line), 1.5 (thin dashed line), 3.0 (dotted line), 4.5 (thick dashed line) and 6.0 (thick line), respectively. In Panel (a), also plotted are the I-V relations measured experimentally (filled circles)[@bj41:119] and simulated by molecular dynamics simulations (filled squares)[@jgp141:619]. \[f:fig02\]](figure02a.pdf "fig:"){width="6cm"} ![The properties of a system of channels with six binding sites inside. The I-V curves of the system, (a), and the mean number of potassium ions in the channel, (b), are plotted as functions of membrane potential. The unit of the membrane potential is arbitrary, but the parameters were chosen so that it seems the unit of the potential is Volt. The five curves in each panel correspond to $b=0$ (thin line), 1.5 (thin dashed line), 3.0 (dotted line), 4.5 (thick dashed line) and 6.0 (thick line), respectively. In Panel (a), also plotted are the I-V relations measured experimentally (filled circles)[@bj41:119] and simulated by molecular dynamics simulations (filled squares)[@jgp141:619]. \[f:fig02\]](figure02b.pdf "fig:"){width="6cm"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The non-exponential features of the I-V curve is better seen if the case of $N=4$ is shown instead. With all other parameters remaining the same, including $K_1$ through $K_4$, the I-V curves and $\bar{n}$-V curves under the same conditions are shown in Fig. \[f:fig03\]. According to Fig. \[f:fig01\], $K_3$ and $K_4$ are much larger than $K_5$ and $K_6$ when all of the parameters except $N$ are the same. Therefore in the $N=4$ case, the channel is much more easily saturated. Even when $b$ is large, unlike the situation in the $N=6$ system, the $\bar{n}$-V curve is quite nonlinear. Compared with Fig. \[f:fig02\](b), it is obvious that this nonlinearity is mainly due to the saturation of the channel. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (a) (b) ![The properties of a system of channels with four binding sites inside. The I-V curves of the system, (a), and the mean number of potassium ions in the channel, (b), are plotted as functions of membrane potential. The five curves in each panel correspond to $b=0$ (thin line), 1.5 (thin dashed line), 3.0 (dotted line), 4.5 (thick dashed line) and 6.0 (thick line), respectively. \[f:fig03\]](figure03a.pdf "fig:"){width="6cm"} ![The properties of a system of channels with four binding sites inside. The I-V curves of the system, (a), and the mean number of potassium ions in the channel, (b), are plotted as functions of membrane potential. The five curves in each panel correspond to $b=0$ (thin line), 1.5 (thin dashed line), 3.0 (dotted line), 4.5 (thick dashed line) and 6.0 (thick line), respectively. \[f:fig03\]](figure03b.pdf "fig:"){width="6cm"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Under high membrane potential, the I-V curves of the present model all become exponential because the channel pore is nearly saturated with the permeating ions so that $\bar{n}$ is nearly constant, but the barium exit rate $k$ increases exponentially with the membrane potential so that $\bar{v}$ also increases exponentially. As the discussion about Eq.  in the previous Section explained, in reality, the current will typically saturate at a value believed to correspond to the diffusion-limited ion transport rate[@bj41:119]. Next, the comparison with the studies on real ion channels is discussed. In Fig. \[f:fig02\]a, the MD-simulated I-V curve[@jgp141:619] and the experimentally measured results[@bj41:119] were shown. The relative scale of these two sets of values were fixed, exactly as shown in Jensen’s comparison[@jgp141:619]. However, to show them side-by-side with the present theory, they were all multiplied by 1.2 times. This made the simulation results comparable with the $b=0$ case in this theory. In that case, it appears that the magnitude of the experimental result is comparable with the $b=6$ curve of this theory. The shapes of the curves are, however, not closed to each other. Especially, in the present model, the $y$-intersect of the I-V curves depends on the binding constants quite strongly. In the experiments done by Andersen[@bj41:119], all of the experiments were done on non-blocked gramicidin A channels with equal salt concentrations on both sides of the membrane (0.1 M) so that the reversal potential is always 0. This difference between the present model with the experiments (and simulations) has also been discussed in the previous Section. Nevertheless, the discussion about the change in magnitudes of the currents is still meaningful. According to the design of this model and the discussion by Jensen, it seems that this similarity in the relative change in magnitude is not accidental. In the molecular dynamics simulation, it was proposed that the low-voltage I-V characteristics is due to the “infrequent ion recruitment into the pore lumen” in the case of gramicidin A channel and that the “formation of the knock-on intermediate occurred too infrequently” in the case of K$_{\rm v}$1.2/2.1[@jgp141:619]. In the present simple model, the flat low-voltage part is simply due to the small number of potassium ions in the channel. However, since it is assumed that two ions are much more preferentially bound, the binding of exactly two ions in the channel may be the analogy of the formation of the knock-on intermediate in K$_{\rm v}$1.2/2.1. In the structures used for gramicidin A channel simulations (PDB 1JNO and 1MAG), there is not any ion shown in the structure, but both structures were obtained from crystals of gramicidin A channel complexed with potassium thiocyanate. In the structures of Kv1.2/2.1 (2R9R and 2A79) or KcsA (3F7V and 1K4C), there are potassium ions at specific positions in the structures. It is reasonable to assume that during the experiments the structures of these channels were stabilized by including permeating ions in them. However, in the relaxed structures in the MD simulations, it seemed that the structures were not stabilized by including these ions. Some artifacts may be there, making it much more difficult for the ions to enter the simulated channels. In Equation , the two factors $b$ and $\delta V$ appear together in the same exponent. It is necessary to further emphasize the different influence of these two terms. Obviously, at low-voltage, the I-V characteristics is mainly determined by $b$, among other constant factors. If $b$ is relatively small, the high-voltage properties of the system is dominated by the membrane potential. To further distinguish the two effects, in Fig. \[f:fig04\], the I-V curves at two different $b$ values are presented. In Panel (a), $b=0$ and $B=1$. The five I-V curves correspond to different values of the sensitivity $\delta$ of the binding constant to voltage: $\delta=0$ for the thin line, 2 for the thin dashed line, 4 for the thick dotted line, 6 for the thick dashed line, and 8 for the thick line. In Panel (b), $b=2$ and $B=100$. The five I-V curves correspond in exactly the same way to the sensitivity factor $\delta$. In both Panels, the maximum current values are about the same, making the comparison easier. When the value of $b$ is large, the effect of the membrane-potential-sensitivity is less obvious. A drastic difference in the two cases is that, when $\delta$ and $b$ are both small, the current is very low even at relatively high voltage; but if $b$ is not so small, even if $\delta$ is zero, the I-V curve is obviously concave upward. This is because when $b$ and $\delta$ are both small, even at relatively high voltage, the mean number of potassium ions in the channel is close to zero. Therefore, the majority part of the barium ion exit rate, that is, the so-called current that is calculated in this model, is from $k_0$, which is pretty small. Of course, if the value of the parameter $\kappa$ is increased, these few I-V curves will not look so flat, but the fact that these curves are flat is exactly the point of this model-system study. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (a) (b) ![The I-V curve of the model system under different conditions. In Panel (a), the parameter $b$ is 0, or $B=1$, in Eq. [«]{}. In Panel (b), $b=\ln 100$ and $B=100$. In each Panel, five different curves correspond to different values of the sensitivity, $\delta$, of the potassium ion binding constant to membrane potential: the thin line for $\delta=0$, thin dashed line for $\delta=2$, thick dotted line for $\delta=4$, thick dashed line for $\delta=6$, and the thick line for $\delta=8$. \[f:fig04\]](figure04a.pdf "fig:"){width="6cm"} ![The I-V curve of the model system under different conditions. In Panel (a), the parameter $b$ is 0, or $B=1$, in Eq. [«]{}. In Panel (b), $b=\ln 100$ and $B=100$. In each Panel, five different curves correspond to different values of the sensitivity, $\delta$, of the potassium ion binding constant to membrane potential: the thin line for $\delta=0$, thin dashed line for $\delta=2$, thick dotted line for $\delta=4$, thick dashed line for $\delta=6$, and the thick line for $\delta=8$. \[f:fig04\]](figure04b.pdf "fig:"){width="6cm"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- By comparing with the experimental and simulation results presented by Jensen[@jgp141:619], it appears that the experimental results are closer to the high-$b$ cases in this model, while the simulation results are closer to the low-$b$ cases. What can be the implications? Since the simulated currents are too weak compared with experiments, Jensen [*et al.*]{} tried to modify the ion-pore interaction to see if there can be more ions in the channel so that the current can also increase. This does not work for K$_{\rm v}$1.2/2.1. For gramicidin A, there is increase in both intra-channel ion number and current when the ion-pore interaction is increased, but still way too low. Beyond these descriptions, there were too little information provided. However, compared with the present model, it seems that the major difference between the low-current and high-current cases are not solely in the attraction interaction between the ions and the channel pore. Indeed, increasing the attraction will lower the energy of the ions in the pore. But according to the present model, the most significant effect would be due to the better stability of the channel structure, instead of just from being able to attract the ions closer to the channel. If the channel structure is ‘relaxed’ without putting a few positive ions inside, extra forces must have been applied somewhere to make the structure more stable than it should be. And, since by doing that the channel is already ‘over’ stabilized, the entrance of more than one positive ions in the channel will most likely be energetically unfavored. In other words, if the simulation results can really be used to calculate whether the whole system can go through a structure reorganization so that the free energy change for binding the first few ions is indeed much more significant, that should help to greatly increase the current level predicted by the simulation. By reading their article, it is not explicitly found out whether those authors included the potassium ions in the relaxation stage or not. By studying their procedures of obtaining the one-dimensional potential of mean force and the simulated I-V curves, it seemed that the initial structure of the channel was devoid of potassium ions, but still this is not sure. Even if potassium ions were included, it was very difficult to judge whether the structure was fully relaxed or not. The practical limitation is that in order to compare the stabilities of the channel structures with different numbers of ions in them, very long simulations have to be carried out. It takes strong motivation for the simulation scientists to devote computational resources on such works. Hopefully the discussions here provide some encouragement. Besides the I-V relation, it is also worthy of examining whether the dependence of ion current on the bulk concentration of potassium ions shows reasonable trend. For this purpose, comparisons were made with the Brownian dynamics simulations done by Gordon and Chung[@bpj101:2671] on Kv1.2. It was argued in their work that the pore region of Kv1.2 is much more hydrophobic and less charged compared with KcsA. The low current level in the simulations on wild-type Kv1.2 was attributed to the higher hydrophobicity and neutrality of the channel pore. To examine whether the conjecture is reasonable, they mutated two proline residues into aspartate residues to increase the Coulomb interaction between the potassium ions and the channel pore. From their figures, the current-concentration data points were extracted. Since the current level is arbitrary in the present model, the ion-current values taken from their data were scaled arbitrarily so that they can be directly compared with the results of the present model. Moreover, since they explicitly used pico-ampere as the unit of the ion current, it is assumed that the relative scale between their data is absolute. The factors used for scaling the two set of experimental data are strictly the same (multiplied by 4, in the following), in order for the comparison to be fair. In Figure \[f:fig05\], these simulation results are plotted together with the current-concentration curve of the present model with $N=4$. Most of the parameters are the same as in the previous figures. The sensitivity factor $\delta$ is adjusted to 10, and the membrane potential is fixed at $V=0.12$, in accordance with the membrane potential used in the simulations by Gordon. Five different values of $b$ were used, namely, 0 (thin solid line), 1 (thin dashed line), 2 (dotted line), 3 (dashed line) and 4 (solid line) respectively. The parameters were not further optimized for fitting of the data. However, it seems that the change of $b$ from 0 to 3 quite nicely reflects the result of adding negatively-charged residues at critical positions in the pore. ![The current-concentration relation of the present model is compared with that obtained through Brownian dynamics and molecular dynamics simulations[@bpj101:2671]. The filled circles are the simulated current-concentration relation of wild-type Kv1.2 channel. The filled squares are the simulated current-concentration relation of mutated Kv1.2 channel in which two of the proline residues were replaced by aspartate residues. The current values of all of these data were multiplied by 6 so that they can be in the same scale as the currents calculated from the present model. The five lines are the current-concentration relation calculated from the present model. The number of binding sites is assumed to be 4. Other parameters are $a=2.5$, $\mu=3$, $\delta=10$, $\kappa=1$, $\beta=1$, and membrane voltage $V=0.12$. The five curves, from the bottom to the top, correspond to $b=0$, 1, 2, 3 and 4, respectively. \[f:fig05\]](figure05.pdf){width="6cm"} Compared with the experiments[@jgp76:425; @bpj101:2671], it was found that the simulated current-concentration of the mutant channel reproduced the characters of the experimental data well. The ion current through the mutant channel saturated much faster than that of the wild type does, and at a much higher level. However, the half-saturation concentration of the simulated mutant channel is “slightly higher than the the experimentally determined values for other potassium channels” and is “somewhat lower than that determined from KcsA using BD simulations”. Figure \[f:fig05\] shows that if the initial structure is even more stabilized by positive ions, the current level will be even higher, and the half-saturation concentration will be lower. In the words of the present model, those BD simulations did include important effects that resulted in the higher ion current, but it is still not enough. According to the present model, there may be a few possibilities. First, they used single-ion static PMF to reflect the difference between the wild-type and mutant channels. As discussed earlier, this picture may not be the most appropriate for long channels like Kir. Second, in their picture, the estimation of the energy difference between the mutant and the wild type was, although standard, not the most deliberate for studying the electrostatic interaction in biological systems. It was found that the dielectric environment in the biological system tends to make the attraction force between opposite charges weaker and the repulsion force between like charges stronger[@pre81:031925]. Therefore, if the empty channels are considered as the initial states, and the energy is taken as the reference point, the free energy difference between the stabilized structures of the mutant and wild-type channels should be even larger than predicted by standard theory. Since the attraction force is generally weaker, it may also imply that even more positive ions than predicted by conventional theory would be in the stable structure. Both possibilities suggest that the $b$ and $\mu$ parameters may be higher in reality than in the model system of Gordon. To limit the length of discussions, the effect of $\mu$ is not systematically presented. Just by considering the effect of increasing the value of $b$, say from 3 to 4, does further lower the half-saturation concentration of the model. Qualitative discussions about the correspondence between the present theory and either other simulations or experiments were all consistent. Nevertheless, it would be more insightful if the structural stabilities of ion-associated and empty ion channels can be compared quantitatively. This is the major assertion of this work. Conclusions {#s:conclusion} =========== In this work, a model system in which positively cooperative transport processes are coupled to negatively cooperative binding processes was proposed, and its behavior was discussed. Two major assertions were made, which are considered as the crucial features that have to be appended into all existing theoretical models. First, many channels can only be well stabilized to their native conducting structures by including a few permeating ions into them. Thermodynamically, this is equivalent to saying that the binding affinities of the first few permeating ions are much higher. It is energetically much less favored for more ions to enter the channel. But if they happen to do so, they are likely to induce efficient ion transport through the knock-on mechanism. Second, since the knock-on mechanism is the most reasonable transport mechanism in long channels, the reaction path used for studying the potential of mean force of the ion transport process is a hyper-curve on a multi-dimensional surface. This was already the way in which many of the simulation scientists analyzed their systems. However, with a chemical kinetics model, the essential meanings of the complicated potential hypersurface, the reaction path, and the potential of mean force, can be more easily encapsulated, investigated, presented and interpreted. The ion-blocking and unblocking process, despite being different from the transport process in several important aspects, can be used to qualitatively discuss the increase and decrease of ion transport rate under the change of the binding and structural stability characters of the system. Moreover, the binding reactions can be assumed to operate in steady state satisfying the pre-equilibrium approximation. Through the careful considerations and designs, the number of arbitrary parameters can be kept as few as possible, while the major features were maintained. By providing the numerical examples of the present model and showing them side by side with the results of some experiments and other simulations, it was demonstrated that the considerations of this model pointed out the direction to improve other theories and simulations in order to diminish the gap between models and reality. The I-V relation of model systems in molecular dynamics simulations, and the current-concentration curve of model systems in Brownian dynamics simulations, were generally lower than that observed experimentally. The present model seems to have identified the underlying problems of the simulations. This work was financially supported by the National Science Council of Taiwan, project NSC 99-2113-M-001-021.

--- abstract: 'Operating conditions and challenging demands of present and future accelerator experiments result in new requirements on detector systems. There are many ongoing activities aimed to develop new technologies and to improve the properties of detectors based on existing technologies. Our work is dedicated to development of Transition Radiation Detectors (TRD) suitable for different applications. In this paper results obtained in beam tests at SPS accelerator at CERN with the TRD prototype based on straw technology are presented. TRD performance was studied as a function of thickness of the transition radiation radiator and working gas mixture pressure.' address: - '$^1$ National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe highway 31, Moscow, 115409, Russia' - '$^2$ P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninsky prospect 53, Moscow, 119991, Russia' - '$^3$ CERN, the European Organization for Nuclear Research, CH-1211 Geneva 23, Switzerland' - '$^4$ Boğaziçi University,34342 Bebek/Istanbul Turkey' - '$^5$ Istanbul Bilgi University, High Energy Physics Research Center, Eyup, Istanbul, 34060, Turkey' - '$^{6}$ Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University, Moscow, Russia' - '$^{7}$ Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland' author: - 'V O Tikhomirov$^{1,2,}$, T Brooks$^{3}$, M Joos$^{3}$, C Rembser$^{3}$, E Celebi$^{4}$, S Gurbuz$^{4}$, S A Cetin$^{5}$, S P Konovalov$^{2}$, K Zhukov$^{2}$, K A Fillipov$^{1,2}$, A Romaniouk$^{1}$, S Yu Smirnov$^{1}$, P E Teterin$^{1}$, K A Vorobev$^{1}$, A S Boldyrev$^{6}$, A Maevsky$^{6}$, and D Derendarz$^{7}$' title: Some results of test beam studies of Transition Radiation Detector prototypes at CERN --- Introduction ============ Particle identification with TRDs is based on the difference of energies deposited in detector module(s) crossed by particles with different gamma factors. Particles with high Lorentz factor, like electrons, produce photons in Transition Radiation (TR) radiator which are absorbed in the detector sensitive volume. For detectors which have small thickness like straw based detector (straw tube diameter of 4 mm) averaged energy deposition due to ionization losses of particle is significantly less than energy of TR photon which is above $\sim$5 keV. With some probability this allows to separate events with TR from events where TR was not absorbed. Typical resulting spectra of energy depositions for different particle types in straw detector are shown in figure \[fig:diff\_spectra\]. Figure \[fig:integ\_spectra\] shows the same spectra in integral form: probability to exceed some energy threshold as a function of threshold. One sees large difference for pions and electrons when TR radiator is used. Very often for the comparison of performances of different detectors with the same structure it is better to use presentation shown in figure \[fig:el\_vs\_pi\]. Each point on this plot correspond to some threshold and projections on axis show probability for electron and pion to exceed this threshold. The larger electron efficiency at the same pion efficiency the better performance of the detector (better electron/pion separation). Optimal operation point of the detector (best separation) is around a “knee” of the dependence for electrons. In our case this point is around 0.05 of pion efficiency. See review in [@PDG] for more details on principles and usage of TRDs. ![\[fig:integ\_spectra\]Integral energy spectra – probability to exceed some energy deposition in a single straw.](diff_spectra.eps){width="18pc" height="15pc"} ![\[fig:integ\_spectra\]Integral energy spectra – probability to exceed some energy deposition in a single straw.](integ_spectra.eps){width="18pc" height="15pc"} ![\[fig:el\_vs\_pi\]Efficiency of electron identification vs efficiency of pion identification.](el_vs_pi.eps){width="18pc" height="15pc"} In order to improve particle identification, several parameters of the TR radiator and detector system can be tuned depending on the desired physics task. The most important ones are radiators parameters, active gas composition and its thickness. One should also mention that the method of data analysis plays significant role in final particle separation process. This paper presents the results on the studies of detector performance with different radiator thicknesses and different working gas pressures. Some other results obtained with the TRD prototype can be also found in reports [@Celebi] and [@Tishchenko], presented at this Conference. Test beam set-up ================ Schematic view of the test beam straw TRD prototype is shown in figure \[fig:strawTRDsetup\]. Beam particle, triggered by 10$\times$10 mm$^2$ scintillators, crosses ten straw layers. The gaps between layers are used to install different TR radiator blocks. Thin-walled proportional chambers (straws) are used to measure ionization and TR photons spectra. Straws with 4 mm diameter are made from a special conductive Kapton film. Straw wall has thickness of 70 $\mu$m. Similar chambers are used in the Transition Radiation Tracker detector [@TRT] of the ATLAS experiment [@ATLAS] at LHC. The straws in the prototype were operated with a gas mixture of 71.8% Xe, 25.6% CO$_2$ and 2.6% O$_2$. The gas gain was of about 2.5$\cdot10^4$ and it was controlled with an accuracy of about 1.5% during the run using Fe$^{55}$ source. Signals from straws were recorded using VME QDC modules. In order to separate signals from noise only energy depositions above 100 eV were considered. ![\[fig:strawTRDsetup\]Schematic view of the TRD prototype exposed to the test beam.](strawTRDsetup.eps){width="27pc"} Each radiator block contains 36 polypropylene foils of 15 $\mu$m thick spaced by 213 $\mu$m air gap. Total radiator thickness along the beam direction is 8.2 mm. Two different radiator configurations were considered. In the first case each of 10 radiators were situated in front of each of 10 straw layers. For the second setup (“double radiators”) 10 radiators were grouped by two and installed in front of last five straw layers. One should note that TR spectrum is changed with an increase of a number radiator-detector layers until it reaches saturation which is defined by equilibrium between produced and absorbed TR photons. That is why all results presented here are related to TR spectra obtained in the last straw layer where the saturation is guaranteed. Ionization spectra of pions are not affected by TR and therefore in order to increase statistics pion spectra from all 10 layer of straws were merged. Results {#sec:Results} ======= As it was mention above energetic TR photons may cross straw layers without absorption and can be absorbed by the following radiator blocks. In order to increase a registration efficiency of the energetic photons one can increase the pressure of the working gas in the straws. It will increase a probability of TR absorption in the detector. However this would also increase ionization losses of particles. That is why the resulting effect of the pressure increase on the particle separation is not evident. Preliminary Monte Carlo simulations predict rather week dependence of probability for electron to exceed registration threshold at fixed probability for pions – figure \[fig:MCpressure\] with maximum close to 1.2 bar (absolute pressure). A special test with the gas pressure of 1.5 bar (abs.) was carried out to verify this behavior. ![\[fig:MCpressure\]Monte Carlo simulation of expected electron registered efficiency at 0.05 pion efficiency as a function of working gas pressure. Line connecting points is guide to the eyes only.](MCpressure.eps){width="20pc" height="18pc"} Figure \[fig:pressure10\] shows comparison of probability for electrons vs probability for pions to exceed certain threshold at 1 bar (abs.) and 1.5 bar (abs.) of working gas pressure for setup with 10 single radiator blocks. The same comparison for setup with five double radiator blocks is presented on figure \[fig:pressure5\]. One sees that in both cases the increase of gas pressure does not significantly change these dependencies even at rather high photon energies which correspond to low pion probabilities and this behavior is what was expected from MC simulations. ![\[fig:pressure5\]Comparison of electron vs pion efficiency at normal and increased gas pressure in straws. Setup with 5 double radiators.](pressure10.eps){width="18pc" height="16pc"} ![\[fig:pressure5\]Comparison of electron vs pion efficiency at normal and increased gas pressure in straws. Setup with 5 double radiators.](pressure5.eps){width="18pc" height="16pc"} Another way to improve particle identification power is to optimize the radiator thickness. The thicker radiator the more TR photons generated but in that case at the same number of detector modules the total detector size and material budget is increased. Figures \[fig:5and10norm\_pressure\] and \[fig:5and10enlarge\_pressure\] give comparison of electron vs pion efficiency dependencies for single and double layer radiator – at 1 and 1.5 bars of the gas pressure respectively. In both cases thicker radiator can provide significantly better identification power but the length of the detector is increased by factor of 1.7. If the total detector length is fixed then the thicker radiator will produce higher number of TR photons, however the reduced number of sensitive layers will lead to larger statistical fluctuations of the registered signal thus decreasing the identification performance. In order to reach better particle identification performance the optimal configuration of radiator thickness and detector sensitive volumes can be obtained taking into consideration all external requirements such as available space and total amount of material which is crossed by particle. ![\[fig:5and10enlarge\_pressure\]Comparison of electron vs pion efficiency for 10 single and 5 double radiator setups at increased gas pressure in straws.](5and10norm_pressure.eps){width="18pc" height="16pc"} ![\[fig:5and10enlarge\_pressure\]Comparison of electron vs pion efficiency for 10 single and 5 double radiator setups at increased gas pressure in straws.](5and10enlarge_pressure.eps){width="18pc" height="16pc"} Conclusion {#sec:Conclusion} ========== Transition Radiation Detector prototype based on straw tubes was tested with 20 GeV pion and electron beams at CERN SPS accelerator. It was shown that an increase of the working gas pressure by 0.5 bar has no advantage for electron/pion separation with respect to normal pressure conditions. Better rejection power can be obtained by increasing radiator thickness due to larger number of generated TR photons at the cost of the increased detector length. We gratefully acknowledge the financial support from Russian Science Foundation – grant No.16-12-10277. References {#references .unnumbered} ========== [9]{} C Patrignani [*et al.*]{} (Particle Data Group) 2016 Chin. Phys. C, 40, 100001, chapter “Transition radiation detectors”. E Celebi [*et al.*]{} Test beam studies of the TRD prototype filled with different gas mixtures based on Xe, Kr, and Ar. *J. Phys.: Conf. Series* (In this Proceedings) A Tishchenko [*et al.*]{} Effect of graphen monolayer on the transition radiation yield of the radiators based on polyethylene foils. This Conference, http://indico.cfr.mephi.ru/event/4/session/18/contribution/317 E Abat [*et al.*]{} 2008 The ATLAS Transition Radiation Tracker (TRT) proportional drift tube: design and performance *JINST* **3** P02013 G Aad [*et al.*]{} The ATLAS Collaboration 2008 The ATLAS Experiment at the CERN Large Hadron Collider. *JINST* **3** S08003

--- abstract: 'In the fast developing countries it is hard to trace new buildings construction or old structures destruction and, as a result, to keep the up-to-date cadastre maps. Moreover, due to the complexity of urban regions or inconsistency of data used for cadastre maps extraction, the errors in form of misalignment is a common problem. In this work, we propose an end-to-end deep learning approach which is able to solve inconsistencies between the input intensity image and the available building footprints by correcting label noises and, at the same time, misalignments if needed. The obtained results demonstrate the robustness of the proposed method to even severely misaligned examples that makes it potentially suitable for real applications, like *OpenStreetMap* correction.' address: | ^1^ Institute of Computer Graphics and Vision, Graz University of Technology, Graz, Austria\ (stefano.zorzi, fraundorfer)@icg.tugraz.at\ ^2^ Remote Sensing Technology Institute, German Aerospace Center (DLR), Wessling, Germany\ ksenia.bittner@dlr.de bibliography: - 'main.bib' title: 'Map-Repair: Deep cadastre Maps Alignment and Temporal Inconsistencies Fix in Satellite Images' --- deep learning, segmentation, building footprint, remote sensing, high-resolution aerial images, cadastre map alignment Introduction {#sec:intro} ============ Semantic segmentation is still a challenging problem in Remote Sensing. Automatic detection and extraction of precise object outlines, such as human constructions, is in the interest of many cartographic and engineering applications. The most effective way to deal with this problem is the use of *Convolutional Neural Networks* trained in a supervised manner. Accurate ground truth annotations allows to achieve great detection and segmentation accuracies, however, these good annotations are hard to come by because they might be misaligned due to multiple causes e.g. human errors or imprecise digital terrain model. Furthermore, the maps may not be temporally synchronized with the satellite images failing to take into account variations in the constructions, i.e. new buildings may have been built or destroyed. Several related works tackle this problem with different approaches. Good alignment performance are achieved in [@girard2018aligning] by training a CNN to predict a displacement field between a map and an image. The same authors proposed in [@girard2019noisy] a multi-rounds training scheme which ameliorates ground truth annotations at each round to fine-tune the model. More recently, a method that performs a sequential annotation adjustment using a combination of consistency and self-supervised losses has been published [@chen2019autocorrect]. In this paper we propose an end-to-end self-supervised deep learning method for the generation of aligned and temporally coherent cadastre annotation in satellite and airborne imagery. The aim of the method is to align in one single shot all the object instances present in the intensity image and, at the same time, detect obsolete footprints and segment constructions that lack annotations. ![*MapRepair* result. Misaligned annotations in red, corrected map in cyan. []{data-label="fig:result_kitsap36"}](imgs/results12.png){width="\linewidth"} Methodology {#sec:method} =========== (-481,55) (-488,115) (-410,113) (-253,53) (-261,174) (-331,145) (-335,85) (-332,14) (-200,25) (-145,85) (-208,202) Our goal is to train a deep neural network which can not only generate an aligned cadastre map, but can also remove obsolete footprints and detect new buildings. The overall network model is shown in Figure \[fig:workflow\], and it is composed of two different branches. The first branch estimates and performs a projection for every building instance in order to produce a map perfectly registered with the intensity image. During this process, obsolete footprints are discarded. If a building does not have a footprint in the map, the second branch segments and regularizes the construction providing an accurate and visually pleasing building boundary. The results from two paths are then merged to produce the final corrected map. Similarity map estimation and instance warping ---------------------------------------------- In order to better align individual object instances to the image content, a generator network $G$ is exploited to predict a similarity transformation map $T \in \mathbb{R}^{4 \times H \times W}$ where the channels store translation (along $x$ and $y$ axis), rotation and scale values for each pixel location. The model receives as input the intensity image $I \in \mathbb{R}^{3 \times H \times W}$ and a binary mask $y = \{0,1\}^{H \times W}$ which represents the noisy or misaligned annotations. $$T = G(I, y)$$ A similarity transformation is then computed independently for every building averaging the values of the tensor $T$ under the area described by the object instance. The transformation for the $i$-th instance can be written as: $$t_i = \frac{1}{N_i} \sum_{p \in \omega_i} T_p$$ where $N_i$ and $\omega_i$ are the number of points and the set of points of the instance $i$-th, respectively. The four values of $t_i$ define a $\mathbb{R}^2 \xrightarrow{} \mathbb{R}^2$ similarity transformation. The refined annotation for the $i$-th object instance $\hat{y}_i$ is expressed as the transformed version $\hat{y}_i=\text{warp}(y_i, t_i)$ of the noisy instance annotation $y_i$ by the predicted transformation $t_i$. The predicted aligned annotations $\hat{y}$ for the binary image $y$ is then calculated as the combination of the single instance transformations and can be expressed as: $$\hat{y} = \sum_{i=1}^M \text{warp}(y_i, t_i)$$ where $y_i$ represents the $i$-th instance of the noisy binary mask and $M$ is the number of object instances in the sample image. The loss function used to train the model to generate the similarity transformation map is a combination of the mean squared error and the mean absolute error between the predicted binary annotations $\hat{y}$ and the ground truth annotations. Segmentations and regularization -------------------------------- Maps may not be temporally synchronized with the satellite or airborne data, failing to take into account variations in human constructions, i.e. removed or new buildings. In order to solve this problem, the model $G$ also predicts two segmentation masks: the first represents footprints of buildings that lack of annotation in $y$, while the second shows the annotations that must be removed because obsolete. The missing footprints predicted by $G$ have round corners and an irregular shape due to the lack of geometric constraints during the prediction. In order to ameliorate the segmentation result we post-process the result with the regularization model proposed in [@8900337]. This network for footprint refinement is capable of generating regular and visually pleasing building boundaries without losing segmentation accuracy. The segmentation of the obsolete annotations is instead used by the warping function to filter out-of-date or wrong instances. During training the ground truth of both the missing instances and the obsolete instances is known and binary cross-entropy losses are computed for these two segmentation branches. The generator network $G$ is used both for the alignment task and for the detection task, therefore it is trained using a linear combination of the alignment losses and the segmentation losses. Network models -------------- The convolutional neural network used as generator $G$ is a recurrent residual U-Net [@alom2018recurrent] modified to produce three outputs: two segmentation masks and the similarity transformation map. The network we adopted is a simple but yet precise segmentation model which guarantees high building segmentation accuracy. The input image has 4 channels, since it is the concatenation of the intensity image $I$ and the noisy annotation mask $y$. The outputs have values that ranges in $[0,1]$ for the segmentation masks and in $[-1,1]$ for the similarity transformation map since we use $\operatorname{sigmoid}$ and $\operatorname{tanh}$ activation functions, respectively. The annotation instances are warped using a *Spatial Transformer Network* [@jaderberg2015spatial] that ensures to have differentiable warping operations and allows gradient flow during back-propagation. The warping function performs scale and rotation with respect to the barycenter of the selected annotation instance. It is noted that the generator $G$ does not receive any information about the separation in instances and about the location of the barycenter of the buildings present in the input mask. The network, in fact, learns to identify building instances and understands the transformation rules during training. The regularization network used to refine the segmentation is pre-trained and it is only used during inference. Experimental Setup {#sec:method} ================== Dataset ------- The generator network $G$ and the regularization model are trained in the Inria Aerial Image Labeling Dataset [@maggiori2017dataset] composed of 180 images ($5000 \times 5000$ resolution) of 5 cities from US to Europe. Two of these images are used as test set. During training we consider the annotation masks provided in the dataset as ground truth even if some of these images contain misalignments. Self-supervised training ------------------------ The network must receive misaligned and incorrect annotations in order to learn. Since the dataset is assumed to be made of aligned image pairs some synthetic misalignments and errors must be introduced to alter the ground truth images. The noise is therefore enhanced by introducing global and instance random translations, rotations and scales. Random translations have a maximum absolute value of $64$ pixels, while random rotations ranges between and . In order to create the ground truth for the segmentation branches some footprints have also been randomly removed and some others have been injected in the annotation masks. Results {#sec:result} ======= ![Alignment result in kitsap36. Synthetic misaligned annotations on the left. *MapRepair* prediction on the right.[]{data-label="fig:result_kitsap36"}](imgs/result.png){width="\linewidth"} ![Alignment result in bloomington22. Noisy OSM annotations are overlaid in red. *MapRepair* prediction is in cyan. Removed annotations are yellow and segmented buildings are pink.[]{data-label="fig:result_bloomington22"}](imgs/results5.png){width="\linewidth"} coordinates [ (8,0.774103)(16,0.602877)(24,0.500426)(32,0.37012)(40,0.334734)(48,0.272036)(56,0.231707)(64,0.186039) ]{}; coordinates [ (8,0.796829)(16,0.76675)(24,0.687254)(32,0.677082)(40,0.576821)(48,0.467999)(56,0.433674)(64,0.395019) ]{}; coordinates [ (8,0.837913)(16,0.82768)(24,0.809527)(32,0.806633)(40,0.799688)(48,0.776678)(56,0.766416)(64,0.635175) ]{}; -- -------- -------- -------- -------- IoU Acc IoU Acc 0.5235 0.9372 0.4739 0.9234 0.8302 0.9813 0.7369 0.9674 0.8281 0.9812 0.7341 0.9673 - - 0.7914 0.9740 -- -------- -------- -------- -------- : Results in bloomington22 using OSM annotations. \[table:osm\_results\] The method has been evaluated in two Inria images: kitsap36 and bloomington22. The two images have a resolution of $5000 \times 5000$ pixels and in order to evaluate the full image we split them into $448 \times 448$ patches. Each patch is individually processed by the network and a $64$ pixels border is discarded due to lack of context information that can lead to the generation of aligned annotations with errors and artifacts. The kitsap36 image contains 1252 building instances having a wide range of shapes and sizes. The ground truth provided by the dataset contains several misalignments that are manually corrected in order to evaluate the algorithm prediction. In this image *MapRepair* correcs the original misaligned ground truth increasing the Intersection over Union (IoU) score from $0.71$ to $0.82$. Several experiments with synthetic misalignments are conducted in the same test image showing the robustness of the method to heavy annotation displacements. Building annotations are randomly rotated between $-30\degree$ and $30\degree$ and translated by increasing absolute displacements from $8$ to $64$ pixels. The results in Figure \[table:synt\] show that all the synthetic annotations aligned by *MapRepair* achieve IoU scores around $0.8$. The best performance in reached with a maximum absolute displacement of $56$ pixels where the network improves the IoU score from $0.23$ to $0.77$ (Figure \[fig:result\_kitsap36\]). The efficiency starts dropping with an annotations misalignment of $64$ pixels. Bloomington22 is an image of the test-set of the Inria dataset, therefore the ground truth is not provided. For this region OSM provides $771$ building footprints, most of them with severe misalignments. Furthermore, several construction do not have an OSM annotation. In order to measure the effectiveness of the correction we manually aligned the footprints and we annotated the unlabelled buildings. The quantitative and qualitative results in this image are shown in Table \[table:osm\_results\] and Figure \[fig:result\_bloomington22\], respectively. Conclusions =========== We presented *MapRepair*, an approach for cadastre map refinement in satellite images composed of a multi-purpose neural network trained in a self-supervised manner. The model is capable of generating an aligned cadastre mask predicting a similarity transformation map and warping each object instance independently. Furthermore, it solves temporal synchronization errors removing unused footprints or segmenting new buildings in the scene. *MapRepair* achieves comparable or even higher alignment performance with respect to state-of-the-art methods, dealing effectively with heavily distorted annotations.

--- abstract: 'We investigate the behaviour of the QCD evolution towards high-energy, in the diffusive approximation, in the limit where the fluctuation contribution is large. Our solution for the equivalent stochastic Fisher equation predicts the amplitude as well as the whole set of correlators in the strong noise limit. The speed of the front and the diffusion coefficient are obtained. We analyse the consequences on high-energy evolution in QCD.' author: - 'C. Marquet' - 'R. Peschanski' - 'G. Soyez[^1]' title: 'Consequences of strong fluctuations on high-energy QCD evolution' --- 1\. The quest for the perturbative high-energy limit of QCD has been the subject of many efforts. It is now well accepted that, due to the strong rise of the amplitude predicted by the linear Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [@bfkl], one has to include saturation effects in order to describe high parton densities and recover unitarity. In the large-$N_c$ limit and in the mean-field approximation, we are led to consider the Balitsky-Kovchegov (BK) equation [@bk]. It has the nice property [@mp] to be mapped, in the diffusive approximation, onto the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation [@fkpp] which has been widely studied in statistical physics and is known to admit traveling waves as asymptotic solutions, translating into geometric scaling in QCD [@geomsc]. It has recently been proven [@fluct; @it] that fluctuation effects have important consequences on the approach to saturation. Practically, the resulting evolution equation, after a coarse-graining approximation [@it], takes the form of a Langevin equation. It is formally equivalent to the BK equation supplemented with a noise term $$\begin{aligned} \partial_Y T(L,Y) & = &\abar\chi(-\partial_L)T(L,Y) - \abar T^2(L,Y) \nonumber\\ & + & \abar\sqrt{\kappa\alpha_s^2 T(L,Y)} \nu(L,Y)\label{eq:langevin}\end{aligned}$$ where $T$ is the event-by-event scattering amplitude, $Y$ is the rapidity, $L=\log(k^2/k_0^2)$ with $k$ the transverse momentum and $k_0$ some arbitrary reference scale. $\chi(\gamma)=2\psi(1)-\psi(\gamma)-\psi(1-\gamma)$ is the BFKL kernel, $\kappa$ is a fudge factor and the noise $\nu(L,Y)$ satisfies ${\left\langle{\nu}\right\rangle}=0$ and $$\label{eq:noisecorel} {\left\langle{\nu(L,Y)\nu(L',Y')}\right\rangle} = \frac{2}{\pi} \delta(\abar Y-\abar Y')\delta(L-L').$$ To obtain equation , we have worked with impact-parameter-independent amplitudes, for which the original non-local, off-diagonal noise term takes [@it; @msw] the form given by equation . Physically, in addition to the BK saturation effects coming from pomeron merging in the target, one also takes into account pomeron splitting in the target. Hence, by combinations of splittings and mergings, pomeron loops are formed. The extra factor $\alpha_s^2$ in the fluctuation term indicates that it is dominant when $T\sim \alpha_s^2$ *i.e.* in the dilute limit. In the same line that the BK equation is equivalent to the F-KPP equation in the diffusive approximation, the Langevin problem corresponds to the stochastic F-KPP (sFKPP) equation [@duality]. To be more precise, let us expand the BFKL kernel to second order around $\gamma_0$ $$\chi(\gamma) = \chi_0 + \chi'_0(\gamma-\gamma_0)+\frac{1}{2}\chi''_0(\gamma-\gamma_0)^2 = A_0 + A_1\gamma + A_2\gamma^2.\label{eq:coefsai}$$ This approximation has proven its ability to exhibit the main properties of the solutions of equation in the limit $\kappa\alpha_s^2 \ll 1$. Unless specified, we shall keep this approximation throughout this paper, leaving the general case for further studies. If we introduce *time* and *space* variables as follows $$t = \abar Y,\quad x = L-A_1\abar Y\quad\text{and }u(x,t)=\frac{T}{A_0},$$ equation gets mapped onto the sFKPP equation [^2] $$\label{eq:sfkpp} \partial_t u = D \partial_x^2 u + \creation u(1-u) + \varepsilon\sqrt{u(1-u)}\eta(x,t),$$ with $$\begin{aligned} &&D=A_2, \quad \creation=A_0, \quad \varepsilon^2=\frac{2\kappa\alpha_s^2}{\pi A_0},\\ &&{\left\langle{\eta(x,t)\eta(x',t')}\right\rangle} = \delta(t-t')\delta(x-x').\end{aligned}$$ At present stage, most of the analytical analysis were performed in the limit where the fluctuations are asymptotically small (the correction being logarithmic [@brunet], it may require a strong coupling constant as small as $\alpha_s \lesssim 10^{-10}$), in which case the relevant quantities, *e.g.* the speed of the wave, can be expanded around the F-KPP solution. In this analysis, the main effects of the noise in the sFKPP equation are, first, to lead to a decrease of the speed of the traveling front and, second, to produce a diffusion of the fronts for each realisation of the noise resulting in violations of geometric scaling for the average amplitude. The numerical studies performed so far show that these effects (decrease of the speed and diffusion of the events) are amplified when the fluctuation coefficient becomes more important. There has been large efforts made to improve the analytical understanding for arbitrary values of the noise strength but many approaches appear to be model-dependent [@panja]. In this paper, we consider the problem of fluctuations through a complementary approach, namely the limit of a strong noise. This limit is tractable with the help of a *duality* property of the sFKPP equation [@duality]. The strong-noise limit then gets related to a coalescence process which can be solved exactly [@coal]. Using these tools from statistical physics, we are able to compute the event-averaged amplitude as well as the higher-order correlators and obtain predictions for the speed of the wavefront as well as for the diffusion coefficient in the limit of strong fluctuations. This knowledge of the strong-noise limit, together with the weak-noise results, can help in further analytical understanding of the QCD evolution in the presence of fluctuations. 2\. Let us now summarise the tools from statistical physics we need for our studies. Our starting point is the duality relation between the sFKPP Langevin equation and the reaction-diffusion process. This duality will allow us to relate the strong noise problem to the coalescence problem which we shall solve. We consider on the one hand amplitudes evolving according to the sFKPP equation and, on the other hand, the reaction-diffusion process of a one-dimensional population $A$ on a lattice of spacing $h$: at each site, one can have particle creation or recombination, and particles can diffuse to a neighbouring site $$\label{eq:partic} A_i \stackrel{\creation}{\to} A_i+A_i, \quad A_i+A_i \stackrel{\varepsilon^2/h}{\longrightarrow} A_i \quad\text{and }\quad A_i \stackrel{D/h^2}{\longrightarrow} A_{i\pm 1}$$ where $A_i$ designs a particle at lattice site $i$. One shows [@duality] that the particle system and the amplitude in the sFKPP equation are related through a duality relation which, in the continuum limit $h\to 0$, can be written $$\label{eq:duality} {\left\langle{\prod_x\left[1-u(x,t)\right]^{N(x,0)}}\right\rangle} = {\left\langle{\prod_x\left[1-u(x,0)\right]^{N(x,t)}}\right\rangle},$$ where $N(x,t)$ is the particle density in the reaction-diffusion system. Physically, this duality equation means that, if one wants to obtain the scattering amplitude at rapidity $t=\abar Y$, one can either keep the target fixed and evolve the projectile wavefunction considered as a particle system, or fix the projectile and consider evolution of scattering amplitudes off the target. One knows [@ddd; @ist] that splitting in the projectile leads to linear growth and saturation in the target while merging in the projectile corresponds to fluctuations in the target. Therefore, to obtain information on the evolution of the average amplitudes for the sFKPP equation, we shall study the dual particle system and then use relation . The limit we are interested in is the strong noise limit (large $\varepsilon$). This corresponds to heavy saturation in the particle system (projectile), *i.e.* two particles at the same lattice site automatically recombine into a single one. This limit is often referred to as the *diffusion-controlled limit*. When $\creation$ is rescaled to give a constant, small, ratio $\creation/\varepsilon^2$, we can alternatively study the *coalescence model*. In this model, one can have at most one particle per lattice site. One particle can diffuse to the neighbouring site at rate $D/h^2$ or give birth to a new one in a neighbouring site at rate $\omega/h$ (with $\omega=2D\creation/\varepsilon^2$ as we shall see later). This system has been studied [@coal] and is fully solvable using the method of *inter-particle probability distribution function*. The main idea is to introduce $E(x, y; t)$ as the probability that sites between $x$ and $y\ge x$ included are empty at time $t$. One obtains that, due to diffusion and creation, $E$ satisfies the following differential equation $$\label{eq:sysevol} \partial_t E = \left\{ D\left(\partial_x^2+\partial_y^2\right) + \omega\left(\partial_y-\partial_x\right) \right\} E.$$ with the boundary condition $\lim_{y\to x} E(x,y;t) = 1$. The particle density can be obtained from the derivative of $E$: $$N(x,t) = \left.\partial_yE(x,y;t)\right|_{y\to x}.$$ Remarkably enough, the evolution equation is linear. It can be solved exactly [@coal] and, for a given initial condition $E(x,y;t)$, introducing the dimensionless variables $$\xi = \frac{\omega}{D}(x+y),\quad \zeta=\frac{\omega}{D}(y-x)\quad\text{ and } \tau = \frac{8\omega^2}{D}t,$$ one finds $$\begin{aligned} \label{eq:syssol} E(x,y;t) & = & e^{-\zeta} + e^{-\tau}\int_{-\infty}^\infty d\xi'\int_0^\infty d\zeta'\\ & & \,G(\xi,\xi',\zeta,\zeta';\tau)\left[E(\xi',\zeta';\tau)-e^{-\zeta'}\right],\nonumber\end{aligned}$$ where the Green function $G(\xi,\xi',\zeta,\zeta';\tau)$ is given by $$\frac{1}{\pi\tau}e^{-(\xi-\xi')^2/\tau} e^{-(\zeta-\zeta')/2} \left[e^{-(\zeta-\zeta')^2/\tau}-e^{-(\zeta+\zeta')^2/\tau}\right].$$ Before considering the solution of this system in the context of the duality relation, let us give the relation between the parameters $\creation$ and $\varepsilon^2$ of the initial system with $\omega$ and $D$ in the coalescence model. The trick is to require that both systems have the same equilibrium density. For the coalescence model, one notice that $\exp(-\frac{\omega}{D}(y-x))$ is a time-independent solution leading to a particle density $N_{\text{eq}}=\omega/D$. In the case of process , at equilibrium, diffusion does not play any role and we have to find equilibrium at each site between creation end annihilation. This is achieved when $N_{\text{eq}} = 2\creation/\varepsilon^2$. Hence, one has $\omega=2D\creation/\varepsilon^2$. 3\. Let us now put together the results from duality and coalescence and derive the sFKPP solution. By carefully choosing the initial condition, equation simplifies. If one starts with one particle at position $x$ *i.e.* $N(\xb,0) = \delta(\xb-x)$, then the l.h.s. of becomes simply $1-{\left\langle{u(x,t)}\right\rangle}$. By starting with $k$ particles at position $x_1,\dots,x_k$, one similarly obtains ${\left\langle{[1-u(x_1,t)]\dots[1-u(x_k,t)]}\right\rangle}$. In addition, let us start with a step function for the amplitude $u(x,0)=\theta(-x)$. Then ${\left\langle{\prod_x\left[1-u(x,0)\right]^{N(x,t)}}\right\rangle}$ is the probability that, in the particle process, all sites $x\le 0$ are empty. For the case of the strong noise *i.e.* when the particle system is the coalescence model, this probability is directly obtained in terms of the density $E$ and the duality relation becomes $${\left\langle{u(x,t)}\right\rangle}=1-E_x(-\infty,0;t),$$ with the initial condition $$E_x(\xb,\yb;0)=1-\theta(x-\xb)\theta(\yb-x).$$ Inserting this initial condition inside the general solution , we get after a bit of algebra $$\begin{aligned} E_x(\xb,\yb;t) & = & \frac{1}{2} \left\{ \erfc\left(\frac{\yb\!-\!\xb\!+\!2\omega t}{\sqrt{8Dt}}\right) \!+\!\erfc\left(\frac{\xb\!-\!\yb\!-\!2\omega t}{\sqrt{8Dt}}\right) \!-\!\erfc\left(\frac{x\!-\!\yb\!-\!\omega t}{2\sqrt{Dt}}\right) \left[1\!-\!\frac{1}{2}\erfc\left(\frac{x\!-\!\xb\!+\!\omega t}{2\sqrt{Dt}}\right) \right] \right\}\\ & + & \frac{1}{2}e^{-\frac{\omega}{D}(\yb-\xb)} \left\{ 2\!-\!\erfc\left(\frac{\yb\!-\!\xb\!-\!2\omega t}{\sqrt{8Dt}}\right) \!-\!\erfc\left(\frac{\xb\!-\!\yb\!+\!2\omega t}{\sqrt{8Dt}}\right) \!+\!\erfc\left(\frac{x\!-\!\xb\!-\!\omega t}{2\sqrt{Dt}}\right) \left[1\!-\!\frac{1}{2}\erfc\left(\frac{x\!-\!\yb\!+\!\omega t}{2\sqrt{Dt}}\right) \right] \right\}\end{aligned}$$ where $\erfc(x)$ is the complementary error function. The limit $\xb\to -\infty$, $\yb\to 0$ in this expression gives $$\begin{aligned} \label{eq:sol} {\left\langle{u(x,t)}\right\rangle} & = & \frac{1}{2}\erfc\left(\frac{x-\omega t}{2\sqrt{Dt}}\right)\\ & = &\frac{1}{2\sqrt{D\pi t}}\int_{-\infty}^\infty d\xb\,\theta(\xb-x) e^{-\frac{(\xb-\omega t)^2}{4Dt}}.\nonumber\end{aligned}$$ This results calls for comments. First, it corresponds to a wave traveling at an average speed $$\omega=\frac{2D\creation}{\varepsilon^2}.$$ This result confirms the decrease of the velocity but contrasts with the speed obtained in the weak noise limit by perturbative analysis around the F-KPP speed $\simeq 2\sqrt{D\lambda}-\pi^2\sqrt{D\lambda} |\log(\varepsilon)|^{-2}.$ The expression shows that the amplitude could be obtained from a superposition of step functions around $x=\omega t$ with a Gaussian noise of width $\sqrt{2Dt}$. The interesting point here lies in the dispersion coefficient: in the weak-noise analysis, it behaves like $|\log(\varepsilon)|^{-3}$. We have proven that this rise goes to $2D$ when the noise becomes strong. In addition, one can probe the correlators of the amplitude by starting with an initial condition with particles at positions $x_{\text{min}}=x_1 < \dots < x_k=x_{\text{max}}$: $$E(x,y;0) = 1-\sum_{i=1}^k \theta(x-x_{i-1})\theta(x_i-x)\theta(y-x_i),$$ with, formally, $x_0=-\infty$. Following the same lines as above, one gets $${\left\langle{[1-u(x_1,t)]\dots[1-u(x_k,t)]}\right\rangle} = 1-\frac{1}{2}\erfc\left(\frac{x_{\text{min}}-\omega t}{2\sqrt{Dt}}\right).$$ We need to use this relation to obtain the correlations of $u$ instead of $1-u$. This is obtained as follows $$\begin{aligned} \lefteqn{{\left\langle{u(x_1,t)\dots u(x_k,t)}\right\rangle}}\\ & = & {\left\langle{\{1-[1-u(x_1,t)]\}\dots \{1-[1-u(x_k,t)]\}}\right\rangle}\\ & = & \sum_{A\subseteq \{1,\dots,k\}} (-)^{\sharp A}{\left\langle{\prod_{i\in A}[1-u(x_i,t)]}\right\rangle}\\ & = & \sum_{A\subseteq \{1,\dots,k\}} (-)^{\sharp A}{\left\langle{1-u(x_{\min(A)},t)}\right\rangle}\end{aligned}$$ where $\sharp A$ is the cardinal of the set $A$ and $\min(A)$ is its minimum. The sum can be reordered to give $$\begin{aligned} \lefteqn{{\left\langle{u(x_1,t)\dots u(x_k,t)}\right\rangle}}\\ & = & 1-\sum_{j=1}^k {\left\langle{1-u(x_j,t)}\right\rangle}\sum_{A\subseteq \{j+1,\dots,k\}} (-)^{\sharp A}\\ & = & 1-\sum_{j=1}^k [1-{\left\langle{u(x_j,t)}\right\rangle}]\sum_{n=0}^{k-j} (-)^n\binom{k-j}{n},\end{aligned}$$ In the last expression, only the term with $j=k$ survives hence (with $x_{\text{max}}=x_k$) $${\left\langle{u(x_1,t)\dots u(x_k,t)}\right\rangle} = \frac{1}{2}\erfc\left(\frac{x_k-\omega t}{2\sqrt{Dt}}\right) = {\left\langle{u(x_k,t)}\right\rangle}.$$ This simple result can again be seen as a superposition of step functions with a Gaussian dispersion. $u(x_1)\dots u(x_k)$ is nonzero provided $u$ does not vanishes at the position with largest coordinate ($x_k$). Hence, the whole dynamics is the same as if only one particle were lying at position $x_k$ in the initial condition [^3]. ![Comparison between the F-KPP and sFKPP wavefronts at $t=0,5,10,15$ and $20$. Solid curve: numerical simulation of the F-KPP equation. Dashed curve: analytic result for $\varepsilon^2=1.5$. Dotted curve: analytic result for $\varepsilon^2=3$.[]{data-label="fig:front"}](fig.eps) In order to illustrate the effect of the noise, we have plotted in figure \[fig:front\] the time evolution of the wavefront ${\left\langle{u(x,t)}\right\rangle}$ for numerical simulations of the F-KPP equation and for our solution ($\varepsilon^2=1.5$ and $3$). In each case, the initial condition is a step function. We clearly see that the effect of the strong noise is to slow down considerably the front and add significant distortion. 4\. Coming back to QCD variables, we find (assuming without loss of generality $L_1\le\dots\le L_n$) $$\begin{aligned} \label{eq:front} {\left\langle{T(L_1,Y)\dots T(L_k,Y)}\right\rangle} & = & A_0^{k-1}{\left\langle{T(L_k,Y)}\right\rangle}\\ & = & \frac{A_0^k}{2}\erfc\left(\frac{L_k-c\abar Y}{\sqrt{2D_{\text{diff}}\abar Y}}\right) \nonumber\end{aligned}$$ with the speed of the traveling front and the diffusion coefficient $$\label{eq:res} c = A_1+\frac{\pi A_2A_0^2}{\kappa\alpha_s^2}\qquad\text{and } D_{\text{diff}} = 2A_2.$$ Let us discuss the physical interpretation of these results. We start by equation . It is remarkable that the strong-fluctuation limit gives, as an analytic solution, the asymptotic result inferred in previous studies [@it; @himst]. It proves the universal feature that high-energy scattering amplitudes are described by a superposition of Heavyside functions with Gaussian dispersion. We confirm analytically that the correlators are driven by the amplitude of the largest momentum in the process. This is the main result of this paper. Equation relates the parameters of the amplitude [*e.g.*]{} the average speed $c$ and the diffusion constant $D_{\text{diff}}$ to the parameters $A_0$, $A_1$ and $A_2$ obtained from the expansion of the BFKL kernel. If one performs this expansion choosing $\gamma_0=\scriptstyle{\frac{1}{2}}$ or $\gamma_0=\gamma_c \approx 0.6275$, as used in the weak-noise limit, the value of $A_1$ turns out to be negative. This would lead to a negative speed which seems unphysical in QCD. The way out of this inconsistency is to insert the solution directly into the exact evolution for ${\left\langle{T(L,Y)}\right\rangle}$. It has been shown [@himst] that it results, as expected, into a vanishing speed. The determination of $c$ through the evolution equation would depend on the corrections to the error function, which disappear in the strong-noise limit. Let us sketch a heuristic argument indicating that a physically meaningful behaviour of these parameters can be obtained in the limit $\gamma_0$ small. Indeed, this is suggested by the fact that, in the strong-noise limit, each front is approaching a Heavyside function which suggests to perform the kernel expansion near $\gamma_0=0$. Considering $\gamma_0$ small, one has $A_0=3\gamma_0^{-1}$, $A_1=-3\gamma_0^{-2}$ and $A_2=\gamma_0^{-3}$, leading to $$c = \frac{3}{\gamma_0^2}\left(\frac{3\pi}{\gamma_0^3\kappa\alpha_s^2}-1\right).$$ If one requires $c\to 0$, one has to choose $$\gamma_0 \approx \left(\frac{3\pi}{\kappa\alpha_s^2}\right)^{1/3}\left\{1-o\left[\left(\frac{3\pi}{\kappa\alpha_s^2}\right)^{\frac{2}{3}}\right]\right\}$$ where $o(x)$ denotes a function falling to zero faster than $x$. The physical parameters, in the strong-noise limit, are then $$\label{eq:newres} c\to 0 \qquad\text{and } D_{\text{diff}} = \frac{2\kappa\alpha_s^2}{3\pi}.$$ It is interesting to notice that, if we choose $\gamma_0$ to satisfy the only requirement that $c\to 0$, the diffusion coefficient is entirely determined, independently of the way this limit is achieved. On a more general ground, it is worth noting that the strong-noise development should not be considered as a strong coupling expansion since the initial equation is derived in a perturbative framework. Indeed, the genuine expansion parameter in the strong noise limit is $\kappa \alpha_s^2$ where $\kappa$ could reach high values in the physical domain of interest [@dt]. Thus, even in the perturbative regime, the noise parameter $\kappa\alpha_s^2$ may be large. These predictions, together with the weak noise results, enclose the physical domain of $\kappa\alpha_s^2$. The knowledge of both these limits, together with information from numerical estimates [@simul], can lead to a better understanding of the physics of fluctuations. An expansion in $1/(\kappa\alpha_s^2)$ could also lead to faster convergence than the logarithmic weak-noise expansion [@bdmm]. 5\. Let us summarise our results. We have used the duality relation between the amplitude given by the stochastic FKPP equation and the particle densities in a reaction-diffusion process. This duality is physically similar to the projectile-target duality noticed recently in high-energy QCD when both saturation and fluctuation effects are taken into account. The saturation in the target is related to splitting in the projectile while fluctuations are mapped to recombination in the particle system. In the case of large fluctuations *i.e.* strong noise in the sFKPP equation, the corresponding particle system can be described as a coalescence problem. This process can be solved exactly using the inter-particle probability distribution function. We use this to compute the average value of the amplitude as well as the correlators. The main result of our analysis is the analytic derivation of the average scattering amplitude as a universal error function which also determines higher-order correlators. This picture proposed in previous studies is thus confirmed and shows that the results obtained in the limit of strong fluctutions possess a physical meaning. The fact that the correlators display the same behaviour as the amplitude itself is physically interesting. This is obtained through a superposition of event-by-event amplitudes which are 0 or 1. The dominant contribution to the scattering process comes when all individual scattering are 1. This picture of a black and white target gives rise to new scaling laws for Deep Inelastic Scattering [@himst]. Since the physically acceptable values of the strong coupling seem to lie in between the strong and the weak noise limits, a knowledge of both approaches is useful. As an outlook, it would be interesting to extend our formalism beyond the diffusive approximation and/or to modified evolution kernels [@param]. This may allow a better determination of the speed and dispersion coefficient, given by in the diffusive approximation. Also, a perturbative approach starting from the strong noise limit could prove useful. These questions certainly deserve further studies. G.S. is funded by the National Funds for Scientific Research (Belgium). [10]{} L. N. Lipatov, [*Sov. J. Nucl. Phys.*]{} [**23**]{}, (1976) 338; E. A. Kuraev, L. N. Lipatov and V. S. Fadin, [*Sov. Phys. JETP*]{} [**45**]{}, (1977) 199; I. I. Balitsky and L. N. Lipatov, [*Sov. J. Nucl. Phys.*]{} [**28**]{}, (1978) 822. I. Balitsky, [*Nucl. Phys.*]{} [**B463**]{} (1996) 99; Y. V. Kovchegov, [*Phys. 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[^3]: Of course, the same argument holds for the product of $1-u(x_i)$ and the fronts around $x_1$.

--- abstract: 'The Gd$_2$Sn$_2$O$_7$ pyrochlore Heisenberg antiferromagnet displays a phase transition to a four sublattice Néel ordered state at a critical temperature $T_c \sim 1$ K. The low-temperature state found via neutron scattering corresponds to that predicted by a classical model that considers nearest-neighbor antiferromagnetic exchange and long-range dipolar interactions. Despite the seemingly conventional nature of the ordered state, the specific heat $C_v$ has been found to be described in the temperature range $350~\mathrm{mK} \le T \le 800~\mathrm{mK}$ by an anomalous power law, $C_v\sim T^2$. A similar temperature dependence of $C_v$ has also been reported for Gd$_2$Ti$_2$O$_7$, another pyrochlore Heisenberg material. Such behavior is to be contrasted with the typical $T^3$ behavior expected for a three-dimensional antiferromagnet with conventional long-range order which is then generally accompanied by an $\exp(-\Delta/T)$ behavior at lower temperature where anisotropy effects induce a gap $\Delta$ to collective spin excitations. Such anomalous $T^2$ behavior in $C_v$ has been argued to be correlated to an unusual energy-dependence of the density of states which also seemingly manifests itself in low-temperature spin fluctuations found in muon spin relaxation experiments. In this paper, we report calculations of $C_v$ that consider spin wave like excitations out of the Néel order observed in Gd$_2$Sn$_2$O$_7$ via neutron scattering. We argue that the parametric $C_v \propto T^2$ does not reflect the true low-energy excitations of Gd$_2$Sn$_2$O$_7$. Rather, we find that the low-energy excitations of this material are antiferromagnetic magnons gapped by single-ion and dipolar anisotropy effects, and that the lowest temperature of 350 mK considered in previous specific heat measurements accidentally happens to coincide with a crossover temperature [*below*]{} which magnons become thermally activated and $C_v$ takes an exponential form. We argue that further specific heat measurements that extend down to at least 100 mK are required in order to ascribe an unconventional description of magnetic excitations out of the ground state of Gd$_2$Sn$_2$O$_7$ or to invalidate the standard picture of gapped excitations proposed herein.' author: - Adrian Del Maestro - 'Michel J. P. Gingras' bibliography: - 'gd2sn2o7-excite.bib' title: 'Low temperature specific heat and possible gap to magnetic excitations in the Heisenberg pyrochlore antiferromagnet Gd$_2$Sn$_2$O$_7$' --- Introduction {#sec:Introductino} ============ Persistent spin dynamics in pyrochlores --------------------------------------- A magnetic system with Heisenberg spins that sit on the vertices of a three-dimensional pyrochlore lattice of corner sharing tetrahedra and interact among themselves via nearest-neighbor antiferromagnetic exchange interactions is highly geometrically frustrated [@Reimers-MFT; @Moessner:Pyro; @Moessner-CJP]. Such a system is theoretically predicted to not develop conventional magnetic long range order at finite temperature for either classical [@Moessner-CJP; @Moessner-PRB] or quantum spins [@Canals:Pyro]. As a result of this frustration, real magnetic materials with antiferromagnetically coupled spins on this pyrochlore structure are highly sensitive to weak perturbative interactions beyond nearest-neighbor exchange which dramatically affect the nature of the low temperature state. It is partially for this reason that the insulating R$_2$M$_2$O$_7$ magnetic pyrochlore oxides have attracted such a great deal of attention in recent years [@Greedan:Review]. Indeed, this family of materials has been found to display a variety of magnetic states and exotic low temperature behaviors that strongly depend on the specific elements R and M considered [@Greedan:Review]. In R$_2$M$_2$O$_7$, the R site is occupied by a trivalent ion, such as diamagnetic Y$^{3+}$ or a magnetic rare earth ion (R=Gd$^{3+}$, Tb$^{3+}$, Dy$^{3+}$, Ho$^{3+}$, Er$^{3+}$, Tm$^{3+}$, Yb$^{3+}$) while the M site is occupied by a tetravalent ion which can be diamagnetic, such as Ti$^{4+}$ or Sn$^{4+}$, or magnetic, such as Mo$^{4+}$ or Mn$^{4+}$. Both the R and M sites form distinct interpenetrating lattices of corner-shared tetrahedra and either site can be magnetic or non-magnetic[@Reimers-MFT; @Greedan:Review]. Such freedom allows for a large variety of phenomenology in the pyrochlore materials. This includes spin glasses like Y$_2$Mo$_2$O$_7$ [@Greedan:SG-YMO; @Gingras:YMO; @Gardner-YMO; @Keren:YMO] and Tb$_2$Mo$_2$O$_7$ [@Gaulin-Tb2Mo2O7], spin ices such as Ho$_2$Ti$_2$O$_7$ [@Harris-PRL; @Bramwell-PRL; @Cornelius] and Dy$_2$Ti$_2$O$_7$ [@Ramirez-Nature], conventional long range ordered materials like Gd$_2$Ti$_2$O$_7$ [@Raju:GTO; @Ramirez-GTO; @Stewart:GTO; @Champion:ETO] and Gd$_2$Sn$_2$O$_7$ [@Wills:GSO] and even possible spin liquids as in the case of Tb$_2$Ti$_2$O$_7$ [@Gardner-PRL; @Gardner-PRB-50mk; @Kao:TTO; @Enjalran-review; @Mirebeau-review; @Molavian]. One common thread throughout these various materials is that several experimental studies have found that, almost without exceptions [@Bonville:GSO1; @Gardner-dynamics-review], [*all*]{} insulating rare-earth pyrochlore materials R$_2$Ti$_2$O$_7$ and R$_2$Sn$_2$O$_7$ display temperature-independent spin dynamics down to $T_0 \sim O(10^1)~{\rm mK}$. Indeed, residual low temperature dynamics has been found in pyrochlore magnetic materials with low temperature states that range from not understood whatsoever [@Gardner-PRL; @YTO-muSR] to seemingly conventional long range ordered [@Raju:GTO; @Ramirez-GTO; @Wills:GSO; @Stewart:GTO; @Champion:ETO]. We note in passing that persistent low-temperature spin dynamics has also been found in the Gd$_3$Ga$_5$O$_{12}$ garnet (GGG) [@Dunsiger:GGG; @Marshall:GGG] and in the SrCr$_8$Ga$_4$O$_{19}$ kagome antiferromagnet [@Uemura:SCGO]. We now briefly review the various experimentally observed behaviors of the R$_2$Ti$_2$O$_7$ and R$_2$Sn$_2$O$_7$ pyrochlore oxides. Strong evidence for fluctuating spins down to extremely low temperatures has been observed in Tb$_2$Ti$_2$O$_7$ where Ti$^{4+}$ at the M site is non-magnetic [@Gardner-PRL]. The reason for the failure of Tb$_2$Ti$_2$O$_7$ to develop long-range magnetic order above 50 mK [@Gardner-PRB-50mk] despite a Curie-Weiss temperature, $\theta_{\rm CW} \sim -14~{\rm K}$ remains to this day largely unexplained [@Enjalran-review; @Mirebeau-review; @Molavian]. Yb$_2$Ti$_2$O$_7$ is perhaps just as intriguing, with specific heat measurements revealing a sharp first order transition at $T_c \approx 0.24$ K [@YTO-muSR; @YTO-neutrons], but with the spins not appearing static below $T_c$ since muon spin relaxation ($\mu$SR) and Mössbauer spectroscopy finds significant spin dynamics down to the lowest temperature [@YTO-muSR]. Hence, the observed first order transition in Yb$_2$Ti$_2$O$_7$ seems rather unconventional. Ho$_2$Ti$_2$O$_7$ [@Harris-PRL; @Bramwell-PRL; @Cornelius] and Dy$_2$Ti$_2$O$_7$ [@Ramirez-Nature] are frustrated ferromagnets [@Harris-PRL] and possess an extensive low-temperature magnetic entropy [@Cornelius; @Ramirez-Nature] similar to that of the common hexagonal I$_{\rm h}$ phase of water ice [@Pauling; @Giauque]. As such, the (Ho,Dy)$_2$(Ti,Sn)$_2$O$_7$ materials are referred to as [*spin ices*]{} [@Bramwell-Science]. Theoretical and numerical studies have shown that the spin ice behavior originates from the long range nature of magnetic dipole-dipole interactions [@Bramwell-Science; @Hertog-PRL]. Numerical Monte Carlo studies using non-local loop dynamics predict that those interactions should lead to long range order at low temperatures [@Melko:JPC]. Yet, at variance with the numerical predictions, experimental studies of the Dy$_2$Ti$_2$O$_7$ [@Fukuyama-DTO] and Ho$_2$Ti$_2$O$_7$ [@Harris-PRL; @Harris-JMMM; @Ehlers] have not found a transition to long range order down to 60 mK. In particular, muon spin relaxation ($\mu$SR) [@Harris-JMMM] and neutron spin echo [@Ehlers] experiments find evidence for Ho$^{3+}$ spin dynamics well below $T_{SI}$ in Ho$_2$Ti$_2$O$_7$. Interestingly, a recent neutron scattering study on Tb$_2$Sn$_2$O$_7$ found a transition to a long-range ordered state at $T_c\approx 0.87~{\rm K}$, with an analysis of the scattering intensity indicating that the observed state is a long-range spin ice state [@Mirebeau-TSO]. However, even more recent $\mu$SR studies find that the state at $T<T_c$ in Tb$_2$Sn$_2$O$_7$ remains dynamic down to the lowest temperature [@Yaouanc-TSO; @Bert-TSO]. Er$_2$Ti$_2$O$_7$, like Tb$_2$Sn$_2$O$_7$, was found via neutron scattering to display long range order below 1.2 K  [@Champion:ETO]. Yet, $\mu$SR found persistent spin dynamics down to the lowest temperature  [@Lago:ETO]. Gd$_2$Ti$_2$O$_7$ displays two consecutive transitions at $T_c^+\sim$ 1 K and $T_c^- \sim 0.7~{\rm K}$ [@Raju:GTO; @Ramirez-GTO]. Neutron scattering experiments [@Stewart:GTO] found that the magnetic state between $T_c^-$ and $T_c^+$ has one site out of four on a tetrahedral unit cell that is paramagnetic and fluctuating. At $T<T_c^-$, the fourth site orders, but remains much more dynamic than the three other sites. The microscopic mechanism giving rise to the two experimentally observed states is still not understood. Here too, in Gd$_2$Ti$_2$O$_7$, $\mu$SR finds considerable spin dynamics persisting down to 20 mK [@Yaouanc-GTO-PRL; @Dunsiger:GTO]. A phenomenological model for the density of states, $g(\epsilon)$, has been proposed for the low-temperature state of Gd$_2$Ti$_2$O$_7$. Most significantly, the proposed model for $g(\epsilon)$ was shown to describe the peculiar temperature dependence of the magnetic specific heat, $C_v(T)$, in Gd$_2$Ti$_2$O$_7$ which was found to be $C_v(T) \propto T^2$ below $T_c^-$. Such $C_v\propto T^2$ behavior is rather unconventional. Indeed, in a conventional long-range ordered three-dimensional antiferromagnet, $C_v \sim T^3$ down to a temperature where the temperature dependence turns to $C_v \sim \exp(-\Delta/T)$ because of a gap $\Delta$ in the excitation spectrum induced by single ion anisotropy or anisotropic spin-spin interactions. In all the pyrochlore systems reviewed above, Tb$_2$Ti$_2$O$_7$, Yb$_2$Ti$_2$O$_7$, (Ho,Dy)$_2$Ti$_2$O$_7$, Tb$_2$Sn$_2$O$_7$, Er$_2$Ti$_2$O$_7$, and Gd$_2$Ti$_2$O$_7$, some theoretical lapses exist in our understanding of the [*equilibrium*]{} thermodynamic low-temperature state. Hence, it is perhaps not completely surprising that the spin dynamics appears unconventional in these materials with, in particular, a temperature independent $\mu$SR spin polarization relaxation rate down to a baseline temperature $T_0\sim 10^1~{\rm mK}$. However, that tentative self-reassured standpoint is put on shaky ground by the $\mu$SR and specific heat measurements on Gd$_2$Sn$_2$O$_7$ that we now discuss. The case of Gd$_2$Sn$_2$O$_7$ ----------------------------- It was first proposed that the aforementioned Gd$_2$Ti$_2$O$_7$ material would be a good candidate for a classical Heisenberg pyrochlore antiferromagnet with leading perturbations coming from long-range magnetic dipole-dipole interactions [@Raju:GTO]. The reason for this is that Gd$^{3+}$ is an S-state ion with half-filled 4f shell, hence orbital angular moment $\mathrm{L}=0$, and spin $\mathrm{S}=7/2$. Spin anisotropy is therefore expected to be much smaller than for the above Ho, Dy and Tb based rare earth materials [@Jensen:REM]. In that context, Gd$_2$Sn$_2$O$_7$ should be similar to Gd$_2$Ti$_2$O$_7$; the main difference being that Gd$_2$Sn$_2$O$_7$ displays only one phase transition observed from a paramagnetic to a long-range ordered phase at $T_c\sim 1$ K  [@Wills:GSO]. Perhaps most interestingly, unlike Gd$_2$Ti$_2$O$_7$, the experimentally observed long-range ordered phase in Gd$_2$Sn$_2$O$_7$ corresponds to the one predicted by Palmer and Chalker for the classical Heisenberg pyrochlore antiferromagnet model with perturbative long-range dipolar interactions [@PC]. It is possible that the experimentally observed transition in Gd$_2$Sn$_2$O$_7$ corresponds to two very close transitions [@Enjalran:GTO; @Cepas:MFT] that are not resolved [@Cepas:MC]. From our perspective, Gd$_2$Sn$_2$O$_7$ is an exemplar of the intriguing behavior discussed above. Yet, it offers itself as a crucial system to understand. The reasons are as follows: (i) as in Gd$_2$Ti$_2$O$_7$, and all the materials previously described, persistent low-temperature spin dynamics have been observed [@Bonville:GSO2] and (ii) again similarly to Gd$_2$Ti$_2$O$_7$, unconventional power-law temperature dependence of the magnetic specific heat has been found, specifically, $C_v \sim T^2$. So here too, there may exists the possibility to relate a dynamical response and a bulk thermodynamic measurements to an unconventional density of states $g(\epsilon)$ $-$ a possible manifestation of the spectral down-shift that corresponds to the hallmark of highly frustrated systems. We see the experimental results on Gd$_2$Sn$_2$O$_7$ as a crucial paradox to contend with. Since the observed ordered state in Gd$_2$Sn$_2$O$_7$ corresponds to the one predicted by the model of Palmer and Chalker [@PC], or a more refined model that includes Gd$^{3+}$ single-ion anisotropy [@Wills:GSO; @Glazkov:ESR] and exchange interactions beyond nearest-neighbor [@Wills:GSO; @Enjalran:GTO; @Cepas:MFT], one could in principle follow the well-trodden road of solid state physics and conventional magnetism: with the Hamiltonian and consequential ground state known, identify the long wavelength excitations and, by second-quantizing them, calculate the low-temperature thermodynamic quantities. It turns out that this program was carried out in a prior work [@DelMaestro:GTO] for a quantum version of a simple pyrochlore lattice model with nearest-neighbor antiferromagnetic exchange plus long-range dipolar interactions [@Raju:GTO; @PC]. What was found in Ref. \[\] is that [*all*]{} spin wave excitations of the Heisenberg pyrochlore antiferromagnet are pushed up in energy by the dipolar interactions and, as a result, all thermodynamic quantities show exponential temperature dependence, $\sim \exp(-\Delta/T)$, at low temperatures[@DelMaestro:GTO]. The following question thus arises: [*Do the $C_v\sim T^2$ results of Ref. \[\] for Gd$_2$Sn$_2$O$_7$ contradict the theoretical prediction of Ref. \[\], and are the magnetic excitations of Gd$_2$Sn$_2$O$_7$ truly unconventional?* ]{} This is the question that we ask, and aim to answer in this paper. ![\[fig:logCvlogT\] (Color online) The specific heat of Gd$_2$Sn$_2$O$_7$ as a function of temperature, with an inset showing an enlargement of the low temperature region from Bonville *et al.*[@Bonville:GSO1] plotted on a logarithmic scale. The dashed line shows the result of a relatively successful $T^2$ fit below 0.75 K. \[Data were generously provided by P. Bonville\].](logCv-vs-logT) To spell out the question above more specifically, we show in Fig. 1 the specific heat data on Gd$_2$Sn$_2$O$_7$ reproduced from Ref. \[\]. The $T^2$ behavior (dashed line in main panel) ranges from $T_{\rm low}\sim 0.5~{\rm K}$ to $T_{\rm up} \sim 0.8 ~{\rm K}$. The $T_{\rm up}$ is very close to the critical temperature, and one does not expect on general grounds (non-critical) power-law behaviors reflecting excitations out of the ground state to extend so close to the phase transition. Secondly, the temperature $T_{\rm low}$, when compared with the results of Ref. \[\], is [*high*]{} compared to the temperature regime where we expect second-quantized (spin wave) excitations to describe this system. Finally, and this is the key aspect of the data that prompted the present work, we note that the $C_v$ data at $T\lesssim 0.5~{\rm K}$ progressively droop below the dashed $C_v \sim T^2$ behavior. This is emphasized in the inset of Fig. 1. Incidentally, we note from this plot that the $C_v\sim T^2$ behavior does not provide a particularly good fit of the data for $T\in [350,800]$ mK. The crux of the argument presented in this paper is that (i) the $T^2$ power law between $T_{\rm low}$ and $T_{\rm up}$ is not a reflection of the low-energy properties of Gd-based antiferromagnetic pyrochlores and, most importantly, (ii) the behavior exhibited by $C_v$ below $T_{\rm low} \sim 0.5 ~{\rm K}$ (inset of Fig. 1) is a signature that the system is progressively entering a low-temperature regime characterized by exponentially activated spin excitations over a gap originating from both magnetic dipole-dipole interactions and single-ion anisotropy. We show below via calculations that expand on the authors’ previous work, Ref. \[\], that the specific heat data of Fig. 1 can be reasonably well described by such gapped magnetic excitations. In other words, we assert that the bulk thermodynamic properties of Gd$_2$Sn$_2$O$_7$, revealed by data like that shown in Fig. 1, are compatible with a conventional semi-classical long-range ordered phase. We suggest that specific heat measurements below $T_{\rm low}$ and down to 100 mK could be used to confirm or disprove our proposal. The rest of the paper is organized as follows. We first present a model for exchange and dipole coupled spins on the pyrochlore lattice in the presence of a crystal field inducing single ion spin anisotropy. The Hamiltonian is decoupled via linear spin wave theory, and expressions for the quantum fluctuations and low temperature thermodynamic properties are calculated. We then investigate the effects of second and third nearest neighbor magnetic exchange on the gap to spin wave excitations. Comparing the specific heat calculated in spin wave theory for Gd$_2$Sn$_2$O$_7$ to that measured in the experiments of Bonville *et al.* [@Bonville:GSO1], we use a maximum likelihood estimator to determine a set of further neighbor couplings which may be present in the material. Finally, we identify the zero temperature quantum fluctuations and present a schematic phase diagram of Gd$_2$Sn$_2$O$_7$. Model Hamiltonian {#sec:ModelHamiltonian} ================= As reviewed in the previous section, the pyrochlore lattice has evinced much experimental and theoretical interest due to the large degree of geometrical frustration arising from a structure consisting of corner sharing tetrahedra. The cases of Gd$_2$Ti$_2$O$_7$ and Gd$_2$Sn$_2$O$_7$ are somewhat special, in that the $\mathrm{S}=7/2$ Gd$^{3+}$ S-state ion should have a relatively small intrinsic anisotropy when compared to other R$_2$Ti$_2$O$_7$ pyrochlore oxides. It is known that the titanate (Gd$_2$Ti$_2$O$_7$) has a complicated low temperature multi-${\boldsymbol{k}}$ magnetic structure [@Stewart:GTO]. However, recent neutron scattering [@Wills:GSO; @Bonville:GSO1] and electron spin resonance [@Glazkov:ESR] experiments performed on gadolinium stanate (Gd$_2$Sn$_2$O$_7$) indicate that this material exhibits a ${\boldsymbol{k}}=0$ long range ordered state below $\sim 1~{\rm K}$. As such, Gd$_2$Sn$_2$O$_7$ should be reasonably well described by a general two-body spin interaction Hamiltonian which includes predominant isotropic magnetic exchange interactions up to at least third nearest neighbor and anisotropy in the form of interactions with the local crystal field as well as long range dipole-dipole interactions [@Raju:GTO; @PC; @Enjalran:GTO; @Wills:GSO; @Cepas:MFT; @Cepas:MC]. Such a Hamiltonian can be written as $$\mathcal{H} = \mathcal{H}_{\rm ex} + \mathcal{H}_{\rm dd} + \mathcal{H}_{\rm cf}, \label{eq:H}$$ where the exchange, dipole-dipole and crystal field terms are given by $$\begin{aligned} \label{eq:Hex} \mathcal{H}_{\rm ex} &=& -\frac{1}{2}\sum_{i,a}\sum_{j,b} J_{ab}({\boldsymbol{R}}^{ij}_{ab}) \, {\boldsymbol{S}}_a({\boldsymbol{R}}^i) \cdot {\boldsymbol{S}}_b({\boldsymbol{R}}^j) \\ \label{eq:Hdd} \mathcal{H}_{\rm dd} &=& \frac{D_{\rm dd}}{2}\sum_{i,a}\sum_{j,b} \left\{ \frac{{\boldsymbol{S}}_a({\boldsymbol{R}}^i) \cdot {\boldsymbol{S}}_b({\boldsymbol{R}}^j)}{|{\boldsymbol{R}}^{ij}_{ab}|^3} \right. - \nonumber \\ && \left. \quad\quad\quad 3 \frac{[{\boldsymbol{S}}_a({\boldsymbol{R}}^i)\cdot {\boldsymbol{R}}^{ij}_{ab}]\; [{\boldsymbol{S}}_b({\boldsymbol{R}}^j)\cdot {\boldsymbol{R}}^{ij}_{ab}]}{|{\boldsymbol{R}}^{ij}_{ab}|^5} \right\} \\ \label{eq:Hcf} \mathcal{H}_{\rm cf} &=& \sum_{i,a} \sum_{\ell,m}B^m_\ell\hat{O}^m_\ell[{\boldsymbol{S}}_a({\boldsymbol{R}}^i)]\end{aligned}$$ with the factors of $1/2$ having been included to avoid double counting. The various conventions used in Eqs. (\[eq:Hex\]) to (\[eq:Hcf\]) are as follows: ${\boldsymbol{S}}_a({\boldsymbol{R}}^i)$ is the spin located on one of $N$ tetrahedra identified by the face centered cubic (FCC) Bravais lattice vector ${\boldsymbol{R}}^i$ and the site by one of four tetrahedral sublattice vectors ${\boldsymbol{r}}_a$. ${\boldsymbol{S}}_a({\boldsymbol{R}}^i)$ is assumed to be a full O(3) operator satisfying ${\boldsymbol{S}}_a({\boldsymbol{R}}^i)\cdot{\boldsymbol{S}}_a({\boldsymbol{R}}^i) = \mathrm{S}(\mathrm{S}+1)$. $J_{ab}({\boldsymbol{R}}^{ij}_{ab})$ gives the value of the isotropic Heisenberg exchange interaction between two spins separated by ${\boldsymbol{R}}^{ij}_{ab} = {\boldsymbol{R}}^j + {\boldsymbol{r}}_b - {\boldsymbol{R}}^i - {\boldsymbol{r}}_a$, with a negative sign corresponding to antiferromagnetic interactions. In this study, we focus on the Gd$_2$Sn$_2$O$_7$ material, and thus consider a fixed value of $J_1 = 3\Theta_{CW} / [z\mathrm{S}(\mathrm{S}+1)] = -0.273$ K where the Curie-Weiss temperature is $\Theta_{CW} = -8.6$ K and $z=6$ is the coordination number on the pyrochlore lattice [@Bonville:GSO1]. We treat the exchange interactions $J_2$ and $J_{31}$ beyond nearest neighbors as parameters to be adjusted below to produce agreement with experimental (specific heat) measurements on Gd$_2$Sn$_2$O$_7$. Following the approach of Wills *et al.*[@Wills:GSO] we treat the two possible third nearest neighbor (NN) exchange paths $J_{31}$ and $J_{32}$, known to be present in the pyrochlores[@Wills:GSO; @Kenedy:TinPyro], separately (see Fig. \[fig:J3ExPaths\]). With the expectation that $J_{32} \ll J_{31}$, we henceforth set $J_{32} = 0$. ![\[fig:J3ExPaths\] (Color online) A schematic portion of the pyrochlore lattice, detailing the paths which correspond to second nearest neighbor and two types of third nearest neighbor exchange interactions.](pyro-exchange) The strength of the dipole interaction is given by $D_{\rm dd} = \mu_0 (g\mu_B)^2 / 4\pi$. At nearest neighbor distance, $R_{\rm nn} = a\sqrt{2}/4 = 3.695$ Å , where $a = 10.45$ Å  is the size of the cubic unit cell, $D_{\rm dd}/{R_{nn}}^3$ is approximately 15% of the exchange energy $J_1$. The crystal field Hamiltonian is written as an expansion of Stevens operators, $\hat{O}^m_\ell$, that transform like the real tesseral harmonics [@Hutchings:CF]. The number of terms in the expansion is strongly constrained by symmetry and, from recent electron spin resonance (ESR) measurements [@Glazkov:ESR], the values of $B_2^0$ and $B_4^0$ have been estimated at $(47 \pm 1)$ mK and $(0.05 \pm 0.02)$ mK, respectively. Here, we only consider the dominant lowest order term in the expansion of $\mathcal{H}_{\rm cf}$, $B_2^0$, which contributes energetically on equal footing with the dipole interactions, and leave the inclusion of higher order corrections to a future study. Writing the Steven’s operators in terms of angular momentum operators [@Jensen:REM], the crystal field part of the Hamiltonian ${\cal H}$, ${\cal H}_{\rm cf}$, is: $$\mathcal{H}_{\rm cf} = -4N B_2^0\mathrm{S}(\mathrm{S}+1) + 3 B_2^0\sum_{i,a} \left[{\boldsymbol{S}}_a(R^i) \cdot \hat{z}_a \right]^2, \label{eq:HcfSpin}$$ where the four unit vectors $\hat{z}_a$ describe the local $\langle 111 \rangle$ direction for each site on a tetrahedron. The conventions and definitions used in this study for all vectors and lengths on the pyrochlore lattice are given in Table 1 of a previous work by one of the authors [@Enjalran:TTO]. We are interested in the effects of the low energy excitations (spin waves) on the thermodynamic properties of a real material described by Eq. (\[eq:H\]). At zero temperature, we assume that the system is in one of the six discrete Palmer-Chalker (PC) ${\boldsymbol{k}} = 0$ ground states [@PC] depicted in Fig. \[fig:PC\]. ![\[fig:PC\] (Color online) Six degenerate Palmer-Chalker (PC) ground states (reverse spins for other three) for spins on a single tetrahedron. The ground states are characterized by each spin being parallel to an edge of the tetrahedron that does not intersect its vertex and all are tangent to the sphere which circumscribes it. The net magnetic moment on each tetrahedron is identically zero.](pcgs){width="3.3in"} We have confirmed by direct numerical simulations that the classical zero temperature ground state in a model with nearest-nearest neighbor antiferromagnetic exchange and long-range dipolar interactions at the level of 10 – 20% of the exchange is the PC ground state. See also Refs. \[\]. The use of the PC state for Gd$_2$Sn$_2$O$_7$ is supported by recent powder neutron scattering experiments [@Wills:GSO] where the magnetic diffraction pattern was compared to the expected result from multiple candidate ground states. Our approach below will be to analyze the stability of this ground state and its accompanying excitations by investigating the role of quantum fluctuations in reducing the fully polarized classical spin value of ${\boldsymbol{S}}_{\rm cl} = (0,0,\mathrm{S})$. This can be accomplished by changing the axis of quantization from the global $z$-direction (an arbitrary choice) to a local axis described by a triad of unit vectors $\{\hat{n}^u_a\}$. This triad is defined such that the locally quantized spin, denoted by a tilde, is related to the spin operator in the Cartesian lab frame via a rotation $$\begin{aligned} {\boldsymbol{S}}_a({\boldsymbol{R}}^i) &=& \mathsf{R}(\theta_a,\phi_a)\tilde{{\boldsymbol{S}}}_a({\boldsymbol{R}}^i) \nonumber \\ &=& \sum_{u} \tilde{S}^u_a({\boldsymbol{R}}^i)\hat{n}^u_a \label{eq:Slocframe}\end{aligned}$$ where $u$ runs over the Cartesian indices $\{x,y,z\}$ and $\hat{n}^z_a$ points in the direction of each classical spin. Using Eq. (\[eq:Slocframe\]) we can rewrite Eq. (\[eq:H\]) in a much more compact form $$\mathcal{H} = -\frac{1}{2}\sum_{a,b}\sum_{i,j}\sum_{u,v} \tilde{S}^u_a({\boldsymbol{R}}^i) \mathcal{J}^{uv}_{ab}({\boldsymbol{R}}^{ij}_{ab}) \tilde{S}^v_b({\boldsymbol{R}}^j), \label{eq:Hlf}$$ where we have neglected a constant term and $$\begin{aligned} \mathcal{J}^{uv}_{ab}({\boldsymbol{R}}^{ij}_{ab}) &=& J_{ab}({\boldsymbol{R}}^{ij}_{ab})\hat{n}^u_a\cdot\hat{n}^v_b \nonumber \\ &&-\ \frac{D_{dd}}{\left|{\boldsymbol{R}}^{ij}_{ab}\right|^3} \left[ \hat{n}^u_a \cdot \hat{n}^v_b - 3(\hat{n}^u_a \cdot \hat{R}^{ij}_{ab})(\hat{n}^v_b\cdot\hat{R}^{ij}_{ab}) \right] \nonumber \\ &&-\ 6B_2^0 (\hat{n}^u_a \cdot \hat{z}_a)(\hat{n}^v_b \cdot \hat{z}_b) \delta_{a,b}\delta_{u,v}. \label{eq:fullJ}\end{aligned}$$ Having manipulated our Hamiltonian into a more manageable form, we employ in the next section the methods of linear spin wave theory to diagonalize Eq. (\[eq:Hlf\]). Linear spin wave theory {#sec:LSWT} ======================= In a previous study[@DelMaestro:GTO], which we henceforth refer to as DG, we presented the diagonalization of $\mathcal{H}_{\rm ex} + \mathcal{H}_{\rm dd}$ on the pyrochlore lattice via a Holstein-Primakoff[@HolstPrim] spin wave expansion to order $1/\mathrm{S}$, through the introduction of bosonic spin deviation (magnon) creation (annihilation) operators $c^\dag_a$ ($c^{\phantom\dag}_a$). The Ewald summation technique [@ewald] was used to calculate the Fourier transform of the infinite range dipole-dipole interaction matrix. The calculations of DG can be straightforwardly generalized to include the effects of the crystal field by shifting the diagonal spin interaction matrix elements ($\mathsf{A}_{\alpha\alpha}({\boldsymbol{k}})$ and $\mathsf{B}_{\alpha\alpha}({\boldsymbol{k}})$ of Eq. (16) in DG) by a term proportional to $B_2^0$ (see Eq. (\[eq:fullJ\])). The result, after the usual Bogoliubov diagonalization procedure, is a Bose gas of non-interacting spin waves described by $$\mathcal{H} = \mathcal{H}^{(0)} + \sum_{{\boldsymbol{k}}}\sum_a \varepsilon({\boldsymbol{k}})\left[ a^\dag_a({\boldsymbol{k}})a^{\phantom\dag}_a({\boldsymbol{k}}) + \frac{1}{2}\right], \label{eq:SWHam}$$ where the summation is over all wavevectors in the first Brillouin zone (BZ) of the FCC lattice. The dispersion relations for the spin wave modes, $\varepsilon_a({\boldsymbol{k}})$, are calculated from the spectrum of the Bogoliubov transformation. Physically, they are identical to $\varepsilon_a({\boldsymbol{k}})=\hbar \omega_a({\boldsymbol{k}})$ where $\omega_a({\boldsymbol{k}})$ are the classical excitation frequencies obtained by linearizing the classical equations of motion for interacting magnetic dipoles or rotors [@DelMaestro:MSC]. In a real magnet, spin wave fluctuations with dispersion $\varepsilon_a({\boldsymbol{k}})$ raise the classical ground state energy and reduce the staggered magnetic moment per spin from its classical value of $\mathrm{S}$. From Eq. (\[eq:SWHam\]), the contribution to the ground state energy is given by $$\Delta \mathcal{H}^{(0)} = \frac{1}{2}\sum_{{\boldsymbol{k}}}\sum_{a} \varepsilon_a ({\boldsymbol{k}}). \label{eq:dH}$$ The full spectrum of the Holstein-Primakoff transformation can be used to calculate the reduction in the staggered magnetization $$\Delta S = \frac{1}{2}\left(\frac{1}{8\mathrm{N}} \sum_{{\boldsymbol{k}}}\mathrm{Tr}[\mathsf{Q}^\dag\mathsf{Q}] - 1 \right), \label{eq:dS}$$ where $\mathsf{Q}$ is the $8 \times 8$ hyperbolically normalized matrix of eigenvectors such that $\mathrm{Tr}\; (\mathsf{Q}^\dag\mathsf{H}\mathsf{Q}) = \sum_a \varepsilon_a ({\boldsymbol{k}})$ and $\mathsf{H}$ is the $8 \times 8$ block matrix Hamiltonian \[see DG Eqs. (16) and (19)\]. $N$ is the number of tetrahedra on a pyrochlore lattice with periodic boundary conditions. At low temperatures ($k_B T < \varepsilon_a({\boldsymbol{k}})$) expressions for the specific heat at constant volume, $C_v$, and staggered magnetization, $m=S-\Delta S$, can be derived from the classical partition function $\mathcal{Z} = \mathrm{Tr}\; [\exp(-\beta \mathcal{H})]$ corresponding to Eq. (\[eq:SWHam\]). Using Eqs. (\[eq:SWHam\]) to (\[eq:dS\]), we find (see DG) $$\begin{aligned} \label{eq:Cv} C_v &=& \beta^2 \sum_{{\boldsymbol{k}}}\sum_{a}[\varepsilon_a({\boldsymbol{k}})n_B(\varepsilon_a({\boldsymbol{k}}))]^2 \exp[\beta\varepsilon_a({\boldsymbol{k}})], \\ \label{eq:M} m &=& \mathrm{S} + \frac{1}{2} \nonumber \\ && -\frac{1}{8\mathrm{N}}\sum_{{\boldsymbol{k}}}\sum_{a}[\mathsf{Q}^\dag\mathsf{Q}]_{aa} [1+n_B(\varepsilon_a({\boldsymbol{k}}))],\end{aligned}$$ where $\beta$ is the inverse temperature and $n_B(\varepsilon_a({\boldsymbol{k}})) = 1/(\mathrm{e}^{\beta\varepsilon_a({\boldsymbol{k}})}-1)$ is the Bose distribution function. Results ======= Having determined the expressions for the zero temperature quantum fluctuations, as well as finite temperature thermodynamic relations of a dipolar Heisenberg model with crystal field interactions on the pyrochlore lattice, we may now study the quantitative effects of perturbative $J_2$ and $J_{31}$ for fixed $J_1$ and $B_2^0$. The limit of stability of the proposed PC classical ground states on the pyrochlore lattice can be investigated by searching for soft modes ($\varepsilon_a({\boldsymbol{k}}) \to 0$) at some wavevector ${\boldsymbol{k}}$. Second and third NN exchange ---------------------------- If we ignore all interactions except nearest neighbor isotropic antiferromagnetic exchange, it has been known for some time[@Villain] that both classical [@Moessner:Pyro] and quantum [@Canals:Pyro] spins on the pyrochlore lattice fail to develop conventional long range magnetic order at nonzero temperature. There exists two unbroken symmetries in the ground state manifold [@Reimers-MFT; @Moessner-CJP] corresponding to lattice translation and time reversal. The excitation spectrum consists of two sets of two degenerate modes, the first being soft over the entire Brillouin zone, (see Fig. \[fig:FCCBZ\]) and the second are acoustic and linear along $\Gamma \to X$ leading to divergent quantum fluctuations. The inclusion of both dipole-dipole, and crystal field interactions break the rotational symmetry and lift *all* excitations to finite frequency [@DelMaestro:GTO]. ![(Color online) The first Brillouin zone of the pyrochlore lattice, showing the five high symmetry points $\Gamma = (0,0,0)$, $X = 2\pi/a (1,0,0)$, $W=2\pi/a(1,1/2,0)$, $L=2\pi/a(1/2,1/2,1/2)$ and $K = 2\pi/a(3/4,3/4,0)$ and the path in $k$-space along which spectra are plotted in this study. \[fig:FCCBZ\]](fccbz) Fig. \[fig:hspSpectra\] shows the dispersion spectrum of the four spin wave modes for three values of $J_2$ and $J_{31}$ along the high symmetry path described in Fig. \[fig:FCCBZ\]. ![(Color online) The spin wave excitation spectrum in kelvin for $(J_2/|J_1|,J_{31}/|J_1|)$ equal to $(0.0, 0.0)$, $( 0.03,-0.03)$ and $(-0.03, 0.03)$ plotted along a high-symmetry path in the first Brillouin zone of the FCC lattice. For the three parameter sets shown there exists a finite gap to spin wave excitations throughout the zone. \[fig:hspSpectra\]](hsp-spectra) We do indeed observe only optical modes, with degeneracy preserved along $X \to W$ for $J_2 = J_{31} = 0$. The qualitative behavior of $\varepsilon_a({\boldsymbol{k}})$ with varying $J_2$ and $J_{31}$ also seems to confirm the naive expectation that even perturbatively small third nearest antiferromagnetic neighbor exchange should reduce the stability of the PC states [@Enjalran:GTO; @Cepas:MFT; @Wills:GSO]. It also appears that the minimum excitation gap does not always occur at ${\boldsymbol{k}}=0$ indicating the underlying presence of a finite wavevector instability as $J_2$ and $J_{31}$ are tuned away from $J_2=J_{31}=0$. The spin wave energy gap can be analyzed more quantitatively by defining $$\begin{aligned} \Delta ({\boldsymbol{k}}) &\equiv& \min_{a} \left[\varepsilon_a({\boldsymbol{k}})\right] \\ \Delta &\equiv& \min_{{\boldsymbol{k}}} \left[ \Delta({\boldsymbol{k}})\right]. \label{eq:gap}\end{aligned}$$ The value of $\Delta({\boldsymbol{k}})$ can be investigated as a function of $J_2$ and $J_{31}$ at each of the high symmetry points described above. As we vary $J_2$ and $J_{31}$ through some critical values, instabilities first appear at these wavevectors of high symmetry. The resulting gap values are shown in Fig. \[fig:hspKPoints\]. ![(Color online) The magnitude of the lowest spin wave excitation energy $\Delta(k)$ in kelvin at various high symmetry points in the first BZ plotted in the $J_2 - J_{31}$ plane. All panels are plotted with the same color scale making the slight variations in the gap at $\Gamma = (0,0,0)$ difficult to discern. A soft mode instability occurs only for antiferromagnetic third NN coupling $J_{31}$ \[fig:hspKPoints\]](hsp-k-points) Although $\Delta(\Gamma) > 0$ for all values of $J_2$ and $J_{31}$ studied here, the region of stability of the PC states is defined by the observed appearance of soft modes, at ${\boldsymbol{k}} = K = 2\pi/a(3/4,3/4,0)$ for ferromagnetic (positive) $J_2$ and antiferromagnetic (negative) $J_{31}$. Performing a search for the minimum value of the gap over the entire Brillouin zone ($\Delta$) at each value of $J_2$ and $J_{31}$ confirms that the instability first appears at the $K$-point. The effect of perturbative second and third NN exchange interactions on the global minimum energy gap (Eq. (\[eq:gap\])) along with the corresponding magnitude of spin fluctuations $\Delta \mathrm{S} / \mathrm{S}$ (Eq. (\[eq:dS\])) is shown in Fig. \[fig:gapMagVer\]. ![(Color online) The spin wave excitation gap in kelvin ($\Delta$, top panel) found by determining the minimum energy over $19^3$ discrete points in the first BZ, and the reduction in sublattice magnetization ($\Delta\mathrm{S} / \mathrm{S}$, bottom panel) for various values of $J_{31}/|J_1|$ plotted against $J_2/|J_1|$. The jump in $\Delta \mathrm{S}/\mathrm{S}$ occurs once the limit of stability of the ${\boldsymbol{k}}=0$ Palmer-Chalker ground state is reached. Both panels share a common legend. \[fig:gapMagVer\]](delta-dS-J2) Here we observe that upon reaching a pair of critical values for $J_2/|J_1|$ and $J_{31}/|J_1|$, the excitation gap is suppressed to zero (top panel), and divergent spin fluctuations ensue (bottom panel). The values of $J_2$ and $J_{31}$ corresponding to $\Delta \to 0$ can be identified, and are best described by the linear relationship $J_{31} = 0.750J_2 - 0.077|J_1|$. This line defines the phase boundary between a sector of stability for the ${\boldsymbol{k}}=0$ PC ground states, and a region characterized by instabilities at finite wavevector. In addition, this line corresponds to the white regions in Fig. \[fig:hspKPoints\] where $\Delta \to 0$, and thus defines the limit of applicability of the spin wave calculation around the PC ground state described in Section \[sec:LSWT\]. Plotting $\Delta({\boldsymbol{k}})$ along $\Gamma \to X \to W \to L \to K \to \Gamma$ with $J_{31}$ pinned to this phase boundary leads to the spectrum shown in Fig. \[fig:J2hspSpectra\]. ![(Color online) The lowest excitation energy $\Delta(k)$ in kelvin plotted along the high symmetry path discussed in the text with $(J_2,J_{31})$ fixed to reside on the phase boundary parameterized by $J_{31} = 0.750 J_2 - 0.077$. The location of the zero modes in $k$-space indicate that the instability out of the PC states occurs via excitations described by ${{\boldsymbol{k}}}=2\pi/a(3/4,3/4,0)$. \[fig:J2hspSpectra\]](J2-hsp-spectra) It is apparent from this result, that once the value of the third NN exchange constant has been set at a suitably antiferromagnetic value, altering the second NN exchange constant, has a relatively limited effect on the gap and on the consequential proliferation of quantum fluctuations about the classical ground state. To summarize, the effects of perturbative second and third NN exchange interactions on the appearance of soft modes and their accompanying quantum fluctuations in a model of a dipolar coupled antiferromagnetic Heisenberg pyrochlore with single-ion anisotropy is globally illustrated in Figs. \[fig:hspSpectra\]-\[fig:J2hspSpectra\]. Such a model should well characterize the low temperature behavior of Gd$_2$Sn$_2$O$_7$, and we next apply these tools with the goal of searching for the unconventional spin excitations believed to be present in this material on the basis of the unconventional $C_v\propto T^2$ specific heat in the temperature range $[350,800]$ mK. The case of Gd$_2$Sn$_2$O$_7$ ----------------------------- As described in the Introduction, recent studies[@Bonville:GSO1; @Bonville:GSO2] of Gd$_2$Sn$_2$O$_7$ have reported, on the basis of $\mu$SR measurements, evidence for Gd$^{3+}$ spin dynamics well below 0.9 K, as well as suggesting that the low temperature specific heat is accurately described by an anomalous $T^2$ power law. This is in stark contrast with the expected $T^3$ behavior for a three dimensional antiferromagnet, with possible exponential suppression at a temperature below a characteristic excitation gap. On the other hand, the long-range ordered state found by neutron scattering in Gd$_2$Sn$_2$O$_7$ is that predicted by the simple model of Eq. \[eq:H\] in Section III and discussed in Refs. \[\] and, consequently, the low-temperature behavior of this material should be well described by linear spin wave theory. Thus, in an attempt to resolve the paradox offered by the $C_v\sim T^2$ behavior, we have calculated the low temperature specific heat via Eq (\[eq:Cv\]) within the $J_2-J_{31}$ plane, and have performed a search for the parameters which best reproduce the reported low temperature specific heat[@Bonville:GSO1]. This was accomplished by performing least squares linear fits of $\log C_v$ vs $1/T$ for $T < 0.5$ K between the experimental data and the spin wave specific heat for approximately 500 values of $J_2$ and $J_{31}$. A characteristic subset of the large number of performed fits are displayed in Fig. \[fig:logCvvs1oT\]. ![(Color online) The low temperature specific heat from Ref. \[\] plotted against inverse temperature on a logarithmic scale. The lines are the calculated value of the specific heat using the ten best fit values for $(J_2/|J_1|, J_{31}/|J_1|)$ (as seen in the inset). \[fig:logCvvs1oT\]](logCv-vs-1oT) ![(Color online) The value of the maximum likelihood estimator ($\chi^2$) in the $J_2 - J_{31}$ plane for a fit of the logarithm of the specific heat at low temperatures ($T < 0.5$ K) calculated using linear spin wave theory and compared with the experimental data of Ref. \[\]. The parameters $(J_2/|J_1|,J_{31}/|J_1|)$ with the smallest $\chi^2$, giving the most effective fit, fall along the indicated line $J_{31} = 0.760 J_2 - 0.014|J_1|$. \[fig:chi2J2J31\]](chi2-J2-J31) The values of $J_2$ and $J_{31}$ which provided the best fit to the experimental data can be quantified by defining a maximum likelihood estimator $\chi^2$ which is shown in Fig. \[fig:chi2J2J31\]. It is important to note that the fits of the specific heat, $C_v$, discussed here, were done with an [*absolute*]{} dimensionfull scale, and thus no vertical adjustment of the experimental data was allowed. The comparisons allow only for adjustments of $J_2$ and $J_{31}$ which are therefore [*fine-tuning*]{} effects. As such, it appears that a model which possesses solely nearest-neighbor exchange, long-range dipolar interactions and single-ion anisotropy [*already*]{} leads to a reasonable semi-quantitative description of the $C_v$ data below 500 mK. This indicates that the temperature $T\sim 500$ mK corresponds to the upper temperature below which magnetic excitations become thermally activated. The minimum of $\chi^2$ in the $J_2-J_{31}$ plane falls along the straight line $J_{31} = 0.760 J_2 - 0.014|J_1|$. This line of best fit also falls in a region of large stability (highly gapped spin wave excitations) for the classical PC ground states with respect to quantum fluctuations (see Fig. \[fig:hspKPoints\]). The poorness of fit for simultaneously strong ferromagnetic second NN and antiferromagnetic third NN interactions or vice versa (top left, or lower right of Fig. \[fig:chi2J2J31\]), seems to indicate that is quite unlikely that Gd$_2$Sn$_2$O$_7$ resides in these portions of the phase diagram. The parameters $J_2 = 0.02$ and $J_{31} = 0.0$ provide the *best* empirical fit to the experimental specific heat data, although qualitatively similar fits are seen for all parameters which satisfy $J_{31} = 0.760 J_2 - 0.014|J_1|$. ![(Color online) The low temperature specific heat (data from Ref. \[\]) versus temperature on a logarithmic scale. The calculated specific heat is plotted for the parameter set $J_2 = 0.02$, $J_{31} = 0.0$ which had the smallest $\chi^2$, although any values close to the line mentioned in the text give a qualitatively very similar result. The agreement is quite good at low temperatures, where spin wave theory should be applicable. The low temperature suppression of $C_v$ is characteristic of gapped excitations, in contrast with previously reported power law behavior[@Bonville:GSO1; @Bonville:GSO2]. The inset displays the temperature dependence of the Gd$^{3+}$ moment calculated using a Bose gas of excitations along with the value measured from $^{155}$Gd Mössbauer measurements [@Bonville:GSO1]. \[fig:CvvsT\]](logCv-vs-T) Setting the parameters to these particular values, we display the spin wave specific heat, as well as the temperature dependence of the order parameter $m$ in Fig. \[fig:CvvsT\]. Again, as shown in Fig. \[fig:logCvvs1oT\], it is clear that at low temperatures, ($T\le 370$ mK), the experimental specific heat data systematically falls below the gapped spin wave results. This behavior is tentatively consistent with the fact that spin wave theory produces a smaller value for the magnetization $m$ than what is measured from Mössbauer experiments. However, we note that due to the intrinsic short dynamical time scale probed by Mössbauer measurements, the experimental data in the inset of Fig. \[fig:CvvsT\] may not reflect the true value of the infinite-time order parameter. It is perhaps worthwhile to make a few comments on the physical meaning of the above fits. Firstly, we note that because of the weakly dispersive nature of the two lowest lying gapped magnon excitations (see Fig. \[fig:hspSpectra\]), the temperature dependence of thermodynamics quantities in the pyrochlore Heisenberg antiferromagnet plus dipolar interactions do not display the typical $C_v\sim T^3$ behavior as the temperature reaches approximately 0.5 K and exits its characteristic low-temperature exponential behavior. In fact, such an observation was already made in Ref. \[\] independently of any attempt to describe $C_v$ for Gd$_2$Sn$_2$O$_7$. Secondly, the calculations presented here constitute a standard procedure for a system with conventional long range magnetic order. In this context, it is therefore interesting to note that, contrary to the reported $T^2$ behavior, the experimental specific heat data are not only relatively well fit using the exponential spin wave form of Eq. (\[eq:Cv\]), but it appears to fall off even faster than the exponentials considered at low temperatures. This would lend credence to the view that analyzing experimental data on a log-log scale over a limited range, can lead to specious power law fits. Hence, and on the basis of specific heat measurements *alone* (i.e. without consideration of the $1/T_1$ $\mu$SR spin-lattice relaxation rate), it would therefore appear that the suggestion of unconventional excitations in Gd$_2$Sn$_2$O$_7$ should be challenged by the principle of “Ockham’s razor”. We are therefore led to suggest that the description of the specific heat $C_v$ in terms of an anomalous power law, $C_v\sim T^2$, in a [*reduced*]{} and [*intermediate*]{} temperature range $T\in [350,800]$ mK does not provide a convincing indicator for anomalous excitations out of the ground state of Gd$_2$Sn$_2$O$_7$. Unlike the suggestion made in Ref. \[\] on the basis of the temperature independence of the $1/T_1$ muon spin relaxation rate below $T_c\sim 1~{\rm K}$, we have found a fully gapped spin wave spectrum with no density of states at zero energy. Hence, at this time, the microscopic origin of the temperature independence of $1/T_1$ found below $T_c$ in Gd$_2$Sn$_2$O$_7$ remains to be understood. Ground state properties ----------------------- In the previous section, we have identified a relationship between the second $J_2$ and third $J_{31}$ NN exchange constants which best reproduce the low temperature thermodynamic behavior in Gd$_2$Sn$_2$O$_7$. We now investigate the role of quantum fluctuations. Fig. \[fig:bestFitGapMag\] displays both the minimum spin wave energy gap $\Delta$ and the reduction in the staggered moment $\Delta \mathrm{S} / \mathrm{S}$ along the line of best fit $J_{31} = 0.760 J_2 - 0.014|J_1|$. This result details the complicated relationship between the value of the gap, and the stability of the ground state, i.e. a decrease in the global spin wave energy gap (which may only occur at a single ${\boldsymbol{k}}$-point) does not immediately trigger an increase of moderate quantum fluctuations. Indeed, the opposite behavior is seen in Fig. \[fig:bestFitGapMag\]. In an exchange coupled Heisenberg antiferromagnet on a non-Bravais lattice, the specifics of all relative energy scales come into play, and one must not neglect the effects of weakly dispersing optical modes. ![(Color online) The global spin wave energy gap (left scale) and the reduction in the staggered moment (right scale) along the line in the $J_2 - J_{31}$ plane most closely corresponding to the parameter set of Gd$_2$Sn$_2$O$_7$. \[fig:bestFitGapMag\]](best-fit-gap-mag) All the results presented here can be compiled into an effective schematic phase diagram for Gd$_2$Sn$_2$O$_7$. Fig. \[fig:J2J31PhaseDiagram\] depicts the separatrix in the $J_2-J_{31}$ plane (solid line) that delineates the limit of stability of the $k=0$ PC state against a soft mode characterized by wavevector $2\pi/a(3/4,3/4,0)$, i.e. the $K-$point. Possibly relevant values of $J_2$ and $J_{31}$ for which $\chi^2$ reaches it minimum value are shown as the parametric dashed line $J_{31}=0.760J_2-0.014 |J_1|$. We note that Fig. \[fig:J2J31PhaseDiagram\] shows a zero temperature phase diagram that delineates the limit of stability of the ${\boldsymbol{k}}=0$ Palmer-Chalker ground state against an instability at $2\pi/a(3/4,3/4,0)$. A similar phase diagram presented in Ref. \[\] (with the signs corresponding to FM and AF interactions reversed) gives the ordering wave vector of the long range magnetic ordered state that first develops as the system is cooled down from the paramagnetic phase. ![(Color online) The $J_2 - J_{31}$ ground state phase diagram showing regions with order characterized by zero and non-zero wavevectors. The solid black line delineates the limit of stability of the Palmer-Chalker ground state. The inset tetrahedron is tiled with one of the PC states, and the dashed line $J_{31} = 0.760 J_2 - 0.014|J_1|$ corresponds to the values of $J_2$ and $J_{31}$ which produced the best fits to experimental data. \[fig:J2J31PhaseDiagram\]](J2-J31-phase-diagram) Discussion ========== We have considered a Heisenberg model that includes isotropic exchange interactions up to third nearest-neighbors, single-ion anisotropy and long-range magnetic dipole-dipole interactions to describe the long-range ordered state of the Gd$_2$Sn$_2$O$_7$ pyrochlore antiferromagnet. The ground state of this system, as found by neutron scattering experiments, corresponds to the (classical, PC) ground state described by Palmer and Chalker [@PC] for the classical Heisenberg pyrochlore with nearest-neighbor antiferromagnetic exchange and long-range dipolar couplings [@Raju:GTO; @Cepas:MC; @Cepas:MFT]. We used a long wavelength ($1/\mathrm{S}$ spin wave) expansion to describe the low-energy excitations about the PC ground state and to calculate the low-temperature behavior of the specific heat, $C_v$, and order parameter, $m$, for this material. By fitting the available specific heat data in the low-temperature range (0.35 K $<T<$ 0.5 K), we were able to procure an estimate of the exchange interactions beyond nearest neighbors. We obtained evidence that Gd$_2$Sn$_2$O$_7$ is in a region of exchange coupling with large stability against quantum fluctuations. Our main result (which *does not* rely on excruciatingly fine-tuned exchange constants beyond nearest-neighbor) is that the experimental temperature range 0.35 K $<T<$ 0.5 K corresponds to the upper temperature range below which the thermodynamic quantities become thermally activated above an excitation gap $\Delta \sim 1$ K [@DelMaestro:GTO]. In other words, the independently experimentally determined microscopic nearest-neighbor exchange (on the basis of DC magnetic susceptibility), single-ion anisotropy (on the basis of ESR) and dipolar coupling strength already predict a temperature dependence for $C_v$ in Gd$_2$Sn$_2$O$_7$ that is in rough agreement with the experiment without significant adjustment. The excitation gap takes its origin from the combination of single-ion anisotropy and magnetic dipolar anisotropy. From our fits of the experimental specific heat, we tentatively conclude that the real gap is actually even *larger* than the one we have determined. Specifically, considering the lower temperature range in Fig. \[fig:logCvvs1oT\] and Fig. \[fig:CvvsT\] (and the inset of Fig. \[fig:CvvsT\] for $m$), it appears that the specific heat is dropping faster in the lower temperature range than the calculations predict. We speculate that this may indicate that the sub-leading anisotropy terms neglected in ${\cal H}_{\rm cf}$ in Eq. (5) (and which correspond to crystal field terms $B_{l,m}$ with $l=4,6$) would further increase the effective gap. In particular, those corrections would resign to further limit the spin fluctuations perpendicular to the local three-fold axis. However, at this time, experimental measurements of $C_v$ below 0.3 K are required to ascertain quantitatively the detail of the microscopic parameters for Gd$_2$Sn$_2$O$_7$ and to determine with better precision the exchange parameters $J_1$, $J_2$ and $J_{31}$. As in other Gd$^{3+}$–based insulating magnetic materials [@Cone-1; @Cone-2], it is also possible that anisotropic exchange interactions ultimately need to be included in a complete description of Gd$_2$Sn$_2$O$_7$. It therefore appears that a rejoinder to the question posed in the Introduction is that the low temperature specific heat observed in gadolinium stanate (Gd$_2$Sn$_2$O$_7$) may possibly be well described by the conventional gapped spin wave excitations of Ref. \[\]. We believe that either a confirmation or rebuttal of our suggestion of gapped excitations in Gd$_2$Sn$_2$O$_7$ via specific heat ($C_v$) measurements down to $\sim 100$ mK would tremendously help focus the discussion about the pervasive low energy excitations in insulating magnetic rare-earth pyrochlore oxides. However, a possible confirmation of such gapped excitations in Gd$_2$Sn$_2$O$_7$ via specific heat measurements would ultimately have to be rationalized within the context of the perplexing and persistent temperature-independent spin dynamics found in muon spin relaxation studies on this, and other geometrically frustrated pyrochlores. Acknowledgments =============== We are indebted to Pierre Bonville for kindly providing us with the low temperature specific heat and Mössbauer data used for all the fits reported in this study. We thank Matt Enjalran, Tom Fennell and Mike Zhitomirsky for useful and stimulating discussions. Support for this work was provided by the NSERC of Canada and the Canada Research Chair Program (Tier I) (M.G), the NSERC of Canada Grant PGS D2-316308-2005 (A.D.), the Canada Foundation for Innovation, the Ontario Innovation Trust, and the Canadian Institute for Advanced research. M.G. acknowledges the University of Canterbury for an Erskine Fellowship and the hospitality of the Department of Physics and Astronomy at the University of Canterbury where part of this work was completed.

--- author: - 'Yvonne Choquet-Bruhat' title: 'Beginnings of the Cauchy problem.' --- Introduction. ============= I was asked to write a short article on the early works on the Cauchy problem for the Einstein equations, in honor of the hundredth anniversary of General Relativity. I accepted with pleasure, but I realized when I started to work on this project that it was more difficult than I thought. I have never been really interested in who did something first, and in fact it is often difficult to ascertain. Ideas have almost always one or several preliminaries, attribution of a name to the final flower is therefore somewhat arbitrary. The work of a true historian is long and difficult, I am not an historian. I often quote, not the first note touching a subject, but a later paper more complete and easier to find. The shortness of this article does not enable me to enter into details. Of course what I know best is my own work, it it is part of my excuse for often quoting it. I apologize to all live or dead authors to whom I did not make enough deserved references. Other sources of information, including my own articles, can compensate my shortcomings. Preliminary definitions ======================= The **Einstein equations** are a geometric system for a pair $(V,g)$, with $V$ an $n+1$ dimensional differentiable manifold, $n=3$ in the classical case, and $g$ a pseudo-Riemannan metric of Lorentzian signature. In vacuum they express the vanishing of the Ricci tensor $$\text{Ricci}(g)=0,$$ equivalently the vanishing of the Einstein tensor ($R(g)$ is the scalar curvature of $g)$$$\text{Einstein}(g):=\text{Ricci}(g)-\frac{1}{2}gR(g)=0.$$ These equations are invariant under diffeomorphisms of $V$ and associated isometries of $g$. The Bianchi identities satisfied by the Riemann tensor imply, by two contractions, identities for the Einstein tensor which read[^1] in local coordinates $x^{\alpha },$ $\alpha =0,1,...n,$$$\nabla _{\alpha }S^{\alpha \beta }\equiv \nabla _{\alpha }(R^{\alpha \beta }-\frac{1}{2}g^{\alpha \beta }R)\equiv 0,$$ where $\nabla $ is the covariant derivative in the metric $g$ The vacuum Einstein equations constitute, from the analyst’s point of view, a system of $\frac{(n+1)(n+2)}{2}$ second order quasilinear[^2] partial differential equations for the $\frac{(n+1)(n+2)}{2},$ $10$ in the classical case $n=3$, coefficients $g_{\alpha \beta }$ of the metric $g$ in local coordinates. However these equations are not independent because of the above identities. The **Cauchy problem** for a system of $N$ second order quasilinear partial differential equations with unkown $u$ a set of $N$ functions $u_{I}, $ $I=1,...,N$ on $R^{n+1},$ $$A_{J}^{I,\alpha \beta }(u,\partial u)\partial _{\alpha \beta }^{2}u_{I}=f_{J}(u,\partial u),\text{ \ \ }\partial _{\alpha }:=\frac{\partial }{\partial x^{\alpha }},$$ is the search for a solution $u$ which takes, together with its set $\partial u$ of first order partial derivatives, given values $\bar{u},$ $\overline{\partial u}$ on a given $n$ dimensional submanifold $M$. The elements of the characteristic determinant of this system, for a function $u$ at a point $x,$ are the second order polynomials in a vector $X:$ $$D_{J}^{I}(u,\partial u,X):=A_{J}^{I,\alpha \beta }(u,\partial u)X_{\alpha }X_{\beta },\text{ \ \ }$$ A submanifold with equation $$\phi (x^{\alpha })=0$$ is called characteristic at a point for the considered system and initial values $\bar{u},$ $\overline{\partial u}$ if the determinant with elements $(D_{J}^{I})(\bar{u},\overline{\partial u},\partial \phi ),$ polynomial of order $2N,$ vanishes at that point. The Cauchy-Kovalevski theorem says that if this system has analytic coefficients the Cauchy problem with analytic given initial data has one and only one analytic solution in a neighbourhood of $M$ if this sumanifold is everywere non characteristic. The Cauchy-Kowalevski theorem does not apply directly to the Einstein equations: their characteristic determinant is identically zero for any metric as can be foreseen from the identities satisfied by the Enstein tensor. The Cauchy problem for the Einstein equations is non standard and has led to interesting and difficult works. Analytic results. ================= Hilbert by Lagrangian methods and Einstein himself by approximation studies had been interested in what Einstein called ”the force” of his equations, that is the generality of their solutions. However the history of exact results on the general Cauchy problem for the Einstein equations starts only in 1927 with the 47 pages book ”Les équations de la gravitation Einsteinienne” by Georges Darmois, a professor of mathematics in the University of Paris[^3]. Darmois considers (case $n=3)$ a submanifold $M$ with equation $x^{0}=0$ and data on $M$ functions of the $x^{i},$ $i=1,2,3,$ which will be the values on $M$ of $g_{\alpha \beta }$ and $\partial _{0}g_{\alpha \beta }.$ The values on $M$ of the first and second partial derivatives $\partial _{\lambda \mu }^{2}g_{a\beta }$ are then determined by derivation of the data on $M$ except for the second transversal derivatives $\partial _{00}^{2}g_{a\beta }.$ Darmois finds by straightforward computation the identities: $$R_{ij}\equiv -\frac{1}{2}g^{00}\partial _{00}^{2}g_{ij}+f_{ij}(g_{\alpha \beta },\partial _{\lambda }g_{a\beta },\partial _{\lambda h}^{2}g_{a\beta })$$ $$R_{i0}\equiv \frac{1}{2}g^{j0}\partial _{00}^{2}g_{ij}+f_{io}(g_{\alpha \beta },\partial _{\lambda }g_{a\beta },\partial _{\lambda h}^{2}g_{a\beta })$$ $$R_{00}\equiv -\frac{1}{2}g^{ij}\partial _{00}^{2}g_{ij}+f_{00}(g_{\alpha \beta },\partial _{\lambda }g_{a\beta },\partial _{\lambda h}^{2}g_{a\beta }$$ The derivatives $\partial _{00}^{2}g_{\alpha 0}$ do not appear in any of these equations, as Darmois already foresaw because a change of coordinates preserving $M$ pointwise does not change $\partial _{00}^{2}g_{ij}$ on $M,$ but permits to give arbitrary values to $\partial _{00}^{2}g_{\alpha 0}.$ If $g$ satisfies the vacuum Einstein equations, the equations $R_{ij}=0$ determine $\partial _{00}^{2}g_{ij}$ on $M$ if $g^{00}$ does not vanish there. Darmois concludes that significant discontinuities of the second derivatives of the gravitational potentials can occur across the submanifold $M,$ $\phi (x^{\alpha })\equiv x^{0}=0,$ only if $g^{00}=0$ on $M,$ that is if the hypersurface $M$ is tangent to the null cone of the Lorentzian metric $g,$ whose normals in the cotangent space satisfy the equation $g^{\alpha \beta }\partial _{\alpha }\phi \partial _{\beta }\phi =0.$ This result, though not a proof of it, is in agreement with the propagation of gravitation with the speed of light, fact already deduced by Einstein from approximations. Darmois continues his study by remarking that, if $g^{00}\not=0,$ it is possible to extract $\partial _{00}^{2}g_{ij}$ from the equation $R_{ij}=0$ and, replacing these in the other equations by the so calculated expressions, obtain four equations which depend only on the initial data, equations which we now call the constraints; he mentions that they are the Gauss - Codazzi equations known from geometers and indicates that a solution of the equations $R_{ij}=0$ with data satisfying the constraints will satisfy the whole set, at least in the analytic case, due to the contracted Bianchi identities. Darmois recognizes that an analyticity hypothesis is physically unsatisfactory, because it hides the propagation properties of the gravitational field. Darmois also studies, again in the analytic case, what initial data to give on a characteristic hypersurface $S_{0}$. He shows that they are the trace of the spacetime metric on the hypersurfacc and proves, in the analytic case, the existence of a local solution to the vacuum Einstein equation which is uniquely determined if one gives also its value on a 3-dimensional manifold $T$ transversal to $S_{0}$ or, what is equivalent for analytic functions, the values of all its derivatives at points of the intersection of $T$ and $S_{0}$. To show this,  he uses adapted coordinates to decompose the problem into an evolution of some components of the metric to satisfy part of the Einstein equations, and the Bianchi identities to show that the remaining equation is also satisfiedThe method used by Darmois does not extend to the non analytic case, though the introduction of the data of the trace of the metric on a second hypersurface, transversal to the characteristic one, but also characteristic in the non analytic case, has been successfully used, in particular in the nineties by Rendall. Before that, inspired by the general theorems of Leray for data with support ”compact towards the past ”[^4], the non analytic Cauchy problem was treated for data supported by a characteristic conoïd[^5]. In the remainder of his book, after quoting the works of Droste and Schwarzchild on the solution with spherical symmetry, Darmois studies solutions with axial symmetry[^6]. Darmois had mentioned the geometric character of the constraints but he had worked in special coordinates, namely in Gaussian coordinates; that is, with timelines geodesics normal to the initial manifold, the quantities that we call now lapse and shift are then equal respectively to one and zero. Lichnerowicz, a bright student of the Ecole Normale Supérieure, had asked from Elie Cartan a subject for his thesis, and Cartan had proposed the proof of a conjecture he had on a property of symmetric spaces that himself had not been able to prove for some time. Lichnerowicz proved it thirty years later, but when he met Darmois by chance in 1937 he was rather discouraged, and followed the suggestion of Darmois to work instead of the too difficult problem proposed by Cartan to problems on mathematical relativity which were many and little considered at the time. Lichnerowicz who was a man of varied interests, from algebra and differential geometry to theoretical physics and philosophy, followed Darmois suggestion and completed quickly a thesis which appeared as a book[^7]. In this book the Darmois computations on the Cauchy problem are extended to the case of a non constant lapse but the shift is kept zero. Lichnerowicz proposed the extension of the 3+1 decomposition to a non zero shift to one of his two first students[^8], the author of this article, who did it through the  use of the Cartan calculus in orthonormal frames, giving thus the general geometric formulas of the $n+1$ decomposition of the Ricci and Einstein tensor on a sliced manifold $M\times R$ in terms of the geometric elements: $t$ dependent induced metric and extrinsic curvature of the slices $M\times \{t\}$. This led to a preliminary publication[^9]. However this formulation did not lead to a new existence theorem for the solution of the evolutionary Cauchy problem and the full detailed article was written only later[^10]. Lichnerowicz, as a student of Elie Cartan, had a formation of geometer. He insisted on geometric formulations and started the study of global problems[^11]. He stated two what he called[^12] ”propositions”, A and B, for the Einsteinian spacetimes which he called regular (The metric had to be $C^{2}$ by pieces with first derivatives satisfying ”junction conditions”)[^13], A: introduction of matter sources in a vacuum spacetime can be done only in domains where this spacetime has singularities; B: the only complete vacuum spacetime with compact or asymptotically Euclidean space sections is flat. He proved B in the case of stationary[^14] spacetimes; that is, the non existence of gravitational solitons. The Lichnerowicz result was much appreciated by Einstein who believed for physical reason that any complete asymptotically Euclidean vacuum Einsteinian spacetime should be Minkowski[^15]. Non analytic local existence, causality and gravitational waves. ================================================================ It had already been stressed by Darmois on the one hand that analyticity was a bad physical hypothesis, on the other hand that a choice of coordinates was necessary to construct solutions of the Cauchy problem. The problem had interested Einstein himself and already[^16] in 1918 he had used coordinates satisfying the flat spacetime wave equation to construct approximated solutions of the vacuum Einstein equations near the Minkowski spacetime. Darmois[^17] considers coordinates $x^{\lambda }$ which he calls ”isothermes”; they satisfy the wave equations $$\square _{g}x^{\lambda }\equiv g^{\alpha \beta }\nabla _{\alpha }\partial _{\beta }x^{\lambda }=0;$$ that is $$F^{\lambda }\equiv g^{\alpha \beta }\Gamma _{\alpha \beta }^{\lambda }=0.$$ Such coordinates are now called ”harmonic” by analogy with solutions of the Laplace equations, or ”wave” as suggested by Klainerman as being more appropriate. By a straightforward concise and precise computation Darmois obtains the decomposition of the Ricci tensor of a pseudo Riemannian general metric as the sum of a second order system for the components of the metric and a term which vanishes identically in harmonic coordinates $$R_{\alpha \beta }\equiv R_{\alpha \beta }^{(h)}+L_{\alpha \beta },\text{ \ with \ }L_{\alpha \beta }\equiv {\frac{1}{2}}\{g_{\alpha \lambda }\partial _{\beta }F^{\lambda }+g_{\beta \lambda }\partial _{\alpha }F^{\lambda }\}.$$ where $$R_{\alpha \beta }^{(h)}\equiv -\frac{1}{2}g^{\lambda \mu }\partial _{\lambda \mu }^{2}g_{\alpha \beta }+P_{\alpha \beta }(g,\partial g).$$ with $P$ a quadratic form in the components of $\partial g$ whose coefficients are polynomials in the components of $g$ and its contravariant associate. In harmonic coordinates the Einstein equations in vacuum reduce to the quasilinear quasi diagonal second order system $R_{\alpha \beta }^{(h)}=0.$ In the years shortly before the second world war great names in mathematics were working on the Cauchy problem for a second order equation of the type then called ”hyperbolic normal”, that is principal coefficients of Lorentzian signature. The Hadamard method of parametrix for solution of linear equations seemed difficult to use for non linear equations. On the other hand a new method, energy estimates, introduced by Friedrichs and Lewy, was a subject of active research. An application of the energy estimates to the reduced vacuum Einstein equations enabled Stellmacher[^18] to prove an uniqueness theorem for a local solution of the Cauchy problem for the reduced equations, with domain of dependence determined by the light cone; that is, a causality property. However Stellmacher did not prove an existence theorem, in spite of a paper of  Schauder[^19] where was sketched an existence proof for a solution of one quasilinear second order equation equation by using energy estimates. I was encouraged to look for the solution of the non analytic Cauchy problem for the Einstein equations in 1947 by Jean Leray who was giving a series of lectures on Cartan exterior differential systems, of which I was one of the few attendants. Leray gave me the name of Schauder as a reference but I found only his paper on hyperbolic system in two variables[^20] written later, which I tried somewhat painfully to read, knowing no german. By chance I fell on a paper by Sobolev[^21], in french, which gives a construction of an elementary kernel for a second order linear hyperbolic equation in dimension 3+1 without to have to resort to a ”finite part” parametrix nor to the method of descent for the case of even spacetime dimension, as did Hadamard. The Sobolev parametrix, whose definition extends to quasi diagonal second order systems, is constructed by solution of a system of integral equations on the characteristic conoïd. These equations, together with those defining the characteristic conoïd can be used to prove the existence of a solution of the Cauchy problem for the quasilinear reduced vacuum Einstein equations in a space of smooth functions[^22]. I showed that the obtained solution satisfies the full Einstein equations if the initial data satisfy the constraints and that it is locally geometrically unique[^23]. This was the subject of my thesis, its jury included Lichnerowicz, Leray[^24] and Marcel Riesz[^25]. I returned later to the construction of the elementary kernel of a tensorial linear system of second order hyperbolic differential equations[^26] motivated by works on quantization in curved spacetime by A. Lichnerowicz who used a propagator, difference of the advanced and retarded elementary kernels[^27]. I pointed out that, being obtained by solving an integral equation on the light cone, the elementary kernel can be split into the sum of a measure supported by the light cone and a smooth function in its causal interior sum of a series of ”diffusion terms”, determined by integrations over characteristic cones with vertices at points of the previously considered cones[^28]. I studied the asymptotic behaviour of these terms. Einstein, whom I met in 1951 at the Institute for Advanced Study in Princeton where I was a postdoc at the invitation of J. Leray, made me explain my thesis on the blackboard of his office; he congratulated me and invited me to knock at his door whenever I felt like it. I regret to have done it only a few times, in spite of his always kind welcome. Einstein was then working with his assistant Bruria Kaufmann on his last unified theory. His comments were very interesting, but the computations, which himself enjoyed to do, were quite complicated and the theory rather deceptive[^29]. I prefered to work at the extension to higher dimensions[^30] of the formulas I had obtained in spacetime dimension 4 and follow the course of Leray on general hyperbolic systems. Though I also attended the Oppenheimer seminar on theoretical physics, where Einstein never came, I did not find there inspiration for personnal work[^31]. Equations with sources. ======================= The existence for classical sources in Special Relativity of a symmetric 2-tensor $T$ representing energy, stresses and momentum densities which satisfy conservation laws was a motivation for Einstein in the choice[^32] of its non vacuum equations $$S_{\alpha \beta }=\kappa T_{\alpha \beta },\text{ }$$ with $\kappa $ a constant usually normalized to $1$ by mathematicians. The problem is then the resolution of the coupled system of Einstein with sources and the conservation laws for these sources, $$\nabla _{\alpha }T^{\alpha \beta }=0,$$ with also eventually equations for fields other than gravitation, for instance Maxwell equations in presence of an electromagnetic field. The Cauchy problem  for the Einstein equations with sources splits again as constraints on initial data and an evolution problem for reduced Einstein equations with sources. The treatment of the electrovacuum case is similar to vacuum and was solved simultaneously[^33]. Solution in the cases of dust and perfect fluids without or with charge and zero conductivity were shown to admit a well posed Cauchy problem[^34] using the Leray theory of hyperbolic systems[^35]; relativistic fluids with infinite conductivity were analysed[^36]. They were seen to be what is now called Leray-Ohya[^37] hyperbolic when Leray and Ohya proved well posedness of the Cauchy problem in Gevrey classes for some systems of differential equations with multiple characteristics. All these results obtained for barotropic fluids were extended by Lichnerowicz to fluids whose equation of state depends also on the entropy[^38] and assembled by him into a book[^39] after a series of lectures he gave in Dallas at the invitation of Ivor Robinson. Relativistic fluids with finite conductivity were proved to be also Leray-Ohya hyperbolic[^40]. Isotropic relativistic elasticity has been proved by Pichon to obey a Leray-Ohya hyperbolic system[^41]. The hyperbolic character, Leray or Leray Ohya, holds for the Einstein equations coupled with any of the quoted sources. The equations of charged fluids with electromagnetic inductions are also Leray- Ohya hyperbolic, but their natural Maxwell tensor being non symmetric their coupling with Einstein equations is problematic[^42]. Well posedness of the Cauchy problem was proved to be true for sources satisfying a Vlasov[^43], or a Boltzman[^44] equation with appropriate cross section. Constraints =========== We said that it has long been known that geometric initial data for the vacuum Einstein equations on a spacelike submanifold $M$ are the two fundamental forms, induced metric $\bar{g}$ and extrinsic curvature $K,$ and they must satisfy $n$ equations, the constraints. Surprisingly it took a long time to split these data into arbitrarily given quantities and unknowns which satisfy elliptic equations, as it was however reasonable to expect for unknowns on a space manifold and the Newtonian approximation of the Einstein equations. The first result in this direction was due to Racine[^45]. He assumed, for $n+1=4,$ the metric $\bar{g}$ to be conformally flat $$\bar{g}:=\phi ^{4}e,\text{ \ \ \ \ \ }e\text{\ the Euclidean metric}$$ and remarked that, if the trace $\bar{g}^{ij}K_{ij}$ of the extrinsic curvature $K$ vanishes and one sets $$P_{ij}=\phi ^{2}K_{ij},$$ the system of constraints for the equations with source of zero momentum splits into a first order linear system for $P$, independent of $\phi ,$ and a semi linear second order equation for $\phi $ with principal term the Laplacian $\Delta \phi .$ The study was taken anew by Lichnerowicz[^46], replacing the Euclidean metric by a general Riemannian metric $\gamma .$ He defines the traceless tensor $\tilde{K}_{ij}$ by $$\tilde{K}_{ij}=\varphi ^{2}(K_{ij}-\frac{1}{3}\bar{g}_{ij}\tau ),\text{ \ \ \ }\tau :=\bar{g}^{ij}K_{ij}.$$ The momentum constraint reads then as the linear system for $\tilde{K}$ $$D_{i}\tilde{K}^{ij}=\frac{2}{3}\varphi ^{6}\gamma ^{ij}\partial _{i}\tau +\varphi ^{10}J^{j},$$ independent of $\varphi $ if the initial surface is maxima (he says ”minima”) i.e. $\tau =0,$ and if the momentum $J$ of the sources is zero. The Hamiltonian constraint reads then as a second order elliptic equation with only unknown $\varphi $ when $\tilde{K}$ and the matter density $\rho $ are known $$8\Delta _{\gamma }\varphi -R(\gamma )\varphi +|\tilde{K}|_{\gamma }^{2}\varphi ^{-7}+(\rho -\frac{2}{3}\tau ^{2})\varphi ^{5}=0.\text{ \ }$$ Lichnerowicz constructs a class of exact initially static data for the $N$ body problem with supports in domains $D_{I},$ $I=1,...n$, and matter densities $\mu _{I}$ by taking $\gamma =e,$ assuming $J=0,\tau =0$ and taking $\tilde{K}=0$ as a solution of the momentum constraint. The system of constraints reduces then to the elliptic non linear equation with pricipal term the Euclidean Laplace operator $$\Delta \varphi =f(\varphi )\text{ \ \ with }f(\varphi )=0\text{\ \ in vacuum and \ }f(\varphi )\equiv -\frac{1}{8}\mu _{I}\varphi ^{5}\text{ \ in \ }D_{I}.$$ Using the potential formula Lichnerowicz solves this equation by iteration, showing the convergence of the series for small enough $\mu _{I}.$ The problem of meaningful non static solutions of the momentum constraint remained unsolved. It is only in 1961, writing an article on the Cauchy problem, for the book edited by Louis Witten[^47] and inspired by a paper of D. Sharp[^48] on possible constraints for the arbitrary quantities of the ”thin sandwich conjecture” of J. A. Wheeler, namely lapse and shift, which did not lead to an elliptic system, that I realized that such an elliptic system can be written for the corresponding spacetime densities $\mathcal{G}^{00}$ and $\mathcal{G}^{i0}$ by using the splitting of the Einstein equations obtained through the harmonic gauge. A. Vaillant-Simon[^49] constructed a solution of this system near from the Minkowski spacetime. In 1971 I wrote an elliptic, but not quasidiagonal[^50], system for geometric data on an arbitrary spacelike manifold which stimulated the interest of J. York, then a student of J. A. Wheeler, in the constraint problem. York remarked that the assumption ”maximal” on the initial manifold made for conformally formulated constraints can be replaced by constant mean extrinsic curvature[^51] and he introduced weights for the sources $\rho $ and $J,$ physically justified at least for electromagnetic sources and dimension $n=3. $ He thus obtained the linear momentum constraint independent of $\varphi $$$D_{i}\tilde{K}^{ij}=\tilde{J}^{i}.$$ The Hamiltonian constraint becomes then the nonlinear elliptic equation with only unknown $\varphi $ when $\gamma $ is chosen, $\tilde{K}$ computed and $\tilde{\rho}$ is known $$\text{\ }8\Delta _{\gamma }\varphi -R(\gamma )\varphi +|\tilde{K}|_{\gamma }^{2}\varphi ^{-7}+\tilde{\rho}\varphi ^{-3}-\frac{2}{3}\tau ^{2}\varphi ^{5}=0.\text{\ }$$ A decomposition theorem for symmetric 2-tensors, linked to the fact that Lie derivatives of vector fields span the $L^{2}$ dual of divergence free symmetric 2-tensors, had been known for some time. It leads to the writing[^52] of the general solution of the momentum constraint, when $\tau $ is constant, under the form, with $U$ an arbitrary given traceless symmetric 2 tensor $$\tilde{K}^{ij}=(\mathcal{L}_{\gamma ,conf}X)^{ij}+U^{ij}+\frac{1}{3}\gamma ^{ij}\tau ,\text{ \ \ \ }(\mathcal{L}_{\gamma ,conf}X)^{ij}:=D^{i}X^{j}+D^{j}X^{i}-\frac{2}{3}\gamma ^{ij}D_{k}X^{k}.$$ The vector field $X$ is then solution of the linear system $$\Delta _{\gamma ,conf}X:=D.(\mathcal{L}_{\gamma ,conf}X)=-D.U+\tilde{J}.$$ which can be shown to be equivalent to a linear elliptic second order operator for $X,$ to which known theorems can be applied. The conformal method gave to the Hamiltonian constraint on a manifold a geometric comparatively simple form but non linear with no known generic solution. I thought of applying to it the Leray-Schauder degree theory. I brought to Leray in 1962  a Note about its solution in Hölder spaces for publication in the C.R. of the french Academy of sciences. Leray remarked that my result would hold for more general equations, and suggested we publish jointly the general result. It was for me a great honor. Leray wrote[^53] very fast for compact manifolds this Note which introduces sub and super solutions, and refused to cosign the Note solving the particular case of the Hamiltonian constraint which I wrote shortly afterwards[^54]. I obtained later the result for asymptotically Euclidean manifolds in weighted Hölder spaces[^55]. A large amount of work has been devoted since that time to the solution of the constraints expressed as the elliptic semilinear system obtained by the conformal method on a constant mean curvature initial manifold, and using weighted Sobolev spaces. Progress has been made in lifting the constant mean curvature hypothesis and weakening the regularity, but there is space for further work. Local existence and global uniqueness. ====================================== In the beginning of the seventies the geometric character of the Cauchy problem for the Einstein equations was well understood[^56]. The global in space, local in time, existence was known for the classical sources mentioned above, at least for compact or asymptotically Euclidean manifolds, apart from lessening the regularity required of data and abandoning the constant mean curvature hypothesis of the initial manifold. Local uniqueness, up to diffeomorphisms, of a solution of the evolution of geometric data satisfying the constraints was also known, but the question of a global isomorphism between solutions was open. In fact, though known of specialists, the geometric local existence and uniqueness did not have in the litterature a concise and precise formulation. I convinced Geroch we write up proper definitions for the geometric local theorem[^57] before publishing our global geometric uniqueness proof. A fundamental notion for the study of global solutions of linear hyperbolic systems of arbitrary order had been introduced in 1952 by J. Leray. He had defined what he called ”time like paths” for such systems and defined global hyperbolicity as compactness (in the space of paths) of any, non vacuum, set of timelike paths joining two arbitrary points. This general definition applies in particular to Lorentzian manifolds. In this case global hyperbolicity was proved (Choquet-Bruhat 1967) to be equivalent to the strong causality defined by Penrose 1967 added to the compactness of the spacetime domaines defined by intersections of past and future of any two spacetime points (see definitions and proof in the Hawking and Ellis book of 1973). In 1969, the local existence was completed[^58] by a geometric global uniqueness result, namely existence and uniqueness, , up to isometries, in the class of globally hyperbolic spacetimes, of a maximal[^59] Einsteinian development of given initial data. Geroch and myself had met and discussed at the ”Batelle rencontres 1967” organized by J. A. Wheeler and C. DeWitt. We obtained a complete proof during a visit we both made in London[^60]. In 1970 Geroch proved the following very useful criterium: global hyperbolicity is equivalent to the existence of a ”Cauchy surface”, 3 manifold cut once and only once by any timelike curve without end point. The geometric global uniqueness result, first proved in the vacuum case, extends to Einstein equations with sources which have a well posed causal Cauchy problem; that is, in particular, satisfy Leray or Leray-Ohya hyperbolic systems. Global existence and singulariries. =================================== Generic problems of global existence or formation of singularities were, towards the end of the sixties, mainly open. They became for analysts and geometers interested in the modeling of the world we live in, a vast field of research. It led to new definitions, conjectures, remarkable results, and new open problems. **References[^61]** D. Bancel *”Problème de Cauchy pour l’équation de Boltzman en Relativité Générale”* Ann. Inst. Poincaré XVIII n$^{0}$3 263-284 1973 D. Bancel and Y. Choquet-Bruhat *”Existence, uniqueness and local stability for the Einstein-Botzman system”* Com. Math. Phys.1-14 1973. Y. Bruhat ”*Fluides relativistes de conductivité infinie”* Astronautica Acta 6 354-355 1961 Y. Bruhat ”*The Cauchy problem*” in ”*Gravitation, an introduction to current research”* Louis Witten ed. Wiley 1962 Y. Bruhat *”Problème des conditions initiales sur un conoïde caractéristique* C.R. Acad. Sci **256**, 371-373, 1963. Y. Bruhat ”*Sur la théorie des propagateurs*” Annali di Matematica Serie IV, tomo LXIV -1964. F. Cagnac ”*Problème de Cauchy sur les hypersurfaces caractéristiques des équations d’Einstein du vide”* C.R. Acad Sci **262** 1966 F. Cagnac ”*Problème de Cauchy sur un conoïde caractéristique pour les équations d’Einstein du vide”* Annali di matemetica pura e applicata 1975. Y. Choquet-Bruhat ”*Etude des équations des fluides chargés relativistes inductifs et conducteurs*” Com. Math. Phys **3** 334-357 1966. Y. Choquet-Bruhat ”Hyperbolic partial differential equations on a manifold” in C. DeWitt and J.A. Wheeler ed *”Batelle rencontres 1967 in Mathematics and Physics”* Y. Choquet-Bruhat ”*Espaces Einsteiniens généraux, chocs gravitationnels*” Ann. Inst. Poincaré, 8 n${{}^\circ}4$, 327-338, 1968. Y. Choquet-Bruhat ”*Un théorème global d’unicité pour les solutions des équations d’Einstein*” Bul. Soc. Math. France **96** 181-192, 1968. Y. Choquet-Bruhat *”New elliptic system and global solutions for the constraints equations in General Relativity”* Com. Math. Phys. **21** 211-218,** **1971. Y. Choquet-Bruhat ”S*olution globale du problème des contraintes sur une variété compacte”* C.R. Acad Sci. **274** 682-684 1972. Y. Choquet-Bruhat ”*Solution of the problem of constraints on open and closed manifolds*” J. Gen. Rel and Grav.** 5** 45-64 1974. Y. Choquet-Bruhat and R. Geroch ”*Probleme de Cauchy intrinsèque en Relativité générale*” C. R. Acad. Sci. 269 746-748, 1969. Y. Choquet-Bruhat and R. Geroch ”*Global aspects of the Cauchy problem in General Relativity*” Comm. Math. Phys. **14**, 329-335, 1969. Y Choquet-Bruhat et J. Leray ”*Sur le problème de Dirichlet quasi-linéaire d’ordre 2”* C R. Acad. Sci **274** 81-85 1972 G. Darmois ”*Les equations de la gravitation Einsteinienne*” Memorial des Sciences mathématiques Gauthier Villars 1927. B. DeWitt.*”Quantization of geometry”* in *Les Houches 1963* Gordon and Breach De Donder *”La gravifique Einsteinienne”* Memorial des Sciences mathématiques Gauthier Villars 1925. Y. Fourès-Bruhat ”*Théorèmes d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires*” Acta Mathematica **88, 42-225,** 1952. Y. Fourès-Bruhat ”*Théorème d’existence et d’unicité dans les théories relativistes de l’électromagnétisme”* C.R. Acad.Sci. **232** 1951. Y. Fourès-Bruhat ”*Résolution du problème de Cauchy pour des équations hyperboliques du second ordre non linéaires”* Bull. Soc. Math. France **81**, 225-288 1953 Y. Fourès-Bruhat *”Solution élémentaire d’équations ultrahyperboliques”* J. Math. Pures et Ap **121** 277-289 1955. Y. Fourès-Bruhat ”*sur l’intégration des équations de la Relativité Générale*” J. Rat. Mech. and Anal. **5** 951-966 1956. Y. Fourès-Bruhat ”*Theorèmes d’existence en mécanique des fluides relativistes”*  Bull. Soc. France **86**, 155-175, 1958. R. Geroch ”*The domain of dependence*” J. Math. Phys **11** 437-439 1970 S. W. Hawking and G.R.F Ellis *”The large scale structure of spacetime”* Cambridge university press 1973. J. Leray *”Hyperbolic differential systems”* Mimeographed Notes, IAS Princeton 1953. J. Leray and Y. Ohya Math Annalen **162**, 228-236 1968. A. Lichnerowicz ”*Problèmes globaux en mécanique relativiste*”, Hermann 1939. A. Lichnerowicz. ”*L’intégration des équations de la gravitation relativiste et le problème des* $n$* corps”* J. Math. pures et App. 37-63, 1944. A. Lichnerowicz *” Théories relativistes de la gravitation et de l’électromagnétisme”* Masson, Paris 1955. A. Lichnerowicz ”*Propagateurs et commutateurs en Relatitité Générale”,* publications mathématiques de l’IHES, 1$,$ 1961* * N. O’Murchada and J. W. York ”*The initial value problem of General Relativity”* Phys. Rev. D **10** 428-446 1974. R. Penrose *”structure of spacetime”* in C. DeWitt and J.A. Wheeler ed *”Batelle rencontres 1967 in Mathematics and Physics”* M. Q. Pham *Etude électrodynamique et thermodynamique d’un fluide relativiste chargé J. Rat. Mech. Anal.* **5**, 473-538 ,1956. G. Pichon. ”*Etude relativiste de fluides visqueux et chargés*” Ann. Inst. Poincaré II n$^{0}1$ 1965. G. Pichon * ”Théorèmes d’existence pour les équations des milieux élastiques”* J. Math. Pures. et App  **45** 3395-409 1966 Ch. Racine C.R. Acad Sci. Paris 1931. Ch. Racine ”*Le problème des* $n$* corps dans la théorie de la Relativité* ” Thèse Paris 1934, Gauthier Villars.  J. Schauder Fundamenta mathematicae, 24 213-246 1935. D. Sharp ”*One and two surfaces formulation of the boundary value problem for the Einstein- Maxwell equations”* thesis Princeton University 1961 S. Sobolev  *”Methode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales”* Rec. Math. Moscou N. s. 1936. K. Stellmacher. Math. Annalen 115, 1938. A.H. Taub Phys. Rev **74** 328-334 1948 A.H. Taub J. Rat. Mech. Anal. **3** 312-324 1959. A. Vaillant-Simon, J. maths pures et App** 48**; 1-90, 1969. J. A. Wheeler in ”Relativity, Groups and Topoloy”, B. and C. DeWitt ed. Gordon and Breach 1964 J. W. York ”*Role of conformal 3 geometry in the dynamics of gravitation”* Phys. Rev. lett. **28** 1082.1972. J W. York ”*Decomposition of symmetric 2-tensors in the theory of gravitation*” Annales de l’IHP A **4**, 319-331 1974. [^1]: We denote $S$ the Einstein tensor. It is denoted $G$ by some authors. [^2]: i.e. linear with respect to second derivatives [^3]: In those times there was only one university of Paris. Sciences, letters, law and arts were housed in a building called the Sorbonne, G. Darmois was a man of varied interests. In 1948 he taught a course on probabilities which I attended. He was a member of the french Academy in the section ”Astronomy”. [^4]: That is itersected along a compact set by the past of any point. [^5]: Y. Bruhat *”Problème des conditions initiales sur un conoïde caractéristique* C.R. Acad. Sci **256**, 371-373, 1963. F. Cagnac ”*Problème de Cauchy sur les hypersurfaces caractéristiques des équtions d’Einstein du vide”* C.R. Acad Sci **262** 1966 For more recent works see papers by Cagnac and his students, in particular Dossa. Still more recent, Choquet-Bruhat, Chrusciel and Martin-Garcia, also Chrusciel and collaborators [^6]: For other early work see J. Delsarte ”*Sur les ds*$^{2}$*d’Einstein à symétrie axiale’* Hermann 1934. [^7]: Lichnerowicz A. ”problèmes globaux en mécanique relativiste”, Hermann 1939 [^8]: The other wasYves Thiry who worked on geometrical aspects and physical interpretation of the five dimensional unitary theory of Jordan, extension of Kaluza and Klein work. [^9]: Fourès-Bruhat C. R. Acad Sci. Paris 1948 [^10]: Y. Fourès-Bruhat ”*sur l’intégration des équations de la Relativity Générale*” J. Rat. Mech. and Anal. **5** 951-966 1956 [^11]: A. Lichnerowicz *”Problèmes globaux en mécanique relativiste*” Hermann et Cie, 1939 and A. Lichnerowicz ”*Théories relativistes de la gravitation et de l’électromagnétisme”* Masson 1955 which contains also a study of the 5 dimensional and the non symmetric unitary theories. [^12]: Better named ”conjectures”. [^13]: The relevant condition is in fact that the Einstein equations are satisfied in a generalized sense: see YCB ”Espaces Einsteiniens généraux, chocs gravitationnels” Ann. Inst. Poincaré, 8 n$^{0}4$, 327-338, 1968. [^14]: That is invariant under a timelike one parameter isometry group. The static case (timelines orthogonal to space sections) had been proved earlier by Racine, C. R. Acad. Sciences 192, 1533 1931., another student of Darmois [^15]: The Christodoulou-Klainerman global existence theorem (1989) has proven that the conjecture was false without stronger hypothesis than the ones originally made on the decay at infinity; that is, vanishing of the ADM mass as shown by Shoen and Yau. [^16]: Einstein A. Sitzgsb 1918. [^17]: Darmois quotes as sources: De Donder ”La gravifique Einsteinienne ”Mem. Sci. Math. 1925 Gauthier Villars This article can be found on numdam. It uses the Lagrangian formulation of Einstein equations . Lanczos K. Physzeitshrift p.137 1922 [^18]: K. Stellmacher. Math. Annalen 115, 1938. [^19]: Schauder J. Fundamenta mathematicae, 24 1935, p213-246. Schauder, a collaborator and friend of J. Leray was a victim of the holocaust, having refused to follow Leray advice to leave Germany when it was still possible for jews. [^20]: J. Schauder Comm. Math. Helv.**9** 1936-1937. [^21]: S. Sobolev  *”Methode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales”* Rec. Math. Moscou N. s. 1936. [^22]: See Fourès-Bruhat Y. ”Théorèmes d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires” Acta Mathematica **88, 42-225,** 1952 and references therein. [^23]: See also Y. Bruhat ”The Cauchy problem” in ”Gravitation, an introduction to current researc” Louis Witten ed. Wiley 1962 [^24]: In fact, while I was still working on my thesis Leray was completing his momentum work on energy estimates and existence theorems for general hyperbolic systems, from which I could have deduced the result for the reduced Einstein equations, but Leray encouraged me warmly to pursue in the constructive direction I was following in the second order case. [^25]: Present in Paris at that time. Darmois, an emeritus, could not belong to a thesis jury. [^26]: Y. Choquet-Bruhat *Sur la théorie des propagateurs*” Annali di Matematica Serie IV, tomo LXIV -1964. [^27]: A. Lichnerowicz ”*Propagateurs et commutateurs en Relatitité Générale”,* publications mathématiques de l’IHES, 1$,$ 1961* * See also Bryce DeWitt.*”Quantization of geometry”* in *Les Houches 1963* Gordon and Breach [^28]: For a lowering of the assumed regularity of the Lorentzian metric see S. Klainerman and I. Rodnianski ”*The Kirchoff-Sobolev formula” *Arxiv.math 2006, [^29]: Einstein tried at that time to interpret the antisymmetric part of the second rank tensor as electromagnetism. It appears that this last Einstein unified theory finds a renewal of interest with another intepretation (see Damour and Deser) [^30]: Y. Fourès-Bruhat ”*Résolution du problème de Cauchy pour des équations hyperboliques du second ordre non linéaires”* Bull. Soc. Math. France **81**, 225-288 1953 [^31]: My main memory of this seminar is a discussion of quantum vacua and the intervention of Wigner ”but in vacuum there is nothing, nothing, there can be only one vacuum”, Wigner, and also Einstein, lived in a time where the observed world could be thought to obey human scale logic. [^32]: With the help of his friend the mathematcian Grossman. [^33]: Y. Fourès-Bruhat ”*Théorème d’existence et d’unicité dans les théories relativistes de l’électromagnétisme”* C.R. Acad.Sci. **232** 1951 [^34]: Y. Fourès-Bruhat ”*Theorèmes d’existence en mécanique des fuides relativistes”*  Bull. Soc. France **86**, 155-175, 1958. [^35]: J. Leray *”hyperbolic differential equations” * Mimeographed Notes IAS, 1953 [^36]: Y. Bruhat *Fluides relativistes de conductivity infinie* Astronautica Acta **VI**, 354-365, 1961. [^37]: J. Leray and Y. Ohya Math Annalen **162**, 228-236 1968. It was shown later by K.O. Friedrichs using general Lagrangian methods that fluids with infinite conductivity satisfy a first order symmetric hyperbolic system. See for instance A. M. Anile Relativistic fluids and magnetofluids  Cambridge University press 1989 [^38]: As suggested by A.Taub, on physical grounds. [^39]: A. Lichnerowicz *Relativistic fluids and magneto fluids”* Benjamin 1967. [^40]: Y  Choquet-Bruhat *”Etude des équations des fluides chargés relativistes inductifs et conducteurs”* Comm. Math. Phys. **3** 334-357 1966 [^41]: G. Pichon * ”Théorèmes d’existence pour les équations des milieux élastiques”* J. Math. Pures. et App  **45** 3395-409 1966 [^42]: See M. Q. Pham *Etude électrodynamique et thermodynamique dun fluide relativiste chargé J. Rat. Mech. Anal.* **5**, 473-538 ,1956. Various symmetrizations have been proposed along the years, but their conservation laws lead to very unpleasant equations with unphysical interpretations. The physical answer -seems to be that at the scale where inductions play a role the gravational field is negligible. [^43]: Case of particles with a given rest mass and no charge: Y. Choquet-Bruhat ”*Solution du problème de Cauchy pour le système intégro-differentiel d’Einstein Liouville”*  Ann..Inst. Fourier **XXI 3** 181-203 1971. Case of particles with electric charge:and arbitrary (positive) rest masses using a weight factor to obtain convergent integrals: Y. Choquet-Bruhat ”*Existence and uniqueness for the Einstein-Maxwell- Liouville” system”* Volume in honor of Professor Petrov,60th birthday Kiev* 1971.* [^44]: D. Bancel *”Problème de Cauchy pour l’équation de Boltzman en Relativité Générale”* Ann. Inst. Poincaré XVIII n$^{0}$3 263-284 1971 D. Bancel and Y. Choquet-Bruhat *”Existence, uniqueness and local stability for the Einstein-Botzman system”* Com. Math. Phys.1-14 1973. [^45]: Ch. Racine ”*Le problème des* $n$* corps dans la théorie de la Relativit*é ” Thèse Paris 1934, Gauthier Villars.  [^46]: A. Lichnerowicz ”L’intégration des équations de la gravitation relativiste et le problème des $n$ corps; J. Math. pures et App. 37-63, 1944. [^47]: Y. Bruhat ”*The cauchy problem”* in ”*Gravitation, an introduction to current research*”, L. Witten ed Wiley 1962 [^48]: D. Sharp ”*One and two surfaces formulation of the boundary value problem for the Einstein- Maxwell equations”* thesis Princeton University 1961 [^49]: A. Vaillant-Simon, J. maths pures et App** 48**; 1-90, 1969. [^50]: Y. Choquet-Bruhat Com. Math. Phys. **21** 211-218,** **1971. [^51]: J. W. York ”*Role of conformal 3 geometry in the dynamics o gravitation”* Phys. Rev. lett. **28** 1082.1972. The constant non zero situation was neglected by previous authors who were only interested in the asymptotically Euclidean manifolds where it does not occur. [^52]: See J W. York ”*Decomposition of symmetric 2-tensors in the theory of gravitation*” Annales de l’IHP A **4**, 319-331 1974  and references therein. [^53]: Y Choquet-Bruhat et J. Leray ”*Sur le problème de Dirichlet quasi-linéaire d’ordre 2”* C. R. Acad. Sci **274** 81-85 1972 [^54]: Y. Choquet-Bruhat ”S*olution globale du problème des contraintes sur une variété compacte”* C.R. Acad Sci. **274** 682-684 1972. [^55]: Y. Choquet-Bruhat ”*Solution of the problem of constraints on open and closed manifolds*” J. Gen. Rel and Grav.** 5** 45-64 1974. [^56]: Let us quote in addition to previously quoted papers: J. A. Wheeler in ”Relativity, Gorups and Topoloy”, B. and C. DeWitt ed. Gordon and Breach 1964 R. Penrose Phys Rev. lett. 14 57 1965, S. Hawking Proc.Roy. Soc **294** A 511, 1966 [^57]: Y. Choquet-Bruhat and R. Geroch ”*Problème de Cauchy intrinsèque en Relativité Générale*” C . R. Acad. Sci. A **269** 746-748, 1969. [^58]: Y. Choquet-Bruhat and R. Geroch ”*Global aspects of the Cauchy problem in General Relativity*” Comm. Math. Phys. **14**, 329-335, 1969. [^59]: i.e. which cannot be isometrically embedded into a larger Einsteinian spacetime. [^60]: I had obtained in 1968 a global uniqueness theorem, but restricted to complete Einsteinian spacetimes. [^61]: Y Bruhat, Y. Choquet-Bruhat and Y. Fourès-Bruhat are the same person, daughter of the physicist Georges Bruhat..

--- abstract: | Aerial Cable Towed Systems (ACTS) are composed of several Unmanned Aerial Vehicles (UAVs) connected to a payload by cables. Compared to towing objects from individual aerial vehicles, an ACTS has significant advantages such as heavier payload capacity, modularity, and full control of the payload pose. This paper presents a novel ACTS with variable cable lengths, named Variable Aerial Cable Towed System (VACTS). Winches are embedded on the UAVs for actuating the cable lengths similar to a Cable-Driven Parallel Robot to increase the versatility of the ACTS. The general geometric, kinematic and dynamic models of the VACTS are derived, followed by the development of a centralized feedback linearization controller. The design is based on a wrench analysis of the VACTS, without . Additionally, the performance of the VACTS and ACTS are compared . A prototype confirms the feasibility of the system. author: - 'Zhen Li$^{1}$, Julian Erskine$^{2}$, St[é]{}phane Caro$^{3}$ and Abdelhamid Chriette$^{4}$[^1][^2][^3][^4]' bibliography: - 'root.bib' title: Design and Control of a Variable Aerial Cable Towed System --- Aerial Systems: Mechanics and Control; Tendon/Wire Mechanism; Parallel Robots INTRODUCTION ============ the agility of aerial vehicles with the manipulation capability of manipulators makes aerial manipulation an attractive topic[@literature]. Unmanned Aerial Vehicles (UAVs) are becoming more and more popular during last decades, not only in research fields but also in commercial applications. Amongst UAVs, quadrotors are widely used in photographing, inspection, transportation and (recently) manipulation because of their agility, versatility, and low cost. However, the coupling between their translational and rotational dynamics complicates tasks requiring fine pose control. Moreover, most have limited payload capacity, which makes it difficult to transport large objects by an individual quadrotor. Multi-quadrotor collaboration is studied as a solution to compensate the previous two drawbacks inherent to individual quadrotors. In this paper, we study a payload suspended via cables, which reduce the couplings between the platform and quadrotors attitude dynamics, allow reconfigurability, and reduce aerodynamic interference between the quadrotor downwash and the payload. Moreover, Cable-Driven Parallel Robots (CDPRs) are well-studied, providing a part of theoretical support, such as tension distribution algorithms in [@bookCDPR], [@li2013optimal], [@lamaury2013tension], [@gouttefarde2015versatile], controllers [@RCDPR], [@adaptive1] and wrench performance evaluations [@wrenchbasic]. Reconfigurable CDPRs with both continuously moving [@RCDPR; @fastkit] and discrete [@DRCDPR] cable anchor points are proposed to solve problems such as wrench infeasiblility and self or environmental collisions. For the past ten years, the Aerial Cable Towed System (ACTS) composed of several UAVs, a payload, and cables to cooperatively manipulate objects, has attracted researchers’ attention and has been well developed from a controls viewpoint. An ACTS prototype for 6-dimensional manipulation “FlyCrane” and a motion planning approach called Transition-based Rapidly-exploring Random Tree (T-RRT) are proposed in [@flycrane]. In [@cone] an ACTS prototype for 3-dimensional manipulation and a decentralized linear quadratic control law are proposed and [@RCDPR] develops a general feedback linearization control scheme for over-actuated ACTS with a 6-DOF payload. An ACTS prototype with three quadrotors and a point mass is implemented in [@lastone] and the performance was evaluated using capacity margin, a wrench-based robustness index adapted from the CDPR in [@wrench]. ![The sketch structure of a VACTS with 4 quadrotors, 6 winches and cables, and a platform[]{data-label="fig:acts"}](figure1){width="1\linewidth"} The ACTS shows promise both in aerial manipulation and sharing heavy burden, however there are still some limitations for current designs. Therefore, a novel Aerial Cable Towed System with actuated cable lengths, the Variable Aerial Cable Towed System (VACTS) is proposed in this paper to make up for these shortcomings. The actuated cable lengths can reshape the size of overall system, which implies the possibility of passing through a constrained environment or limited space[^5]. Moreover, it is conceivable that combining force control of the quadrotors with velocity control by actuating cable lengths might improve payload positioning precision. Section \[sec:model\] derives the geometric, kinematic and dynamic models. Section \[sec:control\] illustrates the architecture of a centralized feedback linearization controller. The design of the VACTS and its performance relative to the ACTS are compared in Sec. \[sec:wrench\]. Section \[sec:exp\] discusses the experimental results. Conclusions and future work are presented in Sec. \[sec:con\]. Modelling {#sec:model} ========= The geometric, kinematic and dynamic models of the VACTS are derived in this section. The sketch structure of a VACTS is shown in Fig. \[fig:acts\]. It is composed of $n$ quadrotors, $m$ cables and winches ($m\geq n$), and a payload. There are $s_j \in [1,2]$ winches mounted on the $j^{\text{th}}$ quadrotor. Geometric Modelling of the VACTS -------------------------------- The geometric parametrization is presented in Fig. \[fig:actsparam\] with symbolic interpretation listed in Table \[table:param\]. Matrix $^a{\bm{T}}_b$ is the homogeneous transformation matrix from frame $\mathcal{F}_a$ to $\mathcal{F}_b$, and consists of a rotation matrix ${}^a{\bm{R}}_b$ and a translation vector ${}^a{\bm{x}}_b$. Matrix $^j{\bm{T}}_{wi}$ expresses the transform of the $i^{\text{th}}$ winch from the $j^{\text{th}}$ quadrotors frame $\mathcal{F}_j$ located at it’s center of mass (COM). The cable length is $l_i$, and the unit vector along cable direction is expressed in the payload frame $\mathcal{F}_p$ as ${}^p{\bm{u}}_i={\begin{bmatrix} c_{{\phi}_i}s_{{\theta}_i} & s_{{\phi}_i}s_{{\theta}_i} & c_{{\theta}_i} \end{bmatrix}}^T$ via azimuth angle ${\phi}_i$ and inclination angle ${\theta}_i$, as used in [@lastone]. The gravity vector is $\bm{g}=[0 \; 0 \; -9.81]\text{~ms}^{-2}$. The $i^{\text{th}}$ loop closure equation can be derived considering the forward and backward derivation of ${}^0{\bm{x}}_{Ii}$. $$\label{eq:GM} {}^0{\bm{x}}_p + {}^0{\bm{R}}_p {}^p{\bm{x}}_{B_i} + l_i {}^0{\bm{R}}_p {}^p{\bm{u}}_i = {}^0{\bm{x}}_j + {}^0{\bm{R}}_j {}^j{\bm{x}}_{Ii}$$ Kinematic Modelling of the VACTS -------------------------------- After differentiating (\[eq:GM\]) with respect to time, the first order kinematic model for each limb can be derived as (\[eq:KM\]), where the payload has translational velocity ${}^0{\dot{\bm{x}}}_p$ and angular velocity ${}^0{\bm{\omega}}_p$, and the $j^{\text{th}}$ quadrotor has translational velocity ${}^0{\dot{\bm{x}}}_j$ and angular velocity ${}^0{\bm{\omega}}_j$. $$\label{eq:KM}\footnotesize \begin{aligned} {}^0{\dot{\bm{x}}}_{p} + {}^0{\bm{\omega}}_p \times ({}^0{\bm{R}}_p {}^p{\bm{x}}_{B_i}) + l_i {}^0{\bm{R}}_p {}^p{\dot{\bm{u}}}_i + l_i {}^0{\bm{\omega}}_p \times ({}^0{\bm{R}}_p {}^p{{\bm{u}}}_i) \\ = {}^0{\dot{\bm{x}}}_{j} - {\dot{l}}_i {}^0{\bm{R}}_p {}^p{\bm{u}}_i + {}^0{\bm{\omega}}_j \times ({}^0{\bm{R}}_j {}^j{\bm{x}}_{Ii}) + {}^0{\bm{R}}_j {}^j{\dot{\bm{x}}}_{Ii} \end{aligned}\normalsize$$ (image) at(0,0)[![[]{data-label="fig:actsparam"}](figure2 "fig:"){width="\linewidth"}]{}; (.65, .15)–(0.95,0.45)..controls (0.9,0.35) and (0.99,0.2)..(0.97,0.1)–(.65, .15); (.65, .15)–(0.97,0.1)–(0.97,0.05)–(.65, .1)–(.65, .15); (0.1, 0.1)–node \[above left\] [$x_0$]{}(.05, .05); (0.1, 0.1)–node \[above right\] [$y_0$]{}(.2, .1); (0.1, 0.1)–node \[above right\] [$z_0$]{}(.1, .2); (.15, .4) node \[below\] [$\mathcal{F}_0$]{}; (.1,.1)circle(0.05cm) node \[below right\] [$O$]{}; (.44, .8) node \[above\] [$\mathcal{F}_j$]{}; (.435, .68)circle(0.05cm) node \[above left\] [$J$]{}; ; (.5, .65) node [$\mathcal{F}_{w_i}$]{}; (.47, .52) circle(0.05cm) node \[below left\] [$W_i$]{}; (.552, .56) circle(0.05cm) node \[right\] [$I_i$]{}; (.9, .3)–node\[xshift=-0.3cm, yshift=-0.1cm\][$x_p$]{}(.85, .25); (.9, .3)–node\[above right\][$y_p$]{}(.97, .3); (.9, .3)–node\[above\][$z_p$]{}(.9, .4); (.9, .6) node \[below\] [$\mathcal{F}_p$]{}; (.9, .3) circle(0.05cm) node \[below right\] [$P$]{}; (.05,.5)–node \[above left\] [$\bm{g}$]{} (.05,.3); (.65, .15)–node \[above right\] [$l_i$]{}(.552, .56); (.65, .15) circle(0.05cm) node \[below\] [$B_i$]{}; (.65, .15)–node\[above right\][$z_p$]{}(.65, .35); (.65, .15)–node\[yshift=-0.2cm\][$x_p$]{}(.55, .1); (.552, .56)–(.552, .25); (.552, .25)–(.65, .15); (.552,.27)–(.56,.26); (.56,.26)–(.56,.24); (.65, .25) to \[out=180,in=60\] node \[xshift=-0.05cm, yshift=0.25cm\][${\theta}_i$]{} (.63, .22); (.6, .12) to \[out=110,in=-130\] node \[left\][${\phi}_i$]{} (.61, .2); (.6,.3)–node \[left\] [${}^{p}{\bm{u}}_i$]{} (.58,.38); (.1,.1) to \[out=110,in=170\] node \[left\][$^0{\bm{T}}_j$]{} (.435, .68); (.435, .68) to \[out=-90,in=140\] node \[left\][$^j{\bm{T}}_{w_i}$]{} (.47,.52); (.1,.1) to \[out=-10,in=-135\] node \[xshift=-1.5cm,yshift=0.4cm\][$^0{\bm{T}}_p$]{} (.9,.3); (.47,.52) to \[out=0,in=-140\] node \[below\][$^w{\bm{x}}_{Ii}$]{} (.552, .56); (.9,.3) to \[out=180,in=30\] node \[above\][$^p{\bm{x}}_{B_i}$]{} (.65, .15); (.57,.65) circle (0.1cm) node \[above right\] [$G$]{}; (.57,.65) – (.585,.65) arc (0:90:0.1cm); (.57,.65) – (.555,.65) arc (180:270:0.1cm); (.8,.2) circle (0.1cm) node \[below left\] [$C$]{}; (.8,.2) – (.815,.2) arc (0:90:0.1cm); (.8,.2) – (.785,.2) arc (180:270:0.1cm); symbol physical meaning -------- ---------------------------------------------------------------------- $O$ Origin of world frame $\mathcal{F}_0$ $J$ The $j^{\text{th}}$ quadrotor centroid and origin of $\mathcal{F}_j$ $W_i$ The $i^{\text{th}}$ winch centroid $I_i$ Coincident point of the $i^{\text{th}}$ cable-winch pair $B_i$ Attachment point of the $i^{\text{th}}$ cable-payload pair $P$ Origin of payload frame $\mathcal{F}_p$ $C$ COM of the payload $G$ COM of the $j^{\text{th}}$ quadrotor with embedded winches : []{data-label="table:param"} The first order kinematic model can be re-expressed as (\[eq:1KM\]), where the Jacobian matrix ${\bm{J}}_{(6+3m) \times 3n}$ can be obtained from the forward Jacobian matrix ${\bm{A}}_{3m \times (6+3m)}$ and the inverse Jacobian matrix ${\bm{B}}_{3m \times 3n}$. ${\bm{A}}^{+}$ is the pseudo-inverse of $\bm{A}$, which minimizes the 2-norm of ${\dot{\bm{x}}}_{\bm{t}}$ since the system is under-determined. For example, if $s_1=1$ and $s_2=2$, the matrix $\bm{B}$ will be expressed as (\[eq:B\]). The task space vector is ${\bm{x_t}}_{(6+3m) \times 1}$ and the joint space vector is ${\bm{q_a}}_{3n \times 1}$. Note that ${\dot{\bm{x}}}_{\bm{t}}$ is not the time derivative of $\bm{x_t}$ considering that the angular velocity ${}^0{\bm{\omega}}_p$ is not the time derivative of payload orientation, which can be expressed in the form of Euler angles or quaternions. The form ${[\bm{x}]}_{\times}$ is the cross product matrix of vector ${\bm{x}}$ and ${\bf{I}}_3$ is a $3\times 3$ identity matrix. $$\label{eq:1KM} {\dot{\bm{x}}}_{\bm{t}} = \bm{J} {\dot{\bm{q}}}_{\bm{a}} + \bm{a} \text{, with } \bm{J} = {\bm{A}}^{+} \bm{B}$$ $$\label{eq:Xdot}\small {\dot{\bm{x}}}_{\bm{t}}={\begin{bmatrix} {}^0{\dot{\bm{x}}}_{p}^T & {}^0{\bm{\omega}}_p^T & {\dot{\phi}}_1 & {\dot{\theta}}_1 & {\dot{l}}_1 & \cdots & {\dot{\phi}}_m & {\dot{\theta}}_m & {\dot{l}}_m \end{bmatrix}}^T \normalsize$$ $$\label{eq:qadot} {\dot{\bm{q}}}_{\bm{a}}={\begin{bmatrix} {}^0{\dot{\bm{x}}}_{1}^T & \cdots & {}^0{\dot{\bm{x}}}_{n}^T \end{bmatrix}}^T$$ $$\label{eq:A}\footnotesize \setlength{\arraycolsep}{1pt} \begin{aligned} \bm{A}=&\begin{bmatrix} {\mathbb{I}}_3 & {\bm{A}}_{12} & l_1 {}^0{\bm{R}}_p {\bm{C}}_1 & {}^0{\bm{R}}_p {}^p{\bm{u}}_1 \\ \vdots & \vdots &&& \ddots& \ddots\\ {\mathbb{I}}_3 & {\bm{A}}_{m2} &&&&& l_m {}^0{\bm{R}}_p {\bm{C}}_m & {}^0{\bm{R}}_p {}^p{\bm{u}}_m \end{bmatrix} \\ &\text{with } {\bm{A}}_{i2} = {[{}^0{\bm{R}}_p ({}^p{\bm{x}}_{B_i} + l_i {}^p{\bm{u}}_i)]}_{\times}^T \end{aligned}\normalsize$$ $$\label{eq:B} \bm{B}=\begin{bmatrix} {\bf{I}}_3 \\ & {\bf{I}}_3 \\ & {\bf{I}}_3 \\ && \ddots\\ &&& {\bf{I}}_3 \end{bmatrix} \begin{matrix} \Rightarrow s_1 \\ \\ \Rightarrow s_2 \\ \vdots \\ \Rightarrow s_n \end{matrix}$$ $$\label{eq:C} {\bm{C}}_i = \begin{bmatrix} -s_{{\phi}_i}s_{{\theta}_i} & c_{{\phi}_i}c_{{\theta}_i} \\ c_{{\phi}_i}s_{{\theta}_i} & s_{{\phi}_i}c_{{\theta}_i} \\ 0 & -s_{{\theta}_i} \end{bmatrix}$$ $$\label{eq:a}\\ \setlength{\arraycolsep}{1pt} \bm{a} = {\bm{A}}^{+} \begin{bmatrix} {}^0{\bm{\omega}}_1 \times ({}^0{\bm{R}}_1 {}^1{\bm{x}}_{I_1}) + {}^0{\bm{R}}_1 {}^1{\dot{\bm{x}}}_{I_1} \\ {}^0{\bm{\omega}}_2 \times ({}^0{\bm{R}}_2 {}^2{\bm{x}}_{I_2}) + {}^0{\bm{R}}_2 {}^2{\dot{\bm{x}}}_{I_2} \\ \vdots \\ {}^0{\bm{\omega}}_n \times ({}^0{\bm{R}}_n {}^n{\bm{x}}_{I_m}) + {}^0{\bm{R}}_n {}^n{\dot{\bm{x}}}_{I_m} \end{bmatrix}$$ Additionally, the second order kinematic model is also derived in matrix form as (\[eq:2KM\]), where $\bm{b}$ is the component related to the derivatives of $\bm{A}$ and $\bm{B}$. ${\bm{B}}^{+}$ is the pseudo-inverse of $\bm{B}$, which will minimize the residue of the equation system if the system is over-determined, i.e. $m>n$. $$\label{eq:2KM} {\ddot{\bm{q}}}_{\bm{a}} = {\bm{B}}^{+} \bm{A} {\ddot{\bm{x}}}_{\bm{t}} + {\bm{B}}^{+} \bm{b} \text{, with }\bm{b}={\begin{bmatrix} {\bm{b}}_1^T & \cdots & {\bm{b}}_m^T \end{bmatrix}}^T$$ $$\label{eq:bi} \small \setlength{\arraycolsep}{2pt} \begin{aligned} {\bm{b}}_i= & {}^0{\bm{\omega}}_p \times ({}^0{\bm{\omega}}_p \times {}^0{\bm{R}}_p ({}^p{\bm{x}}_{B_i}+l_i {}^p{{\bm{u}}}_i)) \\ & + 2{\dot{l}}_i {}^0{\bm{\omega}}_p \times ({}^0{\bm{R}}_p {}^p{{\bm{u}}}_i) + 2{\dot{l}}_i {}^0{\bm{R}}_p {\bm{C}}_i {\begin{bmatrix} {\dot{\phi}}_i & {\dot{\theta}}_i \end{bmatrix}}^T \\ & + 2 l_i {}^0{\bm{\omega}}_p \times ({}^0{\bm{R}}_p {\bm{C}}_i {\begin{bmatrix} {\dot{\phi}}_i & {\dot{\theta}}_i \end{bmatrix}}^T) + l_i {}^0{\bm{R}}_p {\dot{\bm{C}}}_i {\begin{bmatrix} {\dot{\phi}}_i & {\dot{\theta}}_i \end{bmatrix}}^T \\ & - {}^0{\dot{\bm{\omega}}}_j \times ({}^0{\bm{R}}_j {}^j{\bm{x}}_{Ii}) - {}^0{\bm{\omega}}_j \times ({}^0{\bm{\omega}}_j \times {}^0{\bm{R}}_j {}^j{\bm{x}}_{Ii}) \\ & -2{}^0{\bm{\omega}}_j \times {}^0{\bm{R}}_j {}^j{\dot{\bm{x}}}_{Ii} - {}^0{\bm{R}}_j {}^j{\ddot{\bm{x}}}_{Ii} \end{aligned}\normalsize$$ Dynamic Modelling of the VACTS ------------------------------ ### For the payload By using the Newton-Euler formalism, the following dynamic equation is presented, taking into account of an external wrench ${\bm{w}}_e$ , consisting of force ${\bm{f}}_e$ and moment ${\bm{m}}_e$. . Eqns. (\[eqn:platform\_translation\_dynamics\],\[eqn:platform\_rotation\_dynamics\]) express the dynamics of the platform. $$\label{eqn:platform_translation_dynamics} {\bm{f}}_e + m_p\bm{g}+{}^0{\bm{R}}_p \sum_{i=1}^{m} t_i {}^p{\bm{u}}_i = m_p {}^0{\ddot{\bm{x}}}_p$$ $$\label{eqn:platform_rotation_dynamics} \begin{aligned} {\bm{m}}_e + ({}^0{\bm{R}}_p {}^p{\bm{x}}_C) \times m_p\bm{g} + {}^0{\bm{R}}_p \sum_{i=1}^{m}{}^p{\bm{x}}_{B_i} \times t_i {}^p{\bm{u}}_i \\ ={\bm{I}}_p {}^0{\dot{\bm{\omega}}}_p + {}^0{\bm{\omega}}_p \times ({\bm{I}}_p {}^0{\bm{\omega}}_p) \end{aligned}$$ If these equations are expressed in matrix form, the cable tension vector $\bm{t}={\begin{bmatrix}t_1 & \cdots & t_m\end{bmatrix}}^T$ can be derived as (\[eq:DM1\]) considering a wrench matrix $\bm{W}$ and a mass matrix ${\bm{M}}_p$. ${\bm{W}}^{+}$ is the pseudo-inverse of $\bm{W}$ and $\mathcal{N}(\bm{W})$ represents the null-space of $\bm{W}$, used in some tension distribution algorithms [@lamaury2013tension], [@gouttefarde2015versatile]. $$\label{eq:DM1} \bm{t} = {\bm{W}}^{+} \left( {\bm{M}}_p {\begin{bmatrix} {}^0{\ddot{\bm{x}}}_p^T & {}^0{\dot{\bm{\omega}}}_p^T \end{bmatrix}}^T + \bm{c} \right) + \mathcal{N}(\bm{W})$$ $${\bm{M}}_p = \begin{bmatrix} m_p {\mathbb{I}}_3 & {\bm{0}} \\ {\bm{0}} & {\bm{I}}_p \end{bmatrix}$$ $$\label{eq:W} \setlength{\arraycolsep}{2pt} \bm{W}=\begin{bmatrix} {{}^0{\bm{R}}_p {}^p\bm{u}}_1 &\cdots& {}^0{\bm{R}}_p {}^p{\bm{u}}_m\\ {}^0{\bm{R}}_p ({}^p{\bm{x}}_{B_1}\times {}^p{\bm{u}}_1) &\cdots& {}^0{\bm{R}}_p ({}^p{\bm{x}}_{B_m}\times {}^p{\bm{u}}_m) \end{bmatrix}$$ $$\label{eq:c} \setlength{\arraycolsep}{2pt} \bm{c}= \begin{bmatrix} {\bm{0}}\\ {}^0{\bm{\omega}}_p \times ({\bm{I}}_p {}^0{\bm{\omega}}_p) \end{bmatrix} - \begin{bmatrix} m_p\bm{g} \\ ({}^0{\bm{R}}_p {}^p{\bm{x}}_C) \times m_p\bm{g} \end{bmatrix} - {\bm{w}}_e$$ ### For the winch All winches are assumed to have the same design, specifically the drum radius of all winches is $r_{d}$, the mass is $m_w$, and the inertia matrix is ${\bm{I}}_w$. The $x$-axis of the winch frame is along the winch drum . The rotational rate of the $i^{\text{th}}$ winch is ${\omega}_{r_i}$. It relates to the change rate of cable length as (\[eq:lengthrate\]) considering that the radius of cable is small . $$\label{eq:lengthrate} {\dot{l}}_i= r_d {\omega}_{r_i}$$ By using the Newton-Euler formalism, the following dynamic equation is expressed in $\mathcal{F}_{wi}$, where the torque is ${\tau}_i$. Additionally, we define a constant vector $\bm{k}=\begin{bmatrix} 1&0&0 \end{bmatrix}$ to represent the torque generated from cable tension orthogonal to the drum surface. $${{\tau}}_{i} - \bm{k} ({}^w{\bm{x}}_{Ii} \times t_i{}^w{\bm{u}}_i) = {{I}}_{xx} {\dot{\omega}}_{r_i}$$ ### For the quadrotor By using the Newton-Euler formalism, the following dynamic equations can be derived taking into account of the thrust force ${\bm{f}}_j$ and thrust moment ${\bm{m}}_j$ generated by the quadrotor’s motors. The mass of the $j^{\text{th}}$ quadrotor with $s_j$ embedded winches is $m_j= m_q + s_j m_w$ and the inertia matrix ${\bm{I}}_j$. $$\label{eq:quadforce} {\bm{f}}_j + m_j\bm{g} - {}^0{\bm{R}}_p \sum_{k=1}^{s_j} t_{k} {}^p{\bm{u}}_{k} = m_{j} {}^0{\ddot{\bm{x}}}_j$$ $$\label{eq:quadmoment} \begin{aligned} {\bm{m}}_j + \underbrace{({}^0{\bm{R}}_j{}^j{\bm{x}}_G) \times m_j \bm{g}}_{\text{gravity}} - {}^0{\bm{R}}_j \sum_{k=1}^{s_j} \underbrace{{}^j{\bm{x}}_{I_{k}} \times t_{k} {}^p{\bm{u}}_{k}}_{\text{cable tension}} \\ = {\bm{I}}_j {}^0{\dot{\bm{\omega}}}_j+{}^0{\bm{\omega}}_j\times({\bm{I}}_j {}^0{\bm{\omega}}_j) \end{aligned}$$ The dynamic model is expressed in matrix form, where the thrust force of all quadrotors $\bm{f}={\begin{bmatrix} {\bm{f}}_1^T {\bm{f}}_2^T & \cdots & {\bm{f}}_n^T \end{bmatrix}}^T$ are derived as (\[eq:DM3\]), with the mass matrix ${\bm{M}}_q$, and a wrench matrix $\bm{U}={\begin{bmatrix} {\bm{U}}_1^T & \cdots & {\bm{U}}_n^T \end{bmatrix}}^T$ such that ${\bm{U}}_j=\begin{bmatrix} \bm{0}& {}^0{\bm{R}}_p {}^p{\bm{u}}_i & \cdots &\bm{0} \end{bmatrix}$ has $s_j$ non-zero columns. $$\label{eq:DM3} \bm{f} = {\bm{M}}_q {\ddot{\bm{q}}}_{\bm{a}} + \bm{U}\bm{t}+\bm{d}$$ $$\label{eq:d}\scriptsize \setlength{\arraycolsep}{1pt} {\bm{M}}_q=\begin{bmatrix} m_1 {\bf{I}}_3 \\ & m_2{\bf{I}}_3 \\ &&\ddots \\ &&& m_n {\bf{I}}_3 \\ \end{bmatrix} \text{ and } {\bm{d}}= \begin{bmatrix} m_1\bm{g} \\ m_2\bm{g} \\ \vdots \\ m_n\bm{g} \end{bmatrix}\normalsize$$ The inverse dynamic model can be derived as (\[eq:IDM\]) from (\[eq:DM1\],\[eq:DM3\],\[eq:2KM\]). $$\label{eq:IDM} {\bm{f}} = {\bm{D}}_q {\ddot{\bm{x}}}_{\bm{t}} + {\bm{G}}_q$$ $${\bm{D}}_q = {\bm{M}}_q {\bm{B}}^{+} \bm{A} + \bm{U} {\bm{W}}^{+} \begin{bmatrix} {\bm{M}}_p & \bm{0} \end{bmatrix}$$ $${\bm{G}}_q = {\bm{M}}_q {\bm{B}}^{+} \bm{b} + \bm{U}{\bm{W}}^{+} \bm{c} + \bm{d}$$ For each quadrotor, the thrust force and thrust moment can be derived in the following way [@4SP]. As shown in Fig. \[fig:quadrotor\], each propeller produces a thrust force $f_{p_k}$ and a drag moment $m_{p_k}$, for $k=1,2,3,4$. Considering that the aerodynamic coefficients are almost constant for small propellers, the following expression can be obtained after simplification: $$f_{p_k}=k_f{\Omega_k}^2$$ $$m_{p_k}=k_m{\Omega_k}^2$$ where $k_f$, $k_m$ are constants and $\Omega_k$ is the rotational rate of the $k^{\text{th}}$ propeller. The total thrust force and moments expressed in the $j^{\text{th}}$ quadrotor frame $\mathcal{F}_j$ are $\setlength{\arraycolsep}{1.5pt}{\begin{bmatrix}0&0&f_z\end{bmatrix}}^T$ and $\setlength{\arraycolsep}{1.5pt}{\begin{bmatrix} m_x&m_y&m_z \end{bmatrix}}^T$, respectively, where $r$ is the distance between the propeller and the center of the quadrotor. Importantly, we could indicate that quadrotors have four DOFs corresponding to roll, pitch, yaw and upward motions. $$\label{eq:quad} \begin{bmatrix} f_z\\m_x\\m_y&\\m_z \end{bmatrix}=\begin{bmatrix} 1&1&1&1\\ 0&r&0&-r \\ -r&0&r&0 \\ -\frac{k_m}{k_f}&\frac{k_m}{k_f}&-\frac{k_m}{k_f}&\frac{k_m}{k_f} \end{bmatrix}\begin{bmatrix} f_{p_1} \\ f_{p_2} \\ f_{p_3} \\ f_{p_4} \end{bmatrix}$$ ![Free Body Diagram of the quadrotor $j$[]{data-label="fig:quadrotor"}](figure3.pdf){width="0.995\linewidth"} Centralized controller design {#sec:control} ============================= The overview of a centralized feedback linearization VACTS control diagram is shown in Fig. \[flowchart:vactscontroller\], which includes the control loop of quadrotors, as well as winches. Once the thrusts and the cable velocities are decided by the PD control law (\[eq:vactscontrollaw1\]) and the P control law (\[eq:vactscontrollaw2\]), respectively, the quadrotors can be stabilized by attitude controller and the cable lengths can be actuated by servo motors. The poses of quadrotors and the payload can be tracked by the Motion Capture System (MoCap). Therefrom the task space state vector $\bm{x_t}$ and its derivative are derived based on the geometric and kinematic models. Desired Thrust -------------- The control law is proposed based on the inverse dynamic model of the VACTS. $$\label{eq:vactscontrollaw1} {\bm{f}}^d = {\bm{D}}_q({{\ddot{\bm{x}}}_{\bm{t}}}^d + k_d({{\dot{\bm{x}}}_{\bm{t}}}^d-{\dot{\bm{x}}}_{\bm{t}}) + k_p({\bm{x}}_{\bm{t}}^d-{\bm{x_t}}) ) +{\bm{G}}_q$$ Velocity Control Loop --------------------- The velocity control loop represents the control loop of winches for actuating cable lengths. ${\dot{\bm{l}}}^d$ is obtained from the desired trajectory. Moreover, the function of saturation in the control diagram will be introduced in detail. Let us assume that there is a linear relationship between the maximum speed of winch ${\omega}_{r_{max}}$ and its torque, which is called the speed-torque characteristics. For a certain torque, the servo motor can tune the winch speed from $-{\omega}_{r_{max}}$ to $+{\omega}_{r_{max}}$. Therefore, the output signal can be determined based on the knowledge of cable tension and desired cable velocity. Furthermore, the output signal is limited within the $70\%$ of its working range for ensuring safety. $$\label{eq:vactscontrollaw2} \dot{\bm{l}}={\dot{\bm{l}}}^d+k_c({\bm{l}}^d-\bm{l})$$ Attitude Control ---------------- The aim of the attitude controller [@ETH] is to provide the desired thrust force and stabilize the orientation of quadrotor. The yaw angle is set to be a given value such as $0\degree$ in order to fully constrain the problem. In [@lee2010geometric], a second-order attitude controller with feed-forward moment is proposed, . = \[rectangle, minimum width=0.2cm, minimum height=0.2cm, text centered, draw=black\] = \[-&gt;,&gt;=angle 60,thick\] (image) at(0,0)[![[]{data-label="flowchart:vactscontroller"}](figure4 "fig:"){width="1\linewidth"}]{}; ; (0.0,0.45) rectangle (0.5,0.85); (0.0,0.0) rectangle (0.75,0.5); (0.8,0.0) rectangle (1.0,1.0); (DT) at (0.08,0.9) [Desired\ Trajectory]{}; (AC) at (0.65,0.9) [$1^{\text{\tiny st}}$ Attitude\ Controller]{}; (qr)[$1^{\text{\tiny st}}$\ Quadrotor]{}; (ACn) [$n^{\text{ th}}$ Attitude\ Controller]{}; (qrn)[$n^{\text{\tiny th}}$\ Quadrotor]{}; (w1) [$1^{\text{\tiny st}}$\ Winch]{}; (wm)[$m^{\text{\tiny th}}$\ Winch]{}; (pay)[Payload]{}; (mocap)[MoCap]{}; (GKM)[Geometric and\ Kinematic Models]{}; (D) at (0.2,0.75) [$k_d$]{}; (P) at (0.2,0.5) [$k_p$]{}; (C) at (0.14,0.4) [$k_c$]{}; (DFC) at (0.37,0.75) [$\bm{D_q}$]{}; (TDA) at (0.46,0.6) [$\bm{G_q}$]{}; (norm) at (0.37,0.3) [Saturation]{}; pic at (0.1,0.75) [sum block=[-]{}[+]{}]{}; pic at (0.28,0.75) [sum block=[+]{}[+]{}[+]{}]{}; pic at (0.46,0.75) [sum block=[+]{}[+]{}]{}; pic at (0.1,0.5) [sum block=[-]{}[+]{}]{}; pic at (0.14,0.3) [sum block=[+]{}[+]{}]{}; ($(qr.north west)+(-5pt,5pt)$) rectangle ($(pay.south east)+(9pt,-5pt)$); ($(qr.north west)+(-2pt,2pt)$) rectangle ($(qrn.south east)+(2pt,-2pt)$); ($(w1.north west)+(-4pt,2pt)$) rectangle ($(wm.south east)+(3pt,-2pt)$); (DT)-|node \[above left\] [${{\ddot{\bm{x}}}_{\bm{t}}}^d$]{}(0.28,0.78); (0.03,0.84)|-node \[above right\] [${{\dot{\bm{x}}}_{\bm{t}}}^d$]{}(0.07,0.75); (0.015,0.84)|-node \[above right\] [${\bm{x}}_{\bm{t}}^d$]{}(0.07,0.5); (0.125,0.75)–node \[above\] [$\bm{\dot{e}}$]{}(D); (D)–(0.255,0.75); (0.305,0.75)–(DFC); (DFC)–(0.435,0.75); (0.795,0.089)–(mocap); (mocap)–node\[above\][${\bm{x}}_p, {\bm{q}}_a$]{}(GKM); (GKM)-|node\[above right\][$\bm{x_t},{\dot{\bm{x}}}_{\bm{t}}$]{}(0.005,0.6); (0.025,0.6)-|(0.1,0.72); (0.1,0.6)–(0.1,0.53); (0.005,0.6) rectangle (0.025,0.61); (0.015,0.6) circle(0.01); (0.125,0.5)–node\[above\][$\bm{e}$]{}(P); (P)-|(0.28,0.72); (0.14,0.5)–(C); (C)–(0.14,0.33); (0.16,0.3)–node \[above\][$\bm{\dot{l}}$]{}(norm); (0.1,0.6) circle(0.01); (0.1,0.6)–(0.27,0.6); (0.27,0.6) rectangle (0.29,0.61); (0.28,0.6) circle(0.01); (0.29,0.6)-|(DFC); (TDA)–(0.46,0.72); (0.37,0.6) circle(0.01); (0.37,0.6)–(TDA); (0.37,0.6)–node\[xshift=0.5cm,yshift=-0.5cm\][Tension]{}(norm); (0.03,0.75)circle(0.01); (0.03,0.75)–(0.03,0.62); (0.03,0.58) rectangle (0.32,0.62); (0.03,0.6) circle(0.02); (0.03,0.58)–(0.03,0.51); (0.03,0.49) rectangle (0.31,0.51); (0.03,0.5) circle(0.01); (0.03,0.49)|-(0.12,0.3); (0.48,0.75)–node\[xshift=0.3cm\][$\bm{\vdots}$]{}(0.51,0.75); (0.51,0.75)|-node\[above right\][${\bm{f}}_1^d$]{} (AC); (0.51,0.75)|-node\[above right\][${\bm{f}}_n^d$]{}(ACn); (0.45,0.3)–node\[xshift=0.4cm,yshift=0.3cm\][$\bm{\vdots}$]{}(0.51,0.3); (0.51,0.3)|-node\[above right\][${\omega}_{r_1}$]{} (w1); (0.51,0.3)|-node\[above right\][${\omega}_{r_m}$]{}(wm); (AC)–node \[above\] [${\bm{\omega}}_1^d$]{}(qr); (0.81,0.88)–node \[below\][${\bm{q}}_1$]{}(0.735,0.88); (ACn)–node \[above\] [${\bm{\omega}}_n^d$]{}(qrn); (0.81,0.56)–node \[below\][${\bm{q}}_n$]{}(0.735,0.56); Performance evaluation {#sec:wrench} ====================== Wrench Analysis --------------- (image) at(0,0); (.21,.95) to \[out=20,in=160\](.32,.95); (.46,.95) to \[out=20,in=160\](.57,.95); (.72,.95) to \[out=20,in=160\](.81,.95); The available wrench set ${\mathcal{W}}_a$ is the set of wrenches that the mechanism can generate and depends on the state. ${\mathcal{W}}_a$ can be obtained by three space mappings, which is an extension of that introduced in [@wrench] for a general classical ACTS. This paper details the differences, which is caused 1) by the winch preventing cables from passing through the COM of the quadrotor. ### Propeller space Generally, for an UAV with $n_p$ propellers, its propeller space $\mathcal{P}$ can be determined by the lower and upper bounds of thrust force that each propeller can generate. Considering $n$ UAVs, the propeller space is (\[eq:propellerspace\]). $$\label{eq:propellerspace} \mathcal{P}=\left\{{\bm{f}}_p\in{\rm I\!R}^{(n\cdot n_p)}: \underline{{\bm{f}}_p}\leq{\bm{f}}_p\leq\bar{{\bm{f}}_p}\right\}$$ For a certain set of cable tensions, the compensation moment can be obtained according to (\[eq:quadmoment\]), which is risen from non-coincident COM and geometric center of quadrotors, as well as the cables not passing through the geometric center. Then the maximum value of thrust force of the $j^{\text{th}}$ quadrotor can be found by analyzing an optimization problem as (\[eq:optimf\]). $$\label{eq:optimf} \bar{f_z}=\underset{f_{p_k}\in \mathcal{P}}{\arg\max} f_z \text{ s.t. (\ref{eq:quad})}$$ ### Thrust, tension and wrench spaces The thrust space $\mathcal{H}$ is obtained by space mapping from the propeller space $\mathcal{P}$ as (\[eq:optimf\]). It represents the set of available thrusts of quadrotors. The tension space $\mathcal{T}$ shows the available cable tensions, which considers not only the space mapping from the thrust space, but also the torque capacity of winch. The available wrench set ${\mathcal{W}}_a$ can be obtained by mapping tension space $\mathcal{T}$ to wrench space $\mathcal{W}$ [@wrench] by using the convex hull method [@wrenchbasic]. The capacity margin $\gamma$ is a robustness index that is used to analyse the degree of feasibility of a configuration [@arachnis]. It is defined as the shortest signed distance from ${\mathcal{W}}_t$ to the boundary of the ${\mathcal{W}}_a$ zonotope. The payload mass, the cable directions and the capability of UAVs and winches all affect the value of capacity margin. For example, these four spaces can be presented visually as Fig. \[fig:4space\] for the VACTS with three quadrotors, three winches and cables, and a point-mass Comparison ---------- symbol physical meaning value -------------------- ----------------------------------------- ------------------------------------------------------ $m_q$ the mass of quadrotor only 1.05kg $m_w$ the mass of winch only 150g $m_p$ the mass of payload 1kg $\bar{f_p}$ the maximum thrust of propellers 4.5N ${}^j{\bm{x}}_Q$ the COM of quadrotor in $\mathcal{F}_j$ ${\begin{bmatrix}0 & 0 & -2\end{bmatrix}}^T$cm ${}^w{\bm{x}}_A$ the COM of winch in $\mathcal{F}_{wi}$ ${\begin{bmatrix}0 & 0 & 0\end{bmatrix}}^T$cm ${\bar{\tau}}_{w}$ the maximum torque of winches 6kg$\cdot$cm $\bm{\phi}$ the azimuth angles of quadrotors $\begin{bmatrix} 0 & 120 & -120\end{bmatrix}\degree$ : wrench analysis: general parameters[]{data-label="table:generalparam"} The most notable difference of wrench analysis between ACTS and VACTS is that $\mathcal{H}$ of the ACTS is constant, while $\mathcal{P}$ of the VACTS is constant and $\mathcal{H}$ is variable. The case study parameters for a VACTS with three quadrotors, three cables and winches, and one point-mass are shown in Table \[table:generalparam\]. shows the effect of different parameters related to winch (drum radius, offset, and orientation). From this case study, we conclude that (i) For the same system configuration, the VACTS needs extra energy to compensate the moment generated by cable tension because of the offset of winch. Therefore the winch should be mounted to the quadrotor as close to its centroid as possible; (ii) Additionally, the drum radius should be set less than a certain value according to the simulation results, such as $2cm$ for this case study; (iii) The optimal inclination angle is different after embedding winches; (iv) Last but not least, the servo motor selection is also under constraints. The stall torque of servo motor should be large enough to support the force transmission. {width=".9\linewidth"} The performance of ACTS and VACTS can be evaluated from different points of view: ($i$) For the same system configuration for ACTS and VACTS, the VACTS needs extra energy to compensate the moment generated by cable tensions. Therefore the VACTS has a smaller thrust space than the ACTS. So we can come up with the conclusion that the ACTS has a better performance in terms of available wrench set than the VACTS in unconstrained environments; ($ii$) In constrained environments, maybe the ACTS can not achieve the optimal system configuration because of its large size. For example, the ACTS can not achieve the optimal inclination angle ($50\degree$) because of the big width of the overall system which can not be implemented in an environment with limited width. While the VACTS can reshape its size and achieve an optimal configuration. Therefore, the VACTS can behave better than the ACTS in constrained environments and it will reshape the available wrench set; ($iii$) The VACTS has a better manipulability than ACTS from the fact that the VACTS has a larger velocity capacity along cable directions than ACTS because of the actuated cable lengths, which can also be proven by simulation results as Fig. \[fig:invw\]. The manipulability index [@stock2003optimal] $w_s$ as defined in (\[eq:manipulability\]) is proportional to the volume of the ellipsoid of instantaneous velocities and that it can be calculated as the product of the singular values of the normalized Jacobian matrix ${\bm{J}}_{norm}$. For example, for the VACTS with three quadrotors, three cables and winches, and one point-mass, ${\bm{J}}_{norm}$ is derived as (\[eq:normJ\]) considering the actuated cable lengths in joint space rather than task space; $$\label{eq:manipulability} w_s = \sqrt{det({\bm{J}}_{norm}{\bm{J}}_{norm}^T)}\text{, or } w_s = {\lambda}_1{\lambda}_2\cdots{\lambda}_r$$ $$\label{eq:normJ} {\bm{J}}_{norm} = diag(1,1,1,l_1,l_1,l_2,l_2,\cdots,l_m,l_m) \bm{J}$$ ![The reciprocal of the manipulability index for ACTS and VACTS with three quadrotors and a point-mass. The lower, the better.[]{data-label="fig:invw"}](figure7.pdf){width=".69\linewidth"} \(iv) On one hand, the actuated cable lengths increase the control complexity from $4n$ to $4n$+$m$. On the other hand, they improve the reconfigurability from $3n$-$m$ to $4n$-$m$ at the same time, which means the system is more flexible. Specifically, each quadrotor has four control variables, while each actuated cable length has one control variable. Therefore the control complexity is $4n$ for an ACTS with $n$ quadrotors, while it’s $4n$+$m$ for a VACTS with $n$ quadrotors and $m$ cables. In the meanwhile, a single cable has two reconfigurable variables (azimuth and inclination angles) and a pair of coupled cables has one reconfigurable variable (inclination angle). In an ACTS with $n$ quadrotors and $m$ cables, there are $2n$-$m$ single cables and $m$-$n$ pairs of coupled cables. So an ACTS has $3n$-$m$ reconfigurable variables. If the cable lengths are actuated, extra $n$ independent reconfigurable variables appear, i.e., a VACTS has $4n$-$m$ reconfigurable variables. Experimental results {#sec:exp} ==================== The VACTS prototype with three quadrotors, winches and cables, and a point-mass $m_p=670g$ is shown in Fig. \[fig:prototype\]. The Quadrotor body is a Lynxmotion Crazy2fly as Fig. \[fig:drone\_exp\] because of low cost and mass, while the embedded winch is shown in Fig. \[fig:winch\_exp\]. The servo motor winch is FS5106R. The poses of quadrotors and the payload are tracked by the MoCap as we mentioned in Sec. \[sec:control\]. Therefrom the cable lengths and the unit vectors along cable directions are computed. The translation vector ${}^j{\bm{x}}_{Ii}$ is between the coincident point $I_i$ and the origin of the quadrotor frame, as shown in Fig. \[fig:actsparam\]. It is assumed to be a constant value , thanks to the existence of guide hole in the V-shape bar of winch mount piece as shown in Fig. \[fig:winch\_exp\]. (image) at(0,0)[![The VACTS prototype with 3 quadrotors, winches and a point-mass[]{data-label="fig:prototype"}](figure8 "fig:"){width="1\linewidth"}]{}; ; (0.5,0.1)–node \[xshift=1.2cm\] [point-mass]{}(0.45,0.12); (.56,.4) circle (0.3cm) node \[xshift=0.8cm,yshift=-0.4cm\] [MoCap]{}; (0.345,0.79)–(0.425,0.15); (0.26,0.7)–(0.43,0.16); (0.72,0.67)–(0.44,0.15); ![Winch prototype and support[]{data-label="fig:winch_exp"}](figure9){width="1\linewidth"} (image) at(0,0)[![Winch prototype and support[]{data-label="fig:winch_exp"}](figure10 "fig:"){width="1\linewidth"}]{}; (0.6,0.3)–node\[xshift=0.8cm,yshift=-0.7cm\][V-shape bar]{}(0.55,0.45); The motion planning for a VACTS can be divided into several parts based on the task space state vector: the motion of payload, the system configuration of cable directions, and the cable lengths: (i) The motion of payload can be designed under the knowledge of environment; (ii) The design methodology for system configuration varies from quasi-static to dynamic case. In short, it is designed under capacity margin and cost optimization; (iii) The cable lengths are determined on the basis of limited volume space. In order to prove that this novel system is feasible , a motion planning is considered as following. This system takes off firstly and then hover for several seconds. Then the cable lengths are changed from initial values to 1.4m, followed by decreasing from 1.4m to 1.0m by using a fifth-order polynomial trajectory planning method [@gasparetto2010optimal]. During the period of cable displacements, the desired payload position and system configuration are kept as constants. In other words, the quadrotors move along the cable direction while the payload stays at the same position. ![The payload tracking of VACTS with 3 quadrotors, 3 winches and a point-mass $m_p=670$ g[]{data-label="fig:realtracking"}](figure11){width="1\linewidth"} ![The cable lengths tracking of VACTS with 3 quadrotors, 3 winches and a point-mass $m_p=670$ g[]{data-label="fig:finalexp_l"}](figure12){width="1\linewidth"} The experimental results of payload tracking and cable lengths tracking are shown in . Moreover the corresponding video that shows the experimental results step by step can be found in the link[^6]. From the experimental results, it’s shown that the desired cable lengths are followed well and the system is stable under control. The payload tracking mean errors along $x$, $y$ and $z$ axis are $1.90cm$, $9.41cm$ and $10.67cm$, respectively. The standard deviations are $3.15cm$, $3.24cm$ and $3.29cm$, respectively. The cable lengths tracking mean errors for $l_1$, $l_2$ and $l_3$ are $-2.25cm$, $0.24cm$ and $0.10cm$, respectively. The standard deviations are $3.09cm$, $0.73cm$ and $0.23cm$, respectively. Even though there was a cable length that did not follow the trajectory well, it is acceptable. Specifically, the cable length velocity is limited by the winch capability. In the experiments, we define the limitation that the maximum speed of servo motor is 70% of its actual maximum speed for security. At some point, the servo motor reaches its safety limit which leads to the small slope of $l_1$ () in Fig. \[fig:finalexp\_l\]. Experimental results show that the VACTS is feasible and that the change of cable lengths can reshape the VACTS, which implies the possibility of passing through a constrained environment or limited space. Moreover, the precision of cable lengths control is much higher than the torque control of ACTS, which also implies the potential for improving precision of the VACTS when the payload position is fine-tuned by changing cable lengths. CONCLUSIONS {#sec:con} =========== This paper dealt with a novel aerial cable towed system with actuated cable lengths. Its non-linear models were derived and a centralized controller was developed. The wrench analysis was extended to account for propeller saturation resulting from the offset of the cable connections with the quadrotor and the winch was designed based on the wrench performance. The advantage of this novel system is in reshaping the overall size and wrench space in constrained environments, while it comes at the price of lower peak performance in unconstrained situations. The feasibility of the system is experimentally confirmed, showing the system to be capable of resizing while hovering. Later on, ($i$) a decentralized control method will be designed and applied in order to increase the flexibility and precision, with better fault tolerance; ($ii$) cameras will be used to obtain the relative pose of the payload instead of motion capture system as in [@bookquad] and [@vb] to develop a system that can be deployed in an external environment; ($iii$) some criteria will be investigated to quantify the versatility of VACTS; ($iv$) for the experimental demonstrations, a VACTS prototype with a moving-platform instead of point-mass will be developed to perform more complex aerial manipulations in a cluttered environment$^1$; ($v$) a quadrotor attitude controller with feed forward moments from cable tension measurements or estimations will be implemented to improve performance. [^1]: $^{1}$[É]{}cole Centrale de Nantes (ECN), Laboratoire des Sciences du Num[é]{}rique de Nantes (LS2N), UMR CNRS 6004, 1 rue de la Noe, 44321 Nantes, France [zhen.li@eleves.ec-nantes.fr]{} [^2]: $^{2}$ECN, LS2N [julian.erskine@ls2n.fr]{} [^3]: $^{3}$LS2N, Centre National de la Recherche Scientifique (CNRS), France [stephane.caro@ls2n.fr]{} [^4]: $^{4}$ECN, LS2N [abdelhamid.chriette@ls2n.fr]{} [^5]: [](https://drive.google.com/file/d/1IO3qvFyWSTnUMLefXeysFpoqPMR585ud/view?usp=sharing) [^6]: [](https://drive.google.com/open?id=139fZmalOvZGxuETJf_h39mbEexZ2P5At)

--- abstract: 'We study pseudoscalar and scalar mesons using a practical and symmetry preserving truncation of QCD’s Dyson-Schwinger equations. We investigate and compare properties of ground and radially excited meson states. In addition to exact results for radial meson excitations we also present results for meson masses and decay constants from the chiral limit up to the charm-quark mass, e.g., the mass of the $\chi_{c0}(2P)$ meson.' address: - | Institut für Physik, University of Graz, Universitätsplatz 5\ A-8010 Graz, Austria\ andreas.krassnigg@uni-graz.at - | Physics Division, Argonne National Laboratory, 97000 South Cass Avenue\ Argonne, IL 60439, USA\ cdroberts@anl.gov and svwright@anl.gov author: - 'A. KRASSNIGG' - 'C.D. ROBERTS and S.V. WRIGHT' title: | $\;$\ \ Meson spectroscopy and properties using Dyson-Schwinger equations --- Introduction ============ Dyson-Schwinger equations (DSEs) are a nonperturbative continuum approach to quantum chromodynamics (QCD).[@Roberts:1994dr] They provide a means to study properties of the Green functions of QCD[@Alkofer:2000wg; @Fischer:2006ub] as well as hadrons as bound states of quarks and gluons.[@Roberts:2000aa; @Maris:2003vk; @Holl:2006ni] Hadrons are studied in this framework of infinitely many coupled integral equations with the help of a symmetry preserving truncation scheme. In the case of mesons discussed here, one solves the Bethe-Salpeter equation (BSE) for a quark-antiquark pair. For calculations of baryon properties, e.g. electromagnetic, weak, and pionic form factors[@Hellstern:1997pg; @Bloch:2003vn; @Alkofer:2004yf; @Holl:2005zi; @Holl:2006zw] one uses a covariant set of Faddeev equations.[@Hellstern:1997pg; @Oettel:1998bk; @Oettel:1999gc] The Dyson-Schwinger equation framework has numerous features, amongst them: first, it is a Poincaré-covariant framework and thus ideally suited for the study of hadron observables such as, e.g., electromagnetic form factors. Secondly, symmetries are represented by Ward-Takahashi or Slavnov-Taylor identities, which are then built into the scheme used to truncate the infinite tower of coupled integral equations. If a truncation scheme respects such an identity at every step, then one can i) prove exact (model-independent) results and ii) use sophisticated models to calculate physical quantities which illustrate these results and automatically reflect the properties of the corresponding symmetry. One such truncation is the so-called rainbow-ladder truncation, which has been used extensively and successfully to study meson ground states for more than a decade (see, e.g., Refs. ). Despite this success it has become obvious that certain states and phenomena, such as axial-vector mesons, exotic mesons,[^1] or heavy-light systems are not well-described in the most sophisticated calculations available to date. As a consequence, efforts are being made to go beyond this truncation,[@Bender:1996bb; @Bhagwat:2003vw; @Bhagwat:2004hn; @Watson:2004kd; @Fischer:2005wx] but these efforts are considerable, forcing present sophisticated calculations to remain at an exploratory stage. QCD Gap and Bethe-Salpeter Equations {#gapbse} ==================================== The homogeneous BSE is[^2] $$\label{bse} [\Gamma(p;P)]_{tu} = \int^\Lambda_q [\chi(q;P)]_{sr}\, K_{rs}^{tu}(p,q;P)\,,$$ where $p$ is the relative and $P$ the total momentum of the constituents, $r$,…,$u$ represent color, Dirac and flavor indices, $$\label{definechi} \chi(q;P)= S(q_+) \Gamma(q;P) S(q_-)\,,$$ $q_\pm = q\pm P/2$, and $\int^\Lambda_q$ represents a Poincaré invariant regularization of the integral, with $\Lambda$ the regularization mass-scale. In Eq.(\[bse\]), $S$ is the renormalized dressed-quark propagator and $K$ is the fully amputated dressed-quark-antiquark scattering kernel; for details, see Refs. . The dressed-quark propagator appearing in the BSE’s kernel is determined by the renormalized gap equation $$\begin{aligned} S(p)^{-1} & =& Z_2 \,(i\gamma\cdot p + m^{\rm bm}) + \Sigma(p)\,, \label{gap} \\ \Sigma(p) & = & Z_1 \int^\Lambda_q\! g^2 D_{\mu\nu}(p-q) \frac{\lambda^a}{2}\gamma_\mu S(q) \Gamma^a_\nu(q,p) , \label{sigma}\end{aligned}$$ where $D_{\mu\nu}$ is the dressed gluon propagator, $\Gamma_\nu(q,p)$ is the dressed quark-gluon vertex, and $m^{\rm bm}$ is the $\Lambda$-dependent current-quark bare mass. The quark-gluon-vertex and quark wave function renormalization constants, $Z_{1,2}(\zeta^2,\Lambda^2)$, depend on the gauge parameter, the renormalization point, $\zeta$, and the regularization mass-scale. The leptonic decay constant of a pseudoscalar meson is calculated from the solution of Eq. (\[bse\]) via $$\begin{aligned} \label{fpi} f_{\mathrm{PS}} \,\delta^{ij} \, P_\mu &=& Z_2\,{\rm tr} \int^\Lambda_q \frac{1}{2} \tau^i \gamma_5\gamma_\mu\, \chi^j_{\mathrm{PS}}(q;P) \,,\end{aligned}$$ which is gauge invariant, and cutoff and renormalisation-point independent. Rainbow-ladder truncation ------------------------- The first step in the symmetry-preserving truncation scheme described in Refs.  is the rainbow approximation to the gap equation combined with a ladder truncation in the BSE. The interaction kernels of Eqs. (\[bse\]) and (\[gap\]) then take the form $$\begin{aligned} K^{tu}_{rs}(p,q;P) = - \,4\pi\alpha(Q^2) \, D_{\mu\nu}^{\rm free}(Q)\, \left[\gamma_\mu \frac{\lambda^a}{2}\right]_{ts} \, \left[\gamma_\nu \frac{\lambda^a}{2}\right]_{ru} \label{ladderK}\end{aligned}$$ and $$\Sigma(p)=\int^\Lambda_q\! 4\pi\alpha(Q^2) D_{\mu\nu}^{\rm free}(Q) \frac{\lambda^a}{2}\gamma_\mu S(q) \frac{\lambda^a}{2}\gamma_\nu \;, \label{rainbowdse}$$ where $Q=p-q$, $D_{\mu\nu}^{\rm free}(Q)$ is the free gauge boson propagator[^3] and $\alpha(Q^2)$ is an effective running coupling. The ultraviolet behavior of this coupling can be taken from perturbative QCD, while in the infrared one makes an [*Ansatz*]{} with sufficient enhancement to correctly reproduce the phenomenology of dynamical symmetry breaking, i.e., enhancement of the quark mass function on a domain $p^2\lesssim 1$GeV$^2$ and correspondingly a correct value of the chiral condensate. Such an [*Ansatz*]{} is[@Maris:1997tm; @Maris:1999nt] $$\label{calG} \frac{4\pi\alpha(s)}{s} = \frac{4\pi^2}{\omega^6} \, D\, s\, {\rm e}^{-s/\omega^2}+ \frac{8\pi^2 \gamma_m}{\ln\left[ \tau + \left(1+s/\Lambda_{\rm QCD}^2\right)^2\right]} \, {\cal F}(s)\,,$$ with ${\cal F}(s)= [1-\exp(-s/[4 m_t^2])]/s$, $m_t=0.5\,$GeV, $\ln(\tau+1)=2$, $\gamma_m=12/25$ and $\Lambda_{\rm QCD} = \Lambda^{(4)}_{\overline{MS}} = 0.234\,$GeV. The free parameters of this model are the range $\omega$ and the strength $D$. One feature of the model is a specific parameter dependence of ground-state meson masses and properties, namely that they remain constant over a range of $\omega$, if $\omega\,D=$ const. is satisfied.[@Maris:2002mt] This results in a one-parameter model, where the free parameter $\omega$ is varied in the interval $[0.3,0.5]$ GeV. In a calculation, $\omega$ and $D$ as well as the current-quark masses are fixed to pion mass and decay constant as well as the chiral condensate. That being done, further results are predictions. Chiral symmetry --------------- One expression of the chiral properties of QCD is the axial-vector Ward-Takahashi identity $$\begin{aligned} P_\mu \Gamma_{5\mu}^j(p;P) = S^{-1}(p_+) i \gamma_5\frac{\tau^j}{2} + i \gamma_5\frac{\tau^j}{2} S^{-1}(p_-)- \, 2i\,m(\zeta) \,\Gamma_5^j(p;P)\;, \label{avwtim}\end{aligned}$$ which is written here for two quark flavors, each with the same current-quark mass: $\{\tau^i:i=1,2,3\}$ are flavor Pauli matrices. $\Gamma_{5\mu}^j(k;P)$ and $\Gamma_5^j(k;P)$ are the axial-vector and pseudoscalar vertices (for details, see Ref. ). Equation (\[avwtim\]) is satisfied by relating the kernels of the Bethe-Salpeter and gap equations (\[bse\]) and (\[gap\]), e.g., (\[ladderK\]) and (\[rainbowdse\]). A direct consequence of this identity is the relation $$\label{massformula} f_\mathrm{PS}\,m_\mathrm{PS}^2=2m(\zeta)\rho_\mathrm{PS}(\zeta)\;,$$ which relates the pseudoscalar meson mass $m_\mathrm{PS}$ and decay constant $f_\mathrm{PS}$ to the current quark mass $m(\zeta)$ and the residue of the pseudoscalar vertex at the pion pole $\rho_\mathrm{PS}(\zeta)$ at the renormalization point $\zeta$. It has been shown[@Holl:2004fr] that the implications of Eq. (\[massformula\]) in the chiral limit are different for the pion ground and excited states; namely, in the presence of dynamical chiral symmetry breaking the mass of the ground state vanishes, whereas for the excited states it is the leptonic decay constant that vanishes instead. Mesons ====== Meson ground states ------------------- Ground state mesons and their properties have been studied in rainbow-ladder truncation using different [*Ansätze*]{} of various sophistication for the effective running coupling[@Maris:1997tm; @Maris:1999nt; @Munczek:1983dx; @Munczek:1991jb; @Alkofer:2002bp] over a range of quark masses including the heavy-quark domain.[@Bhagwat:2004hn; @Alkofer:2002bp; @Krassnigg:2004if; @Maris:2005tt; @Ivanov:1997yg; @Ivanov:1997iu; @Ivanov:1998ms] The [*Ansatz*]{} of Ref. , Eq. (\[calG\]), has been used successfully to calculate a large number of pseudoscalar and vector meson properties. We also used this [*Ansatz*]{}, since it has the correct ultraviolet behavior and thus yields reliable results not only for spectroscopy, but also for dynamical observables like form factors. Radial meson excitations ------------------------ It was natural to study radial excitations of pseudoscalar mesons first. An estimate of the excited pion mass and leptonic decay constant[@Krassnigg:2003wy] and the structure of excited-state Bethe-Salpeter amplitudes, which shows similar characteristics to a quantum mechanical wave function[@Krassnigg:2003dr] were followed by more detailed studies of pseudoscalar meson radial excitations [@Holl:2004fr] and their electromagnetic properties.[@Holl:2005vu] The present work includes a study of scalar mesons and their first radial excitations. Masses ------ Figure \[mesonmasses\] shows the ground and first radially excited states of pseudoscalar and scalar mesons as functions of the current-quark mass. Figs. \[mesonmasses\] and \[piondecayconstants\] are generated from results for $\omega=0.38$ GeV. In contrast to the ground states, the masses and properties of radial excitations do depend on the value of $\omega$, even if $\omega\,D=$ const. Since $r=1/\omega$ corresponds to a range of the infrared part of the interaction, this means that radial meson excitations provide a means to study the long-range part of the strong interaction. To obtain results independent of the choice for $\omega$, one can make use of ratios of calculated observables, which remain constant over a range of $\omega$, to estimate properties of states on the basis of experimental values of other states that are known. An example for such an estimate is that of the mass of the $K(1460)$, the $K$ radial excitation, where the ratio $M_{K_{ex}}/M_{\pi_{ex}}$ is calculated to be $1.167$; using the experimental number $M_{\pi_{ex}}=1.3\pm 0.1$ GeV this yields $M_{K_{ex}}=1.52\pm 0.12$ GeV.[@Holl:2004un] Meanwhile, we have performed the analogous calculation for the leptonic decay constants of these states and found $f_{K_{ex}}/f_{\pi_{ex}}\simeq 10$ in agreement with an estimate via sum rules.[@Maltman:2001sv; @Maltman:2001gc] Figure \[charmomegadependence\] illustrates the same procedure to estimate the mass of the first scalar radial $\bar{c}c$ excitation $\chi_{c0}(2P)$. The ratio of the calculated masses for the $\chi_{c0}(2P)$ to the $\eta_c(2S)$ is $1.066$. Via the experimental value of $3.64$GeV (all experimental data are taken from Ref. ) for the mass of the $\eta_c(2S)$ we predict the $\chi_{c0}(2P)$ mass to be $3.88$ GeV. This compares well to quark-model predictions,[@Godfrey:1985xj; @Ebert:2002pp; @Barnes:2005pb; @Lakhina:2006vg] which lie somewhat below the corresponding estimates from lattice QCD.[@Okamoto:2001jb; @Chen:2000ej] We note that the $\chi_{c0}(2P)$ has not yet been observed experimentally. Decay constants --------------- For the pseudoscalar ground and first radially excited states we plot $f_\mathrm{PS}$ and $-f_\mathrm{PS}$, respectively, as functions of the current-quark mass in Fig. \[piondecayconstants\]. While $f_\mathrm{PS}$ is not necessarily directly accessible experimentally for $\bar{Q}Q$ systems, where $Q$ is a heavy-quark, it is always a well-defined axial-vector moment of the meson’s Bethe-Salpeter amplitude. Its evolution with current-quark mass is therefore a useful tool with which to probe QCD and models thereof. One can see that both curves have a maximum at about the $c$-quark mass; for higher quark masses, the size of $f_\mathrm{PS}$ decreases for both ground and excited states. It is remarkable that this “turning point” occurs around the same quark mass as for the heavy-light case.[@Maris:2005tt; @Ivanov:1998ms] At the values for the $u/d$-, $s$-, and $c$-quark masses we extract the values for $f_\mathrm{PS}$ for the ground and $-f_\mathrm{PS}$ for excited states as well as their ratio. The results are summarized in Table \[decayconstanttable\]. We note here that for the $s$ quark our pseudoscalar ground state does not correspond to an actual meson, since it consists merely of an $\bar{s}s$ component. However, a radial $\bar{s}s$ excitation can be identified with the $\eta(1475)$.[@Holl:2004un] Furthermore, the ground-state $\bar{s}s$ properties can also be studied on the lattice, where recent efforts have begun to study ratios of ground- and excited-state leptonic decay constants.[@McNeile:2006qy] Conclusions and Outlook ======================= We have extended previous studies of radial meson excitations by studying scalar excitations and quark masses up to the charm quark. While in the model we used ground-state meson properties do not depend on variations of the model parameter $\omega$, the specific parameter dependence of radial excitation properties allows investigations of the long-range part of the strong interaction between quarks. Without fixing $\omega$ to a particular value, we used ratios of properties of different excited states to make estimates for, e.g., the leptonic decay constant of the $K(1460)$ and the mass of the $\chi_{c0}(2P)$. This is made an efficacious procedure by the fact that ratios of excited-state properties remain constant over the domain of $\omega$ under investigation to a very good level of approximation. For both the ground and excited equal-mass pseudoscalar states we have calculated $f_\mathrm{PS}$ and observed a rise to a maximal size around the charm-quark mass and a decrease for higher quark masses. Further efforts in this direction will include studies of the radial excitations of vector mesons. Acknowledgments {#acknowledgments .unnumbered} =============== A. K. is grateful to his colleagues at the Physics Division of Argonne National Laboratory for their hospitality during a research visit, where part of this work was completed. We acknowledge useful discussions with M.S. Bhagwat and A. Höll. 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--- abstract: 'I present galaxy models resulting from violent relaxation in the presence of a pre-existing black hole. The models are computed by maximizing the entropy of the stellar dynamical system. I show that their properties are very similar to those of adiabatic growth models for a suitable choice of parameters. This suggests that observations of nuclear light profiles and kinematics alone may not be sufficient to discriminate between scenarios where a black hole grows adiabatically in the core of a galaxy, and scenarios where the black hole formation preceeds galaxy formation.' author: - 'M. Stiavelli [^1]' title: Violent Relaxation Around a Massive Black Hole --- [*subject headings*]{}: galaxies: elliptical and lenticular, cD — galaxies: nuclei — galaxies: formation Introduction ============ The search for massive black holes in the nuclei of galaxies began with the pioneering work of Young and collaborators (1978, 1980), who computed the effect of the adiabatic growth of a black hole on the underlying stellar population of a galaxy, represented by an isothermal sphere, and applied these models to M87. Further progress in stellar dynamics uncovered a rich variety of possible stellar equilibria (see, e.g., Binney and Mamon 1982), and urged for more sophisticated studies to certify the validity of black hole identifications based on simple models. The existence of black holes in the cores of galaxies is now unambiguously established for the nucleus of NGC 4258, where the rotation curve from water maser emission has been measured down to the parsec scale (Miyoshi et al. 1995). This has changed the prevalent attitude from that of [*proving*]{} that black holes existed to that of [*understanding*]{} what is the interplay between their properties and those of the nuclei where they reside. On the observational side, the refurbished Hubble Space Telescope has provided data on the cores of galaxies with unprecedented angular resolution (e.g., Harms et al. 1994; van der Marel et al. 1997a,b; Kormendy et al. 1995, 1996; Macchetto et al. 1997). On the theoretical side, models constructed with the Schwarzschild (1979) technique have allowed one to confirm the generality of the conclusions based on simpler models (van der Marel et al. 1998). The combination of photometry and spectroscopy with HST, and state-of-the-art dynamical modeling, has revealed that adiabatic black hole growth in isothermal cores may provide a fairly good representation of the data (van der Marel et al. 1998; van der Marel 1998). The adiabatic growth scenario implies that black holes are grown mostly from external gas rather than from disrupting stars. This has wide-ranging consequences on the way black holes and active galactic nuclei, and ultimately galaxies, form and evolve. The question remains whether the adiabatic black hole growth is a [*unique*]{} representation of the observed galactic nuclear properties, i.e., whether a generally good fit by adiabatic models does provide a decisive constraint about how galactic nuclei formed. To explore this issue, one can investigate the physically opposite formation scenario where the massive black hole pre-exists, and the galaxy forms around it via a process of violent relaxation. Numerical studies have been inconclusive for a variety of reasons, including, e.g., their small number of particles (Dekel, Kowitt, and Shaham 1981), grid effects in the neighbourhood of the central mass concentration (Udry 1993), or because focussed mostly on the isophotal effects of BH growth (Norman, May and van Albada 1985). In this letter I follow a different approach. In Section 2 I present a brief summary of how the statistical mechanics of violent relaxation can be used to derive the core properties of a galaxy. In Section 3 I describe the violent-relaxation models, which I compare to the adiabatic models. Section 4 sums up. Statistical Mechanics of Violent Relaxation =========================================== Violent relaxation allows a collisionless stellar dynamical system to reach equibrium by means of strong fluctuations of the mean gravitational potential. In the absence of [*degeneracy*]{} (i.e., if the phase space density of the system is everywhere well below the maximum allowed by Liouville’s theorem), the end-product of violent relaxation is an isothermal distribution function (Lynden-Bell 1967, hereafter LB67; Shu 1978). Given that such distribution function is characterized by an unphysical infinite mass, suitable truncations have been investigated. In his seminal paper, LB67 already pointed out the astrophysical problem of the [*incompleteness*]{} of violent relaxation as the solution to the problem of the unphysical infinite mass of the maximum entropy distribution function. Other authors since then have investigated the incompleteness paradigm with a variety of methods (e.g., Tremaine, H[é]{}non and Lynden-Bell 1986; Stiavelli & Bertin 1987; Stiavelli 1987; Madsen 1987, Shu 1987; Hjort and Madsen 1991;). A second important aspect of the violent relaxation distribution function, which was also pointed out by LB67 but has received less attention, is that of [*degeneracy*]{}. Here I focus on this issue. To do so, I first rederive the expression of the maximum entropy distribution function by including also an external potential. In the notation of Stiavelli & Bertin (1987; see also Shu 1978) I define a microstate as a set of occupation numbers in 6-dimensional microcells small enough that at most one star will occupy each microcell. This is readily ensured by selecting $g=1/p_{max}$ as the microcell size, with $p_{max}$ the maximum initial value of the distribution function before collapse, merging, or any other violent relaxation mechanism which is provided by nature at galaxy formation. In a collisionless formation scenario, Liouville’s theorem guarantees that in its subsequent evolution, the distribution function will never exceed this value. One can now partition the phase space into coarser macrocells, with each macrocell containing $\nu_a$ microcells. The observable state of the system will depend only on the occupation numbers ${n_a}$ of these macrocells, which we term a macrostate. In general $0 \leq n_a \leq min({\nu_a, N})$, with $N$ the total number of stars in the system. Let $M$ be the total number of macrocells. The entropy of the system is $S=log W( {n_a} )$ where $$W( {n_a} ) = \frac{N!}{n_1!...n_M!} \frac{\nu_1!...\nu_M!} {(\nu_1-n_1)!...(\nu_M-n_M)!}$$ is the number of microstates associated with a given macrostate. By using the Stirling approximation one finds: $$\label{entropyeq} S = -\sum_{a=1}^{M} n_a \log{n_a} - \sum_{a=1}^{M} (\nu_a-n_a) \log{(\nu_a-n_a)} + const.$$ The entropy is maximized under the constraints of the conservation of the total energy and number of particles, namely: $$\label{energyeq} E_{tot} = \sum_{a=1}^M m n_a (\frac{v_a^2}{2}+\frac{1}{2}\Phi_a^{(int)} +\Phi_a^{(ext)})$$ and $$\label{numbereq} N = \sum_{a=1}^M n_a$$ where in the expression for the energy I have separated the gravitational potential in an internal component due to self-gravity, $\Phi_a^{(int)}$, and an external component, $\Phi_a^{(ext)}$, and have indicated with $m$ and $v_a$ the particle masses and velocities, respectively. The resulting maximum entropy distribution function is given by: $$\label{distribfuneq} n_a = \frac{\nu_a}{1+\exp{(\beta E_a -\mu)}}$$ with $E_a = \frac{1}{2}{m v_a^2}+ m \Phi_a^{(int)}+m \Phi_a^{(ext)}$ the energy per particle, and $\beta$ and $\mu$ the Lagrange multipliers for the energy and number of particles, respectively. The ratio $\mu/\beta$ is the chemical potential of the system. For a Fermi-Dirac type distribution function such as that of Eq. \[distribfuneq\], the chemical potential can be either positive or negative. Its value represents the so-called Fermi energy, i.e. the upper limit to the individual particle energy of completely degenerate systems. Whenever $1 << \exp{(\beta E_a -\mu)}$, one has $n_a << \nu_a$, i.e. most of the microcells are not occupied, and the system is non-degenerate. In this limit the distribution function is that of the isothermal sphere. It is unclear whether the degenerate limit, where $n_a \simeq \nu_a$, is relevant for the cores of elliptical galaxies in the absence of black holes. However, this limit is relevant in the presence of a central black hole, as already pointed out by Stiavelli (1987). In the presence of a massive black hole of mass $M_{BH}$, one has that $\Phi^{(ext)}(r_a) = -G M_{BH}/r_a$, having indicated with $r_a$ the radial coordinate of macrocell $a$. This component dominates the underlying gravitational potential at sufficiently small radii. Consequently, for large portions of phase space $\exp{(\beta E_a)}<<1$, and $n_a \simeq \nu_a$, i.e. the system becomes degenerate at small radii. I present the resulting models in the next section. It should be noted that incompleteness does not affect my conclusions, since we are dealing with the nuclear galactic regions, where relaxation is expected to be complete. Violent Relaxation in the Presence of a Massive Black Hole ========================================================== Let’s rewrite for convenience the distribution function in continuous representation by dropping the macrocell indices: $$\label{feq} f(v,r) = \frac{A}{1+\exp{(\beta E -\mu)}}$$ with $A$ a normalization constant. At large radii the non-degenerate limit applies, and the density and velocity dispersion are those of the isothermal sphere: $$\label{densityeq} \rho = 4 \pi \int_0^{\infty} f(v,r) v^2 dv \propto r^{-2}$$ and $$\label{sigmaeq} \sigma^2 = \frac{4 \pi}{ 3 \rho} \int_0^{\infty} f(v,r) v^4 dv = \beta^{-1}.$$ At radii sufficiently small so that the black hole dominates the gravitational potential, one can estimate the integrals in Eq. (\[densityeq\]) and (\[sigmaeq\]) with the saddle point method to find $\rho \propto r^{-3/2}$ and $\sigma^2 \propto r^{-1}$. These slopes are equal to those obtained for models where the black hole grows adiabatically in the core of an isothermal sphere. Black holes grown adiabatically in non-isothermal models produce steeper cusps (Quinlan, Hernquist and Sigurdsson 1995), which are therefore distinguishable from those of the models presented here. In order to carry out a more detailed comparison, I have computed a number of self-consistent models from the distribution function of Eq. (\[feq\]) for a variety of degeneracy parameters $\mu$ and of black hole masses. In Figure 1 and 2 I show the comparison between the projected surface brightness and velocity dispersion for [*(i)*]{} violent-relaxation, degenerate models (solid lines), and for [*(ii)*]{} adiabatic black hole growth models (squares). The latter refer to an initial cored isothermal sphere and have been computed by using software kindly made available to me by G.D. Quinlan, and described in Quinlan et al. (1995; for the model properties see also Young 1980; Lee and Goodman 1989; Cipollina and Bertin 1994). I have assumed black hole to galaxy core mass ratios of $M_{BH}/M_{core} = 0.03,0.1,0.3,0.5$. In the violent relaxation models, the value of the degeneracy parameter $\mu$ depends on the initial conditions prior to the formation of each galaxy, and therefore it is an additional free parameter. Numerical experiments of dissipationless galaxy formation (without a central black hole) indicate that the maximum final phase space density is close to that of the initial conditions (e.g., Londrillo, Messina, and Stiavelli 1991). This suggests that $\mu$ is of order unity. The models of Figures 1 and 2 have been obtained for $\mu=0.5$. No best fit to the adiabatic growth models has been attempted. If $\mu$ varies, the slope of the inner surface brightness profile is affected: larger values of $\mu$ provide flatter slopes, and smaller values provide steeper slopes. This ensures that the agreement between the models presented here and the adiabatic models could be improved if $\mu$ were treated as a free parameter. For the case of $M_{BH}/M_{core}=0.3$, I show in Figure 3 the projected surface brightness profiles for $\mu=0,0.2,0.5,0.7,1$. The effect of degeneracy increases with increasing $\mu$. For values of $\mu << 0$ the models become increasingly similar to the isothermal sphere without a black hole but show a bi-modal density profile, with an overluminous cusp, when a black hole potential is included. Models with $\mu >> 1$ are degenerate even in the absence of a black hole. Both the adiabatic models and the maximum entropy models ignore two physical effects: [*i)*]{} the fact that the black hole may be not at rest in the center and, [*ii)*]{} stellar disruption by the black hole. For the models presented here, the former should probably not affect significantly the end result which does not depend on the detailed fluctuations of the gravitational potential. The latter would introduce a tangential anisotropy in the orbital distribution close to the black hole and alter the cusp profile at very small radii. A precise numerical evaluation of these effects goes beyond the scope of this paper. The main conclusion that one can draw from this study is that there is a fairly good agreement between adiabatic growth models and degenerate, violent relaxation models, for the [*same*]{} values of black hole to galaxy core mass ratio. Conclusions =========== Based on statistical mechanical arguments, I have derived the distribution function which results from violent relaxation in the presence of a pre-existing massive black hole, i.e. in the case where the formation of the black hole acts as a seed for the formation of the galaxy. The corresponding stellar dynamical models are qualitatively similar to models of adiabatic growth of a central black hole from an isothermal core, i.e. the physically opposite scenario, where the formation of the galaxy precedes that of the central black hole. For appropriate values of the degeneracy parameter $\mu$, which is arbitrary in the description that I have presented, the two scenarios become essentially indistinguishable. It is unlikely that the models could be distinguished on the basis of high spatial resolution line profiles since the central tangential anisotropy of adiabatic models is small and, in any case, stellar distruption could produce a similar anisotropy also in the maximum entropy models. This implies that on the basis of fits to the observed black hole cusps in galaxies, we are unable to infer the formation mechanism of the black hole and of its cusp. Acknowledgements ================ I thank Giuseppe Bertin, Marcella Carollo and Roeland van der Marel for useful discussions, and Gerald Quinlan for kindly providing the code to compute adiabatic growth models. The anonymous referee provided useful comments. This work has been partially supported by MURST of Italy and by the Italian Space Agency (ASI). Binney J.J., and Mamon G.A., 1982, MNRAS, 200, 361. Cipollina M., and Bertin G., 1994, AA, 288, 43. Dekel A., Kowitt M., and Shaham J., 1981, ApJ, 250, 561. Harms R.J., et al., 1994, ApJ, 435L, 35. Hjorth J., and Madsen J., 1991, MNRAS, 253, 703. Kormendy J., et al., 1996, ApJ, 473L, 91. Kormendy J., et al., 1995, ApJ, 459L, 57. Lee M.H., and Goodman J., 1989, ApJ, 343, 594. Londrillo P., Messina A., and Stiavelli M., MNRAS, 250, 54. Lynden-Bell D., 1967, MNRAS, 136, 101 (LB67). Macchetto, F., Marconi A., Axon D.J., Capetti A., Sparks W., Crane P., 1997, ApJ, 489, 579. Madsen J., 1987, ApJ, 316, 497. 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Young P.J., Westphal J.A., Kristian J., Wilson C.P., 1978, ApJ, 221, 721. van der Marel R.P., de Zeeuw P.T., Rix H.-W., 1997a, ApJ, 488, 119. van der Marel R.P., de Zeeuw P.T., Rix H.-W., Quinlan G.D., 1997b, Nature, 385, 610. van der Marel R.P., 1998, in ‘Galaxy Interactions at Low and High Redshift’ Proc. IAU Symp. 186, D. B. Sanders, J. Barnes, eds., Kluwer Academic Publishers, in press. van der Marel R.P., Cretton N., de Zeeuw P.T., Rix H.W., 1998, ApJ, 493, in press Figure Captions =============== [**Figure 1:**]{} Comparison of the projected surface brightness between my models ($\mu=0.5$, solid lines), and adiabatic black hole growth models (squares). Plotted are curves for black hole to galaxy core mass ratios $M_{BH}/M_{core} = 0.03,0.1,0.3,0.5$. There is a fairly good agreement between the two families of models for large $M_{BH}/M_{core}$ values. The radius $R$ is measured in units of the core radius. [**Figure 2:**]{} Comparison of the projected projected velocity dispersion between my models ($\mu=0.5$, solid lines), and models of adiabatic black hole growth (squares). The black hole to galaxy core mass ratios are $M_{BH}/M_{core} = 0.03,0.1,0.3,0.5$. The radius $R$ is in units of the core radius. The model without a black hole, not shown in the figure, has a velocity dispersion profile characterized by the presence of a small step, which occurs at the radius where degeneracy starts affecting the distribution function. A residual of this effect is seen for small values of $M_{BH}/M_{core}$. It disappears for $M_{BH}/M_{core} \simeq 0.3$. [**Figure 3:**]{} Projected surface brightness profiles for degenerate, violent relaxation models with $M_{BH}/M_{core} = 0.3$, and degeneracy parameter $\mu=0$ (dot-dashed), $0.2$ (long dashed), $0.5$ (solid), $0.7$ (short dashed), and $1$ ( dotted). [^1]: on assignment from the Space Science Department of the European Space Agency

--- abstract: 'Visualizing the trajectory of multiple runners with videos collected at different points in a race could be useful for sports performance analysis. The videos and the trajectories can also aid in athlete health monitoring. While the runners unique ID and their appearance are distinct, the task is not straightforward because the video data does not contain explicit information as to which runners appear in each of the videos. There is no direct supervision of the model in tracking athletes, only filtering steps to remove irrelevant detections. Other factors of concern include occlusion of runners and harsh illumination. To this end, we identify two methods for runner identification at different points of the event, for determining their trajectory. One is scene text detection which recognizes the runners by detecting a unique ’bib number’ attached to their clothes and the other is person re-identification which detects the runners based on their appearance. We train our method without ground truth but to evaluate the proposed methods, we create a ground truth database which consists of video and frame interval information where the runners appear. The videos in the dataset was recorded by nine cameras at different locations during the a marathon event. This data is annotated with bib numbers of runners appearing in each video. The bib numbers of runners known to occur in the frame are used to filter irrelevant text and numbers detected. Except for this filtering step, no supervisory signal is used. The experimental evidence shows that the scene text recognition method achieves an F1-score of 74. Combining the two methods, that is - using samples collected by text spotter to train the re-identification model yields a higher F1-score of 85.8. Re-training the person re-identification model with identified inliers yields a slight improvement in performance(F1 score of 87.8). This combination of text recognition and person re-identification can be used in conjunction with video metadata to visualize running events.' author: - 'Y. Napolean\*' - 'P.T. Wibowo\*' - 'J.C. van Gemert' bibliography: - 'sample-base.bib' title: | Running Event Visualization\ using Videos from Multiple Cameras --- [^1] ![Envisioned result of proposed approach - Shown here is a visualization of a frame of the 2D video with runners represented by markers. The blue line represents the race track where cameras are located at certain points, the flag represents the start and the finish line. The proposed model has the ability to retrieve pictures of runners (at any point of time) given their unique bib number.[]{data-label="fig:teaser"}](concept.png) [.45]{} {height="4cm"} [.45]{} {height="4cm"} Introduction ============ Running events have gained more popularity in recent times due to increasing awareness regarding health and the accessibility of such events for all people regardless of their gender, age, or economic status. In some races, the athlete’s location data is collected using GPS tracker for live tracking [@bib:pielichaty_2017]. This data is used by event organizers to publish statistics or records of the race, or sometimes for the participants themselves to track their performance. Several vendors for GPS trackers provide a service for data analysis and running race visualization from the race data [@bib:racemap; @bib:tech4race]. However, athletes usually would like to view images/visualizations of themselves retrieved from the event. Also, GPS trackers are personalized to each athlete while cameras can provide an overview for multiple athletes at once. Videos also provide additional information that can be used for athlete health monitoring.\ While the video streams are time-stamped and also have GPS information on where they are recorded, there is no prior knowledge as to which runner appears in each of the videos. Given that each runner has a unique ’bib number’ identifying them, attached to the shirt and with the assumption that any two random people are likely to be dissimilar in appearance, computer vision models can be used for recognizing them. Scene text recognition models can be used to identify the bib number and ultimately the athlete. However, there are potential issues as the bib numbers or the athletes themselves can be fully or partially occluded, as shown in Figure \[fig:bib\_number\]. Person re-identification is an approach that utilizes a similarity measure (like Euclidean distance), training a model to retrieve instances of the same person across different cameras [@bib:survey_personreid_1; @bib:deep_personreid_survey_3]. This approach can potentially improve runner identification in scenarios where the bib number is not visible but the runner’s features are.\ In this work, we use scene text recognition both as a baseline and to collect training samples for a person re-identification method downstream in our pipeline for runner identification. The envisioned result of the model is shown in Figure 1, the race track map with the trajectories of athletes with the ability to retrieve instances of detection of each athlete. The only annotations required are bib numbers that appear in the video to evaluate the scene text recognition model. For the scene text recognition method, we choose deep-learning based text spotter baselines [@bib:deeptextspotter; @bib:he_textspotter]. To evaluate the performance of the proposed approach, we created a ground truth database where the videos are annotated with the bib numbers of appearing runners and the frame interval which they belong. Our main contributions are: - A method for visualizing running events using videos recorded at the event. Computer vision based runner detection methods are used with video metadata to find video files and frames where each of the runners appears. The only supervision used in runner identification is the bib numbers known to occur in the video. However, for the whole running event visualization GPS correction and additional filtering steps are used. - Evaluation of relevant computer vision models for the detection of runners in videos. Related Work ============ **Scene Text Recognition**  Scene text recognition is the detection, localization and identification of text in images. Recognizing the text in images of scenes can be useful for a wide range of computer vision applications, e.g. robot navigation or industry automation [@bib:navigation; @bib:automation]. Traditionally, handcrafted features (e.g. HOG, color, stroke-width, etc.) are utilized to implement a text recognition system [@bib:handcraft_1; @bib:handcraft_2]. However, because of limited representation ability of handcrafted features, these methods struggle to handle more complex text recognition challenges [@bib:survey_textdetection_1], like the ICDAR 2015 dataset [@bib:icdar_2015]. Deep learning-based methods [@bib:deeptextspotter; @bib:mask_spotter; @bib:he_textspotter; @bib:fots] are more successful in this regard because they can learn features automatically. @bib:mask_spotter modified the Mask R-CNN to have shape-free text recognition ability. @bib:fots and @bib:deeptextspotter adopted EAST [@bib:east] and YOLOv2 [@bib:yolov2] as their detection branch and developed CTC-based text recognition branch. @bib:he_textspotter adopted EAST [@bib:east] as its text detection branch and developed the attention-based recognition branch. Since in our work, the bib number text appears in unconstrained images of scenes (e.g. dynamic background textures, varying size of text due to distance to camera, illumination variation, etc.) as shown in Figure \[fig:bib\_number\], deep learning based scene text recognition methods would be more well suited to detect the bib numbers because it can perform better with the varying scenarios than traditional (handcrafted features) methods.\ **Bib Number Detection**  Using scene text recognition methods is an intuitive choice to detect the bib number and consequently identify the runner who wears it. @bib:racing_bibnum proposed a pipeline consisting of face detection and Stroke Width Transform (SWT) to find the location and region of bib number tag, and then applied digit pre-processing and an Optical Character Recognition (OCR) engine [@bib:tesseract] to recognize the bib number text. @bib:svm_bibnumber proposed to use torso detection and a Histogram of Oriented Gradient (HOG) based text detector to find the location and region of bib number tag, and then they use text binarization and OCR [@bib:tesseract] to recognize the bib number text. Since there is no ’learning’ component, the aforementioned models cannot adapt to varying environmental conditions. Existing and publicly available text recognition models [@bib:deeptextspotter; @bib:he_textspotter] are used as our scene text recognition baseline to detect the bib numbers (so as to not reinvent the wheel - focusing on implementing a text recognition system).\ **Person Re-identification**  Person re-identification (or re-id) is the task of recognizing an image of a person over a network of video surveillance cameras with possibly non-overlapping fields of view [@bib:survey_personreid_1]. In previous research, authors [@bib:classification_personreid_2; @bib:classification_personreid_1; @bib:classification_personreid_3] have investigated classification-based CNN models for person re-id. However, if the dataset lacks training samples per individual, the classification-based model is prone to overfitting [@bib:survey_personreid_1]. To counter this, the task is cast as a metric learning problem with Siamese networks [@bib:siamese_network]. @bib:pairwise_model_1 uses a Siamese network with pairwise loss that takes a pair of images to train the model and learn the similarity distance between two images. @bib:triplet_loss showed that using the Siamese network model and triplet loss are a great tool for a person re-identification task because it learns to minimize the distance of a positive pair and maximize the distance of a negative pair at once. @bib:bagoftricks improved the previous work (Siamese network model with triplet loss) performance with several training tricks, such as center loss or label smoothing. This work [@bib:bagoftricks], is what we adopt as our person re-id model for runner detection. Methodology =========== Running Event Visualization --------------------------- This section outlines steps for visualization of the running event. ### Running track To visualize computed trajectories of athletes, a map of the event running track is required. The track is formed by an array of GPS coordinates on an interactive map. The sequence of GPS coordinates are created using *geojson.io*, a web-based tool to create geo-spatial data in GeoJSON file format [@bib:geojson_file]. It provides an interactive editor to create, view, and share maps [@bib:geojson_io] similar to Figure \[fig:teaser\]. ### Runner timestamp Another important element to visualize the athletes trajectory is the timestamp where runners appear at different locations. The individual timestamp is determined by the video timestamp and the frame where the runner appears in the video. From video metadata, we use the date last modified $t_{V}$ and Duration $T_{V}$ to get the time the video starts recording. Then we convert the frame $f$ where the runner appears into seconds by dividing it with video frame rate $r$. To get the individual timestamp, we use linear interpolation, adding the video start time and the converted frame. Individual timestamp is given by: $$\label{eq:timestamp} t_{R} \, = \, t_{V} \, - \, T_{d} \, + \, \frac{f}{r} \ , $$ where $t_{R}$ is the individual athlete’s timestamp, $t_{V}$ is the video timestamp, $f$ is the frame where the runner appears, $r$ is the frame rate of the video, and $T_{d}$ is the total duration of the video. Since runners can appear in multiple frames, we also take into account where the relative position of runners to the camera to determine the best frame to use. Based on our observation, in most videos, the position of the camera and the athletes resembles Figure \[fig:cam\_runner\_illustration\], in which the runners ran towards the camera. So the last frame where a runner appears would be the best choice (closest to the camera), given that the GPS coordinates of the camera will be used to estimate runner location. ![Illustration of how video recording was configured . In most videos, the runners move towards the camera. At first, there is considerable distance between the camera and the runners, but the runners progressively get closer to the camera.[]{data-label="fig:cam_runner_illustration"}](camera_view.png) ### Filtering raw GPS data The Figure \[fig:raw\_geo\] shows the trajectory of raw GPS data of videos collected by nine cameras, and it can be seen that the cameras did not stay at one location. Although the videos were taken on the running track, the GPS coordinates seem to stray away. Some of the GPS coordinates deviate too far from the original location; sometimes it is on a highway, a river or a building. The raw GPS data needs to be filtered so that it can be used for visualization. The first filtering step of raw GPS data is quite simple, i.e. replacing the raw GPS coordinates with the nearest running track points, so the video GPS coordinates stay in the running track. Cosine law is used to filter stray points based on the angle and distance formed by the stray point and two neighbouring ones. After we apply the filters on raw GPS data, we have a better-looking camera trajectory, as shown in Figure \[fig:filter\_geo\]. However, there are still a few stray points from camera 6 that intersects with the trajectory of camera 5 and 7. We manually predict and replace the remaining stray points with new GPS coordinates with estimated points after viewing and analyzing the recorded videos. [.5]{} ![Visualization of camera GPS data. (a) Raw camera trajectory. (b) Filtered camera trajectory. There are nine lines with different color representing different cameras. The cameras moved according to the line drawn on the map.[]{data-label="fig:geotag_data"}](raw_geo.png "fig:"){width="4.5cm"} [.5]{} ![Visualization of camera GPS data. (a) Raw camera trajectory. (b) Filtered camera trajectory. There are nine lines with different color representing different cameras. The cameras moved according to the line drawn on the map.[]{data-label="fig:geotag_data"}](filter_geo.png "fig:"){width="3.1cm"} ### Start and finish location A piece of information missing from the video data is the individual timestamp and GPS coordinates at the start and finish because there is no video recorded at both locations. So the GPS coordinates and the individual timestamp at the start and finish location are determined by ourselves. The GPS coordinates at the start and finish location are predicted based on the start and finish locations provided by the event organizer. Meanwhile, the individual timestamp of the start location is determined by setting to zero the seconds of the individual timestamp at the location of the first camera. For example, if the individual timestamp at camera 1 is 16:00:10, then the start timestamp is 16:00:00. We assumed that the camera 1 location is only a few meters away (10 - 20 meters) from the start location. The individual timestamp at the finish location is defined by the start time plus the duration required by a runner to finish the race. This duration is obtained from scraping the Campus Run result website. After the timestamps and the GPS coordinates of the start, finish, and camera locations are obtained, the runner’s motion can be interpolated between those locations. Then the javascript library, D3[^2] and Leaflet[^3] are used to create the visualization. Runner Detection ---------------- The model needs to retrieve the video and frame information where the athletes are detected. We are interested in that information because they are used to visualize the runner’s trajectory; the video provides GPS coordinates and video timestamp information, and the frame represents the individual timestamp. ### Scene text recognition The steps to use the text spotter model [@bib:deeptextspotter; @bib:he_textspotter] for our runner detection task are : - the frames of a video are fed into text spotter model, - all texts in the image are detected by the text spotter, - if the bib number we know from scrapping the marathon website is detected, we retrieve the video and frame information for further evaluation. Text spotter also collects training samples for person re-id which are cropped person images from the video dataset. The cropping occurs around a specified region if the text spotter detects a bib number on the image. Then the bib number is assigned as the label of that cropped image. Although there are two text spotters, we do not merge the training samples collected by the two text recognition models. Instead, the person re-id model is trained with each training sample separately so as to compare the results of both models. ### Person Re-identification Given the effectiveness of metric learning [@bib:triplet_loss], we choose to adopt the work of @bib:bagoftricks which proposes a Siamese network with triplet loss and cross entropy loss. Since we do not have the ground truth identities of athletes annotated, to evaluate the person re-id model, an object detector is used to detect people in the video dataset and these detections are cropped out. Those cropped images do not have a label because the object detector cannot detect the bib numbers, but the video and information are still stored. The person re-id model is evaluated to see if it can recognize the people detected by the object detector. It is possible that the object detector results are noisy and might include pedestrians in addition to athletes. So, a *k-NN* is used as an outlier detector, it has competitive performance compared to other outlier detection methods [@bib:survey_outlier_detection]. The idea of using *k-NN* as an outlier detector is that the larger the distance of a query point from its neighbor points, higher the likelihood that it is an outlier [@bib:knn_distance]. We adopt the definition of an outlier that considers the average of the distance from its $k$ neighbors [@bib:knn_outlier]. If the average distance of an image from its $k$ neighbors is larger than a threshold, then it is considered as an outlier. Performance Evaluation ---------------------- This section describes the mathematical formulation of the evaluation metrics and their implementation in our problems. ### Video-wise metric The video-wise metric is useful to check the number of videos where a runner is detected at least once. Evaluating the performance on the retrieved video information is important because we use the video GPS coordinates to visualize the runners trajectory, so we want to retrieve as many relevant videos as possible. At the same time, it is also undesirable to retrieve irrelevant videos because the irrelevant videos might have GPS coordinates and timestamps that do not agree with GPS coordinates and timestamps that the relevant videos have. Therefore, it could hinder the runner’s trajectory visualization. True positives are when a positive class is correctly predicted to be so, and false positives are when a negative class is detected as positive [@bib:olson_delen_2008]. In our problem, we collect a ground truth database consisting of the video filename, bib number, and frame interval. So true positive could be defined as correctly retrieved information (video filename, bib number, frame), and the false positive for incorrectly retrieved ones. To evaluate the video-wise metric, the retrieved information we need is only the bib number and its video. We use F1-score for the video-wise metric to evaluate on both relevant and irrelevant videos retrieved by runner detection methods. The video-wise F1-score formula is defined as followed: $$\label{eq:recall} \text{Recall}_{i}^{v} \, = \frac{|\text{TP}|_{i}^{v}}{|\text{GT}|_{i}^{v}} \ ,$$ $$\label{eq:precision} \text{Precision}_{i}^{v} \, = \frac{|\text{TP}|_{i}^{v}}{|\text{TP}|_{i}^{v} + |\text{FP}|_{i}^{v}} \ ,$$ $$\label{eq:f1_score} \text{F1-score}_{i}^{v} \, = 2 \ . \ \frac{\text{Recall}_{i}^{v} \ . \ \text{Precision}_{i}^{v}}{\text{Recall}_{i}^{v} + \text{Precision}_{i}^{v}} \, ,$$ where $i$ denotes the runner, $v$ denote a video-wise metric, $|GT|$ denotes the number of runner’s video in ground truth, $|TP|$ denotes the total true positive (relevant) videos and $|FP|$ denotes the total false positive (irrelevant) video. The final score is defined as the average over $M$ classes, as shown below : $$\label{eq:average_f1} \text{F1-score}^{v} \, = \frac{1}{M} \sum_{i=1}^{M} \, \text{F1-score}_{i}^{v} \ .$$ ### Frame-wise metric Since it is possible that every one of the runners are seen at least once during the event, a naive method that predicts every athlete to be visible in every camera at all times, could achieve a high F1-score. So we need a frame-wise metric. Also, runner detection methods might produce false positives at an irrelevant frame in a relevant video. So, it is necessary to evaluate at a frame level in addition to the video level evaluation. We use temporal Intersection of Union (IoU) for the frame-wise metric, which measures the relevant interval of frames where the runner appears. The formula of temporal IoU is similar to regional IoU used in object detection [@bib:iou_jaccard], except it is only one-dimensional, as shown in Figure \[fig:temporal\_iou\_1d\]. The formula for calculating temporal IoU is defined as followed: ![Visualization of frame interval intersection. $S$ denotes the frame start, and $E$ denotes the frame end. And $d$ denotes that the frame interval belongs to detection result and $g$ denotes that the frame interval belongs to the ground truth. The temporal IoU is the ratio between the width of overlap and union.[]{data-label="fig:temporal_iou_1d"}](temporal_iou_1d.png){width="1.\linewidth"} $$\label{eq:overlap} \text{overlap}_{ij} = \min(E_{ij}^{g} \, , \, E_{ij}^{d}) - \max(S_{ij}^{g} \, , \, S_{ij}^{d}) \, ,$$ $$\label{eq:union} \text{union}_{ij} \, = (E_{ij}^{g} - S_{ij}^{g}) \, + \, (E_{ij}^{d} - S_{ij}^{d}) - \text{overlap}_{ij} \ ,$$ $$\label{eq:iou_class} \text{IoU}_{ij} = \begin{cases} 0 \, ,& \text{if overlap}_{ij} < 0 \\ \frac{\text{overlap}_{ij}}{\text{union}_{ij}} \, ,& \text{otherwise} \end{cases}$$ $$\label{eq:iou} \text{IoU}_{i} = \frac{1}{N_i} \sum_{j=1}^{N_i} \, \text{IoU}_{ij} \ ,$$ where $i$ denotes the runner, $j$ denotes the video file, and $N_{i}$ is the total of video files where the runner detected. $S_{ij}^{g}$ and $E_{ij}^{g}$ are the frame start and the frame end of runner $i$ at video $j$ from ground truth database. $S_{ij}^{d}$ and $E_{ij}^{d}$ is the first frame and the last frame which the runner $i$ is detected at video $j$. The temporal IoU per runner is the average of the temporal IoU over the total of retrieved video. If the detection happens at an irrelevant video $j$, then $IoU_{ij}$ is zero, so it will lower the $IoU_i$. The final score is calculated as average over $M$ classes, expressed as : $$\label{eq:average_iou} \text{mIoU} = \frac{1}{M} \sum_{i=1}^{M} \, \text{IoU}_{i} \ .$$  Experiments =========== Datasets -------- ### Ground truth database {#sec:ground_truth} Ground truth is necessary for evaluating the performance of the model. It is obtained by manual video analysis, defining a range of frames for each runner where they are recognizable by the human eye. Due to the massive amount of data and time constraints, only runners from the 5 km category are fully annotated. Consequently, only videos with 5 km runners (a total of 127 people) are annotated as ground truth for evaluation. There are a total of 127 runners for the 5km category. 187 videos of these runners were recorded across 9 different locations. The average length of these videos are 35.44 seconds. The ground truth is established by a range of frames where the runners appear. The range is defined by the “frame start” and “frame end” for every runner in the video. The frame start is annotated when a runner is first recognizable by human-eye. Meanwhile, the frame end is annotated when a runner exits the camera field of view. Figure \[fig:labeling\_2\] shows that runner 156 is first seen at the right edge of the screen at frame 98, so this frame is annotated as frame start for runner 156. Then at frame 232, runner 156 is seen for last time at the video, then this frame is annotated as frame end for runner 156. Meanwhile, for runner 155, the frame start is 96, and the frame end is 248. ### Evaluation set for person re-id The person re-id model can not localize and recognize a person simultaneously from a given frame containing multiple objects and the background. It requires cropped images with one person in each image. The training and evaluation set thus must be a collection of cropped images of people, not videos. To create this evaluation set for the person re-id model , images of people from the videos are collected by an object detector and cropped. The difference between the person images collected by the object detector and that collected by text spotters is that the images from the object detector do not have a label. So, the object detector identifies any images with people from a video regardless of whether the person has a bib number or runners with occluded bib number text. The video and frame information of the extracted images of people are also stored for performance evaluation. There are many object detection methods, such as Faster R-CNN [@bib:faster_rcnn] R-FCN [@bib:rfcn] and single shot detector [@bib:ssd]. However, experiments comparing them [@bib:survey_object_detectors] show that Faster R-CNN has better accuracy compared to the other two, so Faster R-CNN is chosen. Implementation Details ---------------------- For text spotter [@bib:deeptextspotter; @bib:he_textspotter] and the object detector Faster R-CNN, available pre-trained models are used. For the person re-id model, we use the same configuration options as the author [@bib:bagoftricks], using the Adam optimizer [@bib:gradient_descent] and the same number of total epochs (120). The output from the object detector (cropped images of people) with the output of scene text recognition as annotation (bib numbers identify runners) is fed into the person re-id model. Additionally, we collect new training samples to retrain the person re-id model from the images of people obtained using the object detector. We choose the cropped images that are considered as inliers by the previous round of training with the person re-id model. For *k-NN* model, we choose $k$ = 5 and use cosine distance as its distance metric during the inference stage as it is shown to be better than using Euclidean distance for the person re-id task. [@bib:bagoftricks]. The *k-NN* model is trained on the embedding features extracted by the person re-id model. The person re-id model is used only as a feature extractor. Performance Analysis -------------------- ### Bib number detection The model proposed by @bib:he_textspotter is better at detecting the bib numbers than @bib:deeptextspotter. @bib:he_textspotter has larger recall but lower precision compared to @bib:deeptextspotter, as shown in Table \[tab:textspotter\_recall\_precision\]. Consequently, although @bib:he_textspotter detects more runners, it produces a larger number of false positives than @bib:deeptextspotter, as shown in Figure \[fig:num\_runner\_textspotter\]. Even if a few digits of bib number is occluded, the text spotter [@bib:he_textspotter] will detect it. But because the occluded bib number could look like another number, it will count as false positive. These methods are also compared against naive baselines (Table (1)), that predict all runners to be visible at all times in each video (Baseline (all)) and a random prediction given the number of visible people in the frame (Baseline (random)). Per video, the Baseline(all) has a high recall owing to the fact that all runners are predicted to be visible at all times, this resulting in zero false negatives. However, there are a lot of false positives as not all runners are visible in every video at all times. The number of detected runners by text spotter [@bib:deeptextspotter] from camera 8 is quite low probably because videos recorded by 8 are recorded under harsher illumination, which could interfere with the text spotters [@bib:deeptextspotter] performance. **Text Spotter** **Recall$^{v}$** **Precision$^{v}$** **F1-score$^{v}$** ---------------------- ------------------ --------------------- -------------------- @bib:deeptextspotter 61.03 76.85 66.64 @bib:he_textspotter 86.41 68.41 74.05 Baseline (all) 100 40.52 57.66 Baseline (random) 12.76 14.49 13.57 : The average performance of text spotter on our dataset. @bib:he_textspotter has higher recall, but lower precision compared to @bib:deeptextspotter. \[tab:textspotter\_recall\_precision\] ![The comparison of the number of runners detected per camera on the event track. The stacked bars with lighter colors are the false positives produced by the text spotter. @bib:he_textspotter has a higher number of true positives and false positives than @bib:deeptextspotter.[]{data-label="fig:num_runner_textspotter"}](num_runner_textspotter.png){width="1\linewidth"} ### Distance threshold for person re-id A threshold is determined to separate the inliers and outliers for the person re-id methods. We use F1-score$^v$ as a comparison metric between different thresholds. Then we choose a threshold that gives the highest F1-score$^v$. For person re-id with training samples from @bib:deeptextspotter, we choose a threshold of 0.21 that gives an average F1-score$^v$ of 79.00. Also, for person re-id with training samples from @bib:he_textspotter, we choose a threshold of 0.22 that gives an average F1-score$^v$ of 85.77. Since the person re-id with training samples collected by @bib:he_textspotter has higher performance, the detection results from this person re-id model are used to train the model for the second time. For this retrained person re-id model, with 0.05 as the distance threshold an average F1-score$^v$ of 87.76 is obtained. ### Comparative results: F1-score$^v$ We also report the F1-score$^v$ between two text recognition models and also the person re-id model trained in three different scenarios. Figure \[fig:video\_f1\_plot\] shows that person re-id models generally achieves higher performance compared to the scene text recognition models. It validates the hypothesis that using the whole appearance of a person is better for runner detection as compared to using just the bib number. This could be because of occlusion or blurring of bib numbers (due to athlete motion) hindering the performance of text recognition models. Meanwhile, based on visual inspection of detection results, partial occlusion on runner’s body or bib number tag, different runner’s pose, or different camera setting (e.g. camera viewpoint or illumination) does not have any significant effect on the performance of the person re-id model. As long as the appearance of runners is distinguishable, the person re-id model can detect them. Person re-id fails when the recording setup is not ideal (e.g. the distance between the camera and runners is too far, the runner facing against the camera), or a considerable part of the runner’s body is occluded. Figure \[fig:video\_f1\_plot\] also shows a comparison between person re-id models with different training samples. Person re-id with training samples from @bib:he_textspotter has higher performance compared to another with training samples from @bib:deeptextspotter because @bib:he_textspotter has higher recall so it collects more true positive runners for training. Meanwhile, retrained person re-id only improves a little because most remaining undetected runners are the challenging ones (e.g. they are recorded farther away from the camera, and not visually distinguishable from one another). Another important observation is that runners with zero F1-score$^v$ at the lower right side of the plot. They are the runners whose bib numbers have less than three digits, which the text spotter struggles to detect. Consequently, the person re-id model does not have training samples and thus it also has a performance of zero for those particular runners. ![The comparison of F1-scores$^{v_{i}}$ between person re-identification and text spotter models per runner. The x-axis presents runners sorted on F1-score$^{v_{i}}$. The x-axis values are not shown because there are five different axes. Based on F1-score$^{v_{i}}$, person re-id models outperform the text spotters.[]{data-label="fig:video_f1_plot"}](video_f1_plot_slim.png){width="\linewidth"} ### Comparative results: temporal IoU To show the performance in retrieving the relevant frames, we present the temporal IoU plot for scene text recognition and three person re-id methods, as shown in Figure \[fig:temporal\_iou\_plot\]. The person re-id model still outperforms the scene text recognition ones. It happens because the person re-id uses the training samples from the text spotter, and it can expand the frame interval by detecting runner from earlier frames where the text spotter might find difficulties in detecting the bib number; in the earlier frames, the bib number images are much smaller, but the appearance of the person can be distinct. It is important to notice that some runners have zero temporal IoU. They are the same runners that have zero F1-score$^v$. ![The comparison of temporal IoU between person re-identification and text spotter [@bib:he_textspotter] per runner. The x-axis is runners sorted on its temporal IoU. The x-axis values are not shown because there are five different x-axes. Person re-id models also exceed the performance of text spotters in terms of temporal IoU.[]{data-label="fig:temporal_iou_plot"}](temporal_iou_plot.png){width="\linewidth"} The analysis of the comparison between person re-id models with different training samples is more or less similar to the analysis on F1-score$^v$, except it is analyzed on a frame by frame basis. @bib:he_textspotter is better at detecting bib number with a smaller size in the earlier frames; thus person re-id with training samples from this text spotter has a better chance at expanding the frame interval. Meanwhile, the retrained person re-id only improves the average temporal IoU only by 5%. It happens because the runner’s images in remaining frames are the hard ones to detect (e.g. the small image at earlier frames due to the distance) and some false positives remain in the second training set. ![The comparison of the number of detected runners per camera between text spotter [@bib:he_textspotter] and the corresponding person re-id models. The stacked bars with lighter color are false positives. Person re-id models produce less false positives compared to the text spotter [@bib:he_textspotter].[]{data-label="fig:num_runner_reid_he"}](num_runner_reid_he.png){width="\linewidth"} ### Outlier detection In Figure \[fig:num\_runner\_reid\_he\], we show the comparison of number of detected runners per camera between @bib:he_textspotter and the corresponding person re-id models. It can be seen that the number of false positives produced by @bib:he_textspotter is quite high. However, the person re-id method can significantly improve producing lesser false positives, although it uses the training samples from the text spotter that has many false positives. It validates that *k-NN* in person re-id performs well enough as an outlier detector. Person re-id model with triplet loss minimize the similarity distance between images of the same person and maximize the similarity distance between images of different persons, so the false positives of a runner will have further similarity distance from the training samples of a runner. Then the false positives with larger distances could be rejected. However, in the second round of training the person re-id model, the reduction in false positives is not significant. This could be because some of sampling more false positives, some of which may be similar in appearance to the runners, and thus are also not rejected as an outlier. ### 2D timeline visualization Figure \[fig:timeline\_paper\] shows the true positives and false positives in our research problem clearly. It can be seen that runners 280, 253, and 456 have all their blue strips aligned with their green strips; using retrained person re-id, they have perfect F1-score$^v$ of 100. Meanwhile, using retrained person re-id, runners 336, 150, and 450 still have many false positives. It occurs because of the false positives from text spotter [@bib:he_textspotter] that are used as training samples; the false positives from blue strips are aligned with the false positive from brown strips. Another interesting observation is that sometimes the person re-id can detect a runner, although it does not have the training samples at that video. For example, the fourth blue strips at runner 253 do not have aligned brown strips. Limitations =========== It is important to note that the performance of the person re-id method depends heavily on the performance of scene text recognition as the latter collects the training samples for the former. For example, person re-id cannot detect the runners that have the bib number with less than three digits. Another thing to note is that there are false positives in the images retrieved by the text spotter. This hinders the performance of person re-id model. Conclusions and Future Work =========================== In this study, we have proposed an automatic approach to create a running event visualization from the video data. We use scene text recognition and person re-identification models to detect the runners and retrieve the videos and frames information where the runners appear so that we can use the individual timestamp and filtered GPS coordinates from the retrieved data for visualization. The experiments show that the scene text recognition models encounter many challenges in runner detection task, which can be mitigated by a person re-id model. The results also show that the performance of the person re-id method outperforms the scene text recognition method. The person re-id method can retrieve the relevant video information almost as good as the ground truth. This research focused on creating 2D visualizations of athletes with timeline charts and running track visualizations with runners represented by moving markers. This can be further extended to 3D using human pose estimation and spatio-temporal reconstruction [@mustafa2016temporally]. Gait information could also be used additionally for identification. Such reconstruction is not only useful for sports performance analysis and health monitoring, but can also be used for forensic investigations [@chen2016videos]. It would also be interesting to investigate if using another means to collect training samples for person re-id, such as crowdsourcing labeling, can produce a better performance. Acknowledgement =============== This study is supported by NWO grant P16-28 project 8 Monitor and prevent thermal injuries in endurance and Paralympic sports of the Perspectief Citius Altius Sanius - Injury-free exercise for everyone program. [^1]: \*These authors contributed equally [^2]: https://d3js.org/ [^3]: https://leafletjs.com/

--- abstract: 'Results for the inter-quark potential and low-lying $SU(3)$ glueball spectrum from simulations using a new improved action are presented. The action, suitable for highly anisotropic lattices, contains a two-plaquette term coupling with a negative coefficient as well as incorporating Symanzik improvement.' address: - 'Dept. of Physics, University of California at San Diego, La Jolla, California 92093-0319, USA' - 'NIC, Forschungszentrum Jülich, Jülich D-52425, Germany' author: - 'Colin Morningstar and Mike Peardon[^1]' title: 'The glueball spectrum from novel improved actions.' --- INTRODUCTION ============ The QCD glueball spectrum has been investigated in low-cost simulations using anisotropic lattices [@glueballs_su3]. To reduce the computational overhead, the spatial lattice was kept rather coarse (0.2-0.4 fm) while the temporal spacing was made much finer. The fine temporal grid allows adequate resolution of the Euclidean-time decay of appropriate correlation functions which, for gluonic states are rather noisy and fall too rapidly on coarse lattices. In these simulations, the scalar glueball suffered from large finite cut-off effects. The mass in units of $r_0$ fell sharply until the spatial lattice spacing, $a_s$ was about 0.25 fm when the mass rose again; the “scalar dip”. At the conference last year, we presented results from simulations with an anisotropic Wilson “two-plaquette” action which included a term constructed from the product of two parallel plaquettes on adjacent time-slices [@in_search]. This was found to reduce the scalar dip significantly. Here, we report on the status of simulations in progress using a Symanzik-improved action including a similar two-plaquette term. In this study, we tune the anisotropy parameter in the lattice action to recover Euclidean invariance in the “sideways” potential. With these parameters fixed, we investigate the inter-quark potential for this action as an initial test that the benefits of the Symanzik program are preserved by the addition of the extra term. We are currently computing the glueball spectrum for this action. THE ACTION ========== Following Ref. [@in_search], we begin with the plaquette operator, $$P_{\!\mu\nu}(x) = \frac{1}{N} \mbox{ReTr } U_\mu(x) U_\nu(x\!+\!\hat{\mu}) U^\dagger_\mu(x\!+\!\hat{\nu}) U^\dagger_\nu(x).$$ The Wilson (unimproved) discretisation of the magnetic field strength is then constructed from the spatial plaquette. $$\begin{aligned} \Omega_s & = & \sum_{x,i>j} \left\{ 1 - P_{ij}(x) \right\} \nonumber \\ & = & \frac{\xi_0}{\beta} \int\!\! d^4\!x \mbox{ Tr } B^2 + {\cal O}(a_s^2),\end{aligned}$$ where $i,j$ are spatial indices and $\xi_0$ is the anisotropy, $a_s/a_t$ at tree-level in perturbation theory. We introduce a term which correlates pairs of spatial plaquettes separated by one site temporally $$\Omega_{s}^{(2t)} = \frac{1}{2}\sum_{x,i>j} \left\{ 1 - P_{ij}(x) P_{ij}(x+\hat{t})\right\}.$$ The separation of the two plaquettes allows the standard Cabibbo-Marinari and over-relaxation gauge-field update methods to be applied. Including two-plaquette terms adds a computational overhead of only $10\%$ to our improved action workstation codes. It can be shown that for all $\omega$, the operator combination, $$\tilde{\Omega}_s = (1+\omega) \; \Omega_s - \omega \; \Omega_{s}^{(2t)} \label{eqn:twoplaq}$$ has an identical expansion in powers of $a_{s,t}$ (at tree-level) to $\Omega_s$ up to ${\cal O}(a_s^4)$. Thus, starting from the improved action $S_{I\!I}$ used in Refs. [@glueballs_su3; @glueballs_su2], it is straightforward to construct a Symanzik improved, two-plaquette action by simply replacing the spatial plaquette term in $S_{I\!I}$ with the linear combination $\tilde{\Omega}_s$ of Eqn. \[eqn:twoplaq\]. In full, this action is $$\begin{aligned} && S_\omega = \nonumber \\ && \frac{\beta}{\xi_0} \left\{ \frac{5(1+\omega)}{3 u_s^4} \Omega_s - \frac{5\omega}{3 u_s^8} \Omega_{s}^{(2t)} - \frac{1}{12 u_s^6} \Omega^{(R)}_{s} \right\} \nonumber \\ && + \beta \xi_0 \left\{ \frac{4}{3 u_s^2 u_t^2} \Omega_t - \frac{1}{12 u_s^4 u_t^2} \Omega^{(R)}_{t} \right\}, \label{eqn:twoplaq-action}\end{aligned}$$ with $\Omega_t$ the temporal plaquette and $\Omega^{(R)}_{s}$,$\Omega^{(R)}_{t}$ the $2\times1$ rectangle in the $(i,j)$ and $(i,t)$ planes respectively. This action has leading ${\cal O}(a_s^4,a_t^2, \alpha_s a_s^2)$ discretisation errors and only connects sites on adjacent time-slices, ensuring the free gluon propagator has only one real mode. The free parameter $\omega$ is chosen such that the approach to the QCD continuum is made on a trajectory far away from the critical point in the plane of fundamental-adjoint couplings. Close to the QCD fixed point, physical quantities should be weakly dependent on $\omega$. This provides us with a consistency check, however the data presented here are for one value only, $\omega=3$. TUNING THE ANISOTROPY ===================== At finite coupling, the anisotropy measured using a physical probe differs from the parameter in the action at ${\cal O}(\alpha_s)$ [@aniso]. In previous calculations, we relied upon the smallness of these renormalisations for the (plaquette mean-link improved) action $S_{I\!I}$. For the action of Eqn. \[eqn:twoplaq-action\], these renormalisations are larger and thus we chose to tune the input parameter in the action to ensure that the potentials measured along anisotropic axes matched. We follow a similar procedure to Ref. [@klassen]. The potentials between two static sources propagating along the z-axis for separations on both fine and coarse axes, $V_t$ and $V_s$ respectively, are measured using smeared Wilson loops. Since the UV divergences due to the static sources are the same, tuning $\xi_0$ such that the ratio $$\rho_n = \frac{a_s V_s(n a_s)}{ a_s V_t (m n a_t)} \equiv 1, \label{eqn:rho-def}$$ implies the anisotropy $\xi_V = m \; (m \in Z)$. A consistency check is provided by studying different coarse source separations, $n a_s$. Fig. \[fig:xi\_bare\] shows this tuning for $n=3,4,5$, where the desired anisotropy is 6. Consistency is observed for $n=4$ and 5 and the appropriate $\xi_0$ is found to better than $1\%$. SIMULATION RESULTS ================== The inter-quark potential ------------------------- The replacement of the spatial plaquette in $S_{I\!I}$ with $\tilde{\Omega}_s$ of Eqn. \[eqn:twoplaq\] should lead only to changes in the irrelevent operators responsible for ${\cal O}(a_s^4, \alpha_s a_s^2)$ errors. To test this replacement still generates an improved action with the good rotational invariance of $S_{I\!I}$, the inter-quark potential was computed for a variety of different inter-quark lattice orientations. The potential is shown in \[fig:V\], and shows excellent rotational invariance. We conclude that the benefits of the Symanzik improvement programme are preserved by including the two-plaquette term for a typical value of $\omega$ useful for glueball simulation. The glueball spectrum --------------------- At present, we are computing the glueball spectrum on the $\xi_V=6$ tuned lattices. Preliminary data are presented in Figs. \[fig:glue-1\] and \[fig:glue-2\]. In Fig. \[fig:glue-1\], the finite-lattice-spacing artefacts in the scalar glueball mass for the new action are compared to those of $S_{I\!I}$. The lattice cut-off dependence is seen to be significantly reduced and for the range of lattice spacings studied here, the mass rises monotonically with lattice spacing rather than falling first to a minimum. Fig. \[fig:glue-2\] shows the lattice spacing dependence on the tensor and pseudoscalar glueballs. Their lattice spacing dependence is similar to the form for $S_{I\!I}$ and consistent with leading ${\cal O}(a_s^4)$ behaviour. CONCLUSIONS =========== Preliminary data from our simulations of the Symanzik improved action of Eqn. \[eqn:twoplaq-action\] suggest the scalar dip is removed by inclusion of a two-plaquette term with negative coefficient, consistent with the argument that the poor scaling of the scalar glueball, even after Symanzik improvement, is caused by the presence of a nearby critical point. The inter-quark potential on this new action exhibits equally good rotational symmetry to the improved actions of Refs. [@glueballs_su3; @glueballs_su2]. [9]{} C. Morningstar and M. Peardon , Phys. Rev. D56 (1997) 4043-4061, Phys. Rev. D60 (1999) 034509. C. Morningstar and M. Peardon , Nucl. Phys. B (Proc Suppl.) 73 (1999) 927-929. H. Trottier and N. Shakespeare, Phys. Rev. D59 (1999) 014502. M. Alford [*et. al*]{}, in preparation. T. Klassen, Nucl. Phys. B533 (1998) 557-575. [^1]: Poster presented by M.P.

--- bibliography: - 'ref.bib' --- QMUL-PH-15-07\ [**Rigid Supersymmetry from Conformal Supergravity in Five Dimensions** ]{} 1.2cm Alessandro Pini^$\spadesuit$,$\clubsuit$,1^, Diego Rodriguez-Gomez^$\spadesuit$,1^, Johannes Schmude^$\spadesuit$,1^ [c]{} ^$\spadesuit$^Department of Physics, Universidad de Oviedo,\ Avda. Calvo Sotelo 18, 33007, Oviedo, Spain\ \ ^$\clubsuit$^Queen Mary University of London, Centre for Research in String Theory,\ School of Physics, Mile End Road, London, E1 4NS, England\ 1.5cm **Abstract** We study the rigid limit of 5d conformal supergravity with minimal supersymmetry on Riemannian manifolds. The necessary and sufficient condition for the existence of a solution is the existence of a conformal Killing vector. Whenever a certain $\operatorname{SU}(2)$ curvature becomes abelian the backgrounds define a transversally holomorphic foliation. Subsequently we turn to the question under which circumstances these backgrounds admit a kinetic Yang-Mills term in the action of a vector multiplet. Here we find that the conformal Killing vector has to be Killing. We supplement the discussion with various appendices. Introduction ============ Over the very recent past much effort has been devoted to the study of supersymmetric gauge theories on general spaces. Part of this interest has been triggered by the development of computational methods allowing to exactly compute certain (supersymmetric) observables, such as the supersymmetric partition function (starting with the seminal paper [@Pestun:2007rz]), indices or Wilson loops. This program has been very successfully applied to the cases of 4d and 3d gauge theories, and it is only very recently that the 5d case has been considered (*e.g.* [@Hosomichi:2012ek; @Kallen:2012va; @Qiu:2013pta; @Kim:2012gu; @Qiu:2014cha; @Imamura:2012bm]). On the other hand, it has become clear that the dynamics of 5d gauge theories is in fact very interesting, as, contrary to the naive intuition, at least for the case of supersymmetric theories, they can be at fixed points exhibiting rather amusing behavior as pioneered in [@Seiberg:1996bd]. In particular, these theories often show enhanced global symmetries which can be both flavor-like or spacetime-like. The key observation is that vector multiplets in 5d come with an automatically conserved topological current $j\sim \star F\wedge F$ under which instanton particles are electrically charged. These particles provide extra states needed to enhance perturbative symmetries, both flavor or spacetime – such as what it is expected to happen in the maximally supersymmetric case, where the theory grows one extra dimension and becomes the $(2,\,0)$ 6d theory. In fact, very recently the underlying mechanism for these enhancements has been considered from various points of view [@Lambert:2014jna; @Tachikawa:2015mha; @Zafrir:2015uaa; @Rodriguez-Gomez:2015xwa]. Five-dimensional gauge theories have a dimensionful Yang-Mills coupling constant which is irrelevant in the IR. Hence they are non-renormalizable and thus *a priori* naively uninteresting. However, as raised above, at least for supersymmetric theories the situation is, on the contrary, very interesting as, by appropriately choosing gauge group and matter content, the $g_{YM}$ coulpling which plays the role of UV cut-off can be removed in such a way that one is left with an isolated fixed point theory [@Seiberg:1996bd]. From this perspective, it is natural to start with the fixed point theory and think of the standard gauge theory as a deformation whereby one adds a $g_{YM}^{-2}F^2$ term. In fact, the $g_{YM}^{-2}$ can be thought as the VEV for a scalar in a background vector multiplet. Hence, for any gauge theory arising from a UV fixed point[^1] we can imagine starting with the conformal theory including a background vector multiplet such that, upon giving a non-zero VEV to the background scalar, it flows to the desired 5d gauge theory. This approach singles out 5d conformally coupled multiplets as the interesting objects to construct. As described above, on general grounds considering the theory on arbitrary manifolds is very useful, as for example, new techniques allow for exact computation of supersymmetric observables. The first step in this program is of course the construction of the supersymmetric theory on the given (generically curved) space, which is *per se* quite non-trivial. However, the approach put forward by [@Festuccia:2011ws] greatly simplifies the task. The key idea is to consider the combined system of the field theory of interest coupled to a suitable supergravity, which, by definition, preserves supersymmetry in curved space. Then, upon taking a suitable rigid limit freezing the gravity dynamics, we can think of the solutions to the gravity sector as providing the background for the dynamical field theory of interest. Note that, since the combined supergravity+field theory is considered off-shell, both sectors can be analyzed as independent blocks in the rigid limit, that is, one can first solve for the supergravity multiplet and then regard such solution as a frozen background for the field theory, where the supergravity background fields act as supersymmetric couplings. Of course, the supergravity theory to use must preserve the symmetries of the field theory which, at the end of the day, we are interested in. Hence, in the case of 5d theories, it is natural to consider conformal supergravity coupled to the conformal matter multiplets described above. Following this approach, in this paper we will consider 5d conformal supergravity [@Fujita:2001kv; @Bergshoeff:2001hc; @Bergshoeff:2004kh] coupled to 5d conformal matter consisting of both vector and hyper multiplets. As remarked above, the Yang-Mills coupling constant is dimensionful. Hence, the action for the vector multiplets is not the standard quadratic one with a Maxwell kinetic term but rather a cubic action which can be thought as the supersymmetric completion of 5d Chern-Simons. As anticipated, in the rigid limit we can separate the analysis of the gravity multiplet as providing the supersymmetric background for the field theory. One is thus prompted to study the most generic backgrounds where 5d gauge theories with $\mathcal{N}=1$ supersymmetry can be constructed by analyzing generic solutions of the 5d $\mathcal{N}=2$ conformal supergravity. Solutions to various 5d supergravities on (pseudo-) Riemannian manifolds have been studied in different approaches in [@Gauntlett:2002nw; @Pan:2013uoa; @Imamura:2014ima; @Kuzenko:2014eqa; @Pan:2014bwa; @Alday:2015lta; @Pan:2015nba]. For $\mathcal{N}=1$ Poincaré supergravity, the necessary and sufficient condition for the existence of a global solution is the existence of a non-vanishing Killing vector. If one considers conformal supergravity this condition becomes the existence of a conformal Killing vector (CKV).[^2] In this paper we analyze Euclidean solutions of 5d conformal supergravity in terms of component fields. Our analysis proceeds along the lines of [@Klare:2013dka]. Interestingly, by studying the conditions under which a VEV for the scalar in the background vector multiplets paying the role of $g_{YM}^{-2}$ can be given in a supersymmetric way, we find that such vector must be in fact Killing. Hence in this case we simply recover the results obtained using Poincaré supergravity. Our results rely on some reality conditions satisfied by the supersymmetry spinors. In the Lorentzian theory, the spinors generally satisfy a symplectic Majorana condition . If one imposes the same condition in the Euclidean case, there are immediate implication for the spinor bilinears that play an important role in the analysis. Namely, the scalar bilinear $s$ is real and vanishes if and only if the spinor vanishes, while the vector bilinear $v$ – the aforementioned CKV – is real. One should note however that the symplectic Majorana condition is not equivalent to these conditions for $s$ and $v$. Instead, is slightly stronger, while our results only depend on the milder assumptions on the bilinears. While the existence of the CKV is a necessary and sufficient condition many of the backgrounds exhibit a more interesting geometric structure – that of a transversally holomorphic foliation (THF). These appeared already in the context of rigid supersymmetry in three dimensions [@Closset:2012ru] and one can think of it as an almost complex structure on the space transverse to the CKV that satisfies a certain integrability condition. A simple example of a five manifold endowed with a THF is given by Sasakian manifolds. Here, the existence of the THF was exploited in [@Schmude:2014lfa] in order to show that the perturbative partition function can be calculated by counting holomorphic functions on the associated Kähler cone. Similar considerations were used in [@Pan:2014bwa] to solve the BPS equations on the Higgs branch. This gives rise to the question whether such simplifications occur in localization calculations on more generic five manifolds admitting rigid supersymmetry. This was addressed in [@Pan:2015nba] in the context of 5d $\mathcal{N}=1$ Poincaré supergravity. Here it was shown that a necessary and sufficient condition for such manifolds to admit a supersymmetric background is the existence of a Killing vector. If an $\operatorname{su}(2)$-valued scalar in the Weyl multiplet is non-vanishing and covarinatly constant along the four-dimensional leaves of the foliation it follows furthermore that the solution defines a THF. Subsequently it was argued that the existence of a THF (or that of an integrable Cauchy-Riemann structure) is sufficient to lead to similar simplifications in the context of localization as in [@Pan:2014bwa; @Schmude:2014lfa]. With this motivation in mind we will address the question under which circumstances generic backgrounds of the conformal supergravity in question admit THFs. Our results are to be seen in the context of the very recent paper [@Alday:2015lta]. We will find that the necessary and sufficient condition for the solution to support a THF is the existence of a global section of an $\operatorname{su}(2)/\mathbb{R}$ bundle that is covariantly constant with respect to a connection $\mathcal{D}^Q$ that arises from the intrinsic torsions parametrizing the spinor. The outline of the rest of the paper is as follows. In section \[sec: SUGRA\_review\] we offer a lightning review of the relevant aspects of superconformal 5d supergravity, with our conventions compiled in appendix \[sec:conventions\] and further details described in appendix \[sec:computation\]. In section \[sec:general\_solutions\] we turn to the analysis of the general solutions of the supergravity, showing that the necessary condition for supersymmetry is the presence of a conformal Killing vector. Moreover, we will see that given a Killing spinor and the related CKV the general solution depends only on an $\operatorname{su}(2)$-valued $\Delta^{ij}$ and a vector $W$ that is orthogonal to the CKV. Both are determined by solving simple ODEs that become trivial if one goes to a frame where the CKV is Killing. In section \[sec:gauge\_theory\] we study under which conditions it is possible to turn on a VEV for scalars in background vector multiplets thus flowing to a standard gauge theory, finding that the requirement is that the vector is not only conformal Killing but actually Killing. In section \[sec:THF\] we derive the conditions for the existence of a THF. In section \[sec:examples\] we show how some particular examples fit into our general structure, describing in particular the cases of $\mathbb{R}\times S^4$ relevant for the index computation of [@Kim:2012gu] and the $S^5$ relevant for the partition function computation of [@Hosomichi:2012ek; @Kallen:2012va]. We finish with some conclusions in section \[sec:conclusions\]. **Note added:** While this work was in its final stages we received [@Alday:2015lta], which has a substantial overlap with our results. Five-dimensional conformal supergravity {#sec: SUGRA_review} ======================================= Let us begin by reviewing the five-dimensional, $\mathcal{N}=2$ conformal supergravity of [@Bergshoeff:2001hc; @Bergshoeff:2004kh][^3]. The theory has $SU(2)_R$ $R$-symmetry. The Weyl multiplet contains the vielbein $e_\mu^a$, the $\operatorname{SU}(2)_R$ connection $V_\mu^{(ij)}$, an antisymmetric tensor $T_{\mu\nu}$, a scalar $D$, the gravitino $\psi_\mu^i$ and the dilatino $\chi^i$. Our conventions are summarised in appendix \[sec:conventions\]. The supersymmetry variations of the gravitino and dilatino are [rCl]{} \_\^i &=& \_\^i + T \_\^i - \_\^i, \[eq:gravitino-variation\]\ \^i &=& \^i D - \^[ij]{}(V) \_j + \^ T\_ \^i - \^\^T\_ \^i - \^ T\_ T\_ \^i\ &&+ T\^2 \^i + T \^i. \[eq:dilatino-variation\] Up to terms $\mathcal{O}(\psi_\mu, \chi^i)$, [rCl]{} \_\^i &=& \_\^i + \_\^[ab]{} \_[ab]{} \^i + b\_\^i - V\_\^[ij]{} \_j,\ \_\^[ij]{}(V) &=& dV\^[ij]{}\_ - 2 V\_[\[]{}\^[k(i]{} V\_[\] k]{}\^[j)]{}. In what follows we will set the Dilation gauge field $b_\mu$ to zero. As usual, taking the $\gamma$-trace of the gravitino equation allows to solve for the superconformal parameters as $\eta^i = - \frac{\imath}{5} \slashed{\mathcal{D}} \epsilon^i + \frac{1}{5} T \cdot \gamma \epsilon^i$. Hence, we can rewrite the equations arising from the gravitino and dilatino as [rCl]{} 0 &=& \_\^i - \_ \^\^i + \_ T\^ \^i - 3 T\_ \^\^i, \[eq:gravitino-variation\_w/out\_eta\]\ 0 &=& \^i (32 D + R) + T\_ T\^ \^i + \^\_\^i + \_ T\^ \^\^i + \^T\_ \^\^i\ &&+ \_ \^T\^ \^i + \^\^T\_ \^i - \^ T\_ T\_ \^i.\[eq:dilatino-variation\_D-squared-form\] Here, $R$ is the Ricci scalar and the rewriting of the dilatino equation uses the gravitino equation. One could also rewrite the latter using $\slashed{\mathcal{D}}^2$ as in [@Klare:2013dka], yet we found the above formulation to be more economical in this case. General solutions of $\mathcal{N}=2$ conformal supergravity {#sec:general_solutions} =========================================================== General solutions to five-dimensional conformal supergravity have been constructed in [@Kuzenko:2014eqa] using superspace techniques. In this section we will provide an alternative derivation of the most general solutions to $\mathcal{N}=2$ conformal supergravity in euclidean signature using component field considerations along the lines of [@Klare:2013dka]. Before turning to the details, let us recall a counting argument from [@Klare:2013dka] regarding these solutions: In general the gravitino yields $40$ scalar equations. Eliminating the superconformal spinor $\eta^i$ removes $8$. As we will see, the gravitino equation then also fixes the $10$ components of the antisymmetric tensor and $8$ of the components of the $\operatorname{SU}(2)$ connection. This leaves us with $14$, which is exactly enough to remove the traceless, symmetric part of a two-tensor $P$ which will appear in the intrinsic torsion. Since the trace is undetermined we will find a CKV; the vector is Killing if the trace vanishes. The remaining $7$ components of the $\operatorname{SU}(2)$ connection and the scalar in the Weyl multiplet will then be determined by the eight equations arising from the dilatino variation. In order to study solutions of and , we introduce the bispinors $$\label{eq:spinor_bilinears} \begin{aligned} s &= \epsilon^i C \epsilon_{i}, \\ v &= (\epsilon^i C \gamma_\mu \epsilon_{i}) dx^\mu, \\ \Theta^{ij} &= (\epsilon^i C \gamma_{\mu\nu} \epsilon^j) dx^\mu \otimes dx^\nu, \end{aligned}$$ In what follows, we will assume the scalar $s$ to be non-zero and the one-form $v$ to be real. These assumptions are implied if one imposes a symplectic Majorana condition such as . Furthermore, note that $v^2 = s^2$. The one-form then decomposes the tangent bundle into a horizontal and a vertical part, with the former being defined as $TM_H = \{ X \in TM \vert v(X) = 0\}$ and $TM_V$ as its orthogonal complement. Due to the existence of a metric we use $v$ to refer both to the one-form and the correspoding vector and an analogous decomposition into horizontal and vertical forms extends to the entire exterior algebra. In turn, the two-forms $\Theta^{ij}$ are fully horizontal and anti self-dual[^4] with respect to the automorphism $\iota_{s^{-1} v} \star : \Lambda^2_H \to \Lambda^2_H$: $$\iota_{v} \Theta^{ij} = 0, \qquad \iota_{s^{-1}v} \star \Theta^{ij} = - \Theta^{ij}.$$ One finds that the spinor is chiral with respect to the vector $s^{-1} v$, $$s^{-1} v^\mu \gamma_\mu \epsilon^i = \epsilon^i.$$ Note that the sign here is mainly a question of convention. Had we defined $v$ with an additional minus sign, we would find the spinor to be anti-chiral and $\Theta^{ij}$ to be self-dual. One can see this by considering the transformation $v \mapsto -v$. In addition, we define the operator $\Pi^\mu_\nu = \delta^\mu_\nu - s^{-2} v^\mu v_\nu$ which projects onto the horizontal space. A number of additional useful identities involving $\Theta^{ij}$ are given in appendix \[sec:conventions\]. Next, we parametrize the covariant derivative of the supersymmetry spinor using intrinsic torsions as in [@Imamura:2014ima], $$\label{eq:intrinsic_torsions_introduced} \nabla_\mu \epsilon^i \equiv P_{\mu\nu} \gamma^\nu \epsilon^i + Q_\mu^{ij} \epsilon_j.$$ Here, $P_{\mu\nu}$ is a two-tensor while $Q_\mu^{ij}$ is symmetric in its $SU(2)_R$ indices. Rewriting the torsions in terms of the supersymmetry spinor one finds $$\label{eq:intrinsic_torsions_in_terms_of_covariant_derivative} s P_{\mu\nu} = \epsilon^i \gamma_\nu \nabla_\mu \epsilon_i = \frac{1}{2} \nabla_\mu v_\nu, \qquad s Q_\mu^{ij} = 2 \epsilon^{(i} \nabla_\mu \epsilon^{j)}.$$ The gravitino equation ---------------------- We now turn to the study of generic solutions of and using the intrinsic torsions. The reader interested in intermediate results and some technical details might want to consult appendix \[sec:computation\]. To begin, substituting and contracting with $\epsilon_i \gamma_\kappa$ as well as $\epsilon^j$ and symmetrizing in $i, j$ one finds that is equivalent to [rCl]{} 0 &=& s ( P\_[()]{} - g\_ P\^\_ ) + s ( P\_[\[\]]{} - 4 T\_ ) + \_ (P\^[\[\]]{} - 4 T\^) v\^\ &&+ \_ (Q - V)\^[ij]{} \^\_[ij]{}, \[eq:gravitino\_via\_torsion\_vector\_contraction\]\ 0 &=& s (Q - V)\_\^[ij]{} + (Q - V)\^[(j]{}\_[k]{} \^[i) k]{}\_ + \_ (P - 4 T)\^ \^[ij]{}. \[eq:gravitino\_via\_torsion\_triplet\_contraction\] Clearly, the symmetric part in has to vanish independently; so we find $$\label{eq:symmetric_part_of_P} P_{(\mu\nu)} = \frac{1}{5} g_{\mu\nu} P^\lambda_{\phantom{\lambda}\lambda}.$$ This implies that $v$ is a conformal Killing vector as can be seen using . By contracting the two remaining equations with $v^\mu$, one finds [rCl]{} 0 &=& 3 s v\^(P - 4 T)\_[\[\]]{} - s \^[ij]{}\_ (Q - V)\^\_[ij]{}, \[eq:QVPT\_identity\_vector\]\ 0 &=& 2 s v\^(Q - V)\_\^[ij]{} - s \^[ij]{}\_ (P - 4 T)\^ \[eq:QVPT\_identity\_triplet\]. Projecting on the horizontal space, we find that $\Pi(P - 4\imath T)$ is anti self-dual. $$0 = (P - 4 \imath T)^+.$$ Contracting with $\Theta_{ij\kappa\lambda}$ and using gives us the horizontal, self-dual part. $$(P - 4\imath T)^- = s^{-2} \Theta^{ij} \imath_v (Q - V)_{ij}.$$ By now we have equations for the self-dual, anti self-dual and vertical components of $(P-4\imath T)_{[\mu\nu]}$, which means that all components of this two-form are determined. Putting everything together, we find [rCl]{}\[eq:gravitino\_solution\_i\] s\^2 (P - 4T)\_[\[\]]{} &=& (Q-V)\^\_[ij]{}. The only equation we have not considered so far is the horizontal projection of . After using , and this simplifies to $$\label{eq:gravitino_solution_ii} s \Pi_\mu^{\phantom{\mu}\nu} (Q-V)_{\nu\phantom{i}j}^{\phantom{\nu}i} = -\frac{1}{2} [(Q-V)^\nu, \Theta_{\mu\nu}]^i_{\phantom{i}j}.$$ In summary, the gravitino is solved by and . Note that one can solve by brute force after picking explicit Dirac matrices. One finds that the equation leaves seven components of $(Q-V)$ unconstrained. Three of these have to be parallel to $v$ as they do not enter in . This suggests that it is possible to package the seven missing components into a triplet $\Delta^{ij}$ (three components) and a horizontal vector $W^\mu$ (four) and parametrize a generic solution of the gravitino equation as $$\label{eq:Q-V_parametrization} (Q - V)_\mu^{ij} = s^{-1} \left( v_\mu \Delta^{ij} + W^\lambda \Theta^{ij}_{\lambda\mu} \right) \qquad \text{s.t.} \qquad v(W) = 0, \Delta^{ij} = \Delta^{ji}.$$ Using one can verify that satisfies . The above implies that $$\label{eq:T-solution} T_{\mu\nu} = \frac{\imath}{4} \left( s^{-1} \Theta^{ij}_{\mu\nu} \Delta^{ij} + s^{-1} v_{[\mu} W_{\nu]} - P_{[\mu\nu]} \right).$$ The dilatino equation {#sec:dilatino_analysis} --------------------- We finally turn to the dilatino equation . To begin, we note that between $\Delta^{ij}, W_\mu, D$ there are eight unconstrained functions remaining while the dilatino equation provides eight constraints. We can thus expect that there will be no further constraints on the geometry. In this respect, similarly to [@Klare:2013dka], supersymmetry is preserved as long as the manifold supports a conformal Killing vector $v$. In what follows we will need to deal with terms involving derivatives of the spinor bilinears . To do so we use the identities [rCl]{} \_s &=& 2 P\_ v\^,\ \_v\_&=& 2 s P\_,\ \_\_\^[ij]{} &=& 3! s\^[-1]{} \^[ij]{}\_[\[]{} v\_[\]]{} P\_\^ - 2 \_\^[k(i]{} Q\^[j)]{}\_[k]{},\ \_[\[]{} P\_[\]]{} &=& - s\^[-1]{} P\_[\[]{} \_[\]]{} s. Contracting with $\epsilon^j$ and symmetrizing over the $SU(2)_R$ indices $i, j$ one finds [rCl]{} 0 &=& \^[(i]{} \^\_\^[j)]{} + \^[(i]{} \_ T\^ \^\^[j)]{} + \^[(i]{} \^T\_ \^\^[j)]{}\ &&+ \^[(i]{} \_ \^[j)]{} \^T\^. Substituting and one finds after a lengthy calculation[^5] [rCl]{}\[eq:dilatino\_solution\_Delta\_ODE\] £\_v \^i\_[j]{} &=& - s P\^\_ \^i\_[j]{} - \[\_v Q + P\^[\[\]]{}\_, \]\^i\_[j]{}. Contracting with $-\epsilon_i \gamma_\mu$ one obtains [rCl]{}\[eq:dilatino\_vector\_equation\] 0 &=& v\_( + T\_ T\^ ) + \^i \_\^\_\_i + \^i \_\_ T\^ \^\_i\ &&+ \^i \_\^T\_ \^\_i + \_\^ v\_\_T\_ + \^T\_ - \_\^ T\_ T\_. The vertical component of this fixes the scalar $D$. [rCl]{} \[eq:D\] 0 &=& 480 s D + 15 s R + 48 s (P\^\_)\^2 - 130 s W\^2 + 60 \_ P\^[\[\]]{} P\^[\[\]]{} v\^ - 160 s \^[ij]{} \_[ij]{}\ &&+ 100 P\_[\[\]]{} (s P\^[\[\]]{} - 2 v\^W\^) - 200 P\^[\[\]]{} \_\^[ij]{} \_[ij]{} + 48 v\^\_P\^\_ - 120 s \^W\_. The horizontal part of yields a differential equation for $W$ [rCl]{} \[eq:W\] £\_v W\_= \_\^ &(& 3 s\^2 P\^\_ W\_- 34 P\^\_ P\_[\[\]]{} v\^ - 20 s \_P\^\_ ). Note that the left hand side is horizontal since $\iota_v \pounds_v W = \iota_v \iota_v dW = 0$. Similar to the discussion in [@Klare:2013dka], we note that one can always solve and locally. Moreover, after a Weyl transformation to a frame where $v$ is not only conformal Killing yet actually Killing, that is, setting $P^\mu_{\phantom{\mu}\mu}=0$, both equations simplify considerably. All the source terms in the latter vanish which is now solved by $W = 0$ while the former becomes purely algebraic, [rCl]{}\[eq:dilatino\_solution\_Delta\_ALGEBRAIC\] 0 &=& \[\_v Q + P\^[\[\]]{}\_, \]\^i\_[j]{}, and is solved by $\Delta = s^{-1} f (\iota_v Q + P^{[\mu\nu]} \Theta_{\mu\nu})$ for a generic, possibly vanishing, function $f$ as long as $\pounds_v f = 0$. The factor $s^{-1}$ is simply included here to render $\Delta$ invariant under $\epsilon^i \to \lambda \epsilon^i$ for $\lambda \in {\mathbb{C}}$. An alternative way to see that and can be solved globally is by direct construction of the solution following the approach of section 5 in [@Pan:2015nba]. Thus, the existence of a non-vanishing CKV is not only necessary, but also sufficient. See also footnote \[fn:caveat\_vanishing\_spinor\]. Yang-Mills theories from conformal supergravity {#sec:gauge_theory} =============================================== The solutions described above provide the most general backgrounds admitting a five-dimensional, minimally supersymmetric quantum field theory arising in the rigid limit of conformal supergravity. In the maximally supersymmetric case a more general class of solutions is possible, since the R-symmetry of maximal supergravity is $\operatorname{SO}(5)$, one can define supersymmetric field theories on generic five manifolds by twisting with the whole $\operatorname{SO}(5)$. Such field theories were considered in [@Bak:2015hba]. An embedding in supergravity should be possible starting from [@Cordova:2013bea]. Of course, in the case at hand our starting point is conformal supergravity, so only conformal multiplets can be consistently coupled to the theory. While the hypermultiplet is conformally invariant *per se*, the vector multiplet with the standard Maxwell kinetic term breaks conformal invariance as the Yang-Mills coupling has negative mass dimensions. Therefore the action for the conformally coupled vector multiplet is a non-standard cubic action which can be thought as the supersymmetric completion of 5d Chern-Simons. Such action contains in particular a coupling of the form $C_{IJK}\,\sigma^I\, F^J\,F^K$, where $F^I$ is the field strength of the $I$-th vector multiplet, $\sigma^I$ its corresponding real scalar and $C_{IJK}$ a suitable matrix encoding the couplings among all vector multiplets (we refer to [@Bergshoeff:2001hc; @Bergshoeff:2004kh] for further explanations). Thus we can imagine constructing a standard gauge theory by starting with a conformal theory and giving suitable VEVs to scalars in background abelian vector multiplets. Of course, such VEVs must preserve supersymmetry. To that end, let us consider the SUSY variation of a background vector multiplet. As usual, only the gaugino variation is relevant, which, in the conventions of [@Bergshoeff:2004kh], reads $$\delta\Omega_B^i=-\frac{\imath}{2}\,\slashed{\nabla}\sigma_B\,\epsilon^i+Y_B^i\,_j\,\epsilon^j+\sigma_B\,\gamma\cdot T\epsilon^i+\sigma_B\,\eta^i\, ,$$ where we have set to zero the background gauge field. The $Y_B^i\,_j$ are a triplet of auxiliary scalars in the vector multiplet. Contracting with $\epsilon_i$ it is straightforward to see that, in order to have a supersymmetric VEV, we must have $$\label{constraint} \pounds_v \sigma_B+\frac{2\,s}{5}\,P^\mu_{\phantom{\mu}\mu} \sigma_B =0\, ,$$ while the other contractions fix the value of $Y_B^i\,_j$. The VEV of $\sigma_B$ is $g_{YM}^{-2}$, and as such one would like it to be a constant. Therefore, equation gives us an obstruction for the existence of a Maxwell kinetic term; namely, that $v$ is Killing and not only conformal Killing. It then follows that all backgrounds admitting standard – *i.e.* quadratic – supersymmetric Yang-Mills theories, involve a $v$ which is a genuine Killing vector. They are thus solutions of the $\mathcal{N}=1$ Poincaré supergravity – see *e.g.* [@Pan:2013uoa; @Imamura:2014ima; @Pan:2014bwa; @Pan:2015nba]. In particular, the case of $\mathbb{R}\times S^4$ is of special interest as the partition function on this space in the absence of additional background fields gives the superconformal index [@Kim:2012gu]. The relevant supersymmetry spinors appearing in the calculation define a vector $v$ which is conformal Killing; and therefore the background is only a solution of conformal supergravity. As we will explicitly see below, it is easy to check that such a solution, which can be easily obtained by a simple change of coordinates in the spinors in [@Rodriguez-Gomez:2015xwa], nicely fits in our general discussion above. If, on the other hand, one studies supersymmetric backgrounds on $S^5$ without additional background fields, one finds $v$ to be Killing (see below as well). Thus such backgrounds can be regarded as a solution to conformal supergravity that are not obstructed by and do thus admit a constant $\sigma_B$. In fact, it is easy to check this nicely reproduces the results of [@Hosomichi:2012ek]. Eq.  shows that backgrounds admitting only a conformal Killing vector cannot support a standard gauge theory with a constant Maxwell kinetic term. As anticipated above, and explicitly described below, this is precisely the case of $\mathbb{R}\times S^4$, relevant for the computation of the index. Of course it is possible to solve if one accepts that the Yang-Mills coupling is now position dependent. This way we can still think of the standard Yang-Mills action as a regulator to the index computation.[^6] While this goes beyond the scope of this paper, one might imagine starting with the Yang-Mills theory on $\mathbb{R}^5$ where can be satisfied for a constant $\sigma_B$. Upon conformally mapping $\mathbb{R}^5$ into $\mathbb{R}\times S^4$ the otherwise constant $\sigma_B=g_{YM}^{-2}$ becomes $\sigma_B=g_{YM}^{-2}\,e^{\tau}$, being $\tau$ the coordinate parametrizing $\mathbb{R}$. In the limit $g_{YM}^{-2}\rightarrow 0$ we recover the conformal theory of [@Kim:2012gu]. One can imagine computing the supersymmetric partition function in this background. As the preserved spinors are just the same as in the $g_{YM}^{-2}\rightarrow 0$ limit, the localization action, localization locus and one-loop fluctuations will be just the same as in the conformal case. While we leave the computation of the classical action for future work, it is clear that the limit $g_{YM}^{-2}\rightarrow 0$ will reproduce the result in [@Kim:2012gu]. Existence of transversally holomorphic foliations {#sec:THF} ================================================= We will now discuss under which circumstances solutions to equations and define transversally holomorphic foliations (THF). Since we assumed $s \neq 0$ and $v$ real, it follows that the CKV $v$ is non-vanishing and thus that $v$ defines a foliation on $M$. Using one can show then that $\Theta^{ij}$ defines a triplet of almost complex structures on the four-dimensional horizontal space $TM_H$. Thus, given a non-vanishing section[^7] of the $\operatorname{su}(2)_R$ Lie algebra $m_{ij}$ we can define an endomorphism on $TM$ $$(\Phi[m])^\mu_{\phantom{\mu}\nu} \equiv (\det m)^{-1/2} m_{ij} (\Theta^{ij})^\mu_{\phantom{\mu}\nu}.$$ This satisfies $\Phi[m]^2 = -\Pi$ and thus induces a decomposition of the complexified tangent bundle $$T_{{\mathbb{C}}}M = T^{1,0} \oplus T^{0,1} \oplus {\mathbb{C}}v.$$ Any such decomposition is referred to as an almost Cauchy-Riemann (CR) structure. If an almost CR structure satisfies the integrability condition $$\label{eq:THF_integrability} [T^{1,0} \oplus {\mathbb{C}}v, T^{1,0} \oplus {\mathbb{C}}v] \subseteq T^{1,0} \oplus {\mathbb{C}}v,$$ one speaks of a THF.[^8] Intuitively, $m_{ij}$ determines how $\Phi$ is imbedded in $\Theta^{ij}$ and thus how $T^{1,0}$ is embedded in $TM_H$. If one forgets about the vertical direction $v$ for a moment, the question of integrability of $\Phi$ is similar to the question under which circumstances a quaternion Kähler structure on a four-manifold admits an integrable complex structure. To address the question of the existence of an $m_{ij}$ satisfying we follow the construction of [@Pan:2015nba] and define the projection operator $$H^i_{\phantom{i}j} = (\det m)^{-1/2} m^i_{\phantom{i}j} - \imath \delta^i_{\phantom{i}j}.$$ One can then show that $$\label{eq:spinorial_holomorphy_condition} X \in T^{1,0} \oplus {\mathbb{C}}v \qquad \Leftrightarrow \qquad X^\mu H^i_{\phantom{i}j} \Pi_\mu^\nu \gamma_\nu \epsilon^j = 0,$$ if the supersymmetry spinor $\epsilon^i$ satisfies a reality condition such as . Acting from the left with $\mathcal{D}_Y$ for $Y \in T^{1,0} \oplus {\mathbb{C}}v$ and antisymmetrizing over $X, Y$, one derives the spinorial integrability condition $$\label{eq:spinorial_integrability_condition} [X, Y] \in T^{1,0} \oplus {\mathbb{C}}v \qquad \Leftrightarrow \qquad X^{[\mu} Y^{\nu]} \mathcal{D}_\mu (H^i_{\phantom{i}j} \Pi_\nu^\rho \gamma_\rho \epsilon^j) = 0.$$ Note that $H^i_{\phantom{i}j}$ satisfies $H^2 = -2\imath H$ and has eigenvalues $0$ and $-2\imath$. Thus, $H^i_{\phantom{i}j} \epsilon^j$ projects the doublet $\epsilon^i$ to a single spinor that is a linear combination of the two. It is this spinor that will define the THF. To proceed, we first consider $X, Y \in T^{1,0}$. After substituting and making repeatedly use of , one finds that the condition reduces to the vanishing of $$X^{[\mu} Y^{\nu]} (\partial_\mu H^i_{\phantom{i}j} + [Q_\mu, H]^i_{\phantom{i}j}) \gamma_n \epsilon^j.$$ Similarly, the case $X \in T^{1,0}$, $Y = v$ leads to the condition that $$X^m (\partial_v H^i_{\phantom{i}j} + [\iota_v Q, H]^i_{\phantom{i}j}) \gamma_m \epsilon^j$$ must be identically zero. Contracting both expressions with $\epsilon^j$ and symmetrizing over $\operatorname{SU}(2)$ indices, we conclude that the integrability condition can only be satisfied if and only if $$\label{eq:THF_integrability_condition_for_connection} \mathcal{D}^Q_\mu H^i_{\phantom{i}j} \equiv \partial_\mu H^i_{\phantom{i}j} + [Q_\mu, H]^i_{\phantom{i}j} = 0,$$ i.e. iff the projection $H^i_{\phantom{i}j}$ is covariantly constant with respect to the connection defined by $Q$. From the condition that $H^i_{\phantom{i}j}$ be covariantly constant we derive the necessary condition that it is also annihilated by the action of the corresponding curvature tensor: $$\label{THF_integrability_condition_for_curvature} [R^Q_{\mu\nu}, H]^i_{\phantom{i}j} = 0,$$ where $R^Q_{\mu\nu} = [\mathcal{D}^Q_\mu, \mathcal{D}^Q_\nu]$. Now, one can only solve if the $\operatorname{SU}(2)$ curvature $R^Q$ lies in a $\operatorname{U}(1)$ inside $\operatorname{SU}(2)_R$. Note that since the curvature $R^Q$ arises from the intrinsic torsions, we can relate it to the Riemann tensor and $P_{\mu\nu}$ using . The resulting expression is not too illuminating however. To conclude we will relate the integrability condition to the findings of [@Pan:2015nba]. There it was found that solutions of the $\mathcal{N}=1$ Poincaré supergravity of [@Kugo:2000hn; @Kugo:2000af; @Zucker:1999ej] define THFs if $m_{ij} = t_{ij}$ and $\forall X \in TM_H, \mathcal{D}_X t_{ij} = 0$. In other words, the unique choice for $m_{ij}$ is the field $t_{ij}$ appearing in the Weyl multiplet of that theory and the latter has to be covariantly constant (with respect to the usual $\operatorname{SU}(2)_R$ connection $V_\mu^{ij}$) along the horizontal leaves of the foliation. To relate our results to this, consider the case where $v$ is actually Killing. It follows that we can assume $\pounds_v H^i_{\phantom{i}j} = 0$ and thus the vertical part of takes the form of the first condition of [@Pan:2015nba], namely $$[\iota_v Q, H]^i_{\phantom{i}j} = 0.$$ Moreover, our general solutions and are solved by $W = 0$; while this solution is not unique, it makes the connection to [@Pan:2015nba] very eveident as it follows now that $\Pi(Q)^{ij} = \Pi(V)^{ij}$ and so the horizontal part of reproduces the second condition from [@Pan:2015nba]: $$\forall X \in T^{1,0} \qquad \mathcal{D}_X H^i_{\phantom{i}j} = X^\mu (\partial_\mu H^i_{\phantom{i}j} + [V_\mu, H]^i_{\phantom{i}j}) = 0.$$ Examples {#sec:examples} ======== Let us now discuss some specific examples illustrating the general results from the previous sections. Flat $\mathbb{R}^5$ {#sec:example_flat_R5} ------------------- Flat space admits constant spinors generating the Poincaré supersymmetries. In addition, we can consider the spinor generating superconformal supersymmetries $\epsilon^i = x_\mu \gamma^\mu \epsilon_0^i$, where $\epsilon_0^i$ is constant. Let us see how these fit into our general set-up. For the Poincaré supersymmetries, it is clear that we just have $Q=P=V=T=0$. For the superconformal spinors on the other hand, the gravitino and dilatino equations are solved by $$\eta^i = - \imath \epsilon_0^i, \qquad T_{\mu\nu} = V_\mu^{ij} = D = 0.$$ The intrinsic torsions are $$Q_\mu^{ij} = - \frac{2}{s x^2} x_\kappa \Theta^{ij\kappa}_{\phantom{ij\kappa}\mu}, \qquad P_{[\mu\nu]} = \frac{1}{s x^2} (x \wedge v)_{\mu\nu}, \qquad P_{(\mu\nu)} = s^{-1} x_\kappa v^\kappa \delta_{\mu\nu}.$$ Note that $\Theta_{\nu\rho}^{ij} Q^\rho_{ij} = - \frac{3s}{x^2} \Pi_\nu^\sigma x_\sigma$ and thus $$\frac{2}{3} 2 v_{[\mu} \Theta^{ij}_{\nu]\rho} Q^\rho_{ij} = \frac{s}{x^2} (\Pi_\mu^\rho v_\nu - \Pi_\nu^\rho v_\mu) x_\rho = \frac{s}{x^2} (x\wedge v)_{\mu\nu} = s^2 P_{[\mu\nu]}.$$ We don’t only see that is satisfied, yet also that the only contribution to the right hand side of that equation comes from $\frac{2}{3} 2 v_{[\mu} \Theta^{ij}_{\nu]\rho} Q^\rho_{ij}$ while it is exactly the term that vanishes, $\Theta^{ij}_{\mu\nu} v_\rho Q^\rho_{ij}$, that contributes in the in the Sasaki-Einstein case to be discussed below. Note that the superconformal supersymmetries involve non-zero trace of $P$. Hence, these supersymmetries are broken by the background scalar VEV corresponding to $g_{YM}^{-2}$. This just reflects the general wisdom that the 5d YM coupling, being dimensionful, breaks conformal invariance. $\mathbb{R}\times S^4$ ---------------------- Consider now $\mathbb{R}\times S^4$, with $\mathbb{R}$ parametrized by $x^5=\tau$ and $v$ not along $\frac{\partial}{\partial\tau}$. As described in [@Kim:2012gu] – where the explicit spinor solutions are written as well, the spinors satisfy $$\nabla_{\mu}\epsilon^q=-\frac{1}{2}\gamma_{\mu}\,\gamma_5\epsilon^q\, ,\qquad \nabla_{\mu}\epsilon^s=\frac{1}{2}\gamma_{\mu}\,\gamma_5\epsilon^s.$$ Here $\epsilon^{q,\,s}$ generate Poincaré and superconformal supersymmetries respectively. It is straightforward to see that these solutions fit in our general scheme with $$Q^{ij}_{\mu}= \pm\frac{1}{2s} w_{\kappa}\Theta^{ij\kappa}_{\phantom{ij\kappa}\mu},\qquad P_{[\mu\nu]}=\mp \frac{1}{2s}\,(w\wedge v)_{\mu\nu}, \qquad P_{(\mu\nu)}=\mp \frac{1}{2s}\,w_{\kappa} v^{\kappa}\,g_{\mu\nu},$$ where upper signs correspond to the $\epsilon_q$ while lower signs correspond to the $\epsilon_s$. In addition we have defined $w=d\tau$. Note that the trace of $P$ does not vanish, implying that $v$ is conformal Killing. Thus this is a genuine solution of superconformal supergravity that cannot be embedded in $\mathcal{N}=1$ Poincaré supergravity. Moreover, as discussed above, this implies that no (constant) Yang-Mills coupling can be turned on on this background (see [@Kim:2014kta] for a further discussion in the maximally supersymmetric case). Topological twist on $\mathbb{R}\times M_4$ ------------------------------------------- Manifolds of the form $\mathbb{R}\times M_4$ can be regarded as supersymmetric backgrounds at the expense of turning on a non-zero $V$ such that the spinors are gauge-covariantly constant $$\mathcal{D}_{\mu}\epsilon^i=0.$$ To show that we consider $v=\partial_{\tau}$, being $\tau$ the coordinate parametrizing $\mathbb{R}$. Then, from , it follows that $P_{\mu\nu}=0$. Furthermore, by choosing $V=Q$ – which translates into $W_{\mu}=0$, $\Delta^{ij}=0$ and implies $T_{\mu\nu}=0$ – all the remaining constraints are automatically solved. This is nothing but the topological twist discussed in [@Rodriguez-Gomez:2015xwa] (see also [@Anderson:2012ck] for the maximally supersymmetric case; twisted theories on five manifolds were also considered in [@Bak:2015hba]). Note that since $P=0$, in these backgrounds the Yang-Mills coupling can indeed be turned on. $\operatorname{SU}(2)_R$ twist on $M_5$ --------------------------------------- If $M_5$ is not a direct product, one can still perform an $SU(2)_R$ twist. For $v$ Killing, the details of this can be found in [@Pan:2015nba]. One can perform an identical calculation for the conformal supergravity in question. In the case of a $\mathbb{R}$ or $\operatorname{U}(1)$ bundle over some $M_4$ for example, one finds $T$ to be the curvature of fibration. Sasaki-Einstein manifolds ------------------------- For a generic Sasaki-Einstein manifold the spinor satisfies $$\nabla_\mu \epsilon_i = -\frac{\imath}{2} \gamma_\mu (\sigma^3)_i^{\phantom{i}j} \epsilon_j.$$ It follows that $$P_{\mu\nu} = -\frac{\imath}{2} s^{-1} (\sigma^3_{ij} \Theta^{ij})_{\mu\nu}, \qquad Q_\mu^{ij} = - \frac{\imath}{2} s^{-1} v_\mu (\sigma^3)^{ij}.$$ Clearly $$s^2 P_{[\mu\nu]} = -\frac{\imath}{2} s (\sigma^3_{ij} \Theta^{ij})_{\mu\nu} = \Theta_{\mu\nu}^{ij} Q^\rho_{ij} v_\rho.$$ Hence, upon taking $V_\mu^{ij} = 0 = T_{\mu\nu}$, we indeed have a solution of and . Note that the trace of $P$ is vanishing, and hence in these backgrounds the Yang-Mills coupling can be turned on. This holds also for Sasakian manifolds. Super Yang-Mills theories on these were considered in e.g. [@Qiu:2013pta]. $S^5$ ----- The $S^5$ case is paticularly interesting as well, as it leads to the supersymmetric partition function [@Hosomichi:2012ek; @Kallen:2012va]. Not surprisingly, since $S^5$ can be conformally mapped into $\mathbb{R}^5$, the solution fits into our general discussion including two sets of spinors, one corresponding to the Poincaré supercharges and the other corresponding to the superconformal supercharges. Writing the $S^5$ metric as that of conformally $S^5$ as $$ds^2=\frac{4}{(1+\vec{x}^2)^2}\,d\vec{x}^2,$$ we find for the Poincare supersymmetries $$Q^{ij}_{\mu}= \frac{1}{2s} x_{\kappa}\Theta^{ij\kappa}_{\phantom{ij\kappa}\mu},\qquad P_{[\mu\nu]}=- \frac{1}{2s}\,(x\wedge v)_{\mu\nu}, \qquad P_{(\mu\nu)}=- \frac{1}{2s}\,x_{\kappa} v^{\kappa}\,g_{\mu\nu}.$$ For the superconformal supercharges on the other hand, we find $$Q^{ij}_{\mu}= -\frac{1}{2sx^2} x_{\kappa}\Theta^{ij\kappa}_{\phantom{ij\kappa}\mu},\qquad P_{[\mu\nu]}=\frac{1}{2sx^2}\,(x\wedge v)_{\mu\nu}, \qquad P_{(\mu\nu)}= \frac{1}{2sx^2}\,x_{\kappa} v^{\kappa}\,g_{\mu\nu}.$$ Note that in both these cases the trace of $P$ is non-zero, so neither of these spinors are preserved if we deform the theory with a Yang-Mills coupling. Nevertheless it is possible to find a combination of supercharges which does allow for that. This can be easily understood by looking at the explicit form of the spinors, which in these coordinates is simply $$\epsilon_q^i=\frac{1}{\sqrt{1+\vec{x}^2}}\epsilon_0^i,\qquad \epsilon_s^i=\frac{1}{\sqrt{1+\vec{x}^2}}\slashed{x}\eta_0^i,$$ being $\epsilon_0^i$ and $\eta_0^i$ constant spinors. Considering for instance $\slashed{\nabla}\epsilon_{q}^i\supset P_{[\mu\nu]}\gamma^{\mu}\gamma^{\nu}\epsilon_q^i+P^{\mu}_{\phantom{\mu}\mu}\epsilon^i_q$, we see that the term with $P_{[\mu\nu]}$ involves a contraction $\slashed{x}\epsilon_q^i$ which is basically $\epsilon_s^i$. This suggests that one might consider a certain combination of $\epsilon_q$ and $\epsilon_s$ for which the effective $P$-trace is a combination of $P_{[\mu\nu]}$ and $P^\mu_{\phantom{\mu}\mu}$ which might vanish. Indeed one can check that this is the case. Choosing for instance the Majorana doublet $\xi^i$ constructed as $$\xi^1=\epsilon_q^1+\epsilon_s^2,\qquad \xi^2=\epsilon_q^2-\epsilon_s^1,$$ it is easy to see that it satisfies $$\nabla_\mu \epsilon_i = -\frac{\imath}{2} \gamma_\mu (\sigma^2)_i^{\phantom{i}j} \epsilon_j;$$ that is, the same equation as that for the Sasaki-Einstein case. Therefore, borrowing our discussion above, it is clear that it admitts a Yang-Mills kinetic term. Indeed, this is corresponds, up to conventions, to the choice made in [@Hosomichi:2012ek; @Kallen:2012va] to compute the supersymmetric partition function. Conclusions {#sec:conclusions} =========== In this paper we have studied general solutions to $\mathcal{N}=2$ conformal supergravity. In the spirit of [@Festuccia:2011ws], these provide backgrounds admitting five-dimensional supersymmetric quantum field theories. The starting point of our analysis, being conformal supergravity, requires that such quantum field theories must exhibit conformal invariance. In particular, the action for vector multiplets must be the cubic completion of 5d Chern-Simons term instead of the standard quadratic Maxwell one. However, since the Yang-Mills coupling can be thought as a VEV for the scalar in a background vector multiplet, we can regard gauge theories as conformal theories conformally coupled to background vector multiplets whose VEVs spontaneously break conformal invariance. From this perspective it is very natural to consider superconformal supergravity as the starting point to construct the desired supersymmetric backgrounds. We have described the most generic solution to $\mathcal{N}=2$ five-dimensional conformal supergravity (see also [@Kuzenko:2014eqa]). By expanding spinor covariant derivatives in intrinsic torsions we have been able to find a set of algebraic equations , , , together with a set of differential constraints , characterizing the most general solution. Interestingly, the solutions admit transverse holomorphic foliations if the $\operatorname{SU}(2)_R$ connection $R^Q$ “abelianizes” by lying along a $\operatorname{U}(1)$ inside $\operatorname{SU}(2)_R$, in agreement with the discussion in [@Alday:2015lta]. On general grounds, the only obstruction to the existence of supersymmetric backgrounds is the requirement of a conformal Killing vector. On the other hand we have showed that only when the vector becomes actually Killing a constant VEV for background vector multiplet scalars can be turned on. This shows that all cases where a Yang-Mills theory with standard Maxwell kinetic term can be supersymmetrically constructed are in fact captured by Poincaré supergravity. On the other hand, on backgrounds admitting only a conformal Killing vector we can still turn on a Yang-Mills coupling at the expense of being position-dependent. While this is certainly non-standard, in particular this allows to think of the quadratic part of the Yang-Mills action as the regulator in index computations. Having constructed all supersymmetric backgrounds of $\mathcal{N}=2$ superconformal supergravity, the natural next step would be the computation of supersymmetric partition functions. In particular, it is natural to study on what data they would depend along the lines of *e.g.* [@Closset:2013vra]. For initial progress in this direction see [@Imamura:2014ima; @Pan:2015nba]. We postpone such study for future work. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank S. Kuzenko, P. Meessen, Y. Pan and D. Rosa for useful conversations and correspondence. The authors are partly supported by the spanish grant MINECO-13-FPA2012-35043-C02-02. In addition, they acknowledge financial support from the Ramon y Cajal grant RYC-2011-07593 as well as the EU CIG grant UE-14-GT5LD2013-618459. The work of A.P is funded by the Asturian government’s SEVERO OCHOA grant BP14-003. The work of J.S. is funded by the Asturian government’s CLARIN grant ACB14-27. A.P. would like to acknowledge the String Theory Group of the Queen Mary University of London and the String Theory Group of the Imperial College of London for very kind hospitality during the final part of this project. Moreover the authors would like to acknowledge the support of the COST Action MP1210 STSM. Conventions {#sec:conventions} =========== We use the standard NE-SW conventions for $\operatorname{SU}(2)_R$ indices $\{ i, j, k, l \}$ with $\epsilon^{12} = \epsilon_{12} = 1$. The charge conjugation matrix $C$ is antisymmetric, hermitian and orthogonal, i.e. $C^* = C^T = -C = C^{-1}$. Its action on gamma matrices is given by $(\gamma^a)^* = (\gamma^a)^T = C \gamma^a C^{-1}$. In general we choose not to write the charge conjugation matrix explicitly; thus $\epsilon^i \eta^j = (\epsilon^i)^T C \eta^j$. Antisymmetrised products of gamma matrices are defined with weight one, $$\gamma_{a_1 \dots a_p} = \frac{1}{p!} \gamma_{[a_1} \dots \gamma_{a_p]},$$ yet contractions between tensors and gamma matrices are not weighted. $$\gamma \cdot T = \gamma^{\mu\nu} T_{\mu\nu}.$$ In general, symmetrization $T_{(\mu_1 \dots \mu_p)}$ and antisymmetrization $T_{[\mu_1 \dots \mu_p]}$ are with weight one however. One can impose a symplectic Majorana condition $$\label{eq:symplectic_Majorana} \epsilon^{ij} (\epsilon^j)^* = C \epsilon^i,$$ yet as we mentioned in the main body of this paper it is generally sufficient for us to assume $s$ to be non-vanishing and $v$ to be real. Using Fierz identities, one finds the following identities involving the spinor bilinear $\Theta^{ij}$: [rCl]{} \^[ij]{}\_ \^[kl]{} &=& s\^2 (\^[ik]{} \^[jl]{} + \^[il]{} \^[jk]{}),\ \_\^[ij]{} \^\_[ij]{} &=& (\_\^\_\^- \_\^\_\^) - \_\^ v\_\[eq:double\_theta\_identity\]\ \_\^[ij]{} \^[kl ]{} &=& - (\^[ik]{} \^[jl]{} + \^[il]{} \^[jk]{}) \_\^ + (\^[jk]{} \^[il]{} + \^[ik]{} \^[jl]{} + \^[jl]{} \^[ik]{} + \^[il]{} \^[jk]{})\_\^ \[eq:double\_theta\_identity\_2\]. Details of the computation {#sec:computation} ========================== In this appendix we summarize the most relevant details of the computation that lead us to the two equations and and to the three differential equations , and , that allow to determine $\Delta^{ij}$, the scalar $D$ and the vector $W_{\kappa}$. Gravitino equation ------------------ In this subsection we furnish further details for the derivation of the equations and . As explained in section \[sec:general\_solutions\] we rewrite the covariant derivative acting on the spinor $\epsilon^{i}$ as $$\label{eq:cov} \mathcal{D}_\mu \epsilon^i = \nabla_\mu \epsilon^i - V^{ij}_{\mu}\epsilon_j = P_{\mu\nu} \gamma^{\nu}\epsilon^i + (Q-V)_\mu^{ij}\epsilon_j.$$ Inserting this expression for the covariant derivative in the gravitino equation we obtain [rCl]{} \[eq:gravitino\_cov\] 0 &=& P\_[\[\]]{} \^\^i + \_ P\^[\[\]]{} \^ \^i + P\_[()]{} \^\^i - \_P\^\_ \^i\ &&+ (Q\_\^[ij]{} - V\_\^[ij]{}) \_j - \_ (Q\^[ij]{} - V\^[ij]{}) \_j - \_ T\^ \^ \^i - 3 T\_ \^\^i\ &=& ( P\_[()]{} - g\_ P\^\_ ) \^\^i + (Q-V)\_\^[ij]{} \_j - \_ (Q - V)\^[ij]{} \_j\ &&+ (P\_[\[\]]{} -4T\_) \^\^i + \_ (P\^[\[\]]{} - 4T\^) \^ \^i. \[eq:gravitino\_variation\_in\_terms\_of\_intrinsic\_torsions\] We manipulate the previous expression, as discussed in section \[sec:general\_solutions\], multiplying it from the left by $\epsilon_i \gamma_\kappa$. In this way we obtain the equation . While we obtain the equation multiplying the equation by $\epsilon^j$ and symmetrizing in the indices $i$ and $j$. In order to recover the equation we have to determine $(P - 4 \imath T)^+$ and $(P - 4 \imath T)^-$. Therefore we project the equation on the horizontal space using the projector operator $\Pi^{\mu}_{\nu} = \delta^{\mu}_{\nu} -s^{2}v^{\mu}v_{\nu}$. We find $$0 = \frac{5}{8} \Pi_\mu^\kappa \Pi_\nu^\lambda \left[ (P - 4 \imath T)_{[\kappa\lambda]} + \frac{1}{2} \epsilon_{\kappa\lambda\sigma\tau\rho} (P - 4 \imath T)^{\sigma\tau} v^\rho \right] = \frac{5}{4} (P - 4 \imath T)^+.$$ This means that $\Pi(P-4iT)$ is anti-self dual. On the other hand contracting the equation with $\Theta_{ij\kappa\lambda}$ and using the identity we get $$0 = s^3 \left( \Pi_{\kappa\mu} \Pi_{\lambda\nu} - \frac{1}{2} s^{-1} \epsilon_{\kappa\lambda\mu\nu\rho} v^\rho \right) (P - 4\imath T)^{[\kappa\lambda]} - 2 s \Theta_{ij\kappa\lambda} (Q -V)^{ij}_\rho v^\rho.$$ Solving the previous expression we obtain $(P - 4\imath T)^- = s^{-2} \Theta^{ij} \imath_v (Q - V)_{ij}$. Therefore we know all the components of $(P-4iT)$, since we have an equation for $(P-4iT)^{+}$, an equation for $(P-4iT)^{-}$ and finally an equation for $\imath_v(P-4\imath T)$. Putting these information together we recover the equation . In order to determine the equation we evaluate the projection of the equation , using the identities and we get $$0 = \frac{1}{2} s \Pi_\mu^\nu (Q - V)_\nu^{ij} + \frac{1}{4} (Q - V)^{\nu (j}_{\phantom{\nu (j}k} \Theta^{i)k}_{\mu\nu} + \frac{1}{6} s^{-1} \Theta_{\mu\nu}^{ij} \Theta^{\nu\rho}_{kl} (Q -V)^{kl}_\rho.$$ Using the identity the previous expression becomes $$0 = s \Pi_\mu^{\phantom{\mu}\nu} (Q-V)_{\nu\phantom{i}j}^{\phantom{\nu}i} + \frac{1}{2} [(Q-V)^\nu, \Theta_{\mu\nu}]^i_{\phantom{i}j}.$$ Obtaining in this way the equation . Dilatino equation ----------------- In this subsection we furnish further details regarding the derivation of the equations , and . The most involved terms that appear in the equation are [rCl]{} \^\_\^i &=& P\^\_ \^i - \^\^P\_[\[\]]{} \^i + (P\^\_)\^2 \^i + P\_[\[\]]{} P\^[\[\]]{} \^i\ &&- \^(Q-V)\_[j]{}\^[i]{} \^j - V\_[j]{}\^[i]{} (Q-V)\_[k]{}\^[j]{} \^k + (Q-V)\_[j]{}\^[i]{} Q\_[k]{}\^[j]{} \^k\ &&+ P\^\_ \^(Q-V)\_\^[ij]{} \_j - 2 \^P\_[\[\]]{} (Q-V)\^[ij]{} \_j and [rCl]{} \_ T\^ \^\^i &=& \_ T\^ P\^[\[\]]{} \^i + P\^\_ T\_ \^ \^i - 2 P\_[\[\]]{} T\^\_ \^ \^i\ &&+ \_ T\^ (Q - V)\^[ij]{} \_j,\ \^T\_ \^\^i &=& - P\_[\[\]]{} T\^ \^i - P\_[\[\]]{} T\^\_ \^ \^i + P\^\_ T\_ \^ \^i + \^T\_ (Q-V)\^[ij]{} \_j. #### The symmetric contraction Multiplying the equation by $\epsilon^{j}$ and symmetrizing in $i$ and $j$ we obtain [rCl]{} 0 &=& \^[(i]{} \^\_\^[j)]{} + \^[(i]{} \_ T\^ \^\^[j)]{} + \^[(i]{} \^T\_ \^\^[j)]{}\ &&+ \^[(i]{} \_ \^[j)]{} \^T\^. The individual components are [rCl]{} \^[(i]{} \^\_\^[j)]{} &=&\ &&+ P\^\_ v\_(Q-V)\^[ij]{} - v\^P\_[\[\]]{} (Q-V)\^[ij]{},\ \^[(i]{} \_ T\^ \^\^[j)]{} &=& P\^\_ T\^ \^[ij]{}\_ - 2 P\_[\[\]]{} T\^\_ \^[ij]{} + \_\^ T\^ \^[k(i]{}\_ (Q-V)\^[j)]{}\_[k]{},\ \^[(i]{} \^T\_ \^\^[j)]{} &=& -P\_[\[\]]{} T\^\_ \^[ij]{} + P\^\_ T\^ \^[ij]{}\_ + v\_T\^ (Q-V)\^[ij]{}\_,\ \^[(i]{} \_ \^[j)]{} \^T\^ &=& - \_\^ \_\^[ij]{} \^T\^. Putting the various terms together we recover the expression . #### The vector contraction Multiplying the equation with $\epsilon^i \gamma_\mu$ and contracting we obtain [rCl]{} 0 &=& v\_( + T\_ T\^ ) + \^i \_\^\_\_i + \^i \_\_ T\^ \^\_i\ &&+ \^i \_\^T\_ \^\_i + \_\^ v\_\_T\_ + \^T\_ - \_\^ T\_ T\_. The most involved terms are given by [rCl]{} \^i \_\^\_\_i &=& \_P\^\_ - s \^P\_[\[\]]{} - P\^\_ \^[ij]{}\_ (Q-V)\^\_[ij]{} + 2 \^[ij]{}\_ P\^[\[\]]{} (Q-V)\_[ij]{}\ &&+ v\_,\ \^i \_\_ T\^ \^\_i &=& s \_ T\^ P\^[\[\]]{} + P\^\_ T\_ v\^- 2 (P\_[\[\]]{} T\^\_ - P\_[\[\]]{} T\^\_) v\^\ &&- (Q-V)\_[ij]{} \_\^[ij]{} T\^ - 2 T\_ \^[ij]{} (Q-V)\_[ij]{},\ \^i \_\^T\_ \^\_i &=& - v\_P\_[\[\]]{} T\^ - (P\_[\[\]]{} T\^\_ - P\_[\[\]]{} T\^\_) v\^+ P\^\_ T\_ v\^\ &&- \^[ij]{}\_ T\^ (Q-V)\_[ij]{}. Finally putting the various terms together and projecting on the vertical component we recover the equation . While projecting on the horizontal component we recover the equation . =.97 [^1]: Note that the theories outside of this class do require a (presumably stringy) UV completion. Hence the class of theories which we are considering is in fact the most generic class of 5d supersymmetric quantum field theories. [^2]: \[fn:caveat\_vanishing\_spinor\] This statement assumes the spinor – and thus the vector – to be non-vanishing. In the case of Poincaré supergravity, this is always the case if the manifold is connected. After all, the relevant KSE is of the form $\partial_\mu \epsilon^i = \mathcal{O}(\epsilon^i)$. If the spinor vanishes at a point, it vanishes on the whole manifold. For conformal supergravity however, the KSE takes the form of a twistor equation, $\partial_\mu \epsilon^i - \frac{1}{4} \gamma_{\mu\nu} \partial^\nu \epsilon^i = \mathcal{O}(\epsilon^i)$, which has non-trivial solutions even if the right hand side vanishes. The simplest example of this is given by the superconformal supersymmetry in $\mathbb{R}^5$. See section \[sec:example\_flat\_R5\]. Here $\epsilon^i \vert_{x^\mu = 0} = v \vert_{x^\mu = 0} = 0$, yet the global solution is non-trivial. In such cases a more careful analysis is necessary. [^3]: A word on notation is in order here. We stress that we are discussing minimal supersymmetry in five dimensions. [^4]: \[fn:Theta\_identities\] Explicitly, the self-duality condition is $$\Theta^{ij}_{\mu\nu} = - \frac{1}{2} s^{-1} \epsilon_{\mu\nu\kappa\lambda\rho} \Theta^{ij\kappa\lambda} v^\rho, \qquad \epsilon_{\lambda\mu\nu\sigma\tau} \Theta^{ij\sigma\tau} = -3! s^{-1} \Theta_{[\lambda\mu}^{ij} v_{\nu]}.$$ [^5]: We found the Mathematica package `xAct` [@martin2007invar; @MartinGarcia:2008qz] very useful. [^6]: One might wonder that the cubic lagrangian theory is enough. However, in some cases such as *e.g.* $\operatorname{Sp}$ gauge theories, such cubic lagrangian is identically zero. [^7]: Since $\Phi$ is invariant under $m_{ij} \mapsto f m_{ij}$ for any non-vanishing function $f:M \to \mathbb{R}$ it might be more appropriate to think of $m_{ij}$ as a ray in the three-dimensional $\operatorname{su}(2)$ vector space. From this point of view, $m_{ij}$ is a map $$m : M \to S^2 \subset \operatorname{su}(2).$$ [^8]: The similar integrability condition $[T^{1,0}, T^{1,0}] \subseteq T^{1,0}$ defines a CR manifold.

--- author: - | Y. Wang$^1$, J.P. Gill$^2$, H.J. Chiel$^{2,3}$, P.J. Thomas$^{2,4}$\ $^1$ Mathematical Biosciences Institute, The Ohio State University, Columbus, OH\ $^2$ Department of Biology, Case Western Reserve University, Cleveland, OH\ $^3$ Depts of Biomedical Engineering, Neurosciences, Case Western Reserve University, Cleveland, OH\ $^4$ Dept. Mathematics, Applied Mathematics, and Statistics,\ Case Western Reserve University, Cleveland, OH title: 'Shape *versus* timing: linear responses of a limit cycle with hard boundaries under instantaneous and static perturbation' --- Introduction {#sec:intro} ============ A runner adjusts her stride and posture as she leans into a hill; a climber adjusts the timing and tension of his grasp as he works along a steepening incline; a frog adjusts the location and intensity of wiping as it tries to remove an irritant from its skin. In each of these scenarios, a neuromechanical motor control system exhibits periodic motions that make and break contact with a physical substrate, and adjust the shape and timing of the motion in response to parametric changes in the environmental conditions. As these examples illustrate, limit cycles with sliding components are commonplace in biological motor control systems. In this paper we develop the variational and phase response curve analysis needed to understand the stability and robustness of such systems under parametric variation. A *limit cycle with sliding component* (LCSC) is a closed, isolated, periodic orbit of an $n$-dimensional, autonomous, deterministic nonsmooth dynamical system, in which the trajectory is constrained to move along a surface of dimension $k<n$ during some portion of the orbit. As an example, consider a person running. The dynamics of neural activity, muscle activation, joint position, and center-of-mass together form a system of nonlinear ordinary differential equations in $n\gg 1$ dimensions. Models of such systems are often constructed to exhibit asymptotically stable limit cycle trajectories [@holmes2006; @revzen2011; @guckenheimer2018]. Each of the runner’s legs passes alternately through a swing phase and a stance phase, in which the foot is respectively free or in contact with the ground. At the points of making contact (the heel strike) and breaking contact (liftoff of the toe) the dynamics makes a nonsmooth transition into and out of a lower dimensional submanifold of the state space. While the foot is in contact with the ground, the dynamics has fewer degrees of freedom than when the foot moves unconstrained. Transitions onto and off of constraint surfaces occur in many motor control systems. For example, in the transition from biting (attempting to grasp food) to swallowing, an organism makes an initial physical contact with the food; any forces exerted by the food on the organism appear abruptly upon first contact. When a mollusk such as *Aplysia californica* swallows a long stipe of seaweed, the grasper organ repeatedly makes and breaks contact as it pulls seaweed into the gut, through movements orchestrated by a central pattern generator (CPG) circuit [@sutton2004; @chiel2007]. In hindlimb scratching, another CPG driven behavior, the foot repeatedly makes and breaks contact with the underbelly [@barajon1992; @gelfand1988; @mortin1989]. The neural dynamics *internal* to central pattern generators may also exhibit nonsmooth sliding components. Consider a firing rate model for a half-center oscillator CPG with inhibition-mediated synaptic connections. Inhibition tends to drive the firing rate of the postsynaptic cell towards zero. Once an inhibited cell shuts off, its firing rate cannot be further reduced: there is a hard boundary at zero firing rate. Recordings of CPG activity in both vertebrate and invertebrate preparations show cells with bursts of activity switching on and switching off during different phases of activity patterns. While one cell’s firing rate is held at zero, other system components continue to evolve. The study of limit cycle motions in such piecewise smooth systems requires analytical tools beyond the existing arsenal of phase response curves and variational analysis, developed for systems with smooth (differentiable) right-hand sides [@spardy2011a; @spardy2011b]. The resilience of neuromotor activity under perturbations of modest size may be studied via variational analysis. This analysis is particularly tractable in two extremes: when perturbations cause small instantaneous dislocations of the trajectory, and when perturbations are sustained over long times (parametric perturbation). For small instantaneous displacements, analysis in terms of infinitesimal phase response curves (iPRC) is well established when the underlying dynamics is smooth, i.e. differentiable [@Park2017]. The iPRC analysis has been extended to nonsmooth dynamics, provided the flow is always transverse to any switching surfaces at which nonsmooth transitions occur [@shirasaka2017; @park2018; @CGL18; @wilson2019]. Limit cycles with sliding components violate the transverse flow condition; to our knowledge, we are the first to extend iPRC analysis to this case. Variational analysis has likewise been extended to nonsmooth dynamics for studying the linearized effect on the shape of a trajectory following a small instantaneous perturbation [@bernardo2008; @LN2013]. Again, this analysis requires the transverse flow condition, excluding limit cycles with sliding components. In the present paper we extend both iPRC and variational methods to the LCSC in nonsmooth systems, for instantaneous as well as for parametric perturbations. Parametric perturbations arise in motor control systems in several circumstances. As a runner advances along a steeper and steeper path, the general pattern of neural activity and biomechanics remains similar, but the timing and intensity of muscle activations changes in detail to accommodate the changing slope of the hill. As the sea slug progressively ingests a stipe of seaweed, its thickness may increase and its pliability may decrease; the motor activation pattern must adjust accordingly. The CPG regulating respiration adjusts the breathing rate as the runner’s metabolic demand increases. In each of these examples, the dynamical system exhibiting the LCSC becomes a family of dynamical systems indexed by one or more parameters. In general, a small fixed change in a parameter gives rise to a new limit cycle, with different shape and timing than the original. This paper develops the mathematical framework required to analyze the changes in shape and timing wrought by parametric changes, such as changing mechanical or metabolic load, on motor control systems with CPG-driven limit cycles with sliding components. We review the classical variational and phase response curve analysis for the response of smooth dynamical systems to instantaneous perturbations in §\[sec:smooth-inst\]. In order to account for the response of an oscillator to parametric perturbations, we derive an *infinitesimal shape response curve* (iSRC) in §\[sec:smooth-sust\]. In contrast to standard variational analysis, which neglects timing changes, the iSRC takes into account both timing and shape changes arising due to a parametric perturbation. The *infinitesimal phase response curve* (iPRC) captures the change in timing of an oscillator due to an instantaneous perturbation, as well as the global change in period in response to a parametric perturbation. However, in many applications, the impact of a perturbation on local timing can be as important as the global effects. For instance, in the feeding of the marine mollusk *Aplysia californica*, a static perturbation such as a force applied to the animal’s food can only be felt when its grasper or jaws are closed on the food. Similarly, any motor control system that operates by making and breaking physical contact (walking, scratching or grasping) would experience perturbations limited to a discrete component of the limit cycle. In these cases, one would need to compute the local timing changes of the trajectory during the phase when the perturbation exists (e.g., the grasper is closed) to understand the robustness of this system. Such local change is often different from the global timing change and hence cannot be obtained using the iPRC. To this end, in §\[sec:smooth-local\] and Appendix \[ap:T1\], we develop a *local timing response curve (lTRC)* that is analogous to the iPRC but measures the local timing sensitivity of a limit cycle within any given local region. Development of the lTRC leads to a piecewise-specified version of the iSRC which often exhibits greater accuracy. While the applicability of the classical perturbative methods from §\[sec:smooth-theory\] is generally limited by the constraint that the dynamics of the system is smooth, some elements of the methods have already been generalized to nonsmooth systems with only transversal crossing boundaries, which will be reviewed in §\[sec:trans\]. Moreover, we extend these methods to the LCSC case in nonsmooth systems, both for instantaneous and parametric perturbations, in §\[sec:sliding\]. Throughout, we consider nonsmooth systems with degree of smoothness one or higher; that is, systems with continuous trajectories, also known as *Filippov* systems [@bernardo2008]. Our main result is Theorem \[thm:main\], which describes the behavior of the variational and infinitesimal phase response curve dynamics when the LCSC of a Filippov system enters and exits a hard boundary. Appendix \[ap:proof\] gives a proof of the theorem. Numerical algorithms for implementing these methods are presented in Appendix \[sec:algorithm\]. In §\[sec:toy-model\], we illustrate both the theory and algorithms using a planar model, comprising a limit cycle with a linear vector field in the interior of a simply connected convex domain with a hard boundary condition. In this example, we show that under certain circumstances (e.g. non-uniform perturbation), the iSRC together with the lTRC provides a more accurate representation of the combined timing and shape responses to static perturbations than using the global iPRC alone. Surprisingly, we discover nondifferentiable “kinks" in the isochron function that propagate backwards in time along an osculating trajectory that encounters the hard boundary exactly at the liftoff point (the point where the limit cycle trajectory smoothly departs the boundary). Lastly, we discuss limitations of our methods and possible future directions in §\[sec:discussion\]. Appendix \[ap:symbols\] provides a table of symbols used in the paper. Linear responses of smooth systems {#sec:smooth-theory} ================================== In this section we review the classical theory for linear approximation of the effects of instantaneous and sustained perturbation on the timing and shape of a limit cycle trajectory in the smooth case. Consider an $n$-dimensional $C^1$ dynamical system (a system of ordinary differential equations in $n$ variables with velocity $F({\mathbf{x}})$ having continuous first derivatives with respect to the components of ${\mathbf{x}}$), $$\label{eq:dxdt=F} \frac{d\textbf{x}}{dt}=F(\textbf{x}),$$ with a period $T$ stable limit cycle (LC) solution $\gamma(t)=\gamma(t+T)$, that is, a closed, isolated periodic orbit attracting all trajectories originating within some open set containing the LC. The effects of small, instantaneous perturbations on the shape and timing of trajectories near the LC are captured by the variational and infinitesimal phase response curve analysis, which we review in §\[sec:smooth-inst\]. In §\[sec:smooth-sust\], we also derive the infinitesimal shape response curve (iSRC) to account for the combined shape and timing response of $\gamma(t)$ under *static* perturbations. To obtain a more accurate iSRC when the limit cycle experiences different timing sensitivities in different regions within the domain, in §\[sec:smooth-local\], we introduce the *local timing response curve* (lTRC). In contrast to the iPRC, which measures the global shift in the period $T$, the lTRC lets us compute the timing change of $\gamma(t)$ within regions bounded between specified Poincaré sections. Shape and timing response to instantaneous perturbations {#sec:smooth-inst} -------------------------------------------------------- Suppose a small, brief perturbation is applied at time $t_0$ such that there is a small abrupt perturbation in the state space. We have $$\label{eq:initial-inst-perturbation} \tilde{\gamma}(t_0)= \gamma(t_0)+\varepsilon P,$$ where $\tilde{\gamma}$ indicates the trajectory subsequent to the instantaneous perturbation, $\varepsilon$ is the magnitude of the perturbation, and $P$ is the unit vector in the direction of the perturbation in the state space. As we show below, the effects of the small brief perturbation $\varepsilon P$ on the shape and timing of the limit cycle trajectory are given, respectively, by the solution of the variational equation , and the iPRC which solves the adjoint equation . The evolution of a trajectory $\tilde{\gamma}(t)$ close to the limit cycle $\gamma(t)$ may be approximated as $\tilde{\gamma}(t)=\gamma(t)+{\mathbf{u}}(t)+O({\varepsilon}^2)$, where ${\mathbf{u}}(t)$ satisfies the *variational equation* $$\label{eq:var} \frac{d{\mathbf{u}}}{dt}=DF(\gamma(t)){\mathbf{u}}$$ with initial displacement ${\mathbf{u}}(t_0)={\varepsilon}P$ given by , for small ${\varepsilon}$. Here $DF(\gamma(t))$ is the Jacobian matrix evaluated along $\gamma(t)$. On the other hand, an iPRC of an oscillator measures the timing sensitivity of the limit cycle to infinitesimally small perturbations at every point along its cycle. It is defined as the shift in the oscillator phase $\theta\in [0, T)$ per size of the perturbation, in the limit of small perturbation size. The limit cycle solution takes each phase to a unique point on the limit cycle, $\textbf{x}=\gamma(\theta)$, and its inverse maps each point on the cycle to a unique phase, $\theta=\phi(\textbf{x})$. We extend the domain of $\phi(\textbf{x})$ to points in the basin of attraction $\mathcal{B}$ of the limit cycle by defining the *asymptotic phase*: $\phi(\textbf{x}): \mathcal{B}\rightarrow [0,T)$ with $$\frac{d\phi(\mathbf{x}(t))}{dt}=1,\quad \phi(\mathbf{x}(t))=\phi(\mathbf{x}(t+T)).$$ If $\textbf{x}_0\in \gamma(t)$ and $\textbf{y}_0 \in \mathcal{B}$, then we say that ${\mathbf{y}}_0$ has the same asymptotic phase as ${\mathbf{x}}_0$ if ${\left\lVert{\mathbf{x}}(t; {\mathbf{x}}_0)-{\mathbf{y}}(t;{\mathbf{y}}_0) \right\rVert}\to 0$, as $t\to \infty$. This means that $\phi({\mathbf{x}}_0)=\phi({\mathbf{y}}_0)$. The set of all points off the limit cycle that have the same asymptotic phase as the point ${\mathbf{x}}_0$ on the limit cycle is the *isochron* with phase $\phi({\mathbf{x}}_0)$. Suppose $\varepsilon P$ applied at phase $\theta$ results in a new state $\gamma(\theta)+\varepsilon P \in\mathcal{B}$, which corresponds to a new phase $\tilde{\theta}=\phi(\gamma(\theta)+\varepsilon P)$. The phase difference $\tilde{\theta}-\theta$ defines the phase response curve (PRC) of the oscillator. One defines the iPRC as the vector function ${{\mathbf z}}:[0,T)\to{{\mathbf R}}^n$ satisfying $$\label{eq:definition_of_iPRC} {{\mathbf z}}(\theta)\cdot{P}=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}\left(\phi(\gamma(\theta)+\varepsilon{P})-\theta\right) = \nabla_{{\mathbf{x}}}\phi(\gamma(\theta))\cdot{P}$$ for arbitrary unit perturbation ${P}$. The first equality serves as a definition, while the second follows from routine arguments [@brown2004; @ET2010; @SL2012; @Park2017]. It follows directly that the vector iPRC is the gradient of the asymptotic phase and it captures the phase (or timing) response to perturbations in any direction $P$ in state space. Since the vector field $F$ is assumed to be $C^1$, the iPRC is a continuous $T$-periodic solution satisfying the *adjoint equation* [@SL2012], $$\label{eq:prc} \frac{d{{\mathbf z}}}{dt}=-DF(\gamma(t))^\intercal {{\mathbf z}},$$ with the normalization condition $$\label{eq:prc-normalization} F(\gamma(\theta))\cdot {{\mathbf z}}(\theta) = 1.$$ By direct calculation, one can show that the solutions to the variational equation and the adjoint equation satisfy ${\mathbf{u}}^\intercal{{\mathbf z}}=\text{constant}$: $$\label{eq:var-prc} \frac{d({\mathbf{u}}^\intercal {{\mathbf z}})}{dt}=\frac{d{\mathbf{u}}^\intercal}{dt} {{\mathbf z}}+{\mathbf{u}}^\intercal\frac{d{{\mathbf z}}}{dt}={\mathbf{u}}^\intercal DF^\intercal{{\mathbf z}}+{\mathbf{u}}^\intercal(-DF^\intercal{{\mathbf z}})=0.$$ This relation holds for both smooth and nonsmooth systems with transverse crossings [@park2018]. For completeness, we note that differences between phase variables, as in , will be interpreted as the *periodic difference*, $d_T(\phi({\mathbf{x}}),\phi({\mathbf{y}}))$. That is, if two angular variables $\theta$ and $\psi$ are defined on the circle $\mathbb{S}\equiv [0,T)$, then we set $$d_T(\theta,\psi)=\begin{cases} \theta-\psi+T,&\theta-\psi<-\frac{T}2\\ \theta-\psi,&-\frac{T}2\le \theta-\psi\le \frac{T}2\\ \theta-\psi-T,&\theta-\psi>\frac{T}2, \end{cases}$$ which maps $d_T(\theta,\psi)$ to the range $[-T/2,T/2]$. In what follows we will simply write $\theta-\psi$ for clarity rather than $d_T(\theta,\psi)$. Shape and timing response to sustained perturbations {#sec:smooth-sust} ---------------------------------------------------- In this section we study the effects of sustained perturbations on the shape and timing of the LC solution. In contrast to the instantaneous perturbation considered in the previous section, changes in each aspect of shape and timing can now influence the other, and hence a variational analysis of the combined shape and timing response of limit cycles under constant perturbation is needed (see ). Suppose a sustained perturbation on (\[eq:dxdt=F\]) leads to the perturbed system $$\label{eq:dxdt=Feps} \frac{d{\mathbf{x}}}{dt}=F_\varepsilon({\mathbf{x}}),$$ with a stable limit cycle solution $\gamma_{\varepsilon}(t)$ with period $T_\varepsilon$ depending smoothly on $\varepsilon$ over some range. In particular, when $\varepsilon=0$ we use the notation $F_0$, $\gamma_0$, and $T_0$ to denote these quantities, each of which are equivalent to $F$, $\gamma$, and $T$, respectively, introduced earlier for the unperturbed system (\[eq:dxdt=F\]). To simplify notation, we will drop the subscript $0$ except where required to avoid confusion. The perturbed periodic solution $\gamma_{\varepsilon}(t)$ can be represented, to leading order, by the single variable system $$\label{eq:phase-reduction} \frac{d\theta}{dt}=1+{{\mathbf z}}(\theta)^\intercal G({\mathbf{x}},t),$$ where $G({\mathbf{x}},t)=\varepsilon\frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}|_{\varepsilon=0}$ represents the $O(\varepsilon)$ perturbation of the vector field, $\theta\in [0, T_0)$ is the asymptotic phase as defined above, and ${{\mathbf z}}:\theta\in[0,T_0]\to{{\mathbf R}}^n$ is the iPRC. Suppose that for $0\le {\varepsilon}\ll 1$ we can represent $T_\varepsilon$ with a uniformly convergent series $$\label{eq:T_eps_expansion} T_\varepsilon=T_0+\varepsilon T_1+O(\varepsilon^2)$$ where $T_1$ is the linear shift in the limit cycle period in response to the static perturbation. $T_1>0$ if increasing $\varepsilon$ increases the period. From , $T_1$ can be calculated using the iPRC as $$\label{eq:T1} T_1=-\int_{0}^{T_0} {{\mathbf z}}(\theta)^\intercal\frac{\partial F_\varepsilon(\gamma(\theta))}{\partial \varepsilon}\Big|_{\varepsilon=0}d\theta.$$ To understand how the static perturbation changes the shape of the limit cycle $\gamma(t)$, we need to rescale the time coordinate of the perturbed solution so that $\gamma(t)$ and $\gamma_{\varepsilon}(t)$ may be compared at corresponding time points. That is, for ${\varepsilon}>0$ we wish to introduce a perturbed time $\tau(t)$ so that we can write the perturbed limit cycle solution uniformly in $t$ as $$\label{eq:x_epsilon_of_tau} \gamma_{\varepsilon}(\tau(t))=\gamma_0(t)+{\varepsilon}\gamma_1(t)+O({\varepsilon}^2).$$ We define the $T_0$-periodic function $\gamma_1(t)$ to be the *infinitesimal shape response curve (iSRC)*. We show next that $\gamma_1(t)$ obeys an inhomogeneous variational equation . This equation resembles , but has two additional non-homogeneous terms arising, respectively, from time rescaling $t\to\tau(t)$, and directly from the constant perturbation acting on the vector field. We require that the perturbed time satisfy the consistency and smoothness conditions $$\frac{d\tau}{dt}>0,\quad\text{ and }\quad\frac1{\varepsilon}\left(\int_{t=0}^{T_0}\left(\frac{d\tau}{dt}\right)dt - T_0\right)=T_1+O({\varepsilon}),\text{ as }{\varepsilon}\to 0.$$ These constraints do not determine the value of the derivative of $\tau$, which we write as $d\tau/dt=1/\nu_\varepsilon(t)$. In general, the iSRC will depend on the choice of $\nu_\varepsilon(t)$. However, some natural choices are particularly well adapted to specific problems, as we will see. Initially, we will make the simple ansatz $\nu_\varepsilon(t)=\text{const}$, *i.e.* we will assume uniform local timing sensitivity. Later we will introduce local timing response curves to exploit alternative time rescalings for greater accuracy. To derive , we assume $\nu_\varepsilon(t)=\text{const}$ and set the scaling factor to be $\nu_\varepsilon=\frac{T_0}{T_\varepsilon}$. Moreover, we assume that $\nu_\varepsilon$ can be written as a uniformly convergent series $$\nu_\varepsilon=1-\varepsilon \nu_1+ O(\varepsilon^2),$$ where $\nu_1=\frac{T_1}{T_0}$ represents the relative change in frequency. In terms of $\nu_\varepsilon$, the rescaled time for $\gamma_{\varepsilon}$ can be written as $\tau(t)=t/\nu_\varepsilon \in [0, T_\varepsilon]$ for $t\in [0, T_0]$. Differentiating with respect to $t$ ($\frac{d\gamma_\varepsilon}{dt} = \frac{d\gamma_\varepsilon}{d\tau} \frac{d\tau}{dt}$), substituting the ansatz ($\frac{d\tau}{dt}=\frac{1}{\nu_\varepsilon}$), and rearranging leads to $$\label{eq:dxdtau1} \begin{split} {\ensuremath{\begin{array}{cccccccccc}\frac{d \gamma_{\varepsilon}}{d\tau}&=& \nu_\varepsilon\,(\gamma'(t)+\varepsilon \gamma_1'( t)+O(\varepsilon^2))\\ &=& (1-\varepsilon \nu_1+ O(\varepsilon^2))\,(\gamma'(t)+\varepsilon \gamma_1'( t)+O(\varepsilon^2))\\ &=& \gamma'( t)-\varepsilon \nu_1 \gamma'( t) +\varepsilon \gamma_1'( t) + O(\varepsilon^2) \\ &=& F_0(\gamma(t)) + \varepsilon(-\nu_1 F_0(\gamma(t)) +\gamma_1'( t))+O(\varepsilon^2).\end{array}}} \end{split}$$ where $'$ denotes the derivative with respect to $t$. On the other hand, expanding the right hand side of gives $$\label{eq:dxdtau2} \begin{split} {\ensuremath{\begin{array}{cccccccccc}\frac{d \gamma_{\varepsilon}}{d\tau} &=& F_\varepsilon(\gamma_{\varepsilon}(\tau))\\ &=& F_0(\gamma(t))+\varepsilon \Big(DF_0(\gamma(t)) \gamma_1(t) +\frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0}\Big)+O(\varepsilon^2).\end{array}}} \end{split}$$ Equating and to first order, we find that the linear shift in shape produced by a static perturbation, i.e. the iSRC, satisfies $$\begin{aligned} \label{eq:src} \frac{d \gamma_1(t)}{dt} &=& DF_0(\gamma(t)) \gamma_1(t) +\nu_1 F_0(\gamma(t)) +\frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0},\end{aligned}$$ with period $T_0$, as claimed before. It remains to establish the initial condition $\gamma_1(0)$. Let $p_0=\gamma(0)$ denote an arbitrary base point chosen along the limit cycle. If $\Sigma$ is any Poincaré section transverse to the unperturbed limit cycle at $p_0$, then let $p_{\varepsilon}$ denote the intersection point where the LC under static perturbation $|{\varepsilon}|>0$ crosses $\Sigma$. We may choose $p_{\varepsilon}$ as a reference point anchoring the comparison of the perturbed and unperturbed limit cycles by setting $\gamma_{\varepsilon}(0)=p_{\varepsilon}$. Then the initial condition for the shape response curve $\gamma_1(0)$ is given by $(\gamma_{\varepsilon}(0)-\gamma_0(0))/\varepsilon$ and lies in the direction of $(d p_{\varepsilon}/d{\varepsilon})|_{{\varepsilon}=0}$. If $\Sigma_a$ and $\Sigma_b$ are two Poincaré sections through $p_0$, with initial shape displacements $\gamma_1^{a,b}(0)$, it can be readily shown that $\gamma_1^b(0)\in\text{Span}[\gamma_1^a(0),F_0(p)]$. Thus the collection of initial infinitesimal displacement vectors $\gamma_1(0)$ lies within a two-dimensional plane, which is a projection of the tangent space to the manifold defined by the family of limit cycles under variations in ${\varepsilon}$. For nonsmooth systems discussed in the balance of the paper, the ambiguity in the choice of the reference section $\Sigma$ is largely removed by setting $\Sigma$ equal to one of the switching or contact boundaries. To our knowledge, we are the first to derive the variational equation of the combined shape and timing response of a limit cycle to sustained perturbation, in either the smooth case (here) or the nonsmooth case (below). The accuracy of the iSRC in approximating the linear change in the limit cycle shape evidently depends on its timing sensitivity, that is, the choice of the relative change in frequency $\nu_1$. In the preceding derivation, we chose $\nu_1$ to be the relative change in the full period by assuming the limit cycle has constant timing sensitivity. It is natural to expect that different choices of $\nu_1$ will be needed for systems with varying timing sensitivities along the limit cycle. This possibility motivates us to consider local timing surfaces which divide the limit cycle into a number of segments, each distinguished by its own timing sensitivity properties. For each segment, we show that the linear shift in the time that $\gamma(t)$ spends in that segment can be estimated using the lTRC derived in §\[sec:smooth-local\]. The lTRC is analogous to the iPRC in the sense that they obey the same adjoint equation, but with different boundary conditions. Local timing surfaces {#sec:smooth-local} --------------------- The iPRC captures the net effect on timing of an oscillation – the phase shift – due to a transient perturbation , as well as the net change in period due to a sustained perturbation . In order to study the impact of a perturbation on *local* timing as opposed to the global timing, we introduce the notion of *local timing surfaces* that separate the limit cycle trajectory into segments with different timing sensitivities. Examples of local timing surfaces in smooth systems include the passage of neuronal voltage through its local maximum or through a predefined threshold voltage, and the point of maximal extension of reach by a limb. In nonsmooth systems, switching surfaces at which dynamics changes can also serve as local timing surfaces. For instance, in the feeding system of *Aplysia californica* as discussed in §\[sec:intro\], the open-closed switching boundary of the grasper defines a local timing surface. Whatever the origin of the local timing surface or surfaces of interest, it is natural to consider the phase space of a limit cycle as divided into multiple regions. Hence we may consider a smooth system $d{\mathbf{x}}/dt=F_0({\mathbf{x}})$ with a limit cycle solution $\gamma(t)$ passing through multiple regions in succession (see Figure \[fig:local-time-response\]). In each region, we assume that $\gamma(t)$ has constant timing sensitivity. To compute the relative change in time in any given region, we define a *local timing response curve* (lTRC) to measure the timing shift of $\gamma(t)$ in response to perturbations delivered at different times in that region. Below, we illustrate the derivation of the lTRC in region I and show how it can be used to compute the relative change in time in this region, denoted by $\nu_1^{\rm I}$. ![\[fig:local-time-response\] Schematic illustration of a limit cycle solution for a system consisting of a number of regions, each with distinct constant timing sensitivities. $\Sigma^{\rm in}$ and $\Sigma^{\rm out}$ denote the local timing surfaces for region I. ${\mathbf{x}}^{\rm in}$ and ${\mathbf{x}}^{\rm out}$ denote the points where the limit cycle enters region I through $\Sigma^{\rm in}$ and exits region I through $\Sigma^{\rm out}$, respectively.](localtimingsurfaces.png){width="3in"} Suppose that at time $t^{\rm in}$, $\gamma(t)$ enters region I upon crossing the surface $\Sigma^{\rm in}$ at the point ${\mathbf{x}}^{\rm in}$; at time $t^{\rm out}$, $\gamma(t)$ exits region I upon crossing the surface $\Sigma^{\rm out}$ at the point ${\mathbf{x}}^{\rm out}$ (see Figure \[fig:local-time-response\]). Denote the vector field under a constant perturbation by $F_\varepsilon({\mathbf{x}})$ and let ${\mathbf{x}}_{\varepsilon}$ denote the coordinate of the perturbed trajectory. Let $T_0^{\rm I} = t^{\rm out}-t^{\rm in}$ denote the time $\gamma(t)$ spent in region I and let $T_{\varepsilon}^{\rm I}$ denote the time the perturbed trajectory spent in region I. Assume we can write $T_{\varepsilon}^{\rm I}=T_0^{\rm I}+\varepsilon T^{\rm I}_1+O(\varepsilon^2)$ as we did before. It follows that the relative change in time of $\gamma(t)$ in region I is given by $$\nu_1^{\rm I}=\frac{T^{\rm I}_1}{T_0^{\rm I}} = \frac{T^{\rm I}_1}{t^{\rm out}-t^{\rm in}} .$$ The goal is to compute $\nu_1^{\rm I}$, which requires an estimate of $T^{\rm I}_1$. To this end, we define the local timing response curve $\eta^{\rm I}(t)$ associated with region I. We show that $\eta^{\rm I}(t)$ satisfies the adjoint equation and the boundary condition . Let $\mathcal{T}^{\rm I}({\mathbf{x}})$ for ${\mathbf{x}}$ in region I be the time remaining until exiting region I through $\Sigma^{\rm out}$, under the unperturbed vector field. This function is at least defined in some open neighborhood around the reference limit cycle trajectory $\gamma(t)$ if not throughout region I. For the unperturbed system, $\mathcal{T}^{\rm I}$ satisfies $$\frac{d\mathcal{T}^{\rm I}({\mathbf{x}}(t))}{dt}=-1$$ along the limit cycle orbit $\gamma(t)$. Hence $$\label{eq:local-time-const} F({\mathbf{x}})\cdot \nabla\mathcal{T}^{\rm I}({\mathbf{x}})=-1$$ for all ${\mathbf{x}}$ for which $\mathcal{T}^{\rm I}$ is defined. We define $\eta^{\rm I}(t):= \nabla \mathcal{T}^{\rm I}({\mathbf{x}}(t))$ to be the local timing response curve (lTRC) for region I. It is defined for $t\in [t^{\rm in}, t^{\rm out}]$. We show in Appendix \[ap:T1\] that $T^{\rm I}_1$ can be estimated as $$\label{eq:local-time-shift} T^{\rm I}_{1} = \eta^{\rm I}({\mathbf{x}}^{\rm in})\cdot \frac{\partial {\mathbf{x}}_{\varepsilon}^{\rm in}}{\partial \varepsilon}\Big|_{\varepsilon=0}+\int_{t^{\rm in}}^{t^{\rm out}}\eta^{\rm I}(\gamma(t))\cdot \frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0}dt,$$ where ${\mathbf{x}}_\varepsilon^{\rm in}$ denotes the coordinate of the perturbed entry point into region I. We may naturally view $\eta^{\rm I}$ as either a function of space, as in , or as a function of time, evaluated e.g. along the limit cycle trajectory. Comparing with , the integral terms have the same form, albeit with opposite signs. In addition, has an additional term arising from the impact of the perturbation on the point of entry to region I. On the other hand, the impact of the perturbation on the exit point, denoted by $\eta^{\rm I}({\mathbf{x}}_\varepsilon^{\rm out})\cdot \frac{\partial {\mathbf{x}}_\varepsilon^{\rm out}}{\partial \varepsilon}\big|_{\varepsilon=0}$, is always zero because the exit boundary $\Sigma^{\rm out}$ is a level curve of $\mathcal{T}^{\rm I}$; in other words, $\mathcal{T}^{\rm I}\equiv 0$ at $\Sigma^{\rm out}$. This indicates that the lTRC vector $\eta^{\rm I}$ associated with a given region is always perpendicular to the exit boundary of that region. Similar to the iPRC, it follows from that $\eta^{\rm I}$ satisfies the adjoint equation $$\begin{aligned} \label{eq:ltrc} \frac{d\eta^{\rm I}}{dt}=-DF(\gamma(t))^\intercal \eta^{\rm I}\end{aligned}$$ together with the boundary (normalization) condition at the exit point $$\label{eq:ltrc0} \eta^{\rm I}({\mathbf{x}}^{\rm out})=\frac{-n^{\rm out}}{n^{\rm out \intercal} F({\mathbf{x}}^{\rm out})}$$ where $n^{\rm out}$ is a normal vector of $\Sigma^{\rm out}$ at the unperturbed exit point ${\mathbf{x}}^{\rm out}$. The reason $\eta^{\rm I}$ at the exit point has the direction $n^{\rm out}$ is because $\eta^{\rm I}$ is normal to the exit boundary as discussed above. To summarize, in order to compute $\nu_1^{\rm I}=T^{\rm I}_{1}/({t^{\rm out}-t^{\rm in}})$, we need numerically to find $t^{\rm in},t^{\rm out}$ and evaluate  to estimate $T^{\rm I}_{1}$, for which we need to solve the boundary problem of the adjoint equation - for the lTRC $\eta^{\rm I}$. The procedures to obtain the relative change in time in other regions $\nu_1^{\rm II}, \nu_1^{\rm III},\cdots$ are similar to computing $\nu_1^{\rm I}$ in region I and hence are omitted. The existence of different timing sensitivities of $\gamma(t)$ in different regions therefore leads to a piecewise-specified version of the iSRC with period $T_0$, $$\begin{aligned} \label{eq:src-multiplescales} \frac{d \gamma_1^j(t)}{dt} &=& DF_0^j(\gamma(t)) \gamma_1^j(t) +\nu_1^{j} F_0^j(\gamma(t)) +\frac{\partial F_\varepsilon^j(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0},\end{aligned}$$ where $\gamma_1^j$, $F_0^j$, $F_{\varepsilon}^j$ and $\nu_1^j$ denote the iSRC, the unperturbed vector field, the perturbed vector field, and the relative change in time in region $j$, respectively, with $j\in\{\rm I, II, III, \cdots\}$. Note that in a smooth system as concerned in this section, $F_0^j\equiv F_0$ for all $j$. In §\[sec:toy-model\] we will show in a specific example that the iSRC with piecewise-specified timing rescaling has much greater accuracy in approximating the linear shape response of the limit cycle to static perturbations than the iSRC using a global uniform rescaling. The derivation of the lTRC in a given region still holds as long as the system is smooth in that region. Hence the assumption that $F({\mathbf{x}})$ is smooth everywhere can be relaxed to $F({\mathbf{x}})$ being piecewise smooth. Linear responses of nonsmooth systems with continuous solutions {#sec:nonsmooth-theory} =============================================================== Nonsmooth dynamics arises in many areas of biology and engineering. However, methods developed for smooth systems (discussed in §\[sec:smooth-theory\]) do not extend directly to understanding the changes of periodic limit cycle orbits in nonsmooth systems, because their Jacobian matrices are not well defined [@CGL18; @wilson2019]. Specifically, nonsmooth systems exhibit discontinuities in the time evolution of the solutions to the variational equations, ${\mathbf{u}}$ and $\gamma_1$ , and the solutions to the adjoint equations, ${{\mathbf z}}$ and $\eta$ . Following the terminology of @park2018 and @LN2013, we call the discontinuities in ${{\mathbf z}}$ and $\eta$ “jumps” and call the discontinuities in ${\mathbf{u}}$ and $\gamma_1$ “saltations”. Qualitatively, we use “jumps" to refer to discontinuities in the *timing* response of a trajectory, and “saltations" in the *shape* response. Since ${{\mathbf z}}$ and $\eta$ satisfy the same adjoint equation, they have the same discontinuities. Similarly, ${\mathbf{u}}$ and $\gamma_1$ obey versions of the variational equation with the same homogeneous term and different nonhomogeneous terms; since the jump conditions arise from the homogeneous terms (involving the Jacobian matrix in , ) we will presume that ${\mathbf{u}}$ and $\gamma_1$ satisfy the same saltation conditions at the transition boundaries. In the rest of this section, we characterize the discontinuities in the solutions to the adjoint equation in terms of ${{\mathbf z}}$, and discuss nonsmoothness of the variational dynamics in terms of ${\mathbf{u}}$. As mentioned in the introduction, we consider nonsmooth systems with degree of smoothness one or higher (Filippov systems); that is, systems with continuous solutions. In such systems, two types of discontinuities can occur in the trajectory: transversal crossing boundaries and hard boundaries. We review the existing methods for computing the saltations in ${\mathbf{u}}$ (and $\gamma_1$) and the jumps in ${{\mathbf z}}$ (and $\eta$) associated with transversal crossing boundaries in §\[sec:trans\], following [@bernardo2008; @LN2013] and [@park2013; @park2018]. In §\[sec:sliding\] we characterize the discontinuous behavior of the iPRC and the variational dynamics for nonsmooth systems with hard boundaries, stated here as Theorem \[thm:main\]. (145,60)(0,0) (0,0)[![\[fig:fillippov\] Examples of trajectories of nonsmooth systems with a boundary $\Sigma$. (A) The trajectory of a two-zone nonsmooth system intersects the boundary $\Sigma$ transversely at ${\mathbf{x}}_p$ (black dot). $F^{\rm I}$ and $F^{\rm II}$ denote the vector fields in the two regions ${{\mathcal R}}^{\rm I}$ and ${{\mathcal R}}^{\rm II}$. Components of $F^{\rm I}$ and $F^{\rm II}$ normal to $\Sigma$ at the crossing point have the same sign, allowing for the transversal crossing. (B) The trajectory of a nonsmooth system hits the hard boundary $\Sigma$ at the landing point (red dot) and begins sliding along $\Sigma$ under the vector field $F^{\rm slide}$. At the liftoff point (blue dot), the trajectory naturally reenters the interior. $F^{\rm interior}$ denotes the vector field in the interior domain. ](Filippov-transversal.png "fig:"){height="5cm"}]{} (-3,45)[****]{} (80,0)[![\[fig:fillippov\] Examples of trajectories of nonsmooth systems with a boundary $\Sigma$. (A) The trajectory of a two-zone nonsmooth system intersects the boundary $\Sigma$ transversely at ${\mathbf{x}}_p$ (black dot). $F^{\rm I}$ and $F^{\rm II}$ denote the vector fields in the two regions ${{\mathcal R}}^{\rm I}$ and ${{\mathcal R}}^{\rm II}$. Components of $F^{\rm I}$ and $F^{\rm II}$ normal to $\Sigma$ at the crossing point have the same sign, allowing for the transversal crossing. (B) The trajectory of a nonsmooth system hits the hard boundary $\Sigma$ at the landing point (red dot) and begins sliding along $\Sigma$ under the vector field $F^{\rm slide}$. At the liftoff point (blue dot), the trajectory naturally reenters the interior. $F^{\rm interior}$ denotes the vector field in the interior domain. ](Filippov-sliding.png "fig:"){height="5cm"}]{} (80,45)[****]{} Transversal crossing boundary {#sec:trans} ----------------------------- It is sufficient to consider a Filippov system with a single boundary to illustrate the discontinuities of ${{\mathbf z}}$ and ${\mathbf{u}}$ at any boundary transversal crossing point. \[def:2zoneFP\] A *two-zone system* with uniform degree of smoothness one (or higher) is described by $$\begin{aligned} \label{eq:2zoneFS} \frac{d\textbf{x}}{dt}=F({\mathbf{x}}):=\left\{ {\ensuremath{\begin{array}{cccccccccc} F^{\rm I}({\mathbf{x}}), & {\mathbf{x}}\in {{\mathcal R}}^{\rm I} \\ F^{\rm II}({\mathbf{x}}), & {\mathbf{x}}\in {{\mathcal R}}^{\rm II} \\ \end{array}}} \right.\end{aligned}$$ where ${{\mathcal R}}^{\rm I}:=\{{\mathbf{x}}| H({\mathbf{x}})<0\}$ and ${{\mathcal R}}^{\rm II}:=\{{\mathbf{x}}| H({\mathbf{x}})>0\}$ for a smooth function $H$, and the vector fields $F^{\rm I,II}:\overline{{{\mathcal R}}}^{\rm I,II}\to \mathbb{R}^n$ are smooth (at least $C^1$). We assume each region ${{\mathcal R}}^{\rm I,\rm II}$ has non-empty interior, and we write $\overline{{{\mathcal R}}}$ for the closure of ${{\mathcal R}}$ in $\mathbb{R}^n$. The *switching boundary* is the $\mathbb{R}^{n-1}$-dimensional manifold $\Sigma:= \overline{{{\mathcal R}}}^{\rm I}\cap \overline{{{\mathcal R}}}^{\rm II}=\{{\mathbf{x}}|H({\mathbf{x}})=0\}$. Suppose at time $t=t_p$ a limit cycle solution $\gamma(t)$ of crosses the boundary $\Sigma$ from ${{\mathcal R}}^{\rm I}$ to ${{\mathcal R}}^{\rm II}$ transversely (see Figure \[fig:fillippov\]A): $$\begin{aligned} \label{eq:FP-trans} {\ensuremath{\begin{array}{cccccccccc}n_{p}\cdot F^{\rm I}({\mathbf{x}}_p)> 0\quad\quad n_{p} \cdot F^{\rm II}({\mathbf{x}}_p)> 0, \end{array}}}\end{aligned}$$ where the boundary crossing point is denoted as ${\mathbf{x}}_p:=\lim_{t\to t_p^-}\gamma(t)=\lim_{t\to t_p^+}\gamma(t)$ and $n_p=\nabla H({\mathbf{x}}_p)$ refers to the vector normal to $\Sigma$ at ${\mathbf{x}}_p$. Below we show how to characterize discontinuities of ${\mathbf{u}}$ and ${{\mathbf z}}$ for at ${\mathbf{x}}_p$. As discussed before, the iSRC $\gamma_1$ and the lTRC $\eta$ experience the same discontinuities as ${\mathbf{u}}$ and ${{\mathbf z}}$, respectively. For a sufficiently small instantaneous perturbation, the displacement ${\mathbf{u}}(t)$ evolves continuously over the domain in which is smooth and can be obtained to first order in the initial displacement by solving the variational equation . As $\gamma$ crosses $\Sigma$ at time $t_p$, ${\mathbf{u}}(t)$ exhibits discontinuities (or “saltations”) since the Jacobian evaluated at ${\mathbf{x}}_p$ is not uniquely defined. The discontinuity in ${\mathbf{u}}$ at ${\mathbf{x}}_p$ can be expressed with the *saltation matrix $S_p$* as $$\begin{aligned} \label{eq:salt-def} {{\mathbf{u}}_p^+=S_p\,{\mathbf{u}}_p^-}\end{aligned}$$ where ${\mathbf{u}}_p^-=\lim_{t\to t_p^-} {\mathbf{u}}(t)$ and ${\mathbf{u}}_p^+=\lim_{t\to t_p^+} {\mathbf{u}}(t)$ represent the displacements between perturbed and unperturbed solutions just before and just after the crossing, respectively. It is straightforward to show (cf. @LN2013 §7.2 or @bernardo2008 §2.5) that $S_p$ can be constructed using the vector fields in the neighborhood of the crossing point and the vector $n_p$ normal to the switching boundary at ${\mathbf{x}}_p$ as $$\label{eq:salt} S_p=I+\frac{(F_p^+-F_p^-)n_{p}^\intercal}{n_{p}^\intercal F_p^-}$$ where $F_p^-=\lim_{{\mathbf{x}}\to {\mathbf{x}}_p^-}F({\mathbf{x}}),\,F_p^+=\lim_{{\mathbf{x}}\to {\mathbf{x}}_p^+}F({\mathbf{x}})$ are the vector fields of just before and just after the crossing at ${\mathbf{x}}_p$. Throughout this paper, $I$ denotes the identity matrix with size $n\times n$. \[remark-salt\] If the vector field $F$ evaluated along the limit cycle is continuous when crossing the boundary $\Sigma$ transversely, so that $F_p^-=F_p^+$ and $n_p^\intercal F_p^-\neq 0$, then the saltation matrix $S_p$ at such a boundary crossing point is the identity matrix, and there is no discontinuity in ${\mathbf{u}}$ or $\gamma_1$ at time $t_p$. Now we consider discontinuous jumps in the iPRC ${{\mathbf z}}$ for . This curve obeys the adjoint equation and is continuous within the interior of each subdomain in which is smooth. When the limit cycle path crosses the switching boundary at the point ${\mathbf{x}}_p$, ${{\mathbf z}}$ exhibits a discontinuous jump which can be characterized by the *jump matrix* $J_p$ $$\begin{aligned} \label{eq:jump-def} {{\mathbf z}}_p^+=J_p{{\mathbf z}}_p^-\end{aligned}$$ where ${{\mathbf z}}_p^-=\lim_{t\to t_p^-} {{\mathbf z}}(t)$ and ${{\mathbf z}}_p^+=\lim_{t\to t_p^+} {{\mathbf z}}(t)$ are the iPRC just before and just after crossing the switching boundary at time $t_p$. As discussed in @park2018, the relation between ${\mathbf{u}}$ and ${{\mathbf z}}$ for smooth systems remains valid at any transversal boundary crossing point. In other words, ${\mathbf{u}}_p^{-\intercal}{{\mathbf z}}_p^-={\mathbf{u}}_p^{+\intercal}{{\mathbf z}}_p^+$ holds at the transversal crossing point ${\mathbf{x}}_p$. This leads to a relation between the saltation and jump matrices at ${\mathbf{x}}_p$ $$\label{eq:jump-salt-relation} J_p^\intercal S_p=I.$$ The saltation matrix $S_p$ given by has full rank at any transverse crossing point. It follows that $J_p$ can be written as $$J_p = ({S_p}^{-1})^\intercal.$$ Sliding motion on a hard boundary {#sec:sliding} --------------------------------- The existence of the saltation and jump matrices discussed in §\[sec:trans\] is guaranteed by the transversal flow condition , which, however, will no longer hold when part of the limit cycle slides along a boundary (e.g., Figure \[fig:fillippov\]B). As an example of a hard boundary at which the transverse flow condition would break down, consider the requirement that firing rates in a neural network model be nonnegative. When a nerve cell ceases firing because of inhibition, its firing rate will be held at zero until the balance of inhibition and excitation allow spiking to resume. At the point at which the firing rate first resumes positive values, the vector field describing the system lies tangent to the constraint surface rather than transverse to it. In this section, we establish the conditions relating ${\mathbf{u}}$ and ${{\mathbf z}}$ at non-transversal crossings, including the *landing point* at which a sliding motion begins, and the *liftoff point* at which the sliding terminates (see Fig. \[fig:fillippov\]B). We gather our main results in Theorem \[thm:main\]. To this end, we begin with precise definitions of hard boundary, sliding region, sliding vector field and the liftoff condition. Consider a system with domain $\mathcal{R}$. We call a surface $\Sigma$ a *hard boundary* if it is part of the boundary of the closure of $\mathcal{R}$. In order to describe motions that are confined to slide along a hard boundary during a component of the trajectory, we will consider two vector fields: $F^\text{interior},$ defined on the closure of the domain $\mathcal{R}$ (that is, the whole of the domain, including its bounding surface $\Sigma$) and a vector field $F^\text{slide}$ that is tangent to the hard boundary (see Definition \[def:sliding\]). The motion along a trajectory is specified differently depending on the location of a point. For a point in the interior, denoted $\mathcal{R}^{\rm interior}$, the dynamics is determined by $F^\text{interior}$. For a point on the boundary, the velocity obeys either $F^\text{interior}$ or else $F^\text{slide}$ that is tangent to $\Sigma$, depending on whether $F^\text{interior}$ is directed inwardly or outwardly at a given boundary point. This dual definition of the vector field has the effect that points driven into the boundary do not exit through the hard boundary, but rather slide along the boundary until the interior vector field allows them to reenter the domain. \[def:sliding\]The *sliding region* (${{\mathcal R}}^{\rm slide}$) is defined as the portion of a hard boundary $\Sigma$ for which $$\begin{aligned} \label{eq:slidingregion} {\ensuremath{\begin{array}{cccccccccc}{{\mathcal R}}^{\rm slide}=\{{\mathbf{x}}\in\Sigma\,|\,n_{\mathbf{x}}\cdot F^{\rm interior}({\mathbf{x}})>0\},\end{array}}}\end{aligned}$$ where $n_{\mathbf{x}}$ is a unit normal vector of $\Sigma$ at ${\mathbf{x}}$ that points away from the interior domain and $F^{\rm interior}({\mathbf{x}})$ denotes the smooth vector field in $\mathcal{R}^{\rm interior}$. By construction, the *sliding vector field* $F^\text{slide}$ for flows confined to $\mathcal{R}^{\rm slide}$ must have a vanishing normal component. While any vector field with vanishing normal component could be considered for $F^\text{slide}$, in this paper we adopt the natural choice of setting $F^\text{slide}$ to be the continuation of the interior vector field in the component tangential to ${{\mathcal R}}^{\rm slide}$, $$\begin{aligned} \label{eq:slidingVF} {\ensuremath{\begin{array}{cccccccccc}F^{\rm slide}({\mathbf{x}})=F^{\rm interior}({\mathbf{x}})-\big(n_{\mathbf{x}}\cdot F^{\rm interior}({\mathbf{x}})\big)n_{\mathbf{x}}.\end{array}}}\end{aligned}$$ The vector field $F^{\rm slide}$ chosen in this way lies tangent to the hard boundary $\Sigma$, and the flow exits the sliding region ${{\mathcal R}}^{\rm slide}$ as the trajectory crosses the *liftoff boundary* $\mathcal{L}$, defined as $$\label{eq:cond-liftoff} {\ensuremath{\begin{array}{cccccccccc}\mathcal{L}=\{{\mathbf{x}}\in\Sigma \,| \,n_{\mathbf{x}}\cdot F^{\rm interior}({\mathbf{x}})=0\}.\end{array}}}$$ Thus the liftoff boundary constitutes the edge of the sliding region of the hard boundary. Our definition of the sliding region and sliding vector field is consistent with that in [@bernardo2008] §5.2.2, except that the system of interest in this paper is only defined on one side of the sliding region. However, our main Theorem \[thm:main\], below, holds in either case. Hence our results also apply to Filippov systems with sliding regions bordered by vector fields on either side, as in the example the stick-slip oscillator (@LN2013 §6.5). Using the preceding notation, in the neighborhood of a sliding region in the hard boundary $\Sigma$, a system with a limit cycle component confined to the sliding region takes the following form $$\begin{aligned} \label{eq:1zoneFP} \frac{d\textbf{x}}{dt}=F({\mathbf{x}}):=\left\{ {\ensuremath{\begin{array}{cccccccccc} F^{\rm interior}({\mathbf{x}}), & {\mathbf{x}}\in {{\mathcal R}}^{\rm interior}\\ F^{\rm slide}({\mathbf{x}}), & {\mathbf{x}}\in{{\mathcal R}}^{\rm slide}\subset \Sigma \\ \end{array}}} \right.\end{aligned}$$ where $F^{\rm slide}$ and $\mathcal{R}^{\rm slide}$ are given by and . Recall that $F^{\rm interior}$ is smooth (cf. Definition \[def:sliding\]). \[def:lcsc\]In a general Filippov system which locally at a hard boundary $\Sigma$ has the form , we call a closed, isolated periodic orbit that passes through a sliding region a *limit cycle with sliding component*, denoted as LCSC. To identify the liftoff point at which the LCSC reenters the interior of the domain, we require the nondegeneracy condition that the trajectory crosses the liftoff boundary $\mathcal{L}$ at a finite velocity. Specifically, the outward normal component of the interior velocity should switch from positive (outward) to negative (inward) at the liftoff boundary, as one moves in the direction of the flow (see Fig. \[fig:sliding-motion-coordinates\]). That is, $$\begin{aligned} \label{eq:nondege-1} {\ensuremath{\begin{array}{cccccccccc}\left.\left[\nabla(n_{\mathbf{x}}\cdot F^{\rm interior}({\mathbf{x}})) \cdot F^{\rm slide}({\mathbf{x}})\right] \right|_{{\mathbf{x}}\in \mathcal{L}}< 0\end{array}}}.\end{aligned}$$ Note that the liftoff condition together with the nondegeneracy condition uniquely defines a liftoff point for the LCSC. Note that at the liftoff point, we have $F^{\rm slide}=F^{\rm interior}$. ![ Trajectory from the interior with vector field $F^{\rm interior}$ making a transverse entry to a hard boundary $\Sigma$, followed by motion confined to the sliding region $\mathcal{R}^{\rm slide}$, then a smooth liftoff at $\mathcal{L}$ back into the interior of the domain. Red dot: landing point (point at which the trajectory exits the interior and enters the hard boundary surface). Blue dot: liftoff point (point at which the trajectory crosses the liftoff boundary $\mathcal{L}$ and reenters the interior). After a suitable change of coordinates, the geometry may be arranged as shown, with the hard boundary $\Sigma$ coinciding with one coordinate plane. Downward vertical arrow: $\mathbf{n}$, the outward normal vector for $\Sigma$. The region $\mathbf{n}\cdot F^\text{interior}>0$ defines the sliding region within $\Sigma$; the condition $\mathbf{n}\cdot F^\text{interior}=0$ defines the liftoff boundary $\mathcal{L}$. []{data-label="fig:sliding-motion-coordinates"}](LCSC-main-text.png){width="70.00000%"} We assume that under an appropriate smooth change of coordinates, we may transform the hard boundary $\Sigma$ into a plane, so that $\Sigma$ has a constant normal vector $n$ throughout $\mathcal{R}^\text{slide}$ (cf. Fig. \[fig:sliding-motion-coordinates\]). Theorem \[thm:main\] gathers together several conclusions about the variational and infinitesimal phase response curve dynamics of a LCSC local to a sliding boundary that follow from these definitions (see Appendix \[ap:proof\] for the proof): \[thm:main\] Consider a general LCSC described locally by  in the neighborhood of a hard boundary $\Sigma$ with a constant normal vector $n$, and with a liftoff point defined by . Assume that within the stable manifold of the limit cycle there is a well defined asymptotic phase function $\phi({\mathbf{x}})$ satisfying $d\phi/dt=1$ along trajectories. Assume that $\phi$ is Lipschitz continuous, and assume that on the constraint surface $\Sigma$, the directional derivatives of $\phi$ with respect to directions tangential to the surface are Lipschitz continuous, except (possibly) at the liftoff and landing points. Finally, assume the nondegeneracy condition holds at the liftoff point. Then the following properties hold for the saltation matrix for ${\mathbf{u}}$, and the jump matrix for ${{\mathbf z}}$: 1. At the landing point, the saltation matrix is $S=I-n n^\intercal$, where $I$ is the identity matrix. 2. At the liftoff point, the saltation matrix is $S=I$. 3. Along the sliding region, the component of ${{\mathbf z}}$ normal to $\Sigma$ is zero. 4. The normal component of ${{\mathbf z}}$ is continuous at the landing point. 5. The tangential components of ${{\mathbf z}}$ are continuous at both landing and liftoff points. We make the following additional observations about Theorem \[thm:main\]: \[rem:Jland=I\] - It follows from (a) in Theorem \[thm:main\] that the component of ${\mathbf{u}}$ normal to $\Sigma$ vanishes when the LCSC hits $\Sigma$. Once on the sliding region, the Jacobian used in the variational equation switches from $DF^{\rm interior}$ to $DF^{\rm slide}$ where $F^{\rm slide}$ has zero normal component by construction . Hence, the normal component of ${\mathbf{u}}$ is stationary over time, and remains zero on the sliding region. - It follows from parts (d) and (e) in Theorem \[thm:main\] that the jump matrix of a LCSC at the landing point is trivial (identity matrix). - The assumption that the asymptotic phase function $\phi({\mathbf{x}})$ is differentiable with respect to the directions forming a basis of the constraint surface is necessary for the proof of part (c). A stable limit cycle arising in a $C^r$-smooth vector field, for $r\ge 1$, will have $C^r$ isochrons [@Wiggins1994book; @JosicShea-BrownMoehlis2006]. In [@park2018] and [@wilson2019] the authors assume differentiability of the phase function with respect to a basis of vectors spanning a switching surface. The assumption we require here is similarly plausible; it appears to hold at least for the model systems we have considered. \[rem:iprc-liftoff\] Theorem \[thm:main\] excludes discontinuities in ${{\mathbf z}}$ *except* at the liftoff point, and then only in its normal component. Since the normal component of ${{\mathbf z}}$ along each sliding component of a LCSC is zero by Theorem \[thm:main\], a discontinuous jump occurring at a liftoff point must be a nonzero instantaneous jump, which cannot be specified directly in terms of the value of ${{\mathbf z}}$ prior to the jump. However, a time-reversed version of the jump matrix at the liftoff point, denoted as $\mathcal{J}$, is well defined as follows: $$\begin{aligned} {\ensuremath{\begin{array}{cccccccccc} {{{\mathbf z}}_{\text{lift}}^{-}}=\mathcal{J} {{{\mathbf z}}_{\text{lift}}^{+}} \end{array}}}\end{aligned}$$ where ${{{\mathbf z}}_{\text{lift}}^{-}}$ and ${{{\mathbf z}}_{\text{lift}}^{+}}$ are the iPRC just before and just after the trajectory crosses the liftoff point in forwards time, and $\mathcal{J}$ at the liftoff point has the same form as the saltation matrix $S$ at the corresponding landing point $$\mathcal{J}=I-n {n}^\intercal .$$ That is, the component of ${{\mathbf z}}$ normal to $\Sigma$ becomes $0$ as the trajectory enters $\Sigma$ in backwards time. \[rem:uztable\] Combining Theorem \[thm:main\], Remarks \[rem:Jland=I\] and \[rem:iprc-liftoff\], we summarize the behavior of the solutions of the variational and adjoint equations ${\mathbf{u}}$ and ${{\mathbf z}}$ in limit cycles with sliding components: Landing Sliding Liftoff ----------------- -------------------- ------------------------- ------------------------------ ${\mathbf{u}}$ $S=I-nn^\intercal$ ${\mathbf{u}}^\perp=0$ $S=I$ ${{\mathbf z}}$ $\mathcal{J}=I$ ${{\mathbf z}}^\perp=0$ $\mathcal{J}=I-nn^\intercal$ where $S$ is the regular saltation matrix and $\mathcal{J}$ is the time-reversed jump matrix. It follows directly from Remark \[rem:uztable\] that the relation between the saltation and jump matrices at a transversal boundary crossing point $J^\intercal S=I$ (see ) is no longer true at a landing or a liftoff point. Instead, the following condition holds $$\mathcal{J}^\intercal_p S_p = I-n n^\intercal$$ where $\mathcal{J}_p$ and $S_p$ denote the time-reversed jump matrix and the regular saltation matrix at a landing or a liftoff point. We illustrate the behavior of a limit cycle with sliding component via an analytically tractable planar model in §\[sec:toy-model\]. In this example, we will see that a nonzero instantaneous jump discussed in Remark \[rem:iprc-liftoff\] can occur in the normal component of ${{\mathbf z}}$ at the liftoff point, reflecting a “kink” or nonsmooth feature in the isochrons (*cf.* Figure \[fig:LC-2d-isochrons\]). In that example system, the discontinuity in the iPRC reflects a curve of nondifferentiability in the asymptotic phase function propagating backwards along a trajectory from the liftoff point to the interior of the domain (Figure \[fig:LC-2d-isochrons\]A). The presence of a discontinuous jump from zero to a nonzero normal component in ${{\mathbf z}}$ in forward time implies that numerical evaluation of the iPRC (presented in Appendix \[sec:algorithm\]) should be accomplished via backward integration along the limit cycle. In addition to the iPRC, we also provide numerical algorithms for calculating the lTRC, the variational dynamics and the iSRC for a LCSC in a general nonsmooth system with hard boundaries, in Appendix \[sec:algorithm\]. Shape and timing response in a planar limit cycle model with sliding components {#sec:toy-model} ================================================================================ In this section, we apply our methods to a two dimensional, analytically tractable model that has a single interior domain with purely linear flow and hard boundary constraints that create a limit cycle with sliding components (LCSC). We find the surprising result that the isochrons exhibit a nonsmooth “kink” propagating into the interior of the domain from the locations of the liftoff points, *i.e.* the points where the limit cycle smoothly departs the boundary. In addition, we show that using local timing response curve analysis gives significantly greater accuracy of the shape response than using a single, global, phase response curve. MATLAB source code for simulating the model and reproducing the figures is available: <https://github.com/yangyang-wang/LC_in_square>. In the interior of the domain $[-1,1]\times [-1,1]$, we take the vector field of a simple spiral source to define the interior dynamics of the planar model $$\label{eq:toy-model} \frac{d\textbf{x}}{dt}=F(\textbf{x})=\begin{bmatrix} \alpha x - y\\x+\alpha y \end{bmatrix}$$ where $\textbf{x}=\begin{bmatrix} x\\y\end{bmatrix}$ and $\alpha$ is the expansion rate of the source at the origin. The rotation rate is fixed at a constant value $1$. The Jacobian matrix $DF$ evaluated along the limit cycle solution in the interior of the domain is $$\label{eq:toy-model-2} DF=\begin{bmatrix} \alpha & -1 \\ 1 & \alpha \end{bmatrix}$$ In what follows we will require $0<\alpha < 1$, so we have a weakly expanding source. For illustration, $\alpha=0.2$ provides a convenient value. Every trajectory starting from the interior, except the origin, will eventually collide with one of the walls at $x=\pm 1$ or $y=\pm 1$ (in time not exceeding $\frac 1{2\alpha}\ln(2/(x(0)^2+y(0)^2)$). As in §\[sec:sliding\], we set the sliding vector field when the trajectory is traveling along the wall to be equal to the continuation of the interior vector field in the component parallel to the wall, while the normal component is set to zero (except where it is oriented into the domain interior). The resulting vector fields of the planar LCSC model $F({\mathbf{x}})$ on the interior and along the walls are given in Table \[tab:natural\], and illustrated in Fig. \[fig:simu-toymodel\]B. $x$ range $y$ range $dx/dt$ $dy/dt$ ----------- ---------------------- ---------------- --------------- $|x|<1$ $|y|<1$ $\alpha x-y$ $x+\alpha y$ $x=1$ $-1\le y<\alpha$ $0$ $1+\alpha y$ $x=1$ $\alpha\le y < 1$ $\alpha - y$ $1+\alpha y$ $y=1$ $1\ge x > - \alpha$ $\alpha x-1$ $0$ $y=1$ $-\alpha \ge x > -1$ $\alpha x - 1$ $x+\alpha$ $x=-1$ $1\ge y>-\alpha$ $0$ $-1+\alpha y$ $x=-1$ $-\alpha\ge y > -1$ $-\alpha - y$ $-1+\alpha y$ $y=-1$ $-1\le x < \alpha$ $\alpha x + 1$ $0$ $y=-1$ $\alpha \le x < 1$ $\alpha x + 1$ $x-\alpha$ : Vector field of the planar LCSC model on the interior and along the boundaries.[]{data-label="tab:natural"} The trajectory will naturally lift off the wall and return to the interior when the normal component of the unconstrained vector field changes from outward to inward, i.e., $(F^{\text{interior}}\cdot n^{\text{wall}})|_{\text{wall}}=0$ (see ). For instance, on the wall $x=1$ with a normal vector $n=[1,0]^\intercal$, we compute $$F^{\text{interior}}|_{\text{wall}}\cdot n^{\text{wall}}=(\alpha x-y)|_{\text{wall}}=\alpha-y=0.$$ It follows that $y=\alpha$ defines the liftoff condition on the wall $x=1$. For this planar model, there are four lift-off points with coordinates $(1,\alpha), (-\alpha,1), (-1,-\alpha), (\alpha,-1)$ on the walls $x=1,\,y=1,\,x=-1, y=-1$, respectively. Denote the LCSC produced by the planar model by $\gamma(t)$, whose time series over $[0, T_0]$ is shown in Figure \[fig:simu-toymodel\]A, where $T_0$ is the period. The projection of $\gamma(t)$ onto the $(x,y)$-plane is shown in the right panel, together with an osculating trajectory that starts near the center and ends up running into the wall $x=1$ at the lift-off point $(1,\alpha)$ (black star). (145,60)(0,0) (-5,0)[![ Simulation result of the planar LCSC model with parameter $\alpha=0.2$. (A): Time series of the limit cycle $\gamma(t)$ generated by the planar LCSC model over one cycle with initial condition $\gamma(0)=[1,\alpha]^\intercal$. (B): Projection of $\gamma(t)$ onto $(x,y)$ phase space (solid black) and the osculating trajectory (dashed black), starting near the center that ends up running into the wall at the liftoff point $[1,\alpha]^\intercal$ (black star). Red arrows represent the vector field. []{data-label="fig:simu-toymodel"}]({LC_in_square_VF_star.pdf} "fig:"){width="6.5in"}]{} (3,45)[****]{} (82,45)[****]{} Next we implement algorithms given in Appendix \[sec:algorithm\] to find the timing and shape responses of the LCSC to both instantaneous perturbations and sustained perturbation. We start with finding the iPRC for the LCSC to understand the timing response, and then solve the variational equation to find the linear shape response of the planar LCSC model to an instantaneous perturbation. Lastly, we compute the iSRC when the applied sustained perturbations are both uniform and nonuniform, to understand the shape response of the planar LCSC model to sustained perturbations. Infinitesimal phase response analysis {#sec:iprc} ------------------------------------- In the case of weak coupling or small perturbations of a strongly stable limit cycle, a linearized analysis of the phase response curve – the iPRC – suffices to predict the behavior of the perturbed system. When trajectories slide along a hard boundary, however, the linearized analysis breaks down. For nonsmooth systems such as the LCSCs, the asymptotic phase function $\phi({\mathbf{x}})$ may itself be nonsmooth at certain locations, even when it remains well defined; its gradient (i.e., the iPRC) may therefore be discontinuous at those locations. Nevertheless, one may be able to derive a consistent first order approximation to the phase response curve notwithstanding that the directional derivative may not be well defined, as discussed in §\[sec:nonsmooth-theory\]. The dynamics of the planar LCSC model are smooth except for the discontinuities when crossing the switching boundaries, that is, entering or exiting the walls. The iPRC, ${{\mathbf z}}(t)$, will be continuous in the interior domain as well as in the interior of the four boundaries. As discussed in §\[sec:sliding\], the discontinuity of iPRCs only occurs at the liftoff point. By Remark \[rem:uztable\], the time-reversed jump matrix at a liftoff point, which takes the iPRC just after crossing the liftoff point to the iPRC just before crossing the liftoff point in backwards time, is given by $$\mathcal{J}={\ensuremath{\left[\begin{array}{rrrrrrrrrrrrrrrrrr} 0 & 0 \\ 0 & 1 \end{array}\right]}}$$ when the trajectory leaves the walls $x=\pm 1$, and is given by $$\mathcal{J}={\ensuremath{\left[\begin{array}{rrrrrrrrrrrrrrrrrr} 1 & 0 \\ 0 & 0 \end{array}\right]}}$$ when the trajectory leaves the walls $y=\pm 1$. (145,60)(0,0) (0,0)[![iPRCs for the planar LCSC model with parameter $\alpha=0.2$. (A): Trajectories and isochrons for the LCSC model. The solid black and dashed black curves are the same as in Figure \[fig:simu-toymodel\]B. The colored scalloped curves are isochrons of the LCSC $\gamma(t)$ (black solid) corresponding to 50 evenly distributed phases $nT_0/50$, $n=1,\cdots, 50$. We define the phase at the liftoff point (black star) to be zero. (B): iPRCs for the planar LCSC model. The blue and red curves represent the iPRC for perturbations in the positive $x$ and $y$ directions, respectively. The intervals during which $\gamma(t)$ slides along a wall are indicated by the shaded regions. While $y=+1$, the iPRC vector is parallel to the wall (${{\mathbf z}}_y\equiv 0$) and oriented opposite to the direction of flow (${{\mathbf z}}_x < 0$). Similarly, on the remaining walls, the iPRC vector has zero normal component relative to the active constraint wall, and parallel component opposite the direction of motion. []{data-label="fig:LC-2d-isochrons"}]({LC_in_square_isochrons_2.pdf} "fig:"){height="5.7cm"}]{} (0,46)[****]{} (75,0)[![iPRCs for the planar LCSC model with parameter $\alpha=0.2$. (A): Trajectories and isochrons for the LCSC model. The solid black and dashed black curves are the same as in Figure \[fig:simu-toymodel\]B. The colored scalloped curves are isochrons of the LCSC $\gamma(t)$ (black solid) corresponding to 50 evenly distributed phases $nT_0/50$, $n=1,\cdots, 50$. We define the phase at the liftoff point (black star) to be zero. (B): iPRCs for the planar LCSC model. The blue and red curves represent the iPRC for perturbations in the positive $x$ and $y$ directions, respectively. The intervals during which $\gamma(t)$ slides along a wall are indicated by the shaded regions. While $y=+1$, the iPRC vector is parallel to the wall (${{\mathbf z}}_y\equiv 0$) and oriented opposite to the direction of flow (${{\mathbf z}}_x < 0$). Similarly, on the remaining walls, the iPRC vector has zero normal component relative to the active constraint wall, and parallel component opposite the direction of motion. []{data-label="fig:LC-2d-isochrons"}](iPRC-numerical-shaded.pdf "fig:"){height="5.5cm"}]{} (70,46)[****]{} Figure \[fig:LC-2d-isochrons\]A shows the limit cycle (solid black curve), the osculating trajectory (dashed black curve) corresponding to the liftoff point (black star, $\theta=0$) and the isochrons computed from a direct method, starting from a grid of initial conditions and tracking the phase of final locations (colored scalloped curves). There appears to be a “kink" in the isochron function, propagating backwards in time along the trajectories that encounter the boundaries exactly at the liftoff points, such as the dashed curve. This apparent discontinuity in the gradient of the isochron function in the *interior* of the domain exactly corresponds, at the boundary, with the point of discontinuity occurring in the iPRC along the limit cycle (cf. Remark \[rem:iprc-liftoff\]). According to Figure \[fig:LC-2d-isochrons\], the isochron curves are perpendicular to the sliding region of the wall at which the interior vector field is pointing outward. That is, the normal component of the iPRC when the trajectory slides along a wall is equal to 0. There is no jump in ${{\mathbf z}}$ when the trajectory enters the wall, but instead a discontinuous jump from zero to nonzero occurs in the normal component of ${{\mathbf z}}$ at the liftoff point. All of these observations are consistent with iPRCs ${{\mathbf z}}$ (Figure \[fig:LC-2d-isochrons\], right) that are computed using **Algorithm for ${{\mathbf z}}$** in §\[sec:algorithm-iprc\] based on Theorem \[thm:main\]. After the trajectory lifts off the east wall ($x=1$) at the point marked $\theta=0$ in Figure \[fig:LC-2d-isochrons\]A (black star), a perturbation along the positive $x$-direction (resp., positive $y$-direction) causes a phase delay (resp., advance). While the timing sensitivity of the LCSC to small perturbations in the $x$-direction reaches a local maximum before reaching the next wall $y=1$, the phase advance caused by the $y$-direction perturbation decreases continuously to $0$ as the trajectory approaches $y=1$. As the trajectory is sliding along the wall ($y=1$), the positive $y$-direction perturbation that is normal to the wall has no effect on the LCSC and hence will not affect its phase. Moreover, we showed in Theorem \[thm:main\] that a perturbation in the negative $y$-direction also has no effect on the phase, since the perturbed trajectory returns to the wall within time $O(\varepsilon)$, with a net phase offset that is at most $O(\varepsilon^2)$, where $\varepsilon$ is the size of the perturbation. As the trajectory lifts off the wall, there is a discontinuous jump in ${{\mathbf z}}_y$, so that a negative $y$-direction perturbation applied immediately after the liftoff point leads to a phase advance. On the other hand, on the sliding region of the wall $y=1$, a perturbation along the positive $x$-direction, against the direction of the flow, results in a phase delay, which decreases in size as the phase increases, and becomes $0$ upon reaching the next wall, $x=-1$. The timing sensitivity of the LCSC to perturbations applied afterwards are similar to what are observed in the first quarter of the period due to the $\mathbf{Z}_4$-symmetry $\sigma (x,y) = (-y,x)$. The linear change in the oscillation period of the LCSC in response to a static perturbation can then be estimated by taking the integral of the iPRC multiplying the given perturbation, as shown in the last step in **Algorithm for ${{\mathbf z}}$** (§ \[sec:algorithm-iprc\]). As noted before, the change in period will be needed to solve for the iSRC to understand how this perturbation affects the shape of the LCSC. In this example, the interior vector field is linear. Therefore its Jacobian is constant, and the iPRC may be obtained analytically [@park2018]. The resulting curves are indistinguishable from the numerically calculated curves shown in Fig. \[fig:LC-2d-isochrons\]B. Variational analysis -------------------- Suppose a small instantaneous perturbation, applied at time $t=0$, leads to an initial displacement ${\mathbf{u}}(0)=\tilde{\gamma}(0)-\gamma(0)$, where $\gamma(0)=[1,\alpha]$ is the liftoff point (black star in Figs. \[fig:simu-toymodel\]B and \[fig:LC-2d-isochrons\]A) as in the previous section. We use the variational analysis to study how this perturbation evolves over time. Similar to the iPRC, ${\mathbf{u}}(t)$ will be continuous everywhere in the domain except when entering or exiting the walls. In contrast to the iPRC, ${\mathbf{u}}$ is continuous at all liftoff points, but exhibits discontinuous saltations when the trajectory enters a wall. According to Theorem \[thm:main\], the saltation matrix $S$, which takes ${\mathbf{u}}$ just before entering a wall to ${\mathbf{u}}$ just after entering the wall in forwards time, for the planar LCSC model is given by $$S={\ensuremath{\left[\begin{array}{rrrrrrrrrrrrrrrrrr} 0 & 0 \\ 0 & 1 \end{array}\right]}}$$ when the trajectory enters the walls $x=\pm 1$, and is given by $$S={\ensuremath{\left[\begin{array}{rrrrrrrrrrrrrrrrrr} 1 & 0 \\ 0 & 0 \end{array}\right]}}$$ when the trajectory enters the walls $y=\pm 1$. Solutions to the variational equation of the planar LCSC model with the given initial condition ${\mathbf{u}}(0)$ can be computed using **Algorithm for ${\mathbf{u}}$** in §\[sec:algorithm-u\]. As discussed in Remark \[rem:fund-matrix\], an alternative way to find the displacement ${\mathbf{u}}(t)$ is to compute the fundamental solution matrix $\Phi(t, 0)$ by running **Algorithm for ${\mathbf{u}}$** twice and then to evaluate ${\mathbf{u}}(t) = \Phi(t,0) {\mathbf{u}}(0)$. The advantage of the latter approach is that once $\Phi(t,0)$ is obtained, it can be used to compute ${\mathbf{u}}(t)$ with any given initial value by evaluating a matrix multiplication instead of solving the variational equation. Here, by taking $[1,0]$ and $[0, 1]$ as the initial conditions for ${\mathbf{u}}$ at the liftoff point A, we apply **Algorithm for ${\mathbf{u}}$** to compute the time evolution of the two columns for the fundamental matrix $\Phi(t,0)$. A simple calculation shows that the monodromy matrix $\Phi(T_0,0)$ has an eigenvalue $+1$, whose eigenvector $[0,1]$ is tangent to the limit cycle at the liftoff point, as expected (Remark \[rem:monodromy\]). It follows that if the initial displacement at the liftoff point is along the limit cycle direction, then the displacement after a full period becomes the same as the initial one. To see this, we take the initial displacement ${\mathbf{u}}(0)=[0,\varepsilon]$ where $\varepsilon=0.1$ to be the tangent vector of the limit cycle at the liftoff point, and compute ${\mathbf{u}}(t)$, $x$ and $y$ components of which are shown in black solid curves in Figure \[fig:var-toy\]B,D. The saltations in ${\mathbf{u}}$ at time when the trajectory hits the walls can be clearly distinguished in the plot. Moreover, ${\mathbf{u}}(T_0)={\mathbf{u}}(0)$ as we expect. To further validate the accuracy of ${\mathbf{u}}$, we solve and plot $\gamma(t)$ with $\gamma(0)=[1,\alpha]$ (black, Figure \[fig:var-toy\]A,C) and the perturbed trajectory $\tilde{\gamma}(t)$ with $\tilde{\gamma}(0)=[1,\alpha]+{\mathbf{u}}(0)$ (red dotted, Figure \[fig:var-toy\]A,C). The differences between the two trajectories along the $x$-direction and the $y$-direction are indicated by the black lines in Figure \[fig:var-toy\]B and D, both showing good agreements with the approximated displacements computed from the variational equation, indicated by the red dotted lines in Figure \[fig:var-toy\]B and D. Such an approximation becomes better as the perturbation size $\varepsilon$ gets smaller (simulation result not shown). ![Linear shape response ${\mathbf{u}}(t)$ of the LCSC trajectory $\gamma(t)$ to an instantaneous perturbation applied at $\gamma(0)=[1,\alpha]$ where $\alpha=0.2$. The initial displacement is ${\mathbf{u}}(0)=[0,\varepsilon]$ where $\varepsilon=0.1$. (A, C) Time series of $\gamma(t)$ (black solid) and $\tilde{\gamma}(t)$ (red dotted) with a perturbed initial condition $\tilde{\gamma}(0)=[1,\alpha+\varepsilon]$. (B, D) The difference between $\tilde{\gamma}(t)$ and $\gamma(t)$ obtained by direct calculation from the left panels (black) and the displacement solution ${\mathbf{u}}(t)$ obtained using the **Algorithm for ${\mathbf{u}}$** (red dotted). (A) and (B) show trajectories and the displacement along the $x$-direction, while (C) and (D) show trajectories and displacements along the $y$-direction. Shaded regions have the same meanings as in Figure \[fig:LC-2d-isochrons\]. []{data-label="fig:var-toy"}](variational-toy-model-3.pdf){width="6in"} Next, we study the effects of static perturbations on the timing and shape using the iPRC and iSRC. Shape response analysis {#sec:planar-src} ----------------------- In this section, we illustrate how to compute the iSRC $\gamma_1$, the linear shape responses of the LCSC to small static perturbations. Recall that we use $\gamma_0(t)$ with period $T_0$ and $\gamma_{\varepsilon}(t)$ with period $T_{\varepsilon}$ to denote the original and the perturbed LCSC solutions. We write $\gamma_1$ for the linear shift in the limit cycle shape in response to the static perturbation as indicated by , which we also repeat here: $$\gamma_{\varepsilon}(\tau(t))=\gamma_0(t)+{\varepsilon}\gamma_1(t)+O({\varepsilon}^2),$$ where the time for the perturbed LCSC is rescaled to be $\tau(t)$ to match the unperturbed time points. The iSRC $\gamma_1$ satisfies the nonhomogeneous variational equation . To solve this equation, an estimation of the timing scaling factor $\nu_1$, determined by the choice of time rescaling $\tau(t)$, is needed. Here we consider two kinds of static perturbations on the planar LCSC model: global perturbation and piecewise perturbation. #### Global perturbation. We apply a small static perturbation to the planar LCSC model by increasing the model parameter $\alpha$ by $\varepsilon$ globally: $\alpha\to \alpha+\varepsilon$. To compare the LCSCs before and after perturbation at corresponding time points, we rescale the perturbed trajectory uniformly in time so that $\tau(t) = T_{\varepsilon}t/T_0$. It follows that $\nu_1 = T_1/T_0$, where the linear shift $T_1:=\lim_{\varepsilon\to 0}(T_{\varepsilon}-T_0)/\varepsilon$ can be estimated using the iPRC as discussed in §\[sec:smooth-sust\] (see ). Using **Algorithm for $\gamma_1$ with uniform rescaling**, we numerically compute the iSRC $\gamma_1(t)$ for $\varepsilon=0.01$. The $x$ and $y$ components of $\varepsilon\gamma_1(t)$ are shown by the red curves in Figure \[fig:change-shape\]A, both of which show good agreement with the numerical displacement $\gamma_{\varepsilon}(\tau(t))-\gamma_0(t)$ (black solid), as expected from our theory. For $\varepsilon$ over a range $[0, 0.01]$, we repeat the above procedure and compute the Euclidean norms of both the numerical displacement vector $\gamma_{\varepsilon}(\tau(t))-\gamma(t)$ (Figure \[fig:change-shape\]B, black solid) and the approximated displacement vector $\varepsilon\gamma_1(t)$ (Figure \[fig:change-shape\]B, red dotted) over one cycle. From the plot, we can see that the iSRC with uniform rescaling of time gives a good first-order $\varepsilon$ approximation to the shape response of the planar LCSC model to a global static perturbation. (145,60)(0,0) (0,0)[![\[fig:change-shape\] iSRC of the LCSC model to a small perturbation $\alpha\to \alpha+{\varepsilon}$ with unperturbed parameter $\alpha=0.2$. (A) Time series of the difference between the perturbed and unperturbed solutions along the $x$-direction (top panel) and the $y$-direction (bottom panel) with ${\varepsilon}=0.01$. The black curve denotes the numerical displacement computed by subtracting the unperturbed solution trajectory from the perturbed trajectory, after globally rescaling time. The red dashed curve denotes the product of $\varepsilon$ and the shape response curve solution. Shaded regions have the same meanings as in Figure \[fig:LC-2d-isochrons\]. (B) The norm of the numerical difference (black) and the product of $\varepsilon$ and the iSRC (red dashed) grow linearly with respect to $\varepsilon$ with nearly identical slope, indicating that the iSRC is very good for approximating the numerical difference over a range of $\varepsilon$ and improves with smaller $\varepsilon$. ](shape_response_curve_plot_2.pdf "fig:"){height="5.5cm"}]{} (-8,46)[****]{} (75,0)[![\[fig:change-shape\] iSRC of the LCSC model to a small perturbation $\alpha\to \alpha+{\varepsilon}$ with unperturbed parameter $\alpha=0.2$. (A) Time series of the difference between the perturbed and unperturbed solutions along the $x$-direction (top panel) and the $y$-direction (bottom panel) with ${\varepsilon}=0.01$. The black curve denotes the numerical displacement computed by subtracting the unperturbed solution trajectory from the perturbed trajectory, after globally rescaling time. The red dashed curve denotes the product of $\varepsilon$ and the shape response curve solution. Shaded regions have the same meanings as in Figure \[fig:LC-2d-isochrons\]. (B) The norm of the numerical difference (black) and the product of $\varepsilon$ and the iSRC (red dashed) grow linearly with respect to $\varepsilon$ with nearly identical slope, indicating that the iSRC is very good for approximating the numerical difference over a range of $\varepsilon$ and improves with smaller $\varepsilon$. ](normdiff-eps-plot-uniform-perturb.pdf "fig:"){height="5.5cm"}]{} (70,46)[****]{} #### Piecewise perturbation. Uniform rescaling of time as used above is the simplest choice among many possible rescalings, and is shown to be adequate in the global perturbation case for computing an accurate iSRC. As discussed in §\[sec:smooth-theory\], in certain cases we may instead need the technique of *local* timing response curves (lTRCs) to obtain nonuniform choices of rescaling for greater accuracy. As an illustration, we add two local timing surfaces $\Sigma^{\rm in}$ and $\Sigma^{\rm out}$ to the planar LCSC model (see Figure \[fig:time-response\]A). We denote the subdomain above $\Sigma^{\rm in}$ and $\Sigma^{\rm out}$ by region I (${{\mathcal R}}^{\rm I}$) and denote the remaining subdomain by region II (${{\mathcal R}}^{\rm II}$). Moreover, we introduce a new parameter $\omega$, the rotation rate of the source at the origin, that has previously been fixed at $1$, and rewrite the interior dynamics of the planar LCSC model as $$\label{eq:toy-model-omega} \frac{d\textbf{x}}{dt}=F(\textbf{x})=\begin{bmatrix} \alpha x -\omega y\\\omega x+\alpha y \end{bmatrix}.$$ The vector fields on a given wall are obtained by replacing the coefficient of $y$ in $dx/dt$ (in Table \[tab:natural\]) by $-\omega$ and replacing the coefficient of $x$ in $dy/dt$ (in Table \[tab:natural\]) by $\omega$ on that wall. We apply a static piecewise perturbation to the system by letting $(\alpha,\omega)\rightarrow (\alpha+\varepsilon,\omega-\varepsilon)$ over region I but not region II. Such a piecewise constant perturbation affects both the expansion and rotation rates of the source in region $\rm I$, and hence will lead to different timing sensitivities of $\gamma(t)$ in the two regions. It is therefore natural to use piecewise uniform rescaling when computing the shape response curve as opposed to using a uniform rescaling. In the following, we first compute the lTRC (see Figure \[fig:time-response\]) and use it to estimate the two time rescaling factors for ${{\mathcal R}}^{\rm I}$ and ${{\mathcal R}}^{\rm II}$, which are denoted by $\nu_1^{\rm I}$ and $\nu_1^{\rm II}$, respectively. We then show the iSRC computed using the piecewise uniform rescaling factors provides a more accurate representation of the shape response to the piecewise static perturbation than using a uniform rescaling (see Figure \[fig:shape-response\] and \[fig:norm-shape-response\]). (145,60)(0,0) (0,0)[![\[fig:time-response\] lTRC of the planar LCSC model under perturbation $(\alpha,\omega)\rightarrow (\alpha+\varepsilon,\omega-\varepsilon)$ over region I with unperturbed parameters $\alpha=0.2$ and $\omega=1$ held fixed in region II. (A) Projection of the limit cycle solution to the planar model with two new added switching surfaces $\Sigma^{\rm in}$ (green dashed line) and $\Sigma^{\rm out}$ (blue dashed line) onto its phase plane. (B) Time series of the lTRC $\eta^{\rm I}$ from $t^{\rm in}$ (the time of entry into region I at ${\mathbf{x}}^{\rm in}$) to $t^{\rm out}$ (the time of exiting region I at ${\mathbf{x}}^{\rm out}$). A discontinuous jump occurs when the trajectory exits the wall $y=1$ indicated by the right boundary of the shaded region, which has the same meaning as in Figure \[fig:LC-2d-isochrons\]. ]({LC-PP-V-new.pdf} "fig:"){height="5.5cm"}]{} (-5,46)[****]{} (75,0)[![\[fig:time-response\] lTRC of the planar LCSC model under perturbation $(\alpha,\omega)\rightarrow (\alpha+\varepsilon,\omega-\varepsilon)$ over region I with unperturbed parameters $\alpha=0.2$ and $\omega=1$ held fixed in region II. (A) Projection of the limit cycle solution to the planar model with two new added switching surfaces $\Sigma^{\rm in}$ (green dashed line) and $\Sigma^{\rm out}$ (blue dashed line) onto its phase plane. (B) Time series of the lTRC $\eta^{\rm I}$ from $t^{\rm in}$ (the time of entry into region I at ${\mathbf{x}}^{\rm in}$) to $t^{\rm out}$ (the time of exiting region I at ${\mathbf{x}}^{\rm out}$). A discontinuous jump occurs when the trajectory exits the wall $y=1$ indicated by the right boundary of the shaded region, which has the same meaning as in Figure \[fig:LC-2d-isochrons\]. ](local-timing-response-curve.pdf "fig:"){height="5.5cm"}]{} (70,46)[****]{} Although the lTRC $\eta$ is defined throughout the domain, estimating the effect of the perturbation localized to region I only requires evaluating the lTRC in this region. Figure \[fig:time-response\]B shows the time series of $\eta^{\rm I}$ for the planar LCSC model in region I, obtained by numerically integrating the adjoint equation (\[eq:ltrc\]) backward in time with the initial condition of $\eta^{\rm I}$ given by its value at the exit point of region I denoted by ${\mathbf{x}}^{\rm out}$ (see **Algorithm for $\eta^j$**). Similar to the iPRC, the $y$ component of the lTRC $\eta^\text{I}$ shown by the red curve in Figure \[fig:time-response\]B is zero along the wall $y=1$, and the only discontinuous jump of $\eta^{\rm I}$ occurs at the liftoff point. Note that $\eta^{\rm I}$ is defined as the gradient of the time remaining in ${{\mathcal R}}^{\rm I}$ until exiting through $\Sigma^{\rm out}$. If the $x$ or $y$ component of $\eta^{\rm I}$ is positive then the perturbation along the positive $x$-direction or $y$-direction increases the time remaining in ${{\mathcal R}}^{\rm I}$, and the exit from ${{\mathcal R}}^{\rm I}$ will occur later. On the other hand, if the $x$ or $y$ component $\eta^{\rm I}$ is negative then the perturbation along the positive $x$-direction or $y$-direction decreases the time remaining in ${{\mathcal R}}^{\rm I}$, and the exit from ${{\mathcal R}}^{\rm I}$ will occur sooner. The relative shift in time spent in ${{\mathcal R}}^{\rm I}$ caused by a static perturbation can therefore be estimated using the lTRC (see ) as illustrated in the last step of **Algorithm for $\eta^j$**. Note that the first term in implies that the timing change in a region generically depends on the shape change at the corresponding entry point, leading to the possibility of bidirectional coupling between timing and shape changes. However, in this planar system, a perturbed trajectory with $\varepsilon\ll 1$ will converge back to the original trajectory, within region II (where the perturbation is absent), in finite time. Under these circumstances, there is no shift between the perturbed and unperturbed trajectories in the entry location to region I. Hence, in this case, the local timing shift does not depend on the shape change. Let $T_{0}^{\rm I}$ denote the time spent in region I, and let $T_{0}^{\rm II}=T_{0}-T_{0}^{\rm I}$ denote the time spent in region II (recall $T_{0}$ is the total period). The linear shift in $T_0^{\rm I}$, denoted by $T_1^{\rm I}$, can be estimated using the lTRC $\eta^{\rm I}$ as discussed above. By definition the two time rescaling factors required to compute the iSRC are given by $\nu_1^{\rm I}=\frac{T_1^{\rm I}}{T_0^{\rm I}}$ and $\nu_1^{\rm II}=\frac{T_1-T_1^{\rm I}}{T_0^{\rm II}}$ where the global relative change in period, $T_1$, can be estimated using the iPRC as discussed before. With $\nu_1^{\rm I}$ and $\nu_1^{\rm II}$ known, we take ${\mathbf{x}}^{\rm in}$, the coordinate of the entry point into ${{\mathcal R}}^{\rm I}$, as the initial condition for $\gamma(t)$ and apply **Algorithm for $\gamma_1$ with piecewise uniform rescaling** to compute the iSRC $\gamma_1$ for $\varepsilon=0.1$. The $x$ and $y$ components of $\varepsilon \gamma_1$ are shown by the red dashed curves in Figure \[fig:shape-response\]B, both of which show good agreement with the numerical displacement $\gamma_{\varepsilon}(\tau(t)) - \gamma(t)$ (black solid curves). Here the rescaling $\tau(t)$ is piecewise uniform: $$\begin{aligned} \label{eq:piecewise-uniform-time-rescaling} \tau(t)=\left\{ {\ensuremath{\begin{array}{cccccccccc} t^{\rm in}+T_{\varepsilon}^{\rm I}(t-t^{\rm in})/T_0^{\rm I}, & \gamma(t) \in {{\mathcal R}}^{\rm I}\\\\ t^{\rm in}+T_{\varepsilon}^{\rm I}+T_{\varepsilon}^{\rm II}(t-t^{\rm out})/T_0^{\rm II}, & \gamma(t) \in {{\mathcal R}}^{\rm II}\\ \end{array}}} \right.\end{aligned}$$ where $T_{\varepsilon}^i$ denotes the time $\gamma_{\varepsilon}$ spends in ${{\mathcal R}}^i$ with $i\in\{\rm I, II\}$. It follows that the exit time of the trajectory from region I before (Figure \[fig:shape-response\]B, vertical blue line) and after (Figure \[fig:shape-response\]B, vertical magenta line) perturbation are the same. As a comparison, for $\varepsilon=0.1$, we also compute the iSRC and the numerical displacement using the uniform rescaling of time as we did in the global perturbation case (see Figure \[fig:shape-response\]A). The difference between the vertical blue and magenta lines (the time when the unperturbed and perturbed trajectories leave region I) indicates region I and region II have different timing sensitivities. As expected, the resulting $\varepsilon \gamma_1$ no longer shows good agreement with the numerical displacement obtained from subtracting the unperturbed solution from the rescaled perturbed solution. (145,60)(0,0) (0,0)[![\[fig:shape-response\] A small perturbation is applied to the planar model over region I in which $(\alpha,\omega)\to(\alpha+\varepsilon,\omega-\varepsilon)$ with unperturbed parameters $\alpha=0.2$, $\omega=1$ and perturbation $\varepsilon=0.1$. Time series of the difference between the perturbed and unperturbed solutions along the $x$-direction (top panel) and the $y$-direction (lower panel) using (A) the global rescaling factor and (B) two different rescaling factors within regions I and II. The vertical blue dashed line denotes the exit time of the unperturbed trajectory from Region I, while the vertical magenta solid line denotes the exit time of the rescaled perturbed trajectory from Region I. Other color codings of lines are the same as in Figure \[fig:change-shape\]A. Shaded regions have the same meanings as in Figure \[fig:LC-2d-isochrons\]. ](shape-response-onecale-alpha-omega.pdf "fig:"){height="5.5cm"}]{} (-8,46)[****]{} (75,0)[![\[fig:shape-response\] A small perturbation is applied to the planar model over region I in which $(\alpha,\omega)\to(\alpha+\varepsilon,\omega-\varepsilon)$ with unperturbed parameters $\alpha=0.2$, $\omega=1$ and perturbation $\varepsilon=0.1$. Time series of the difference between the perturbed and unperturbed solutions along the $x$-direction (top panel) and the $y$-direction (lower panel) using (A) the global rescaling factor and (B) two different rescaling factors within regions I and II. The vertical blue dashed line denotes the exit time of the unperturbed trajectory from Region I, while the vertical magenta solid line denotes the exit time of the rescaled perturbed trajectory from Region I. Other color codings of lines are the same as in Figure \[fig:change-shape\]A. Shaded regions have the same meanings as in Figure \[fig:LC-2d-isochrons\]. ](shape-response-twocales-alpha-omega.pdf "fig:"){height="5.5cm"}]{} (68,46)[****]{} Piecewise uniform rescaling, on the other hand, leads to a more accurate iSRC for the LCSC model than uniform rescaling, when the LCSC $\gamma(t)$ experiences distinct timing sensitivities for $\varepsilon=0.1$. Fig. \[fig:shape-response\] contrasts the accuracy of the linearized shape response using global (A) versus local (B) timing response curves, for $\varepsilon=0.1$. We also show the same conclusion holds for other $\varepsilon$ values. To this end, for $\varepsilon$ over a range of $[0,0.1]$ we repeat the above procedure and compute the Euclidean norms of both the numerical displacement vector and the displacement vector approximated by the iSRC, as illustrated in Figure \[fig:norm-shape-response\]A. The numerical and approximated norm curves using the uniform rescaling are shown in red solid and red dotted lines, while the numerical and approximated norm curves using the piecewise uniform rescaling are shown in blue solid and blue dotted lines. Unsurprisingly, the norms of the displacements between the perturbed and original trajectory grow approximately linearly with respect to $\varepsilon$, and the displacement norms with piecewise uniform rescaling are smaller than that with uniform rescaling. The fact that the difference between the lines in red is much bigger than the difference between the lines in blue suggests that the piecewise uniform rescaling gives a more accurate iSRC than using the uniform rescaling for $\varepsilon\in [0,0.1]$, as we expect. This heightened accuracy is further demonstrated in Figure \[fig:norm-shape-response\]B, where the relative difference between the numerical and approximated norms with uniform rescaling (red curve) is significantly larger than the relative difference when using piecewise uniform rescaling (blue curve). (145,60)(0,0) (0,0)[![\[fig:norm-shape-response\] A small perturbation is applied to the planar model over region I in which $(\alpha,\omega)\to(\alpha+\varepsilon,\omega-\varepsilon)$ with unperturbed parameters $\alpha=0.2$, $\omega=1$. (A): Values of the Euclidean norm of $(\gamma_\varepsilon(\tau(t))-\gamma(t))$ computed numerically (solid curve) versus those computed from the iSRC (dashed curve), as $\varepsilon$ varies. The norms grow approximately linearly with respect to $\varepsilon$. The approximation obtained by the iSRC when using piecewise uniform rescaling (blue) is closer to the actual simulation than using the uniform rescaling (red). (B): The relative difference between the actual and approximated norms with a uniform rescaling (red) is larger than that when piecewise uniform rescaling is used (blue). The difference between the two curves expands as $\varepsilon$ increases. ]({norm-difference-wrt-eps.pdf} "fig:"){height="5.5cm"}]{} (-5,46)[****]{} (75,0)[![\[fig:norm-shape-response\] A small perturbation is applied to the planar model over region I in which $(\alpha,\omega)\to(\alpha+\varepsilon,\omega-\varepsilon)$ with unperturbed parameters $\alpha=0.2$, $\omega=1$. (A): Values of the Euclidean norm of $(\gamma_\varepsilon(\tau(t))-\gamma(t))$ computed numerically (solid curve) versus those computed from the iSRC (dashed curve), as $\varepsilon$ varies. The norms grow approximately linearly with respect to $\varepsilon$. The approximation obtained by the iSRC when using piecewise uniform rescaling (blue) is closer to the actual simulation than using the uniform rescaling (red). (B): The relative difference between the actual and approximated norms with a uniform rescaling (red) is larger than that when piecewise uniform rescaling is used (blue). The difference between the two curves expands as $\varepsilon$ increases. ](relative-norm-difference-wrt-eps.pdf "fig:"){height="5.5cm"}]{} (70,46)[****]{} Discussion {#sec:discussion} ========== Rhythmic motions making and breaking contact with a constraining boundary, and subject to external perturbations, arise in motor control systems such as walking, running, scratching, biting and swallowing, as well as other natural and engineered hybrid systems [@branicky1998]. Dynamical systems describing such rhythmic motions are therefore nonsmooth and often exhibit limit cycle trajectories with sliding components. In smooth dynamical systems, classical analysis for understanding the change in periodic limit cycle orbits under weak perturbation relies on the Jacobian linearization of the flow near the limit cycle. These methods do not apply to nonsmooth systems, for which the Jacobian matrices are not well defined. In this work, we describe for the first time the variational analysis and the infinitesimal phase response curves (iPRC) for limit cycles with sliding components (LCSC). Moreover, we give a rigorous derivation of the saltation matrix associated with the variational dynamics and the closely related jump matrix for the iPRC at the hard boundary crossing point. We also report, for the first time, how the presence of a liftoff point, where a limit cycle leaves a constraint surface, can create a nondifferentiable “kink” in the asymptotic phase function, propagating backwards in time along an osculating trajectory (see Figure \[fig:LC-2d-isochrons\]A). Most significantly, we have defined the infinitesimal shape response curve (iSRC) to analyze the joint variation of both shape and timing of limit cycles with sliding components, under parametric perturbations. We show that taking into account local timing sensitivity *within* a switching region improves the accuracy of the iSRC over global timing analysis alone. This improvement in accuracy is facilitated by our introduction of a novel *local timing response curve* (lTRC) measuring the timing sensitivity of an oscillator within a given local region. Our results clarify an important distinction between the effects of the boundary encounter on the timing and shape changes in limit cycles with sliding components. We have extended both iPRC and variational analysis developed for smooth limit cycle systems to the LCSC case, presented here as Theorem \[thm:main\]. In addition, our analysis yields an explicit expression for the iPRC jump matrix that characterizes the behavior of the iPRC at the landing and liftoff points. Surprisingly, we find that the iPRC experiences no discontinuity when the trajectory first contacts a hard boundary, while the variational equation suffers a discontinuity, captured by the saltation matrix. Even more interesting, at the liftoff point – where the saltation matrix for the variational problem is trivial – the iPRC *does* show a discontinuous change, captured by a nontrivial jump matrix. Specifically, there is a discontinuous jump from zero to a nonzero normal component in the iPRC. Consequently, numerical evaluation of the iPRC must be obtained by backward integration along the limit cycle, as discussed in §\[sec:algorithm-iprc\]. Finally, we find that both the iPRC and the variational dynamics have zero normal components during the sliding component of the limit cycle, due to dimensional compression at the hard boundary. Standard variational and phase response curve analysis typically neglect changes in timing or shape, focusing instead on only one of the two aspects [@kuramoto1975]. However, in many applications such as motor control systems, both the shape and timing of the trajectory are often affected under slow or parametric perturbations. In this paper, we consider both timing and shape aspects using the iSRC. We have discussed two ways of incorporating timing changes into the iSRC: uniform timing rescaling based on the global timing analysis (iPRC) and piecewise uniform timing rescaling based on the local timing analysis (lTRC). As demonstrated in the planar system example in §\[sec:toy-model\], when the trajectory exhibits approximately constant timing sensitivities, the iSRC with global timing rescaling is good enough for approximating the shape change (see Figure \[fig:change-shape\]); otherwise, we need take into account local timing changes to increase the accuracy of the iSRC (see Figure \[fig:shape-response\]). LCSC with piecewise timing sensitivities naturally arise in many motor control systems due to nonuniform perturbations. For instance, the friction of the ground acts as a perturbation during the stance phase of locomotion, when a leg generates ground reaction forces, and is absent during the swing phase; a force applied to the food can only be felt when an animal is biting on the food. Local timing analysis (lTRC) will then provide a better understanding of such systems compared with the global timing analysis (iPRC). Other investigators have also considered variational [@bernardo2008; @LN2013] and phase response analysis in nonsmooth systems [@shirasaka2017; @park2018; @CGL18; @wilson2019], but these studies were subject to transverse flow conditions. Our work extends both variational and iPRC analysis to the LCSC case in which the transversal crossing condition fails. Combined timing and shape responses of limit cycles to perturbations have also been explored in other works. @monga2018 examined energy-optimal control of the timing of limit cycle systems including spiking neuron models and models of cardiac arrhythmia. They showed that when one of the nontrivial Floquet multipliers of an unperturbed limit cycle system has magnitude close to unity, control inputs based solely on standard phase reduction, which neglects the effect on the shape of the controlled trajectory, can dramatically fail to achieve control objectives. They and other authors have introduced augmented phase reduction techniques that use a system of coordinates (related to the Floquet coordinates) transverse to the limit cycle to improve the accuracy of phase reduction and control [@castejon2013; @WM2015; @WM2016; @WE2018; @monga2018b; @wilson2019]. These methods require the underlying dynamics be smoothly differentiable, and rely on calculation of the Jacobian (first derivative) and in some cases the Hessian (second derivative) matrices [@WE2018]. For nonsmooth limit cycle systems with sliding components, our analysis is the first to address the combined effects of shape and timing, an essential element of improved control in biomedical applications as well as for understanding mechanisms of control in naturally occurring motor control systems. For trajectories with different timing sensitivities in different regions, we rely on the local timing response curve (lTRC) to estimate the relative shift in time in each sub-region, in order to compute the full infinitesimal shape response curve (iSRC). Conversely, solving for the lTRC in a given region may also require an understanding of the impact of the perturbation on the entry point associated with that region (see ). Thus, in general, the iSRC and the lTRC are interdependent. While we have not derived a closed-form expression for the shape and timing response in the most general case, we have provided effective algorithms for solving each of them separately, which requires preliminary numerical work to find the trajectory shape shift at the entry point. In the future, it may be possible to derive general closed-form expressions for the iSRC and lTRC in systems with distinct timing sensitivities. While our methods are illustrated using a planar limit cycle system with hard boundaries, they apply to higher dimensional systems as well. For instance, preliminary investigations suggest that the methods developed in this paper are applicable to analyzing the nonsmooth dynamics arising in the control system of feeding movements in the sea slug *Aplysia* [@shaw2012; @shaw2015; @lyttle2017]. More generally, limit cycles with discontinuous trajectories arise in neuroscience (e.g., integrate and fire neurons) and mechanics (e.g., ricochet dynamics). If such systems manifest limit cycles with sliding components, our methods could be combined with variational methods adapted for piecewise continuous trajectories [@coombes2012; @shirasaka2017]. It was observed heuristically by @lyttle2017 that sensory feedback could in some circumstances lead to significant robustness against an increase in applied load, in the sense that although modest relative increases in external load (*c.* 20%) led to comparable changes in both the timing and shape of trajectories, the net effect on the performance (rate of intake of food) was an order of magnitude smaller (*c.* 1%). Similarly, @diekman2017 showed that in a model for control of a central pattern generator regulating the breathing rhythm, mean arterial partial pressure of oxygen (PPO$_2$) remained approximately constant under changing metabolic loads when chemosensory feedback from the arterial PPO$_2$ to the central pattern generator was present, but varied widely otherwise [@diekman2017]. Understanding how rhythmic biological control systems respond to such perturbations and maintain robust, adaptive performance is one of the fundamental problems within theoretical biology. Solving these problems will then require variational analysis along the lines we develop here. Nonsmooth dynamics arise naturally in many biological systems [@AiharaSuzuki2010; @coombes2012], and thus, the approach in this paper is likely to have broad applicability to many other problems in biology. Acknowledgement {#acknowledgement .unnumbered} =============== This work was made possible in part by grants from the National Science Foundation (DMS-1413770, DEB-1654989 and IOS-174869). PT thanks the Oberlin College Department of Mathematics for research support. This research has been supported in part by the National Science Foundation Grant DMS-1440386 to the Mathematical Biosciences Institute. Table of Common Symbols {#ap:symbols} ======================= -------------------------------------------------------------------------------------------------------------------------------------------------- **Symbol** **Meaning** ----------------------------------------------------------- -------------------------------------------------------------------------------------- ${\mathbf{x}}$ state variables $t$ time $\theta(t)$ phase of a limit cycle $\phi({\mathbf{x}})$ asymptotic phase of a stable limit cycle $F({\mathbf{x}})$ unperturbed velocity vector field $\gamma(t)$ unperturbed limit cycle solution $T$ period of the unperturbed limit cycle $\varepsilon P$ small instantaneous perturbation vector $\tilde{\gamma}(t)$ trajectory near limit cycle after instantaneous perturbation ${\mathbf{u}}(t) \simeq \tilde{\gamma}(t) - \gamma(t)$ displacement from limit cycle after instantaneous perturbation $\varepsilon$ sustained (parametric) perturbation $F_\varepsilon({\mathbf{x}})$ perturbed velocity vector field $\gamma_\varepsilon(t)$ perturbed limit cycles solution $T_\varepsilon$ period of the perturbed limit cycle $F_0 = F$ zeroth-order term of Taylor expansion of $F_\varepsilon$ around $\varepsilon=0$ $\gamma_0 = \gamma$ zeroth-order term of Taylor expansion of $\gamma_\varepsilon$ around $\varepsilon=0$ $T_0 = T$ zeroth-order term of Taylor expansion of $T_\varepsilon$ around $\varepsilon=0$ $F_1 = \partial F_\varepsilon / first-order term of Taylor expansion of $F_\varepsilon$ around $\varepsilon=0$ \partial \varepsilon \big|_{\varepsilon=0}$ $\gamma_1 = \partial \gamma_\varepsilon / first-order term of Taylor expansion of $\gamma_\varepsilon$ around $\varepsilon=0$, \partial \varepsilon \big|_{\varepsilon=0}$ also called the infinitesimal shape response curve (iSRC) $T_1 = \partial T_\varepsilon / first-order term of Taylor expansion of $T_\varepsilon$ around $\varepsilon=0$ \partial \varepsilon \big|_{\varepsilon=0}$ $DF$; $D_\mathbf{w}\phi$ Jacobian matrix; directional derivative of $\phi$ in $\mathbf{w}$ direction $I$ identity matrix $S$ saltation matrix (for variation equation) $J$ jump matrix (for adjoint equation) $\mathcal{J}$ time-reversed jump matrix (for adjoint equation) $\Sigma^i$ boundary $i$ ${{\mathcal R}}^j$ region $j$ $F^j({\mathbf{x}})$ velocity vector field in region $j$ $\nu_\varepsilon = T_0/T_\varepsilon$ relative frequency of perturbed limit cycle $\nu_1 = T_1/T_0$ first-order term of Taylor expansion of $\nu_\varepsilon$ around $\varepsilon=0$, also called the relative change in frequency $\mathcal{T}^j$ time remaining in region $j$ along a trajectory ${\mathbf{u}}(t)$ variational dynamics governed by ${{\mathbf z}}(t)=\nabla_{\mathbf{x}}\phi(\gamma(t))$ infinitesimal phase response curve (iPRC) governed by $\gamma_1(t)$ infinitesimal shape response curve (iSRC) governed by $\eta^j(t) = \nabla_{\mathbf{x}}\mathcal{T}^j(\gamma(t))$ local timing response curve (lTRC) governed by -------------------------------------------------------------------------------------------------------------------------------------------------- Derivation of Equation \[eq:local-time-shift\] {#ap:T1} ============================================== This section establishes equation , which specifies the first-order change in the transit time through region I, or $T_1^\text{I}$: $$T^{\rm I}_{1} = \eta^{\rm I}({\mathbf{x}}^{\rm in})\cdot \frac{\partial {\mathbf{x}}_{\varepsilon}^{\rm in}}{\partial \varepsilon}\Big|_{\varepsilon=0}+\int_{t^{\rm in}}^{t^{\rm out}}\eta^{\rm I}(\gamma(t))\cdot \frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0}dt,$$ Recall $\mathcal{T}^{\rm I}({\mathbf{x}})$ is the time remaining until exiting region I through $\Sigma^{\rm out}$, under the unperturbed vector field, starting from location ${\mathbf{x}}$; $\eta^{\rm I}:= \nabla \mathcal{T}^{\rm I}({\mathbf{x}})$ is the local timing response curve (lTRC) for region I, defined for the component of the trajectory lying within region I, i.e. for times $t\in [t^{\rm in}, t^{\rm out}]$; and ${\mathbf{x}}_\varepsilon^{\rm in}$ is the coordinate of the perturbed entry point into region I. We consider a single region $\mathcal{R}$ with entry surface $\Sigma^{\rm in}$ and exist surface $\Sigma^{\rm out}$. We assume that these two surfaces are fixed, independent of static perturbation with size ${\varepsilon}$. The limit cycle solution ${\mathbf{x}}=\gamma_{\varepsilon}(\tau)$ satisfies $$\frac{d{\mathbf{x}}}{d\tau}=F_{\varepsilon}({\mathbf{x}})$$ where $\tau$ is the time coordinate of the perturbed trajectory. Moreover, $\gamma_{\varepsilon}(\tau)$ enters $\mathcal{R}$ at ${\mathbf{x}}^\text{in}_{\varepsilon}\in\Sigma^{\rm in}$ when $\tau=t^\text{in}_{\varepsilon}$ and exits at ${\mathbf{x}}^\text{out}_{\varepsilon}\in\Sigma^{\rm out}$ when $\tau=t^\text{out}_{\varepsilon}$. Since the system is autonomous, we are free to choose the reference time along the limit cycle orbit. For convenience of calculation, we set $t^\text{out}_{\varepsilon}\equiv 0$ for all ${\varepsilon}$. Denote the transit time that $\gamma_{\varepsilon}$ spends in $\mathcal{R}$ by $T^{{\mathcal R}}_\varepsilon$. It follows that $t^\text{in}_{\varepsilon}= -T^{{\mathcal R}}_{{\varepsilon}}$, where ${\varepsilon}$ can be $0$. Assuming that the transit time has a well behaved expansion in ${\varepsilon},$ we write $$\begin{aligned} \label{eq:Teps} T^{{\mathcal R}}_{\varepsilon}&=T^{{\mathcal R}}_0+{\varepsilon}T^{{\mathcal R}}_1+O({\varepsilon}^2)\end{aligned}$$ where $T^{{\mathcal R}}_0$ is the transit time for the unperturbed trajectory and $T^{{\mathcal R}}_1$ is the linear shift in the transit time. In the rest of this section, we drop the superscript ${{\mathcal R}}$ on $T^{{\mathcal R}}_\varepsilon,\,T^{{\mathcal R}}_0$ and $T^{{\mathcal R}}_1$ for simplicity. Our goal is to prove that $T_1$ is given by . We do this in two steps. First, we show that the transit time $T_{{\varepsilon}}$ can be expressed in terms of the perturbed vector field and perturbed local timing response curve (see ). Second, we expand the expression for $T_{{\varepsilon}}$ to first order in $\varepsilon$ to obtain the expression for $T_1$. Since the time remaining to exit, denoted as $\mathcal{T}_\varepsilon$, decreases at a constant rate along trajectories, for arbitrary ${\varepsilon}$ we have $$\label{eq:dTdt=-1} -1=\frac{d\mathcal{T}_\varepsilon}{d\tau}=F_\varepsilon(\gamma_\varepsilon(\tau))\cdot\eta_\varepsilon(\gamma_\varepsilon(\tau)),$$ where $\eta_\varepsilon({\mathbf{x}})=\nabla\mathcal{T}_\varepsilon({\mathbf{x}})$ is defined as the local timing response curve under perturbation. By , the transit time $T_\varepsilon$ is therefore given by $$\label{eq:integral-for-transit-time} T_\varepsilon=\int_{\tau=t^\text{out}_{\varepsilon}}^{t^\text{in}_{\varepsilon}}F_\varepsilon(\gamma_{\varepsilon}(\tau))\cdot \eta_{\varepsilon}(\gamma_{\varepsilon}(\tau))\,d\tau.$$ In this expression, we integrate *backwards in time* along the limit cycle trajectory, from the egress point ${\mathbf{x}}^\text{out}_\varepsilon$ at time $t^\text{out}_\varepsilon$, to the ingress point ${\mathbf{x}}^\text{in}_\varepsilon$ at time $t^\text{in}_\varepsilon$: For ${\varepsilon}=0$, and taking into account , this integral reduces to $$T_0=\int_{\tau=t^\text{out}_0}^{t^\text{in}_0} F_0(\gamma_0(\tau))\cdot\eta_0(\gamma_0(\tau))\,d\tau=\int_{\tau=t^\text{out}_0}^{t^\text{in}_0}(-1)\,d\tau = t^\text{out}_0-t^\text{in}_0=0-(-T_0),$$ since $t^\text{in}_0= -T_0$ and $t_{\varepsilon}^\text{out}\equiv 0$. In order to derive an expression for $T_1$, the first order shift in the transit time, we need to expand to first order in ${\varepsilon}$. To this end, we need to know the Taylor expansions for all terms in . Suppose we can expand $F_{\varepsilon}$, $\mathcal{T}_{\varepsilon}$, and $\eta_\varepsilon$ as follows: $$\begin{aligned} \label{eq:eps} {\ensuremath{\begin{array}{cccccccccc}F_{\varepsilon}({\mathbf{x}})&=&F_0({\mathbf{x}})+{\varepsilon}F_1({\mathbf{x}})+O({\varepsilon}^2),&\text{ as }{\varepsilon}\to 0,\\ \mathcal{T}_\varepsilon({\mathbf{x}})&=&\mathcal{T}_0({\mathbf{x}})+{\varepsilon}\mathcal{T}_1({\mathbf{x}})+O({\varepsilon}^2),&\text{ as }{\varepsilon}\to 0,\\ \eta_\varepsilon({\mathbf{x}})&=&\eta_0({\mathbf{x}})+{\varepsilon}\eta_1({\mathbf{x}})+O({\varepsilon}^2),&\text{ as }{\varepsilon}\to 0, \end{array}}} \end{aligned}$$ where $\eta_0({\mathbf{x}}) = \nabla\mathcal{T}_0({\mathbf{x}})$ is the unperturbed local timing response curve. Following the idea of deriving the infinitesimal shape response curve in §\[sec:smooth-sust\], we write the portion of the perturbed limit cycle trajectory within region $\mathcal{R}$ in terms of the unperturbed limit cycle, plus a small correction, $$\begin{aligned} \label{eq:gammaeps} \gamma_\varepsilon(\tau)&=\gamma\left(\nu_\varepsilon\tau\right)+{\varepsilon}\gamma_1\left(\nu_\varepsilon\tau\right)+O({\varepsilon}^2)\end{aligned}$$ where $-T_\varepsilon\le \tau \le 0$ and $\nu_\varepsilon= \frac{T_0}{T_\varepsilon}$. Now we expand to first order $$\begin{aligned} T_{\varepsilon}&=\int_{\tau=0}^{-T_{\varepsilon}}\Big[ F_0(\gamma_0(\nu_\varepsilon\tau))+{\varepsilon}DF_0(\gamma_0(\nu_\varepsilon\tau))\cdot \gamma_1(\nu_\varepsilon\tau)+{\varepsilon}F_1(\gamma_0(\nu_\varepsilon\tau)) \Big]\cdot\\ \nonumber &\quad\quad\quad\Big[\eta_0(\gamma_0(\nu_\varepsilon\tau))+{\varepsilon}D\eta_0(\gamma_0(\nu_\varepsilon\tau)) \cdot\gamma_1(\nu_\varepsilon\tau)+{\varepsilon}\eta_1(\gamma_0(\nu_\varepsilon\tau)) \Big]d\tau +O({\varepsilon}^2)\\ \nonumber &=\int_{\tau=0}^{-T_{\varepsilon}} F_0(\gamma_0(\nu_\varepsilon\tau))\cdot \eta_0(\gamma_0(\nu_\varepsilon\tau)) d\tau + {\varepsilon}\Big[ F_0(\gamma_0(\nu_\varepsilon\tau)) \cdot\eta_1(\gamma_0(\nu_\varepsilon\tau))+ F_1(\gamma_0(\nu_\varepsilon\tau))\cdot \eta_0(\gamma_0(\nu_\varepsilon\tau)) \Big] d\tau + \\\nonumber & \quad\quad\quad {\varepsilon}\Big[ F_0(\gamma_0(\nu_\varepsilon\tau)) \cdot D\eta_0(\gamma_0(\nu_\varepsilon\tau)) \cdot\gamma_1(\nu_\varepsilon\tau) + DF_0(\gamma_0(\nu_\varepsilon\tau))\cdot \gamma_1(\nu_\varepsilon\tau) \cdot \eta_0(\gamma_0(\nu_\varepsilon\tau)) \Big] d\tau +O({\varepsilon}^2) \\ \nonumber &=\frac{1}{\nu_\varepsilon}\int_{t=0}^{-T_{0}} F_0(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) dt + {\varepsilon}\Big[ F_0(\gamma_0(t)) \cdot\eta_1(\gamma_0(t))+ F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) \Big] dt + \\\nonumber & \quad\quad\quad {\varepsilon}\Big[ F_0(\gamma_0(t)) \cdot D\eta_0(\gamma_0(t)) \cdot\gamma_1(t) + DF_0(\gamma_0(t))\cdot \gamma_1(t) \cdot \eta_0(\gamma_0(t)) \Big] dt +O({\varepsilon}^2) \\ \nonumber\end{aligned}$$ To order $O(1)$, we recover $$T_0=\int_{t=0}^{-T_0} F_0(\gamma_0(t))\cdot\eta_0(\gamma_0(t))\,dt.$$ This leads to $ T_{\varepsilon} = \frac{1}{\nu_{\varepsilon}}T_0, $ as required for consistency. We are therefore left with $$\begin{aligned} \label{eq:nu1_on_one_side} 0&=\int_{t=0}^{-T_{0}} \Big[ F_0(\gamma_0(t)) \cdot\eta_1(\gamma_0(t))+ F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) \Big] dt \\\nonumber & + \int_{t=0}^{-T_{0}} \Big[ F_0(\gamma_0(t)) \cdot D\eta_0(\gamma_0(t)) \cdot\gamma_1(t) + DF_0(\gamma_0(t))\cdot \gamma_1(t) \cdot \eta_0(\gamma_0(t)) \Big] dt +O({\varepsilon}) \\ \nonumber &=\int_{t=0}^{-T_{0}} \Big[ F_0(\gamma_0(t)) \cdot\eta_1(\gamma_0(t))+ F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) \Big] dt \\\nonumber & + \int_{t=0}^{-T_{0}} \Big[F_0(\gamma_0(t))^\intercal D\eta_0(\gamma_0(t)) + \eta_0(\gamma_0(t))^\intercal DF_0(\gamma_0(t))\Big] \cdot\gamma_1(t) dt +O({\varepsilon}) \\ \nonumber\end{aligned}$$ where the second equality follows from rearranging orders of factors in the second integral. Note that since $F_0\cdot \eta_0\equiv -1$ everywhere, we have the identity $$\begin{aligned} 0=\frac\partial{\partial {\mathbf{x}}_j}\left(\sum_i \eta^i F^i\right)=\sum_i\frac{\partial \eta^i}{\partial {\mathbf{x}}_j} F^i+\sum_i\eta^i\frac{\partial F^i}{\partial {\mathbf{x}}_j}\end{aligned}$$ where $F^i$ and $\eta^i$ are the $i$-th components for $F_0$ and $\eta_0$; ${\mathbf{x}}_j$ denotes the $j$th component of ${\mathbf{x}}$ for $j\in \{1,\cdots, n\}$. It follows that $F_0^\intercal(D\eta_0)+\eta_0^\intercal(D F_0)=0$ in , leaving only $$0=\int_{t=0}^{-T_0} \Big[F_0(\gamma_0(t)) \cdot\eta_1(\gamma_0(t))+ F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t))\Big] \,dt.$$ Since $F_0(\gamma_0(t))=d\gamma_0/dt$ and $\eta_1({\mathbf{x}})=\partial \eta_{\varepsilon}({\mathbf{x}})/\partial{\varepsilon}|_{{\varepsilon}=0} = \partial \nabla\mathcal{T}_{\varepsilon}({\mathbf{x}})/\partial {\varepsilon}|_{{\varepsilon}=0}$, $$\begin{aligned} \int_{t=0}^{-T_0} F_0(\gamma_0(t))\cdot \eta_1(\gamma_0(t)) \,dt \nonumber &= \int_{t=0}^{-T_0}\left(\frac{d\gamma_0}{d t}\right)\cdot\left.\left(\frac\partial{\partial{\varepsilon}}\left[\nabla \mathcal{T}_{\varepsilon}(\gamma_0(t)) \right]\right)\right|_{{\varepsilon}=0}\,dt\\\nonumber &=\int_{t=0}^{-T_0}\left(\frac{d\gamma_0}{dt}\right)\cdot\nabla\left.\left(\frac\partial{\partial{\varepsilon}}\left[ \mathcal{T}_{\varepsilon}(\gamma_0(t)) \right]\right)\right|_{{\varepsilon}=0}\,dt\\\nonumber &=\int_{t=0}^{-T_0}\frac{d}{dt}\left.\left(\frac\partial{\partial{\varepsilon}}\left[ \mathcal{T}_{\varepsilon}(\gamma_0(t)) \right]\right)\right|_{{\varepsilon}=0}\,dt\\\nonumber &= \left.\left(\frac\partial{\partial{\varepsilon}}\left[ \mathcal{T}_{\varepsilon}({\mathbf{x}}^\text{in}_0) \right]\right)\right|_{{\varepsilon}=0} -\left.\left(\frac\partial{\partial{\varepsilon}}\left[ \mathcal{T}_{\varepsilon}({\mathbf{x}}^\text{out}_0) \right]\right)\right|_{{\varepsilon}=0} \\\nonumber &=\left.\left(\frac\partial{\partial{\varepsilon}}\left[ \mathcal{T}_{\varepsilon}({\mathbf{x}}^\text{in}_0) \right]\right)\right|_{{\varepsilon}=0} -0\\\nonumber &=\mathcal{T}_1({\mathbf{x}}^\text{in}_0).\end{aligned}$$ Therefore $$\label{eq:T1v1} \mathcal{T}_1({\mathbf{x}}^\text{in}_0) = \int_{t=-T_0}^{0} F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) dt =\int_{t=t_0^{\rm in}}^{t_0^{\rm out}} F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) dt .$$ The second equality follows from our convention that $t^\text{in}_0= -T_0$ and $t_{\varepsilon}^\text{out}\equiv 0$. We notice that $$\begin{aligned} \label{eq:Teps2} T_{\varepsilon}&=\mathcal{T}_{\varepsilon}({\mathbf{x}}_{\varepsilon}^\text{in})=T_0+ {\varepsilon}\left( \mathcal{T}_1({\mathbf{x}}^\text{in}_0)+\nabla\mathcal{T}_0({\mathbf{x}}^\text{in}_0) \cdot{\mathbf{x}}^\text{in}_1 \right), \end{aligned}$$ where we have made use of the Taylor expansion ${\mathbf{x}}^\text{in}_\varepsilon={\mathbf{x}}^\text{in}_0+{\varepsilon}{\mathbf{x}}^\text{in}_1+O({\varepsilon}^2),\text{ as }{\varepsilon}\to 0$. Equating the first order terms in and leads to $$\label{eq:T1v2} T_1=\mathcal{T}_1({\mathbf{x}}^\text{in}_0)+\eta_0({\mathbf{x}}^\text{in}_0)\cdot{\mathbf{x}}^\text{in}_1.$$ Substituting into , we finally obtain $$\label{eq:T1proof} T_1= \eta_0({\mathbf{x}}^\text{in}_0)\cdot{\mathbf{x}}^\text{in}_1+ \int_{t=t_0^{\rm in}}^{t_0^{\rm out}} F_1(\gamma_0(t))\cdot \eta_0(\gamma_0(t)) dt$$ which is , as desired. Proof of Theorem \[thm:main\] {#ap:proof} ============================= In this section we present a proof of Theorem \[thm:main\], which we restate for the reader’s convenience. **Theorem.** Consider a general LCSC described locally by  in the neighborhood of a hard boundary $\Sigma$ with a constant normal vector $n$, and with a liftoff point defined by . Assume that within the stable manifold of the limit cycle there is a well defined asymptotic phase function $\phi({\mathbf{x}})$ satisfying $d\phi/dt=1$ along trajectories. Assume that $\phi$ is Lipschitz continuous, and assume that on the constraint surface $\Sigma$, the directional derivatives of $\phi$ with respect to directions tangential to the surface are Lipschitz continuous, except (possibly) at the liftoff and landing points. Finally, assume the nondegeneracy condition holds at the liftoff point. Then the following properties hold for the saltation matrix for ${\mathbf{u}}$, and the jump matrix for ${{\mathbf z}}$: 1. At the landing point, the saltation matrix is $S=I-n n^\intercal$, where $I$ is the identity matrix. 2. At the liftoff point, the saltation matrix is $S=I$. 3. Along the sliding region, the component of ${{\mathbf z}}$ normal to $\Sigma$ is zero. 4. The normal component of ${{\mathbf z}}$ is continuous at the landing point. 5. The tangential components of ${{\mathbf z}}$ are continuous at both landing and liftoff points. We choose coordinates ${\mathbf{x}}=({\mathbf{w}},v)=(w_1,w_2,\ldots,w_{n-1},v)$ so that within a neighborhood containing both the landing and liftoff points, the hard boundary corresponds to $v=0$, the interior of the domain coincides with $v>0$, and the unit normal vector for the hard boundary is ${\mathbf{n}}=(0,\ldots,0,1)$. Writing the velocity vector ${\mathbf{F}}=(f_1,f_2,\ldots,f_{n-1},g)$ in these coordinates. In addition, we use ${\mathbf{F}}^{\rm slide}$ to denote the vector field for points on the sliding region, whereas the dynamics of other points is governed by ${\mathbf{F}}^{\rm int}$. The transversal intersection condition for the trajectory entering the hard boundary is $g^\text{int}({\mathbf{x}}_\text{land},0)<0$ (cf. eq. ; note that ${\mathbf{n}}$ defined here points in the opposite direction from the *outward* normal vector in ). At points ${\mathbf{x}}\in\mathcal{L}$ on the liftoff boundary, ${\mathbf{F}}^\text{slide}$ and ${\mathbf{F}}^\text{int}$ coincide and we will use whichever notation seems clearer in a given instance. Under the nondegeneracy condition at the liftoff point , we can further arrange the coordinates $(w_1,\ldots,w_{n-1})$ so that the unit vector normal to the liftoff boundary $\mathcal{L}$ at the liftoff point is $\ell=(0,\ldots,0,1,0)$, and $g^\text{int}\gtrless 0 \iff w_{n-1}\gtrless 0$. With these coordinates, the nondegeneracy condition is ${\mathbf{F}}^{\rm slide}({\mathbf{x}}_{\rm lift})\cdot \ell=f^\text{slide}_{n-1}({\mathbf{x}}_\text{lift})>0$. #### (a) At the landing point, the saltation matrix is $S=I-{\mathbf{n}}{\mathbf{n}}^\intercal$, where $I$ is the identity matrix. The saltation matrix at a transition from the interior to a sliding motion along a hard boundary is given in (@bernardo2008, Example 2.14, p. 111) as $$\begin{aligned} \label{eq:salt-filippov} S=I+\frac{({\mathbf{F}}^{\rm slide}-{\mathbf{F}}^{\rm int}){\mathbf{n}}^\intercal}{{\mathbf{n}}^\intercal {\mathbf{F}}^{\rm int}},\end{aligned}$$ provided the trajectory approaches the hard boundary transversally. It follows from the definition of the sliding vector field $F^{\rm slide}$ given by that $$S=I - {\mathbf{n}}{\mathbf{n}}^\intercal,$$ as claimed. #### (b) At the liftoff point, the saltation matrix is $S=I$. We adapt the argument in ([@bernardo2008], §2.5) to our hard boundary/liftoff construction. The essential difference is that the trajectory is not transverse to the hard boundary at the liftoff point, indeed ${\mathbf{n}}^\intercal {\mathbf{F}}=0$ at ${\mathbf{x}}_\text{lift}$, so eq.  does not give a well defined saltation matrix. However, by replacing the vector ${\mathbf{n}}$ normal to the hard boundary with the vector $\ell$ normal to the liftoff boundary, we recover an equation analogous to , as we will show. Since ${\mathbf{F}}^\text{slide}={\mathbf{F}}^\text{int}$ at the liftoff point, we conclude that the saltation matrix at the liftoff point reduces to the identity matrix. Let $\Phi_\text{I}$ and $\Phi_\text{II}$ denote the flow operators on the sliding region and in the domain complementary to the sliding region, respectively. That is, $\Phi_\text{I}({\mathbf{x}},t)$ takes initial point ${\mathbf{x}}\in\mathcal{R}^\text{slide}$ at time zero to $\Phi_\text{I}({\mathbf{x}},t)$ at time $0\le t \le \mathcal{T}({\mathbf{x}})$. So $\Phi_\text{I}$ is restricted to act for times up to the time $\mathcal{T}({\mathbf{x}})$ at which the trajectory starting at ${\mathbf{x}}$ reaches the liftoff point, $\Phi_\text{I}({\mathbf{x}},\mathcal{T}({\mathbf{x}}))\in\mathcal{L}$. Such a trajectory necessarily has initial condition ${\mathbf{x}}=(w_1,\ldots,w_{n-1},0)$ satisfying $w_{n-1}<0$, by our coordinatization. Let ${\mathbf{x}}_\text{a}\in \mathcal{R}^\text{slide}$ be a point on the periodic limit cycle solution, so that $\Phi_\text{I}({\mathbf{x}}_\text{a},\mathcal{T}({\mathbf{x}}_\text{a}))={\mathbf{x}}_\text{lift}$. Write $\tau=\mathcal{T}({\mathbf{x}}_\text{a})$ for the time it takes for the trajectory to reach the liftoff point after passing location ${\mathbf{x}}_\text{a}$. We require a first-order accurate estimate of the effect of the boundary on the displacement between the unperturbed trajectory and a nearby trajectory. If we make a small (size ${\varepsilon}$) perturbation into the domain interior, away from the constraint surface, the normal component of the perturbed trajectory will return to zero within a time interval of $O({\varepsilon})$ duration, before the two trajectories reach the liftoff boundary. Therefore we need only consider perturbations tangent to the constraint surface. Let ${\mathbf{x}}_\text{a}'\in\mathcal{R}^\text{slide}$ denote a point near ${\mathbf{x}}_\text{a}$, and suppose it takes time $\mathcal{T}({\mathbf{x}}_\text{a}')=\tau+\delta$ for the trajectory through ${\mathbf{x}}_\text{a}'$ to liftoff, at some point ${\mathbf{x}}_\text{lift}'\in\mathcal{L}$. There are two cases to consider: either $\delta\ge 0$ or else $\delta \le 0$. The two cases are handled similarly; we focus on the first for brevity. In case $\delta>0$, the original trajectory arrives at the liftoff boundary before the perturbed trajectory, and the point ${\mathbf{x}}_\text{b}'=\Phi_\text{I}({\mathbf{x}}_\text{a}',\tau)\in\mathcal{R}^\text{slide}$. We write ${\mathbf{x}}_\text{b}'={\mathbf{x}}_\text{lift}+\Delta {\mathbf{x}}_\text{b}$ (see Fig. \[fig:proof-parts-b-c-d\]B) and expand the flow operator as follows: $$\begin{aligned} \nonumber \Phi_\text{I}({\mathbf{x}}_\text{b}',\delta)=&{\mathbf{x}}_\text{b}'+\delta\, {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{b}')+\frac{\delta^2}{2}\left(\nabla^\text{slide}{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{b}') \right)\cdot {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{b}')+O(\delta^3)\\ \label{eq:PhiI_expand1} =&{\mathbf{x}}_\text{lift}+\Delta{\mathbf{x}}_\text{b} +\delta\, {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})+\delta\,\left( \nabla^\text{slide}{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift} )\right)\cdot\Delta{\mathbf{x}}_\text{b}\\ \nonumber &+\frac{\delta^2}{2}\left(\nabla^\text{slide}{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{b}') \right)\cdot {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{b}')+O(3),\end{aligned}$$ where $\nabla^\text{slide}$ is the gradient operator restricted to ${\mathbf{x}}=(x_1,\ldots,x_{n-1})$. The Taylor expansion in is justified in a neighborhood of ${\mathbf{x}}_\text{b}'$ contained in the sliding region of the hard boundary. The transversality of the intersection of the reference trajectory with $\mathcal{L}$ (that is, ${\mathbf{F}}_{n-1}({\mathbf{x}}_\text{lift})>0$) means that $\delta$ and $|\Delta{\mathbf{x}}_\text{b}|$ will be of the same order. We write $O(n)$ to denote terms of order $\left(|\Delta{\mathbf{x}}_\text{b}|^p \delta^{n-p}\right)$ for $0\le p \le n$. Next we estimate $\delta$ and the location ${\mathbf{x}}_\text{lift}'$ at which the perturbed trajectory crosses $\mathcal{L}$. To first order, $$\begin{aligned} \ell^\intercal{\mathbf{x}}_\text{b}'&=\ell^\intercal{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{b}')\,\delta\\ \ell^\intercal({\mathbf{x}}_\text{lift}+\Delta{\mathbf{x}}_\text{b})&=\ell^\intercal\left({\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift}+\Delta{\mathbf{x}}_\text{b})\right)\delta \label{eq:ineedabettername}\\ \ell^\intercal\Delta{\mathbf{x}}_\text{b} &=\ell^\intercal\left( {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})+\left(\nabla^\text{slide}{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})\right)\cdot\Delta{\mathbf{x}}_\text{b} \right)\delta\\ \nonumber &=\ell^\intercal{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})\delta+O(2)\\ \delta &= \frac{\ell^\intercal\Delta{\mathbf{x}}_\text{b}}{\ell^\intercal{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})}+O(2).\end{aligned}$$ Combining this result with , the perturbed trajectory’s liftoff location is $$\label{eq:perturbed_liftoff_location} {\mathbf{x}}_\text{lift}'={\mathbf{x}}_\text{lift}+\Delta{\mathbf{x}}_\text{b}+{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})\delta + O(2).$$ Meanwhile, as the perturbed trajectory proceeds to $\mathcal{L}$, during a time interval of duration $\delta$, the unperturbed trajectory has reentered the interior and evolves according to $\Phi_\text{II}$, the flow defined for all initial conditions *not* within the sliding region. At a time $\delta$ after reaching $\mathcal{L}$, the unperturbed trajectory is located, to first order, at a point $$\label{eq:delta-after-liftoff} {\mathbf{x}}_\text{c}={\mathbf{x}}_\text{lift}+{\mathbf{F}}^\text{int}({\mathbf{x}}_\text{lift})\,\delta+O(2).$$ Thus, combining and the displacement between the two trajectories immediately following liftoff of the perturbed trajectory, $\Delta{\mathbf{x}}_\text{c}={\mathbf{x}}_\text{lift}'-{\mathbf{x}}_\text{c}$, is given (to first order) by $$\begin{aligned} \Delta{\mathbf{x}}_\text{c}&={\mathbf{x}}_\text{lift}'-{\mathbf{x}}_\text{c}\\ &={\mathbf{x}}_\text{lift}+\Delta{\mathbf{x}}_\text{b}+{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})\delta-\left( {\mathbf{x}}_\text{lift}+{\mathbf{F}}^\text{int}({\mathbf{x}}_\text{lift})\,\delta \right)\\ &=\Delta{\mathbf{x}}_\text{b} + \left( {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})-{\mathbf{F}}^\text{int}({\mathbf{x}}_\text{lift}) \right)\delta\\ &=\Delta{\mathbf{x}}_\text{b} + \frac{\left( {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})-{\mathbf{F}}^\text{int}({\mathbf{x}}_\text{lift}) \right)\ell^\intercal\Delta{\mathbf{x}}_\text{b}}{\ell^\intercal{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})}\\ &=S_\text{lift}\Delta{\mathbf{x}}_\text{b} + O(2).\end{aligned}$$ Therefore, the saltation matrix at the liftoff point is $$S_\text{lift}=I+\frac{\left( {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})-{\mathbf{F}}^\text{int}({\mathbf{x}}_\text{lift}) \right)\ell^\intercal}{\ell^\intercal{\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})}. \label{eq:Slift}$$ We take the vector field on the sliding region to be the projection of the vector field defined for the interior onto the boundary surface (cf. ). Therefore for our construction ${\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{lift})={\mathbf{F}}^\text{int}({\mathbf{x}}_\text{lift})$, and hence $S_\text{lift}=I,$ as claimed. We note that equation will hold for more general constructions as well. This concludes the proof of part (b). In parts (c) and (d) of the proof, our goal is to show the normal component of the iPRC is zero along the sliding region on $\Sigma$ and is continuous at the landing point. To this end, we compute the normal component of the iPRC using its definition , which in $({\mathbf{w}},v)$ coordinates takes the form $$\label{eq:z-norm}{{\mathbf z}}_v := {{\mathbf z}}\cdot {\mathbf{n}}= \lim_{\varepsilon\to 0}\frac{\phi({\mathbf{x}}+\varepsilon {\mathbf{n}}) -\phi({\mathbf{x}})}{\varepsilon},$$ where $\phi({\mathbf{x}})$ denotes the asymptotic phase at point ${\mathbf{x}}$ on the limit cycle. That is, we apply a small instantaneous perturbation to the limit cycle, either while it is sliding along $\Sigma$ (part c) or else just before landing (part d), in the ${\mathbf{n}}$ direction, and estimate the phase difference between the perturbed and unperturbed trajectories (cf. Fig. \[fig:proof-parts-b-c-d\]). (160,86)(0,0) (0,43)[![\[fig:proof-parts-b-c-d\] Unperturbed trajectory (black curve) and a perturbed trajectory (red curve) near the hard boundary $\Sigma$ (horizontal plane) in the $({\mathbf{w}}, v)$ phase space. Dashed line: intersection of liftoff boundary $\mathcal{L}$ and $\Sigma$. (A) Trajectory moves downward towards the sliding region (the area in $\Sigma$ where $g<0$), hits $\Sigma$ at the landing point ${\mathbf{x}}_{\rm land}$, and exits $\Sigma$ at the liftoff point ${\mathbf{x}}_{\rm lift}$. (B) Construction for the proof of part (b). An instantaneous perturbation tangent to $\Sigma$ is made to the point ${\mathbf{x}}_a$ at $t=0$, pushing it to a point ${\mathbf{x}}_a'\in\Sigma$. The trajectory starting at ${\mathbf{x}}_a$ (resp., ${\mathbf{x}}_a'$) reaches the liftoff point ${\mathbf{x}}_{\rm lift}$ (resp., ${\mathbf{x}}_b'$) after time $\tau$, and reaches ${\mathbf{x}}_c$ (resp., ${\mathbf{x}}_{\rm lift}'$) after additional time $\delta$. The displacements $\Delta{\mathbf{x}}_b={\mathbf{x}}_b'-{\mathbf{x}}_{\rm lift}$ and $\Delta{\mathbf{x}}_c={\mathbf{x}}_{\rm lift}'-{\mathbf{x}}_{c}$ differ by an amount captured, to linear order, by the saltation matrix. (C) Construction for the proof of part (c). An instantaneous perturbation with size $\varepsilon$ in the positive $v$-direction (green arrow) is made to the point ${\mathbf{x}}_a\in \Sigma$, pushing it off the boundary to an interior point ${\mathbf{x}}_a'$. After time $\tau$, the trajectory starting at ${\mathbf{x}}_a'$ (resp., ${\mathbf{x}}_a$) reaches a landing point ${\mathbf{x}}_{\rm land}'$ (resp., ${\mathbf{x}}_b$). (D) The same perturbation (green arrow) as in panel (C) is applied to the point ${\mathbf{x}}_a$ located at a distance of $h$ above $\Sigma$, pushing it to a point ${\mathbf{x}}_a'$. The trajectory starting at ${\mathbf{x}}_a$ lands on $\Sigma$ at ${\mathbf{x}}_{\rm land}$. After the same amount of time, the perturbed trajectory starting at ${\mathbf{x}}_a'$ reaches ${\mathbf{x}}_b'$. After additional time $\tau$, the two trajectories reach ${\mathbf{x}}_c$ and ${\mathbf{x}}_{\rm land}'$, respectively. ]({LCSC-proof-coord-construction.png} "fig:"){width="80mm"}]{} (80,43)[![\[fig:proof-parts-b-c-d\] Unperturbed trajectory (black curve) and a perturbed trajectory (red curve) near the hard boundary $\Sigma$ (horizontal plane) in the $({\mathbf{w}}, v)$ phase space. Dashed line: intersection of liftoff boundary $\mathcal{L}$ and $\Sigma$. (A) Trajectory moves downward towards the sliding region (the area in $\Sigma$ where $g<0$), hits $\Sigma$ at the landing point ${\mathbf{x}}_{\rm land}$, and exits $\Sigma$ at the liftoff point ${\mathbf{x}}_{\rm lift}$. (B) Construction for the proof of part (b). An instantaneous perturbation tangent to $\Sigma$ is made to the point ${\mathbf{x}}_a$ at $t=0$, pushing it to a point ${\mathbf{x}}_a'\in\Sigma$. The trajectory starting at ${\mathbf{x}}_a$ (resp., ${\mathbf{x}}_a'$) reaches the liftoff point ${\mathbf{x}}_{\rm lift}$ (resp., ${\mathbf{x}}_b'$) after time $\tau$, and reaches ${\mathbf{x}}_c$ (resp., ${\mathbf{x}}_{\rm lift}'$) after additional time $\delta$. The displacements $\Delta{\mathbf{x}}_b={\mathbf{x}}_b'-{\mathbf{x}}_{\rm lift}$ and $\Delta{\mathbf{x}}_c={\mathbf{x}}_{\rm lift}'-{\mathbf{x}}_{c}$ differ by an amount captured, to linear order, by the saltation matrix. (C) Construction for the proof of part (c). An instantaneous perturbation with size $\varepsilon$ in the positive $v$-direction (green arrow) is made to the point ${\mathbf{x}}_a\in \Sigma$, pushing it off the boundary to an interior point ${\mathbf{x}}_a'$. After time $\tau$, the trajectory starting at ${\mathbf{x}}_a'$ (resp., ${\mathbf{x}}_a$) reaches a landing point ${\mathbf{x}}_{\rm land}'$ (resp., ${\mathbf{x}}_b$). (D) The same perturbation (green arrow) as in panel (C) is applied to the point ${\mathbf{x}}_a$ located at a distance of $h$ above $\Sigma$, pushing it to a point ${\mathbf{x}}_a'$. The trajectory starting at ${\mathbf{x}}_a$ lands on $\Sigma$ at ${\mathbf{x}}_{\rm land}$. After the same amount of time, the perturbed trajectory starting at ${\mathbf{x}}_a'$ reaches ${\mathbf{x}}_b'$. After additional time $\tau$, the two trajectories reach ${\mathbf{x}}_c$ and ${\mathbf{x}}_{\rm land}'$, respectively. ]({LCSC-proof-part-b.png} "fig:"){width="80mm"}]{} (0,0)[![\[fig:proof-parts-b-c-d\] Unperturbed trajectory (black curve) and a perturbed trajectory (red curve) near the hard boundary $\Sigma$ (horizontal plane) in the $({\mathbf{w}}, v)$ phase space. Dashed line: intersection of liftoff boundary $\mathcal{L}$ and $\Sigma$. (A) Trajectory moves downward towards the sliding region (the area in $\Sigma$ where $g<0$), hits $\Sigma$ at the landing point ${\mathbf{x}}_{\rm land}$, and exits $\Sigma$ at the liftoff point ${\mathbf{x}}_{\rm lift}$. (B) Construction for the proof of part (b). An instantaneous perturbation tangent to $\Sigma$ is made to the point ${\mathbf{x}}_a$ at $t=0$, pushing it to a point ${\mathbf{x}}_a'\in\Sigma$. The trajectory starting at ${\mathbf{x}}_a$ (resp., ${\mathbf{x}}_a'$) reaches the liftoff point ${\mathbf{x}}_{\rm lift}$ (resp., ${\mathbf{x}}_b'$) after time $\tau$, and reaches ${\mathbf{x}}_c$ (resp., ${\mathbf{x}}_{\rm lift}'$) after additional time $\delta$. The displacements $\Delta{\mathbf{x}}_b={\mathbf{x}}_b'-{\mathbf{x}}_{\rm lift}$ and $\Delta{\mathbf{x}}_c={\mathbf{x}}_{\rm lift}'-{\mathbf{x}}_{c}$ differ by an amount captured, to linear order, by the saltation matrix. (C) Construction for the proof of part (c). An instantaneous perturbation with size $\varepsilon$ in the positive $v$-direction (green arrow) is made to the point ${\mathbf{x}}_a\in \Sigma$, pushing it off the boundary to an interior point ${\mathbf{x}}_a'$. After time $\tau$, the trajectory starting at ${\mathbf{x}}_a'$ (resp., ${\mathbf{x}}_a$) reaches a landing point ${\mathbf{x}}_{\rm land}'$ (resp., ${\mathbf{x}}_b$). (D) The same perturbation (green arrow) as in panel (C) is applied to the point ${\mathbf{x}}_a$ located at a distance of $h$ above $\Sigma$, pushing it to a point ${\mathbf{x}}_a'$. The trajectory starting at ${\mathbf{x}}_a$ lands on $\Sigma$ at ${\mathbf{x}}_{\rm land}$. After the same amount of time, the perturbed trajectory starting at ${\mathbf{x}}_a'$ reaches ${\mathbf{x}}_b'$. After additional time $\tau$, the two trajectories reach ${\mathbf{x}}_c$ and ${\mathbf{x}}_{\rm land}'$, respectively. ]({LCSC-proof-part-c.png} "fig:"){width="80mm"}]{} (80,0)[![\[fig:proof-parts-b-c-d\] Unperturbed trajectory (black curve) and a perturbed trajectory (red curve) near the hard boundary $\Sigma$ (horizontal plane) in the $({\mathbf{w}}, v)$ phase space. Dashed line: intersection of liftoff boundary $\mathcal{L}$ and $\Sigma$. (A) Trajectory moves downward towards the sliding region (the area in $\Sigma$ where $g<0$), hits $\Sigma$ at the landing point ${\mathbf{x}}_{\rm land}$, and exits $\Sigma$ at the liftoff point ${\mathbf{x}}_{\rm lift}$. (B) Construction for the proof of part (b). An instantaneous perturbation tangent to $\Sigma$ is made to the point ${\mathbf{x}}_a$ at $t=0$, pushing it to a point ${\mathbf{x}}_a'\in\Sigma$. The trajectory starting at ${\mathbf{x}}_a$ (resp., ${\mathbf{x}}_a'$) reaches the liftoff point ${\mathbf{x}}_{\rm lift}$ (resp., ${\mathbf{x}}_b'$) after time $\tau$, and reaches ${\mathbf{x}}_c$ (resp., ${\mathbf{x}}_{\rm lift}'$) after additional time $\delta$. The displacements $\Delta{\mathbf{x}}_b={\mathbf{x}}_b'-{\mathbf{x}}_{\rm lift}$ and $\Delta{\mathbf{x}}_c={\mathbf{x}}_{\rm lift}'-{\mathbf{x}}_{c}$ differ by an amount captured, to linear order, by the saltation matrix. (C) Construction for the proof of part (c). An instantaneous perturbation with size $\varepsilon$ in the positive $v$-direction (green arrow) is made to the point ${\mathbf{x}}_a\in \Sigma$, pushing it off the boundary to an interior point ${\mathbf{x}}_a'$. After time $\tau$, the trajectory starting at ${\mathbf{x}}_a'$ (resp., ${\mathbf{x}}_a$) reaches a landing point ${\mathbf{x}}_{\rm land}'$ (resp., ${\mathbf{x}}_b$). (D) The same perturbation (green arrow) as in panel (C) is applied to the point ${\mathbf{x}}_a$ located at a distance of $h$ above $\Sigma$, pushing it to a point ${\mathbf{x}}_a'$. The trajectory starting at ${\mathbf{x}}_a$ lands on $\Sigma$ at ${\mathbf{x}}_{\rm land}$. After the same amount of time, the perturbed trajectory starting at ${\mathbf{x}}_a'$ reaches ${\mathbf{x}}_b'$. After additional time $\tau$, the two trajectories reach ${\mathbf{x}}_c$ and ${\mathbf{x}}_{\rm land}'$, respectively. ]({LCSC-proof-part-d.png} "fig:"){width="80mm"}]{} (2,81)[****]{} (82,81)[****]{} (2,38)[****]{} (82,38)[****]{} #### (c) Along the sliding region, the component of ${{\mathbf z}}$ normal to $\Sigma$ is zero. By the normal component of the iPRC for a point on the sliding component of the trajectory, denoted by ${\mathbf{x}}_\text{a} = (w_\text{a},0)$ is given by $$\label{eq:iprc-zero-normcomponent} {{\mathbf z}}_v({\mathbf{x}}_\text{a})=\lim_{\varepsilon\to0}\frac{\phi(w_\text{a},\varepsilon)-\phi(w_\text{a},0)}{\varepsilon}.$$ By ${\mathbf{x}}_\text{a}'=(w_\text{a},\varepsilon)$ we denote a point that is located at a distance of $\varepsilon$ above ${\mathbf{x}}_\text{a}$. Our goal is to show ${{\mathbf z}}_v({\mathbf{x}}_\text{a})=0$. The perturbed trajectory from ${\mathbf{x}}_\text{a}'$ is governed by the interior flow $\Phi_\text{II}$ until it reaches the sliding region at a point ${\mathbf{x}}_\text{b}'\in\Sigma$, after some time $\tau$. Meanwhile the unperturbed trajectory from ${\mathbf{x}}_\text{a}$ is governed by the sliding flow $\Phi_\text{I}$ until it crosses the liftoff point at $\mathcal{L}$ (Fig. \[fig:proof-parts-b-c-d\], dotted line). To first order in $\varepsilon$, the time for the perturbed trajectory ${\mathbf{x}}'(t)$ to return to the constraint surface is $$\begin{aligned} \nonumber \tau(\varepsilon)& =-\frac{\varepsilon}{g^\text{int}({\mathbf{w}}_\text{a},\varepsilon)}+O(\varepsilon^2) =-\frac{\varepsilon}{g^\text{int}({\mathbf{w}}_\text{a},0)+\varepsilon D_v g^\text{int}({\mathbf{w}}_\text{a},0)+O(\varepsilon^2)}+O(\varepsilon^2)\\ &=-\frac{\varepsilon}{g^\text{int}({\mathbf{w}}_\text{a},0)}+O(\varepsilon^2),\text{ as }\varepsilon\to 0.\end{aligned}$$ Because ${\mathbf{x}}_\text{a}=({\mathbf{w}}_\text{a},0)$ is in the sliding region, $g^\text{int}({\mathbf{w}}_\text{a},0)<0$; we conclude that $\tau$ and $\varepsilon$ are of the same order. We use $(p)$ to denote terms of order $p$ in $\varepsilon$ or $\tau$. At time $\tau$ following the perturbation, the location of the perturbed trajectory is $$\begin{aligned} {\mathbf{x}}_\text{b}'&=\Phi_\text{II}({\mathbf{x}}_\text{a}',\tau)\\ \nonumber &={\mathbf{x}}_\text{a}'+\tau {\mathbf{F}}^\text{int}({\mathbf{x}}_\text{a}')+O(2)\\ &={\mathbf{x}}_\text{a}+\varepsilon{\mathbf{n}}+\tau\left( {\mathbf{F}}^\text{int}({\mathbf{x}}_\text{a})+\varepsilon {\mathbf{n}}^\intercal D{\mathbf{F}}^\text{int}({\mathbf{x}}_\text{a})\right)+O(2) \nonumber\\ \nonumber &={\mathbf{x}}_\text{a}+(0,\ldots,0,\varepsilon)-\frac\varepsilon{g^ \text{int}({\mathbf{x}}_\text{a})}(f_1^\text{int}({\mathbf{x}}_\text{a}),\ldots,f_{n-1}^\text{int}({\mathbf{x}}_\text{a}),g^\text{int}({\mathbf{x}}_\text{a}))+O(2)\\ \nonumber &={\mathbf{x}}_\text{a}-\frac\varepsilon{g^ \text{int}({\mathbf{x}}_\text{a})}(f_1^\text{int}({\mathbf{x}}_\text{a}),\ldots,f_{n-1}^\text{int}({\mathbf{x}}_\text{a}),0)+O(2).\end{aligned}$$ Simultaneously, the location of the unperturbed trajectory is $$\begin{aligned} {\mathbf{x}}_\text{b}&=\Phi_\text{I}({\mathbf{x}}_\text{a},\tau)\\ \nonumber &={\mathbf{x}}_\text{a}+\tau {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{a})+O(2)\\ \nonumber &={\mathbf{x}}_\text{a}-\frac{\varepsilon}{g^\text{int}({\mathbf{x}}_\text{a})}(f_1^\text{int}({\mathbf{x}}_\text{a}),\ldots,f_{n-1}^\text{int}({\mathbf{x}}_\text{a}),0) +O(2),\end{aligned}$$ since for ${\mathbf{x}}\in\Sigma$, we have $f^\text{slide}({\mathbf{x}})=f^\text{int}({\mathbf{x}})$ by construction. Comparing the difference in location of the two trajectories at time $\tau$ after the perturbation, we see that $$||{\mathbf{x}}_b'-{\mathbf{x}}_b||=O(\varepsilon^2).$$ By assumption, the asymptotic phase function $\phi({\mathbf{x}})$ is $C^1$ with respect to displacements tangent to the constraint surface. Since both ${\mathbf{x}}_\text{b}$ and ${\mathbf{x}}_\text{b}'$ are on this surface, ${\mathbf{n}}^\intercal({\mathbf{x}}_\text{b}'-{\mathbf{x}}_\text{b})=0,$ and $\phi({\mathbf{x}}_\text{b}')=\phi({\mathbf{x}}_\text{b})+O(\varepsilon^2)$. Therefore ${{\mathbf z}}_v({\mathbf{x}}_\text{a})=0$ for points ${\mathbf{x}}_\text{a}$ on the sliding component of the limit cycle. This completes the proof of part (c). #### (d) The normal component of ${{\mathbf z}}$ is continuous at the landing point. In order to show that the normal component of the iPRC (${{\mathbf z}}_v$) is continuous at the landing point, we prove that ${{\mathbf z}}_v$ has a well-defined limit at the landing point and moreover, this limit equals $0$ which is the value of ${{\mathbf z}}_v$ at the landing point as proved in (c). To this end, consider a point on the limit cycle shortly ahead of the landing point, ${\mathbf{x}}_ \text{a}=({\mathbf{w}}_\text{a},h)$ with $0<h\ll 1$ fixed, (cf. Fig. \[fig:proof-parts-b-c-d\]D). By $$\begin{aligned} \label{eq:prc-d} {\ensuremath{\begin{array}{cccccccccc}{{\mathbf z}}_v({\mathbf{x}}_a)=\lim_{\varepsilon\to 0}\frac{\phi({\mathbf{w}}_\text{a},h+\varepsilon)-\phi({\mathbf{w}}_\text{a},h)}{\varepsilon}.\end{array}}}\end{aligned}$$ Our goal is to show $\lim_{h\to 0}{{\mathbf z}}_v({\mathbf{x}}_a) = {{\mathbf z}}_v({\mathbf{x}}_\text{land}) = 0$. We consider the case $\varepsilon>0$; the treatment for $\varepsilon<0$ is similar. For $\varepsilon>0$, when the unperturbed trajectory arrives at the constraint surface (at landing point ${\mathbf{x}}_\text{land}$), the perturbed trajectory is at a point ${\mathbf{x}}_\text{b}'$ that is still in the interior of the domain. Denote the unperturbed landing time $t=0$; denote the time of flight from initial point ${\mathbf{x}}_\text{a}$ to ${\mathbf{x}}_\text{land}$ by $s$. Through an estimate similar to that in part (c), to first order in $h$, we have $$s(h)=-\frac{h}{g^\text{int}({\mathbf{x}}_\text{land})}+O(h^2).$$ Between $t=-s$ and $t=0$, the displacement between the perturbed trajectory (${\mathbf{x}}'(t)$) and the unperturbed trajectory (${\mathbf{x}}(t)$) satisfies $$\begin{aligned} \frac{d({\mathbf{x}}'-{\mathbf{x}})}{dt}=D{\mathbf{F}}^\text{int}({\mathbf{x}}(t))\cdot({\mathbf{x}}'-{\mathbf{x}})+O({\varepsilon}^2),\end{aligned}$$ with initial condition ${\mathbf{x}}'(-s)-{\mathbf{x}}(-s)=\varepsilon{\mathbf{n}}$. Because the interior vector field is presumed $C^1$, for $h,s\ll 1$ we have $$\begin{aligned} {\mathbf{x}}'_\text{b}-{\mathbf{x}}_\text{land}&= {\mathbf{x}}'_\text{a}-{\mathbf{x}}_\text{a}+s\left( \varepsilon D_v {\mathbf{F}}^\text{int}({\mathbf{x}}_\text{land}) + O({\varepsilon}^2)\right) + O(s^2)\\ &= \varepsilon {\mathbf{n}}-h\left( \varepsilon \frac{D_v {\mathbf{F}}^\text{int}({\mathbf{x}}_\text{land})}{g^\text{int}({\mathbf{x}}_\text{land})} + O({\varepsilon}^2)\right)+O(h^2) \\ &= (0,\cdots,0, \varepsilon) -\frac{h\varepsilon}{g^\text{int}({\mathbf{x}}_\text{land})} (f_{1,v}^\text{int}({\mathbf{x}}_\text{land}),\ldots,f_{n-1,v}^\text{int}({\mathbf{x}}_\text{land}),g^\text{int}_v({\mathbf{x}}_\text{land})) + O(2)\\ &= \left(-h\varepsilon \frac{\mathbf{f}_v^\text{int}({\mathbf{x}}_\text{land})}{g^\text{int}({\mathbf{x}}_\text{land})}, \varepsilon-h\varepsilon\frac{g^\text{int}_v({\mathbf{x}}_\text{land})}{g^\text{int}({\mathbf{x}}_\text{land})}\right) + O(2). \end{aligned}$$ Here $\mathbf{f}_v^\text{int} = (f_{1,v}^\text{int},\ldots,f_{n-1,v}^\text{int})$, where $f^{\rm int}_{k,v}$ denotes $\partial f^{\rm int}_k/\partial v$, and $O(2)$ denotes terms of order 2 in $\varepsilon$ or $h$ as in (c). In the rest of this proof, we drop the dependence of the functions on ${\mathbf{x}}_{\rm land}$ for simplicity. Since ${\mathbf{x}}_\text{land}$ is in the sliding region, it follows that ${\mathbf{x}}_b'$ is $\varepsilon-h\varepsilon\frac{g^\text{int}_v}{g^\text{int}} + O(2)$ above the sliding region. Through a similar estimation as in part (c), to first order in $\varepsilon$ and $h$, the time for the perturbed trajectory to arrive at the sliding region is $$\tau(h,\varepsilon) = \frac{\varepsilon-h\varepsilon\frac{g^\text{int}_v}{g^\text{int}}}{-g^\text{int}}+O(2) = -\frac{\varepsilon}{g^\text{int}} + h\varepsilon \frac{g_v^\text{int}}{(g^\text{int})^2} +O(2).$$ At time $\tau$, the location of the perturbed trajectory is $$\label{eq:D:xland'} \begin{array}{rcl} {\mathbf{x}}_\text{land}' &=& \Phi_\text{II}({\mathbf{x}}_b',\tau)\\ &=& {\mathbf{x}}_b'+ \tau {\mathbf{F}}^\text{int}({\mathbf{x}}_b') + O(2)\\ &=& {\mathbf{x}}_\text{land} + \left(-h\varepsilon \frac{\mathbf{f}_v^\text{int}}{g^\text{int}}, \varepsilon-h\varepsilon\frac{g^\text{int}_v}{g^\text{int}}\right) + \tau {\mathbf{F}}^\text{int}({\mathbf{x}}_\text{land})+O(2)\\ &=& {\mathbf{x}}_\text{land} + \left(-h\varepsilon \frac{\mathbf{f}_v^\text{int}}{g^\text{int}}, \varepsilon-h\varepsilon\frac{g^\text{int}_v}{g^\text{int}}\right) + \left(-\frac{\varepsilon}{g^\text{int}} + h\varepsilon \frac{g_v^\text{int}}{(g^\text{int})^2} \right) (\mathbf{f}^\text{int}, g^\text{int})+O(2)\\ &=& {\mathbf{x}}_\text{land} + \left(-h\varepsilon \frac{\mathbf{f}_v^\text{int}}{g^\text{int}}, 0) + (-\frac{\varepsilon}{g^\text{int}} + h\varepsilon \frac{g_v^\text{int}}{(g^\text{int})^2} \right) (\mathbf{f}^\text{int}, 0)+O(2). \end{array}$$ Simultaneously, the location of the unperturbed trajectory is $$\label{eq:D:xc} \begin{array}{rcl} {\mathbf{x}}_\text{c}&=& \Phi_\text{I}({\mathbf{x}}_\text{land},\tau)\\ &=& {\mathbf{x}}_{\rm land}+\tau {\mathbf{F}}^\text{slide}({\mathbf{x}}_\text{land}) +O(2)\\ &=& {\mathbf{x}}_\text{land} + \left(-\frac{\varepsilon}{g^\text{int}} + h\varepsilon \frac{g_v^\text{int}}{(g^\text{int})^2} \right)(\mathbf{f}^\text{int}, 0) + O(2). \end{array}$$ Comparing the difference between and , we see that $${\left\lVert{\mathbf{x}}_{\rm land}' - {\mathbf{x}}_c\right\rVert} = O(h\varepsilon).$$ Recall that the asymptotic phase is assumed to be $C^1$, with respect to displacements tangent to $\Sigma$. Since ${\mathbf{x}}_{\rm land}'$ and ${\mathbf{x}}_c$ are on $\Sigma$, it follows that $$\phi({\mathbf{x}}_{\rm land}') - \phi({\mathbf{x}}_c) = O(h\varepsilon).$$ Therefore, by , $${{\mathbf z}}_v({\mathbf{x}}_a) = \lim_{\varepsilon\to 0} \frac{\phi({\mathbf{x}}_a')-\phi({\mathbf{x}}_a)}{\varepsilon} = \lim_{\varepsilon\to 0} \frac{\phi({\mathbf{x}}_{\rm land}')-\phi({\mathbf{x}}_c)}{\varepsilon}=O(h)$$ Consequently, $$\lim_{h\to 0} {{\mathbf z}}_v({\mathbf{x}}_a) = 0$$ as required. This completes the proof of part (d). #### (e) The tangential components of ${{\mathbf z}}$ are continuous at both landing and liftoff points. We denote the tangential components of the iPRC by ${{\mathbf z}}_{\mathbf{w}}$, where ${\mathbf{w}}$ represents vectors in the $n-1$ dimensional tangent space of the hard boundary. The $n-1$ dimensional iPRC vector ${{\mathbf z}}_{\mathbf{w}}$ obeys a restricted (*i.e.* reduced dimension) adjoint equation given in terms of $f_{\mathbf{w}}$, the $(n-1)\times (n-1)$ Jacobian derivative of $f$ with respect to the $n-1$ tangential coordinates (${\mathbf{w}}$), and $g_{\mathbf{w}}$, the $1\times(n-1)$ Jacobian derivative of $g$ with respect to the tangential coordinates, and ${{\mathbf z}}_v$, the (scalar) component of ${{\mathbf z}}$ in the normal direction $$\begin{aligned} \label{eq:zu-interior} {\ensuremath{\begin{array}{cccccccccc}\frac{d{{\mathbf z}}_{\mathbf{w}}}{dt}=-f_{\mathbf{w}}({\mathbf{w}},v)^ \intercal {{\mathbf z}}_{\mathbf{w}}-g_{\mathbf{w}}({\mathbf{w}},v)^\intercal {{\mathbf z}}_v\end{array}}} \end{aligned}$$ along the limit cycle in the interior domain. On the other hand, along the sliding component of the limit cycle that is restricted to $\{\Sigma: v=0\}$, ${{\mathbf z}}_u$ satisfies $$\begin{aligned} \label{eq:zu-sliding} {\ensuremath{\begin{array}{cccccccccc}\frac{d{{\mathbf z}}_{\mathbf{w}}}{dt}=-f_{\mathbf{w}}({\mathbf{w}},0)^\intercal {{\mathbf z}}_{\mathbf{w}}.\end{array}}}\end{aligned}$$ By part (c), ${{\mathbf z}}_v$ goes continuously to zero as the trajectory from the interior approaches the landing point. Therefore ${{\mathbf z}}_{\mathbf{w}}$ is continuous at the landing point. ![\[fig:proof-partE-liftoff\] Unperturbed trajectory (black) leaves the hard boundary at the liftoff point ${\mathbf{x}}_{\rm lift}$, in the $(\mathbf{w}, v)$ phase space. An instantaneous perturbation tangent to $\Sigma$ is made to the liftoff point at $t=\tau$, pushing it to ${\mathbf{x}}_a$ on the sliding region or to ${\mathbf{x}}_b$ that is outside the sliding region. The points ${\mathbf{x}}_c$ and ${\mathbf{x}}_{\rm lift}'$ denote the positions of the unperturbed trajectory and the perturbed trajectory at $t=0$. ](LCSC-proof-part-e.pdf){width="4in"} Next we prove the continuity of ${{\mathbf z}}_{\mathbf{w}}$ at the liftoff point ${\mathbf{x}}_{\rm lift}=({\mathbf{w}}_{\rm lift},0)$. Recall that in the coordinates employed, the unit vector tangent to $\Sigma$ and normal to $\mathcal{L}$ at ${\mathbf{x}}_\text{lift}$ is $\ell=(0,\ldots,0,1,0)$ (cf. Fig. \[fig:proof-partE-liftoff\]). Fix an arbitrary tangential unit vector $\hat{{\mathbf{w}}}$ oriented away from the sliding region (such that $\ell^\intercal\hat{{\mathbf{w}}}> 0$). The left and right limits of ${{\mathbf z}}_w$ at ${\mathbf{x}}_{\rm lift}$ are given by $$\label{eq:lift-leftlimit} {{\mathbf z}}_{\hat{{\mathbf{w}}}}^-({\mathbf{x}}_{\rm lift}) = \lim_{\varepsilon\to 0^+} \frac{\phi({\mathbf{w}}_{\rm lift}-\varepsilon \hat{{\mathbf{w}}},0)-\phi({\mathbf{w}}_{\rm lift},0)}{-\varepsilon}$$ and $$\label{eq:lift-rightlimit} {{\mathbf z}}_{\hat{{\mathbf{w}}}}^+({\mathbf{x}}_{\rm lift}) = \lim_{\varepsilon\to 0} \frac{\phi({\mathbf{w}}_{\rm lift}+\varepsilon \hat{{\mathbf{w}}},0)-\phi({\mathbf{w}}_{\rm lift},0^+)}{\varepsilon}.$$ By ${\mathbf{x}}_a=({\mathbf{w}}_{\rm lift}-\varepsilon \hat{{\mathbf{w}}},0)$ and ${\mathbf{x}}_b=({\mathbf{w}}_{\rm lift}+\varepsilon \hat{{\mathbf{w}}},0)$ we denote the two points that are located at a distance of $\varepsilon$ away from ${\mathbf{x}}_{\rm lift}$ along the $-\hat{{\mathbf{w}}}$ and $\hat{{\mathbf{w}}}$ directions, respectively (cf. Fig. \[fig:proof-partE-liftoff\]). We will show that $$\label{eq:defined-limit} {{\mathbf z}}_{\hat{{\mathbf{w}}}}^-({\mathbf{x}}_{\rm lift}) = {{\mathbf z}}_{\hat{{\mathbf{w}}}}^+({\mathbf{x}}_{\rm lift}).$$ The equality of these limits will establish that ${{\mathbf z}}_{{\mathbf{w}}}$ is continuous at the liftoff point. First, we consider ${{\mathbf z}}_{\hat{{\mathbf{w}}}}^+({\mathbf{x}}_\text{lift})$. Given $\hat{{\mathbf{w}}}$, there exists a unique point ${\mathbf{x}}_{\rm lift}'$ at the liftoff boundary $\mathcal{L}\cap\Sigma$, and a time $\tau>0$, such that the trajectory beginning from ${\mathbf{x}}_{\rm lift}'$ at time $0$ passes directly over ${\mathbf{x}}_b'$ at time $\tau$, in the sense that $\Phi_\text{II}({\mathbf{x}}_\text{lift}',\tau)=({\mathbf{w}}_b, h)$, where $\Phi_\text{II}$ is the flow operator in the complement of the sliding region, $h>0$ is the “height” of ${\mathbf{x}}_b'$ above ${\mathbf{x}}_b$, and ${\mathbf{w}}_b$ is the coordinate vector along the tangent space of the hard boundary. Let ${\mathbf{x}}_{\rm lift} = ({\mathbf{w}}_{\rm lift},0)$ and ${\mathbf{x}}_{\rm lift}' = ({\mathbf{w}}_{\rm lift}',0)$. By our construction, ${\mathbf{w}}_b = {\mathbf{w}}_{\rm lift} + \varepsilon\hat{{\mathbf{w}}}$. Hence, the location of the perturbed trajectory at time $\tau$ is $$\begin{array}{rcl} ({\mathbf{w}}_b,h)= ({\mathbf{w}}_{\rm lift}+\varepsilon \hat{{\mathbf{w}}},h) &=&\Phi_{\rm II}({\mathbf{x}}_{\rm lift}',\tau)\\ &=& ({\mathbf{w}}_{\rm lift}',0) + (f^{\rm int}({\mathbf{x}}_{\rm lift}'), 0) \tau + O(\tau^2)\\ &=& ({\mathbf{w}}_{\rm lift}'+f^{\rm int}({\mathbf{x}}_{\rm lift}')\tau+O(\tau^2), O(\tau^2)). \end{array}$$ Hence $$\label{eq:wlift} {\mathbf{w}}_{\rm lift}-{\mathbf{w}}_{\rm lift}' = f^{\rm int}({\mathbf{x}}_{\rm lift}')\tau -\varepsilon \hat{{\mathbf{w}}}+O(\tau^2),$$ $$\label{eq:hlift} h = O(\tau^2),$$ and $$\label{eq:wb} {\mathbf{w}}_b - {\mathbf{w}}_{\rm lift}'= f^{\rm int}({\mathbf{x}}_{\rm lift}')\tau+O(\tau^2)$$ On the other hand, $$\varepsilon \hat{{\mathbf{w}}} + ({\mathbf{w}}_{\rm lift}' - {\mathbf{w}}_{b}) = {\mathbf{w}}_{\rm lift}- {\mathbf{w}}_{\rm lift}'.$$ By and , the above equation becomes $$\varepsilon \hat{{\mathbf{w}}} - f^{\rm int}({\mathbf{x}}_{\rm lift}')\tau =f^{\rm int}({\mathbf{x}}_{\rm lift}')\tau-\varepsilon \hat{{\mathbf{w}}} + O(\tau^2).$$ That is, $$\varepsilon \hat{{\mathbf{w}}} = f^{\rm int}({\mathbf{x}}_{\rm lift}')\tau+ O(\tau^2).$$ Taking the inner product of both sides with the unit vector $\ell$ (normal to $\mathcal{L}$), and noting that for sufficiently small ${\varepsilon}$, $\ell^\intercal f^\text{int}(x_\text{lift}')>0$ (our nondegeneracy condition), we have $$\tau={\varepsilon}\frac{\ell^\intercal \hat{{\mathbf{w}}}}{\ell^\intercal f^\text{int}({\mathbf{x}}_\text{lift}')}+O(\tau^2),$$ and hence $\tau=O(\varepsilon)$. Therefore, becomes $$\label{eq:h-is-O-eps2} h=O(\varepsilon^2)$$ and hence the phase difference between ${\mathbf{x}}_b'$ and ${\mathbf{x}}_b$ is $$\label{eq:phase-diff-b} \phi({\mathbf{x}}_b)-\phi({\mathbf{x}}_b')=O(\varepsilon^2)$$ due to the assumption that $\phi$ is Lipschitz continuous. Next we show holds using and . Let the unperturbed trajectory pass through ${\mathbf{x}}_\text{lift}$ at time $\tau$, and let ${\mathbf{x}}_c$ be the location of the unperturbed trajectory at time $t=0$ (see Fig. \[fig:proof-partE-liftoff\]). Let $\Delta{\mathbf{x}}_c={\mathbf{x}}_{\rm lift}'-{\mathbf{x}}_c$ and $\Delta{\mathbf{x}}_b={\mathbf{x}}_b' -{\mathbf{x}}_{\rm lift}$. Then by part (b), $$\Delta{\mathbf{x}}_b-\Delta{\mathbf{x}}_c= O(|\Delta{\mathbf{x}}_b|^2);$$ since the saltation matrix is equal to the identity matrix at the liftoff boundary. Since ${\mathbf{x}}_{\rm lift},{\mathbf{x}}_b,{\mathbf{x}}_b'$ form a right triangle, $$|\Delta{\mathbf{x}}_b|^2 = \varepsilon^2+ h^2 = \varepsilon^2+ O(\varepsilon^4),$$ which implies that $$\label{eq:DeltaxbDeltaxc} \Delta{\mathbf{x}}_b-\Delta{\mathbf{x}}_c= O(\varepsilon^2).$$ Direct computation shows $$\label{eq:xb-xlift} {\ensuremath{\begin{array}{cccccccccc}\phi({\mathbf{x}}_b)-\phi({\mathbf{x}}_{\rm lift}) &=&(\phi({\mathbf{x}}_b)-\phi({\mathbf{x}}_b')) + (\phi({\mathbf{x}}_b')-\phi({\mathbf{x}}_{\rm lift}))\\ &=& (\phi({\mathbf{x}}_{\rm lift}')-\phi({\mathbf{x}}_c))+O(\varepsilon^2) \\ &=& D_{\mathbf{w}}\phi({\mathbf{x}}_c)\cdot \Delta{\mathbf{x}}_c+O(\varepsilon^2)\\ &=& D_{\mathbf{w}}\phi({\mathbf{x}}_c)\cdot \Delta{\mathbf{x}}_b+O(\varepsilon^2)\\ &=& D_{\mathbf{w}}\phi({\mathbf{x}}_c)\cdot ({\mathbf{x}}_b' - {\mathbf{x}}_{\rm lift})+O(\varepsilon^2)\\ &=& D_{\mathbf{w}}\phi({\mathbf{x}}_c)\cdot ({\mathbf{x}}_b - {\mathbf{x}}_{\rm lift})+O(\varepsilon^2)\\ &=& D_{\mathbf{w}}\phi({\mathbf{x}}_c)\cdot \varepsilon\hat{{\mathbf{w}}}+O(\varepsilon^2)\end{array}}}.$$ To obtain the second equality, we translate the trajectories backward in time by $\tau$ beginning from ${\mathbf{x}}_b'$ and ${\mathbf{x}}_\text{lift}$, respectively; shifting both trajectories by an equal time interval does not change their phase relationship. The $O({\varepsilon}^2)$ difference arises from . The third equality follows from the assumption that $\phi$ is differentiable with respect to displacements tangent to the sliding region. The fourth equality uses ; the fifth and seventh follow from the definitions; the sixth uses . Recall the we assume $\phi$ to have Lipschitz continuous derivatives in the tangential directions at the boundary surface (except possibly at the landing and liftoff points). Under this assumption, taking the limit $\varepsilon\to 0^+$ leads to ${\mathbf{x}}_c\to {\mathbf{x}}_{\rm lift}^-$ and hence $${{\mathbf z}}^+_{\hat{{\mathbf{w}}}}({\mathbf{x}}_{\rm lift}) =D_{\mathbf{w}}\phi({\mathbf{x}}_{\rm lift}^-)\cdot \hat{{\mathbf{w}}}$$ by . On the other hand, $$\label{eq:xa-xlift} {\ensuremath{\begin{array}{cccccccccc}\phi({\mathbf{x}}_a)-\phi({\mathbf{x}}_{\rm lift}) &=& D_{\mathbf{w}}\phi({\mathbf{x}}_a)\cdot ({\mathbf{x}}_a-{\mathbf{x}}_{\rm lift}) +O(\varepsilon^2)\\ &=& -D_{\mathbf{w}}\phi({\mathbf{x}}_a)\cdot \varepsilon\hat{{\mathbf{w}}}+O(\varepsilon^2). \end{array}}}$$ Taking the limit $\varepsilon\to 0+$ results in ${\mathbf{x}}_a\to {\mathbf{x}}_{\rm lift}^-$ and hence together with , implies $${{\mathbf z}}^-_{\hat{{\mathbf{w}}}}({\mathbf{x}}_{\rm lift}) =D_{\mathbf{w}}\phi({\mathbf{x}}_{\rm lift}^-)\cdot \hat{{\mathbf{w}}}.$$ Hence, holds. Numerical Algorithms {#sec:algorithm} ==================== We will now describe how the results presented in §\[sec:smooth-theory\] and §\[sec:nonsmooth-theory\] can be implemented as numerical algorithms. MATLAB code that implements these algorithms for the example system described in §\[sec:toy-model\] is available: <https://github.com/yangyang-wang/LC_in_square>. Consider a multiple-zone Filippov system generalized from , $$\label{eq:multiple-zone-FP-simp} \frac{d{\mathbf{x}}}{dt}=F({\mathbf{x}}),$$ that produces a $T_0$-periodic limit cycle solution $\gamma(t)\subset{{\mathbf R}}^n$. Suppose $\gamma(t)$ includes $k$ sliding components confined to boundary surfaces denoted as $\Sigma^i\subset{{\mathbf R}}^{n-1},\,i\in\{1,...,k\}$. $\gamma(t)$ exits the $i$-th boundary $\Sigma^i$ at a unique liftoff point ${\mathbf{x}}_{\text{lift}}^i$ given that the nondegeneracy condition at ${\mathbf{x}}_{\text{lift}}^i$ is satisfied. We denote the normal vector to $\Sigma^i$ at liftoff, landing, or boundary crossing points by $n^i$. We denote the interior domain by ${{\mathcal R}}^{\rm interior}$, which can now consist of multiple subdomains separated by transversal crossing boundaries, and denote the piecewise smooth vector field in ${{\mathcal R}}^{\rm interior}$ by $F^{\rm interior}$. By , the sliding vector field on the sliding region ${{\mathcal R}}^{\mathrm{slide}_i}\subset\Sigma^i$ is therefore $$\label{eq:multiple-zone-sliding} F^{\mathrm{slide}_i}({\mathbf{x}})= F^{\rm interior}({\mathbf{x}}) - (n^i\cdot F^{\rm interior}({\mathbf{x}}))n^i$$ Using this notation, the vector field can be written as $$\begin{aligned} \label{eq:multiple-zone-FP} F({\mathbf{x}}):=\left\{ {\ensuremath{\begin{array}{cccccccccc} F^{\rm interior}({\mathbf{x}}), & {\mathbf{x}}\in {{\mathcal R}}^{\rm interior}&\\ F^{\mathrm{slide}_i}({\mathbf{x}}), & {\mathbf{x}}\in{{\mathcal R}}^{\mathrm{slide}_i}&\\ \end{array}}} \right.\end{aligned}$$ and we denote the vector field after a static perturbation by $$\begin{aligned} \label{eq:multiple-zone-PFP} F_{\varepsilon}({\mathbf{x}}):=\left\{ {\ensuremath{\begin{array}{cccccccccc} F^{\rm interior}_\varepsilon({\mathbf{x}}), & {\mathbf{x}}\in {{\mathcal R}}^{\rm interior}&\\ F^{\mathrm{slide}_i}_\varepsilon({\mathbf{x}}), & {\mathbf{x}}\in{{\mathcal R}}^{\mathrm{slide}_i}&\\ \end{array}}} \right.\end{aligned}$$ where $i\in\{1, ..., k\}$. Here we assume that the regions are independent of static perturbation with size $\varepsilon$. Notice that the computation of the iSRC requires estimating the rescaling factors, for which we need to compute the iPRC or the lTRC depending on whether a global uniform rescaling or a piecewise uniform rescaling is needed. We hence first present the numerical algorithms for obtaining the iPRC in §\[sec:algorithm-iprc\] and the lTRC in §\[sec:algorithm-lTRC\]; the algorithm for solving the homogeneous variational equation for the linear shape responses of $\gamma(t)$ to instantaneous perturbations (the variational dynamics ${\mathbf{u}}$) is presented in §\[sec:algorithm-u\]; lastly, in §\[sec:algorithm-x1\] we illustrate the algorithms for computing the linear shape responses of $\gamma(t)$ to sustained perturbations (the iSRC $\gamma_1$) with a uniform rescaling factor computed from the iPRC as well as with piecewise uniform rescaling factors computed from the lTRC. For simplicity, we assume the initial time is $t_0=0$. Algorithm for Calculating the iPRC z for LCSCs {#sec:algorithm-iprc} ---------------------------------------------- It follows from Remark \[rem:iprc-liftoff\] that the iPRC ${{\mathbf z}}$ for the LCSCs need to be solved backward in time. While there is no discontinuity of ${{\mathbf z}}$ at a landing point, a time-reversed version of the jump matrix at the liftoff point on the hard boundary $\Sigma^i$, denoted as $\mathcal{J}^i_{\rm lift}$, is given by $$\label{eq:back-jump-def} \mathcal{J}^i_{\rm lift}=I-n^i {n^i}^\intercal,$$ where $I$ is the identity matrix. $\mathcal{J}^i_{\rm lift}$ updates ${{\mathbf z}}$ local to the liftoff point as $$\begin{aligned} {\ensuremath{\begin{array}{cccccccccc} {{{\mathbf z}}_{\text{lift}}^{i^-}}=\mathcal{J}^i_{\rm lift} {{{\mathbf z}}_{\text{lift}}^{i^+}} \end{array}}}\end{aligned}$$ where ${{{\mathbf z}}_{\text{lift}}^{i^-}}$ and ${{{\mathbf z}}_{\text{lift}}^{i^+}}$ are the iPRC just before and just after the trajectory crosses the liftoff point $x_{\text{lift},i}$ in forwards time. We now describe an algorithm for numerically obtaining the complete iPRC ${{\mathbf z}}$ for $\gamma(t)$, a stable limit cycle with sliding components along hard boundaries and transversal crossing boundaries as described before. #### Algorithm for ${{\mathbf z}}$ 1. Fix an initial condition ${\mathbf{x}}_0=\gamma(0)$ on the limit cycle, and integrate  to compute $\gamma(t)$ over $[0, T_0]$. 2. Integrate the adjoint equation backward in time by defining $s=T_0-t$ and numerically solve for the fundamental matrix $\Psi(s)$ over one period $0\le s \le T_0$, where $\Psi$ satisfies 1. $\Psi(0)=I$, the identity matrix. 2. For $s$ such that $\gamma(T_0-s)$ lies in the interior of the domain, $$\frac{d\Psi}{ds}=A^{\rm interior}(T_0-s)\Psi$$ where $A^{\rm interior}(t)=\left(DF^{\rm interior} (\gamma(t))\right)^\intercal$ is the transpose of the Jacobian of the interior vector field $F^{\rm interior}$. 3. For $s$ such that $\gamma(T_0-s)$ lies within a sliding component along boundary $\Sigma^i$, $$\frac{d\Psi}{ds}= A^i(T_0-s)\Psi$$ where $A^i(t)=\left(DF^{\mathrm{slide}_i}(\gamma(t))\right)^\intercal$ is the transpose of the Jacobian of the sliding vector field $F^{\mathrm{slide}_i}$, given in . 4. At any time $t_p$ when $\gamma$ transversely crosses a switching surface with a normal vector $n_p$, $$\Psi^- = \mathcal{J} \Psi^+$$ where $\Psi^-=\lim_{s\to (T_0-t_p)^+}\Psi(s)$ and $\Psi^+=\lim_{s\to (T_0-t_p)^-}\Psi(s)$ are the fundamental matrices just before and just after crossing the surface in forwards time. $\mathcal{J}=S^\intercal$ since $J^\intercal S=I$ as discussed in §\[sec:trans\], where the saltation matrix at any transversal crossing point is $$S=I+\frac{(F_p^+-F_p^-)n_p^\intercal }{n_p^\intercal F_p^-}$$ where $F_p^-, F_p^+$ are the vector fields just before and just after the crossing in forwards time (see ). 5. At a liftoff point on the $i$-th hard boundary $\Sigma^i$ (in backwards time, a transition from the interior to $\Sigma^i$), update $\Psi$ as $$\Psi^- = \mathcal{J}^i \Psi^+$$ where $\mathcal{J}^i=I-n^i n^{i \intercal}$ as defined in , and then switch the integration from the full Jacobian $A^{\rm interior}$ to the restricted Jacobian $A^i$. 6. At a landing point on the $i$-th hard boundary $\Sigma^i$ (in backwards time, a transition from $\Sigma^i$ to the interior) switch integration from the restricted Jacobian $A^i$ to the full Jacobian $A^{\rm interior}$; no other change in $\Psi$ is needed. 3. Diagonalize the fundamental matrix at one period $\Psi(T_0)$; it should have a single eigenvector $v$ with unit eigenvalue. The initial value for ${{\mathbf z}}_\text{BW}$ (represented in *backwards time*) at the point $\gamma(T_0)=\gamma(0)={\mathbf{x}}_0$ is given by $${{\mathbf z}}_\text{BW}(0)=\frac{v}{F({\mathbf{x}}_0)\cdot v}$$ 4. The iPRC in backward time over $s\in [0, T_0]$ is given by ${{\mathbf z}}_\text{BW}(s)=\Psi(s){{\mathbf z}}_{\rm BW}(0)$ and is $T_0$-periodic. Equivalently, one may repeat step (2) by replacing $\Psi(s)$ with ${{\mathbf z}}_\text{BW}(s)$ and replacing the initial condition $\Psi(0)=I$ with ${{\mathbf z}}_{\rm BW}(0)$ to solve for the complete iPRC. 5. The iPRC in forward time is then given by ${{\mathbf z}}(t)={{\mathbf z}}_\text{BW}(T_0-t)$ where $t\in[0,T_0]$. 6. The linear shift in period in response to the static perturbation can be calculated by evaluating the integral (see ) $$T_1=-\int_{0}^{T_0} {{\mathbf z}}^\intercal(t)\frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0}dt$$ An alternative way (in MATLAB) to do backward integration is reversing the time span in the numerical solver; that is, integrate the adjoint equation over $[T_0, 0]$ to compute ${{\mathbf z}}(t)$. Algorithm for Calculating the lTRC for LCSCs {#sec:algorithm-lTRC} -------------------------------------------- The lTRC satisfies the same adjoint equation, , as the iPRC, and hence exhibits the same jump matrix at each liftoff, landing and boundary crossing point. It follows that the algorithm for the iPRC from §\[sec:algorithm-iprc\] can mostly carry over to computing the lTRC. Suppose the domain of $\gamma(t)$ can be divided into $m$ regions ${{\mathcal R}}^1, ..., {{\mathcal R}}^m$, each distinguished by its own timing sensitivity properties. We denote the lTRC in ${{\mathcal R}}^j$ by $\eta^j$. Below we describe the algorithm to compute $\eta^j$ in region ${{\mathcal R}}^j$ bounded by the two local timing surfaces $\Sigma^{\rm in}$ and $\Sigma^{\rm out}$. Following the notations in §\[sec:smooth-theory\], $t^{\rm in}$ and $t^{\rm out}$ denote the time of entry into and exit out of ${{\mathcal R}}^j$, at locations ${\mathbf{x}}^{\rm in}$ and ${\mathbf{x}}^{\rm out}$, respectively. The algorithm for computing $\eta^j$ is described as follows. #### Algorithm for $\eta^j$ 1. Compute $\gamma$, the unperturbed limit cycle, and $T_0$, its period, by integrating  . 2. Compute $t^{\rm in}, t^{\rm out}$ for region $j$. Evaluate ${\mathbf{x}}^{\rm in}=\gamma(t^{\rm in}),\, {\mathbf{x}}^{\rm out}=\gamma(t^{\rm out})$ and $T_0^{j}=t^{\rm out}-t^{\rm in}$. 3. Compute the boundary value for $\eta^j$ at the exit point ${\mathbf{x}}^{\rm out}$ (see ) $$\eta^{j}({\mathbf{x}}^{\rm out})=\frac{-n^{\rm out}}{{n^{\rm out}}^\intercal F({\mathbf{x}}^{\rm out})}$$ where $n^{\rm out}$ is a normal vector to $\Sigma^{\rm out}$. 4. Integrate the adjoint equation backward in time by defining $s=T_0-t$ and numerically solve for $\eta^j_{\rm BW}(s)$ (represented in backwards time) over $[T_0-t^{\rm out}, T_0-t^{\rm in}]$. $\eta^j_{\rm BW}(s)$ satisfies the initial condition $\eta^j_{\rm BW}(T_0-t_{\rm out})=\eta^{j}(t_{\rm out})$ computed from step (3) as well as conditions (b) through (f) from step (2) of **Algorithm for ${{\mathbf z}}$** in §\[sec:algorithm-iprc\]. 5. The lTRC in forward time is then given by $\eta^j(t) = \eta^j_{\rm BW}(T_0-t)$ where $t\in [t_{\rm in}, t_{\rm out}]$. 6. Compute $\gamma_\varepsilon$, the limit cycle under some small static perturbation $\varepsilon\ll 1$, and find ${\mathbf{x}}_{\varepsilon}^{\rm in}$, the coordinate of the intersection point where $\gamma_\varepsilon(t)$ crosses $\Sigma^{\rm in}$. The linear shift in time in region $j$ in response to the static perturbation can be calculated by evaluating the integral (see ) $$T^{j}_{1} = \eta^j({\mathbf{x}}^{\rm in})\cdot \frac{ {\mathbf{x}}_{\varepsilon}^{\rm in}-{\mathbf{x}}^{\rm in}}{\varepsilon}+\int_{t^{\rm in}}^{t^{\rm out}}\eta^j(\gamma(t))\cdot \frac{\partial F_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0}dt.$$ All the local linear shifts in time sum up to the global linear shift in period, that is, $T_1 = \sum_{j=1}^{j=m}T_1^j$. Algorithm for Solving the Homogeneous Variational Equation for LCSCs {#sec:algorithm-u} -------------------------------------------------------------------- Here we describe the algorithm for solving the homogeneous variational equation for linear displacement ${\mathbf{u}}$, the shape response to an instantaneous perturbation. This makes use of Theorem \[thm:main\], which describes different jumping behaviors of ${\mathbf{u}}$ at liftoff, landing, and boundary crossing points. Unlike the iPRC and lTRC which require integration backwards in time, the variational dynamics can be solved with forward integration. This makes the algorithm comparatively simpler by allowing $\gamma(t)$ and ${\mathbf{u}}(t)$ to be solved simultaneously. #### Algorithm for ${\mathbf{u}}$: 1. Fix an initial condition ${\mathbf{x}}_0=\gamma(0)$ on the limit cycle and an initial condition ${\mathbf{u}}_0={\mathbf{u}}(0)$ for the displacement at $\gamma(0)$ of the limit cycle. 2. Integrate the original differential equation  and the homogeneous variational equation  simultaneously forward in time and numerically solve for ${\mathbf{u}}(t)$ over one period $0\le t \le T_0$, where ${\mathbf{u}}$ satisfies 1. ${\mathbf{u}}(0)={\mathbf{u}}_0$. 2. For $t$ such that $\gamma(t)$ lies in the interior of the domain, $$\frac{d{\mathbf{u}}}{dt}=DF^{\rm interior} (\gamma(t)) {\mathbf{u}}$$ 3. For $t$ such that $\gamma(t)$ lies within a sliding component along boundary $\Sigma^i$, $$\frac{d{\mathbf{u}}}{dt}= DF^{\mathrm{slide}_i}(\gamma(t)){\mathbf{u}}$$ where $DF^{\mathrm{slide}_i}$ is the Jacobian of the sliding vector field $F^{\mathrm{slide}_i}$ given in . 4. At any time $t_p$ when $\gamma$ transversely crosses a switching surface with a normal vector $n_p$ separating vector field $F_p^-$ on the incoming side from vector field $F_p^+$ on the outgoing side, $${\mathbf{u}}^+ = S {\mathbf{u}}^-$$ where ${\mathbf{u}}^-=\lim_{t\to t_p^-}{\mathbf{u}}(t)$ and ${\mathbf{u}}^+=\lim_{t\to t_p^+}{\mathbf{u}}(t)$ are the displacements just before and just after crossing the surface. By the definition for the saltation matrix at transversal crossing point , we have $$S=I+\frac{(F_p^+-F_p^-)n_p^\intercal }{n_p^\intercal F_p^-}.$$ 5. At a landing point on the $i$-th hard boundary $\Sigma^i$, update ${\mathbf{u}}$ as $${\mathbf{u}}^+ = S^i {\mathbf{u}}^-$$ where $S^i=I-n^i{n^i}^\intercal$ (recall $n^i$ is the normal vector to $\Sigma^i$) and switch integration from the full Jacobian $DF^{\rm interior}$ to the restricted Jacobian $DF^{\mathrm{slide}_i}$. 6. At a liftoff point on the $i$-th hard boundary $\Sigma^i$, switch integration from the restricted Jacobian $DF^{\mathrm{slide}_i}$ to the full Jacobian $DF^{\rm interior}$; no other change in ${\mathbf{u}}$ is needed. \[rem:fund-matrix\] The *fundamental solution matrix* satisfies $$\frac{d\Phi(t,0)}{dt}=DF\Phi(t,0),\, \text{with}\quad \Phi(0,0)=I$$ and takes the initial perturbation ${\mathbf{u}}(0)$ to the perturbation ${\mathbf{u}}(t)$ at time $t$, that is, $${\mathbf{u}}(t)=\Phi(t,0){\mathbf{u}}(0).$$ Computing $\Phi$ therefore requires applying **Algorithm for ${\mathbf{u}}$** $n$ times, once for each dimension of the state space. Specifically, let $\Phi(t,0)=[\phi_1(t,0)\, ...,\,\phi_n(t,0)]$. The $i$-th column $\phi_i(t,0)$ is the solution of the variational equation with the initial condition $\phi_i(0,0)=e_i$, a unit column vector with zeros everywhere except at the $i$-th row where the entry equals 1. \[rem:monodromy\] Once $\Phi$ is obtained, we can obtain the monodromy matrix, $M= \Phi(T_0,0)$. It follows from the periodicity of $\gamma(t)$ that $M$ has $+1$ as an eigenvalue with eigenvector $v$ tangent to the limit cycle at ${\mathbf{x}}_0$; this condition provides a partial consistency check for the algorithm. Algorithms for computing iSRC, the response to sustained perturbation {#sec:algorithm-x1} --------------------------------------------------------------------- Now we discuss the calculation of iSRC $\gamma_1$, the linear shape response to a sustained perturbation. While $\gamma_1$ shares the same saltation as ${\mathbf{u}}$ at each liftoff, landing and boundary crossing point, $\gamma_1$ satisfies the nonhomogeneous version of the variational equation, or , where one of the nonhomogeneous terms depends on the time scaling factor, $\nu_1$ or $\nu^j_1$. Moreover, the initial condition for $\gamma_1$ depends on the given perturbation and hence needs to be computed in the algorithm whereas the initial value for ${\mathbf{u}}$ is arbitrarily preassigned. In the following, we first describe the algorithm for computing $\gamma_1$ using the global uniform rescaling and then consider using piecewise uniform rescaling. #### Algorithm for $\gamma_1$ with uniform rescaling 1. Fix an initial condition ${\mathbf{x}}_0=\gamma(0)$ on the limit cycle. 2. Compute the linear shift in period $T_1$ using **Algorithm for ${{\mathbf z}}$**, then evaluate $\nu_1=T_1/T_0$. 3. Choose an arbitrary Poincaré section $\Sigma$ (this can be one of the switching boundaries for appropriate ${\mathbf{x}}_0$) that is transverse to $\gamma$ at ${\mathbf{x}}_0$. Compute $\gamma_\varepsilon$, the limit cycle under some fixed small static perturbation, and find ${{\mathbf{x}}_0}_\varepsilon$, the coordinate of the intersection point where $\gamma_\varepsilon(t)$ crosses $\Sigma$. The initial value for $\gamma_1$ at the initial point ${\mathbf{x}}_0$ is then given by $$\gamma_1(0) = \frac{{{\mathbf{x}}_{0}}_\varepsilon - {\mathbf{x}}_0}{\varepsilon}$$ 4. Integrate the original differential equation  with the initial condition ${\mathbf{x}}_0$ and the nonhomogeneous variational equation  simultaneously forward in time and numerically solve for $\gamma_1$ over one period $0\le t \le T_0$, where $\gamma_1$ satisfies 1. $\gamma_1(0)=({{\mathbf{x}}_{0}}_\varepsilon - {\mathbf{x}}_0)/\varepsilon$. 2. For $t$ such that $\gamma(t)$ lies in the interior of the domain, $$\frac{d\gamma_1}{dt}=DF^{\rm interior}(\gamma(t)) \gamma_1 + \nu_1 F^{\rm interior}(\gamma( t)) +\frac{\partial F^{\rm interior}_\varepsilon(\gamma(t))}{\partial \varepsilon}\Big|_{\varepsilon=0}$$ 3. For $t$ such that $\gamma(t)$ lies within a sliding component along boundary $\Sigma^i$, $$\frac{d\gamma_1}{dt}= DF^{\mathrm{slide}_i}(\gamma(t)) \gamma_1 + \nu_1 F^{\mathrm{slide}_i}(\gamma( t)) + \frac{\partial F_\varepsilon^{\mathrm{slide}_i}(\gamma( t))}{\partial \varepsilon}\Big|_{\varepsilon=0}$$ where $DF^{\mathrm{slide}_i}$ is the Jacobian of the sliding vector field $F^{\mathrm{slide}_i}$ given in . 4. For transversal crossings, landing points, and liftoff points, apply (d), (e) and (f), respectively, from step 2) in **Algorithm for ${\mathbf{u}}$** in §\[sec:algorithm-u\], by replacing ${\mathbf{u}}$ with $\gamma_1$. Next we consider the case when $\gamma(t)$ exhibits $m$ different uniform timing sensitivities at regions ${{\mathcal R}}^1, ..., {{\mathcal R}}^m$, each bounded by two local timing surfaces, as discussed in §\[sec:algorithm-lTRC\]. Piecewise uniform rescaling is therefore needed to compute the shape response curve. The procedure for obtaining $\gamma_1$ in this case is nearly the same as described in **Algorithm for $\gamma_1$ with uniform rescaling**, except we now need to compute various rescaling factors using the lTRC. This hence leads to different variational equations that need to be solved. On the other hand, the local timing surfaces naturally serve as the Poincaré sections that are required to compute the initial values for $\gamma_1$ in the uniform rescaling case. #### Algorithm for $\gamma_1$ with piecewise uniform rescaling 1. Take the initial condition for $\gamma(t)$ to be $\gamma(0)={\mathbf{x}}_0\in \Sigma$, where $\Sigma$ is one of the local timing surfaces. Compute $\gamma(t)$, the unperturbed trajectory, and $\gamma_\varepsilon(t)$, the trajectory under some static perturbation $0<\varepsilon\ll 1$, by integrating . 2. For $j\in\{1,...,m\}$, compute $T^j_0$, the time that $\gamma(t)$ spends in region $j$ and $T^j_1$, the linear shift in time in region $j$ using **Algorithm for $\eta^j$**, and then evaluate $\nu^j_1=T^j_1/T^j_0$. 3. Compute ${{\mathbf{x}}_0}_\varepsilon$, the coordinate of the intersection point where $\gamma_\varepsilon(t)$ crosses $\Sigma$. The initial value for $\gamma_1$ at the initial point ${\mathbf{x}}_0$ is given by $$\gamma_1(0) = \frac{{{\mathbf{x}}_0}_\varepsilon- {\mathbf{x}}_0}{\varepsilon}$$ 4. Integrate the original differential equation  with the initial condition ${\mathbf{x}}_0$ and the piecewise nonhomogeneous variational equation  simultaneously forward in time and numerically solve for $\gamma_1$ over one period $0\le t \le T_0$, where $\gamma_1$ satisfies 1. $\gamma_1(0)=({{\mathbf{x}}_{0}}_\varepsilon - {\mathbf{x}}_0)/\varepsilon$. 2. For $t$ such that $\gamma(t)$ lies in the intersection of the interior of the domain and region ${{\mathcal R}}^j$, $$\frac{d\gamma_1}{dt}=DF^{\mathrm{interior}_j}(\gamma(t)) \gamma_1 + \nu^j_1 F^{\mathrm{interior}_j}(\gamma( t)) +\frac{\partial F_\varepsilon^{\mathrm{interior}_j}(\gamma( t))}{\partial \varepsilon}\Big|_{\varepsilon=0}$$ where $DF^{\mathrm{interior}_j}$ is the Jacobian of the interior vector field $F^{\mathrm{interior}_j}$ in ${{\mathcal R}}^j$. 3. 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--- abstract: 'We present *Spitzer*/IRAC Ch1 and Ch2 monitoring of six brown dwarfs during 8 different epochs over the course of 20 months. For four [brown]{} dwarfs, we also obtained simulataneous *HST*/WFC3 G141 Grism spectra during two epochs and derived light curves in five narrow-band filters. Probing different pressure levels in the atmospheres, the multi-wavelength light curves of our six targets all exhibit variations, and the shape of the light curves evolves over the timescale of a rotation period, ranging from 1.4 h to 13 h. We compare the shapes of the light curves and estimate the phase shifts between the light curves observed at different wavelengths by comparing the phase of the primary Fourier components. We use state-of-the-art atmosphere models to determine the flux contribution of different pressure layers to the observed flux in each filter. We find that the light curves that probe higher pressures are similar and in phase, but are offset and often different from the light curves that probe lower pressures. The phase differences between the two groups of light curves suggest that the modulations seen at lower and higher pressures may be introduced by different cloud layers.' author: - Hao Yang - Dániel Apai - 'Mark S. Marley' - Theodora Karalidi - Davin Flateau - 'Adam P. Showman' - Stanimir Metchev - Esther Buenzli - Jacqueline Radigan - Étienne Artigau - 'Patrick J. Lowrance' - 'Adam J. Burgasser' bibliography: - 'hao.bib' title: '*EXTRASOLAR STORMS*: PRESSURE-DEPENDENT CHANGES IN LIGHT CURVE PHASE IN BROWN DWARFS FROM SIMULTANEOUS *HUBBLE* AND *SPITZER* OBSERVATIONS' --- INTRODUCTION ============ [Sharing the same range of effective temperatures with many extrasolar giant planets, brown dwarfs are thought to possess similar atmospheric properties [e.g., @lodders2006]. Compared with planets, brown dwarfs are generally more luminous and have no disburbance from host stars. Therefore, studies of brown dwarf atmospheres can be performed with higher precision, providing important reference and guidance to understanding of the atmospheres of extrasolar planets. ]{} The presence of condensate clouds in brown dwarf atmospheres strongly impacts the chemistry and thermal structure in the atmosphere, removing gas-phase opacity and modifying the emergent spectra [e.g., @ackerman2001; @cooper2003; @helling2008; @marley2015]. Cloud covers are likely to be heterogenous in these ultracool atmospheres, similar to what have been observed for solar system giant planets [e.g., @west2009; @sromovsky2012; @karalidi2015; @simon2015], and complex atmospheric structures such as spots and storms combined with rotational modulation are expected to produce periodic photometric variabilities [@bailerjones2001]. In addition to patch clouds, [ large-scale atmospheric motions and temperature fluctuations, induced by regional and global atmospheric circulation, have also been proposed to explain the brightness variations [@showman2013; @zhang2014; @robinson2014]. ]{} [Starting in the]{} early 2000s, substantial observational efforts have been put into detecting and characterizing periodic flux variations of brown dwarfs. While most early ground-based monitoring campaigns [had]{} no or marginal detections [e.g., @bailerjones2001; @clarke2002; @koen2005; @clarke2008; @khandrika2013], @artigau2009 found that the L/T transition dwarf, SIMP J013656.5+093347 (hereafter SIMP0136), displayed periodic variation in the $J$-band with an amplitude of $50$ mmag, and @radigan2012 measured a peak-to-peak amplitude as large as 26% in the $J$-band light curves of another L/T transition object, 2MASS J21392676+0220226 (hereafter 2M2139). In a large ground-based $J$-band survey of 57 brown dwarfs, @radigan2014a concluded that strong flux variations over 2% were exclusively observed for the L/T transition objects in their sample. With much higher precision than ground-based observations, space-based photometric and spectroscopic monitoring campaigns have revealed that low-amplitude flux variations on a sub-percent level appear to be a common characteristic for all L and T dwarfs [@heinze2013; @buenzli2014; @metchev2015]. While single-band photometric monitoring uncovers the longitudinal cloud structures in the brown dwarfs, multi-wavelength observations probe different depths in the brown dwarf atmospheres and provide valuable information on the vertical cloud structures. So far there have been only a handful of such studies. @buenzli2012 discovered for the late-T dwarf 2MASS J22282889-4310262 (hereafter 2M2228) that simultaneous *HST* narrow-band and *Spitzer* 4.5-$\mu$m light curves have different phases. They derived from models the pressure levels probed by different wavelengths and found the phase shifts increase with decreasing pressure level, indicating complex atmospheric structures in both horizontal and vertical directions. @biller2013 performed ground-based photometric monitoring of the T0.5 dwarf Luhman 16B [@luhman2013; @gillon2013] and their simultaneous multi-band light curves suggested similar pressure-dependent phase shifts. On the other hand, in the *HST* observations of Luhman 16B @buenzli2015a found no phase shifts between light curves of narrow $J$ and $H$ bands as well as the 1.35-1.44 $\mu$m water band. This is consistent with the results reported by @apai2013 that the same *HST* narrow-band light curves showed no phase shifts for the two L/T transition objects, SIMP0136 and 2M2139. @gizis2015 found that the *Kepler* optical light curves of the L1 dwarf WISEP J190648.47+401106.8 are consistent in phase with subsequent *Spitzer* 4.5-$\mu$m light curves obtained about 5 months after the *Kepler* monitoring ended. To date, 2M2228 remains the only brown dwarf found to display unambiguous phase shifts between multi-band light curves, and [observationally it is difficult to probe the cloud structures in the vertical direction.]{} In this paper, we report the first results from the *Spitzer* Cycle-9 Exploration Science Program, *Extrasolar Storms*, on detecting phase shifts between the light curves of six varying brown dwarfs observed in two IRAC channels as well as between simultaneous *HST* and *Spitzer* light curves of four objects. The rest of this paper is organized as follows. In §2, we give the background information on the the six targets in our sample. In §3, we describe our observations and the data reduction procedures. In §4, we discuss our methods for measuring the phase shifts in our *Spitzer* and *HST* observations. The phase shift measurements and other results are presented in §5, which is followed by some discussion in §6. TARGETS ======= 2MASS J15074769-1627386 and 2MASS J18212815+1414010 (L5) -------------------------------------------------------- The L5 dwarf 2MASS J15074769-1627386 (hereafter 2M1507) was discovered by @reid2000 and the L5 dwarf 2MASS J18212815+1414010 (hereafter 2M1821) was discovered by @looper2008. Spectroscopic analysis by @gagne2014 identified 2M1821 as a young field dwarf showing signs of low surface gravity. @sahlmann2015 measured the parallax of 2M1821 to be $106.15 \pm 0.18$ mas, corresponding to a distance of $9.38 \pm 0.03$ pc. Based on the $J-K$ color of 1.78, weak water bands, triangular $H$-band continuum, and strong 9-11 $\mu$m silicate absorption, @looper2008 suggested that the atmosphere of 2M1821 might have unusually thick dust clouds. For 2M1507, weak silicate absorption between 9 and 11 $\mu$m was found @cushing2006, indicating different cloud thickness than 2M1821. With the *HST*/WFC3 G141 grism spectra of 2M1507 and 2M1821, @yang2015 studied the wavelength dependence of their flux density variation between 1.1 and 1.7 $\mu$m. By taking the ratio of the brightest and faintest spectra in an *HST* visit, @yang2015 found that for the two L5 dwarfs the flux density of the 1.4 $\mu$m water band varies at a similar rate as that of the $J$- and $H$-band. This is significantly different from the reduced variation amplitude in the 1.4 $\mu$m water band observed for the two T2 dwarfs (SIMP J013656.5+093347.3 and 2MASS J21392676+0220226) reported by @apai2013. @yang2015 showed that models of an L5 dwarf with a haze layer high ($<$ 50 mbar) in the atmosphere could explain the similar flux variation amplitude in and out of the 1.4 $\mu$m water band observed for 2M1507 and 2M1821. SIMP J013656.5+093347.3 and 2MASS J21392676+0220226 (T2) -------------------------------------------------------- The T2 dwarf SIMP0136 was discovered by @artigau2006 in a proper-motion survey. At an estimated photometric distance of 6.4 $\pm$ 0.3 pc, it is the brightest T dwarf in the northern hemisphere. Ground-based photometric monitoring by @artigau2009 revealed that SIMP0136 is variable with a peak-to-peak amplitude of $\sim$ 50 mmag in $J$-band, marking the first detection of high-amplitude IR flux variations for a T dwarf. @kao2015 detected circularly polarized radio emission with two pulses in the 4-8 GHz band and estimated a magnetic field strength of at least 2.5 kG. The T2 dwarf 2M2139 was discovered by @reid2008. [@radigan2012] reported periodic flux variation of 2M2139 in the $J-$band with a peak-to-peak amplitude as large as 26%, making 2M2139 the most variable brown dwarf discovered to date. *HST* time-resolved spectroscopy by @apai2013 provided high-quality spectral series and light curves for both SIMP0136 and 2M2139. Based on these data and comparison to state-of-the-art atmosphere models the authors concluded that the variations in both T2 dwarfs are caused by thickness variations in the silicate cloud cover (warm thin and cooler thicker clouds). Light curve modeling through a Genetic Algorithm-optimized ray tracer by @apai2013 and an MCMC-optimized pixelized atmosphere model by @karalidi2015 both found that at least three elliptical features are required to fit the light curve of SIMP0136 and 2M2139. Principal components analysis of the HST spectral series showed that $>$97% of the spectral variations can be reproduced with only a single principal component on top of a mean spectrum, arguing for a single type of cloud feature, consistent with the result of the atmospheric modeling. 2MASS J13243553+6358281 (T2) ---------------------------- The T2 dwarf 2MASS J13243553+6358281 (hereafter 2M1324) was discovered independently by @looper2007 and @metchev2008. It has an usually red spectral energy distribution and also exhibits peculiar IR colors [@looper2007; @metchev2008; @faherty2009]. @looper2007, @burgasser2010a, and @geibler2011 discussed the possibility of 2M1324 being a close binary (L9 + T2 or L8 + T3.5). 2MASS J22282889-4310262 (T6) ---------------------------- The T6 dwarf 2M2228 was discovered by @burgasser2003. In a ground-based monitoring campaign searching for variability, @clarke2008 detected a peak-to-peak amplitude of $15.4 \pm 1.4$ mmag for 2M2228. @buenzli2012 analyzed *HST*/WFC3 G141 grism spectra and partially simulatenous *Spitzer* Ch2 photometry of 2M2228, and for the first time found phase shifts between light curves of five narrow spectral bands between 1.1 and 1.7 $\mu$m and 4.5 $\mu$m. The fluxes in these spectral bands probe different depths in the atmosphere, and the measured phase shifts of 2M2228 were found to correlate with the characteristic pressure levels derived from atmospheric models. For lower pressure levels, the phase shift is larger. Such findings by @buenzli2012 revealed atmospheric structures in the vertical direction, adding a new dimension in the studies of the ultracool atmospheres. OBSERVATIONS AND DATA REDUCTION =============================== With the *Spitzer* Cycle-9 Exploration Science Program, *Extrasolar Storms* (hereafter *Storms*; Program ID: 90063, PI: D. Apai), we obtained for six targets 1,144 hours of photometric data in staring mode in channel 1 (Ch1, the $3.6~\mu$m bandpass) and channel 2 (Ch2, the $4.5~\mu$m bandpass) of the *Spitzer* Infrared Array Camera [IRAC; @fazio2004]. Our observations were designed to monitor the light curve evolution of each target over more than 1,000 rotation periods in eight separate *Spitzer* visits, probing flux variations on a number of timescales and studying the evolution and dynamics of the brown dwarf atmospheres and their heterogenous cloud covers. The target properties are provided in Table \[obs1\] and details of the observations are given in Table \[obs2\]. Previously, @metchev2015 have observed all the *Storms* targets except for SIMP0136 for about 20 hr each. In addition to the *Spitzer* observations, for the four shorter-period objects in the sample, we also obtained time-resolved, high-precision *HST*/WFC3 G141 grism spectra simulateneously during two of the eight *Spitzer* visits. As the coordinated *HST* component (Program ID: 13176, PI: D. Apai) of the *Storms* program, the WFC3 observations were acquired between April, 2013 and October, 2013 for a total of 28 orbits. Detailed information of the *HST* observations are also listed in Table \[obs1\]. The data reduction procedures are described in detail in @apai2013 and @yang2015. The reduced G141 grism spectra provide wavelength coverage between 1.05 and 1.7 $\mu$m and a spectral resolution of $\sim$ 130. The signal-to-noise ratio is over 300. Part of the *HST* observations have been published in @yang2015. The *Spitzer* observations were carried out from December, 2012 to August, 2014. The exposure time for each individual image is 10.40 s. The detector arrays in both IRAC channels are 256 $\times$ 256 pixels in size and the pixel size is $1\farcs2 \times 1\farcs2$, providing a field of view of $5\farcm2 \times 5\farcm2$. The eight visits for each target were arranged in pairs of two. The two visits within a pair were separated by $\sim$ 40 rotation periods, and each pair was separated by $\sim$ 100 rotation periods. The first four visits and the last four visits were separated by about 1,000 rotation periods, corresponding to roughly one year. As illustrated in Figure \[fig1schedule\], such scheduling of the observations allowed us to detect flux variations on a broad range of timescales and to probe different physical processes reponsible for these changes. During each visit, we first observed two consecutive rotations in Ch1, then one rotation in Ch2. followed by another rotation in Ch1, allowing us to separate light curve evolution in time from the wavelength dependency of the light curve. Figure \[rawimage\] shows a sample of raw *Spitzer* images for the *Storms* targets. To reduce the *Spitzer* photometric data we first downloaded from the *Spitzer* Science Center the corrected Basic Calibrated Data (cBCD) images, which have been processed through the IRAC calibration pipeline (version: S19.1.0). After flat-fielding and manually masking out bad pixels and bright objects within a 20-pixel radius of the target, we used the IDL routine $box\_centroider$ supplied by the *Spitzer* Science Center to measure the exact location of the object on the detector. After subtracting the background level determined from an annulus between 12 and 20 pixels from the centroid position of the target, we performed photometry for each image using the IDL routine $aper$ with a fixed aperture of 2 pixels. Then we rejected photometric points with centroid positions outside of 5$\sigma$ in $x$ or $y$ from a 25-point median-smoothed values. Flux measurements that are outside of 3$\sigma$ of the 25-point median-smoothed light curve were also rejected. In each visit, less than 3% of the photometric points were rejected. The reduced photometric data displays the intra-pixel sensitivity variation in IRAC [@reach2005], commonly referred to as the pixel phase effect, as the detector sensitivity varies slightly depending on where the exact location (on a sub-pixel scale) of the point source is. This effect manifests itself as flux discontinuities due to re-acquisition of targets between consecutive *Spitzer* Astronomical Observation Requests (AORs) as well as zigzag-shaped flux variations within an AOR with a period around 40 minutes. To correct for the pixel-phase effect, we model the observed light curves as a combination of astrophysical variations and sensitivity variations, and simultaneously fit both model components with Markov Chain Monte Carlo (MCMC) simulations. The best-fit model for sensitivity variations is then removed from the observed light curves. We model the sensitivity variations as a quadratic function of the target’s centroid location, ($x$,$y$), in pixels (e.g., Knutson et al. 2008; Heinze et al. 2013): $$Q(x,y)= 1 + p_1 \times x + p_2 \times y+ p_3 \times x^2 + p_4 \times y^2 + p_5 \times x \times y \ , \,\,\,\,\,\, \eqno(1)$$ To account for simultaneous astrophysical variations of our targets, we fit a third-order Fourier series to the normalized light curve: $$\mathscr{F}(t) = 1 + \sum\limits_{i=1}^3 \left( A_i~\textrm{cos}~(\frac{2{\pi}i}{P}~t) + B_i~\textrm{sin}~(\frac{2{\pi}i}{P}~t) \right) \ , \,\,\,\,\,\, \eqno(2)$$ where $t$ is the observation timestamp and $P$ is the rotation period of the object, which is fixed to the value in Table \[obs1\]. Then we perform MCMC simulations utilizing the Python package PyMC [@patil2008], and simulateneously fit the normalized observations with $Q(x,y)\mathscr{F}(t)$, the product of the quadractic function and the Fourier series. The data in each channel of a visit are fitted separately, and for each run, two million iterations are calculated with additional $0.6$ million burn-in iterations. We have run five chains for multiple visits and found that different chains converge well within one million iterations and return the same results for the parameters. Finally, we corrected for the pixel phase effect by calculating the value of the best-fit quadratic correction function at each observation timestamp and dividing that value from the corresponding aperture photometric measurement. A few example light curves and the corresponding correction functions are shown in Figure \[mcmcexample\]. This correction method is able to effectively remove the flux discontinuities due to target re-acquisition and flux oscillations due to target sub-pixel centroid shift. For our subsequent analysis, all the corrected light curves were binned in 5-minute intervals to reduce noise, and the midpoint of each 5-minute interval was calculated to be the time of observation. Each of the three segments in a visit was normalized to its own mean value. PHASE SHIFT ANALYSIS ==================== Light Curve Shapes and Evolution -------------------------------- [ We found that the light curves in both *Spitzer*/IRAC channels exhibit a variety of shapes, as shown in Figure \[fig2replc\]. In some cases, the light curve shapes could evolve over a timescale as short as one rotation period, and the shape and amplitude of the light curve observed at different wavelengths can be substantially different. ]{} [While our observations reveal differences in the light curve shape as a function of time and, sometimes, as a function of the wavelength of the observations, in 20 out of 48 *Spitzer* visits the differences between the light curves of consecutive rotations are relatively small. In these cases, the light curve shape observed at different wavelengths is either identical or very similar, but potentially delayed (“phase-shifted”) and may have different amplitudes.]{} [In this paper, we performed analyses to quantify phase shifts of only the datasets where the light curves are slowly evolving and the features are similar between different wavelengths, so that simple, robust approaches can be applied to measure phase shifts. A separate publication (Apai et al., in preparation) will fully explore the evolution of the light curves.]{} [Here, we focus on the information available in the wavelength-dependent phase shifts. As different wavelengths of observation probe different pressure levels in the atmosphere, the phase shifts measured here can be used to explore the vertical-longitudinal structure of the atmosphere. This approach has been demonstrated in @buenzli2012 and @apai2013 observationally, and models by @zhang2014 and @robinson2014 have predicted characteristic observational signatures contained in such pressure-dependent phase shifts. ]{} With the *Storms* data set, we adopted different approaches to determine phase shifts for two different data subsets: non-simultaneous *Spitzer* Ch1 and Ch2 observations and simultaneous *HST* and *Spitzer* observations. The *Spitzer* data allow analysis of continuous temporal variations and constrain longitudinal heterogeneities in the atmospheres, while the simultaneous multi-wavelength observations yield pressure-dependent phase shifts and probe atmospheric structures in the vertical direction. Measuring Phase Shifts Between *Spitzer* Ch1 and Ch2 Observations ----------------------------------------------------------------- One of the objectives of our *Spitzer* observations is to measure phase shifts between Ch1 and Ch2 light curves. In a typical *Spitzer* visit, as shown in Figure \[fig2replc\], we first observe two rotations in Ch1, followed by one rotation in Ch2 and then another rotation in Ch1. We employed two methods to determine potential phase shifts between Ch1 and Ch2 observations. The first method derives the phase shift from the times at which the target is at maximum or mininum flux levels, $T_{\rm{max}}$ or $T_{\rm{min}}$. For visits where we can unambiguously identify the flux maxima and minima in Rotations 2, 3, and 4 (R2, R3, and R4, as marked for example in [Figure \[fig1507\]]{}), we first found the midpoint in $T_{\rm{max}}$ or $T_{\rm{min}}$ for R2 and R4, which are both observed in Ch1. This was achieved by fitting a third-order Fourier function (see Eq. \[2\]) and then calculating the maxima and minima from the fits. Next, we calculated the time difference between the $T_{\rm{max}}$ or $T_{\rm{min}}$ in R3 (observed in Ch2) and the midpoint time of R2 and R4. The time difference is expressed as a phase shift in degrees using the rotation period given in Table \[obs1\]. Both the time differences in hours and the phase shifts in degrees are given in Table \[shiftmaxmin\]. We excluded the visits where $T_{\rm{max}}$ or $T_{\rm{min}}$ can not be well determined due to light curve evolution and/or incomplete phase coverage, such as Visit 6 of 2M1324 (see Figure \[fig2replc\]). To estimate the uncertainty in the phase shift measurements, we created $1,000$ synthetic light curves by adding random noise to the Fourier fits. The random noise is drawn from a normal distribution derived from the residuals of the Fourier fit. We performed the same analysis described above on the synthetic light curves, and the standard deviation of the $1,000$ phase shift measurements were adopted as the uncertainties. The second method we used to measure phase shifts betweeen light curves taken at different wavelengths was to directly cross-correlate the light curves in each visit between the two channels. For each visit, we first measured a rotation period by cross-correlating the two rotations observed in Ch1, R2 and R4. The shift in time at which the cross-correlation function reaches maximum plus the time difference between the end times of R2 and R4 are equal to two rotation periods. Then we cross-correlated R2 (in Ch1) and R3 (in Ch2), and also determined the shift in time corresponding to the peak of the cross-correlation function. The phase shift was calculated from the shift in time and the rotation period measured from R2 and R4. The same analysis was applied to R3 (in Ch2) and R4 (in Ch1). We also fit a Gaussian to the cross-correlation function, and the uncertainties in the center location of the fitted Gaussian expressed in phase were taken as the uncertainties of the measured phase shifts. Note that this method does not work well when light curve evolves significantly between consecutive rotations, such as Visit 7 of 2M2139 (see Figure \[fig2replc\]). We list the results from the visits that show little evolution of light curve shape in Table \[shiftcross\]. Measuring Phase Shifts Between *HST* and *Spitzer* Observations --------------------------------------------------------------- For the four objects simultaneously observed with *HST* and *Spitzer*, the *HST* observations have incomplete phase coverages. Therefore, *HST*-*Spitzer* phase shift measurement requires a different approach than the non-simultaneous but longer and better-sampled Ch1-Ch2 phase shift comparisons. We compare *HST* $J$-band light curves with *Spitzer* light curves of the four objects in [Figures \[fig1507\]–\[fig2228\]]{}. [The $J$-band light curves were]{} calculated by integrating the *HST*/WFC3 grism spectra over the 2MASS $J$-band relative spectral response curve [@cohen2003]. To be consistent with the timestamps of the *Spitzer* observations, all *HST* timestamps were converted from Modified Julian Dates to barycentric Modified Julian Dates, using the IDL routine $barycen.pro$. The mid-time of each *HST* exposure is used as the time of observation. The light curves from the two instruments are normalized by their respective median values. To quantify the phase shifts, we fit separate third-order Fourier functions (see Eq.\[2\]) to the simultaneous *HST* J-band and *Spitzer* Ch1 or Ch2 light curves. The rotation periods in the Fourier functions were fixed to the values listed in Table \[obs1\]. We used the best-fit parameters of the Fourier fits to calculate the phases of the first- and second-order Fourier components and then, from their difference, the corresponding phase shifts. Given the high precision of the *HST* data, the main source of uncertainty in the measured phase shifts is from the Fourier fits to the *Spitzer* light curves. We estimated the amplitude of noise by fitting a Gaussian to the residuals of the Fourier fit, and created a synthetic light curve by adding random noise drawn from the fitted Gaussian distribution to the Fourier fit. We then measured the phase shift in the synthetic light curve in the same fashion as the observed light curve. This procedure was repeated 1,000 times for each segment of *Spitzer* light curve analyzed and the standard deviation of the 1,000 measurements was taken as the uncertainties of the phase shift. We report the phase shifts and associated uncertainties for first- and second-order Fourier components in Table \[shift1\]. In the following discussion, we excluded measurements from two visits, *Spitzer* Visit 7 of SIMP0136 and Visit 1 of 2M1507, because for Visit 7 of SIMP0136 (bottom panel of Figure \[fig0136\]), light curve evolution makes the measurement ambiguous, and for Visit 1 of 2M1507 (top panel of Figure \[fig1507\]), the Fourier fit to the *HST* data is poor. RESULTS ======= Variability in all sources and all spectral bands ------------------------------------------------- We found that all six sources in our sample are variable in both IRAC channels in all *Spitzer* visits. Most light curves are not strictly sinusoidal and can evolve on timescales as short as one rotation period, as shown in [Figures \[fig2replc\]–\[fig2228\]]{}. The simultaneous *HST* observations of four short-period sources also exhibit flux variations in the integrated light curves of different narrow spectral bandpasses (Figures \[narrow1507\]–\[narrow2228\]). Rotation Periods ---------------- For most of our targets our Spitzer data represents the longest continuous, high-precision light curves. As such, these data can place powerful constraints on the rotational period of the objects. In order to examine periodicity and estimate the rotational period, we calculated the auto-correlation functions for each visit for the Ch1 observations (approximately three rotations of the targets per visit sampling a time interval corresponding to approximately four rotation periods, see Figure \[fig1schedule\]). We used the standard $\mathrm{IDL}$ script $a_correlate.pro$ for the auto-correlation analysis and identified the first peak (corresponding to non-zero shifts) in the autocorrelation function for each visit. For three of our sources (SIMP0136, 2M2139, and 2M2228), the autocorrelation function was well-defined in most visits and the peaks identified were consistent. [Examples are shown in Figure \[autocorr\].]{} For these objects we adopted, as rotation periods, the mean of the peak auto-correlation values and, as the uncertainties of the rotation periods, the standard deviation of the peak locations in the auto-correlation functions. For the other three targets, however, the light curve evolution was so significant that no peak in the auto-correlation function emerged consistently among the eight visits. For these three objects, therefore, the auto-correlation analysis did not yield reliable rotation period estimates. For these source we adopted previous rotation estimates by @metchev2015, primarily based on Fourier analyses, which are qualitatively consistent with our light curves. The measured and adopted rotation periods are listed in Table \[obs1\]. Phase Shift Between *Spitzer* Ch1 and Ch2 Observations ------------------------------------------------------ We utilized two different methods to measure phase shifts between light curves observed in the two IRAC channels. While both methods are somewhat limited by light curve evolution on short timescales, our measurements showed that there is no detectable phase shift between the Ch1 and Ch2 light curves. As listed in Tables \[shiftmaxmin\] and \[shiftcross\], the majority of measurements are within 1- or 2-$\sigma$ limit of 0$^\circ$, with a typical upper limit of $15^\circ$. Phase Shift Between *HST* and *Spitzer* Observations ---------------------------------------------------- Simultaneous *HST* and *Spitzer* observations of the four targets from the *Storms* program provide the most comprehensive brown dwarf monitoring dataset, with each object covered for at least six *HST* orbits. Previously, only @buenzli2012 have studied 2M2228 with simultaneous *HST* and *Spitzer* observations, which overlapped for two *HST* orbits. With our unique dataset, we found that SIMP0136 (Figure \[fig0136\]) shows a phase shift of $\sim 30^\circ$ between *HST* $J$-band and *Spitzer* light curves, while the other three objects (2M1507, 2M1821, and 2M2228) all show substantial phase shifts within $10-20^\circ$ of $180^\circ$ (Table \[shift1\]). Phase Shift Between *HST* Narrow-Band Light Curves -------------------------------------------------- @buenzli2012 discovered phase shifts among several integrated light curves of characteristic wavelength regions in the *HST* and *Spitzer* observations of 2M2228 obtained in July, 2011. The selected bandpasses, including the $J$- and $H$-band windows as well as water and methane absorption, probe different pressure levels in the atmosphere, and @buenzli2012 found that the phase shifts increase monotonically with decreasing pressure level, indicating vertical atmospheric structures. We applied the same analysis procedure as described in @buenzli2012 to the *Storms* observations of 2M2228, and found that the light curves of various narrow bandpasses still display phase shifts even after two years, corresponding to over 12,000 rotations. We fit sine waves to the narrow-band light curves and compared the phases of the sine waves. As shown in Figure \[narrow2228\], with respect to the $J$- and $H$-band light curves, the water and methane bands shows phase shifts close to 180$^\circ$ in both *HST* visits. We extended this analysis to all four *Storms* targets observed with *HST*. The T2 dwarf SIMP0136 (Figure \[narrow0136\]) shows little phase shift, typically $4^\circ\pm 2^\circ$. We discuss this finding in the next section. DISCUSSION ========== Atmospheric Models and Pressure-dependent Flux Contribution ----------------------------------------------------------- The greatest advantage of multi-band monitoring of brown dwarfs is that emergent fluxes at different wavelengths probe different depths in the atmospheres. To investigate the characteristic pressure levels that different wavelength regions probe, we employ state-of-the-art radiative transfer and atmospheric chemistry models and calculate relative flux contributions from each model pressure levels. We first performed least-squares fits to available *HST* grism spectra to find the best-fit atmospheric model for each target. The atmosphere models we use are the most up-to-date versions of those published in @saumon2008. Besides the four targets observed with the *Storms* program, we also used *HST*/WFC3 G141 spectra of 2M2139 from @apai2013. The observations were resampled according to the model spectral grid. The model spectra were normalized to match with the flux peak in the $J$-band between 1.25 and 1.28 $\mu$m. The observed and the best-fit model spectra are shown in Figure \[modelfits\], and the best-fit model parameters are given in Table \[modelparam\]. The observed spectrum beyond 1.55 $\mu$m for 2M1821 is contaminated by a background object and thus not well fit by the model. As 2M1324 has not been observed with the WFC3 G141 grism, we use the best-fit model for the other T2 dwarfs. To compute the contribution functions, we first converged a standard radiative-convective equilibrium atmosphere thermal structure model following the approach of @saumon2008. Since the [time]{} of that model description, there have been substantial updates to the opacity database employed, which [were utilized in the present analysis and]{} will be described more fully in an upcoming paper (Marley et al., in prep.). Once the model converged, a temperature perturbation was iteratively applied to quarter-scaleheight subregions of the atmosphere. Given this new, artificial temperature profile a new emergent spectra was computed. The perturbation was then removed, a new perturbation applied to the next overlying region, and the process repeated. By [computing the ratio of]{} each perturbed thermal emission spectrum to the baseline case, the sensitivity of each spectral region to temperature perturbations at depth could be computed. We repeated this procedure for 37 model pressure levels that cover pressures from $1.8 \times 10^{-4}$ bars to $\sim$ 23 bars, and essentially obtained the relative flux contributions from a range of pressures at the wavelengths covered by our *HST* and *Spitzer* observations (left panels of Figures \[contrifunc1700\]–\[contrifunc950\]). To find the characteristic pressure level where most of the flux emerges from for a spectral bandpass of interest, we integrated the relative flux contributions over the wavelength bandpass and calculated a cumulative flux contribution function starting from the top of the atmosphere. We identified the two pressure levels between which the cumulative flux reaches 80% of the total flux and determined the exact pressure level by linear interpolating between the two model pressure levels. The 80%-cumulative-flux pressure levels were calculated for 8 spectral bandpasses, including 5 *HST* narrow bands as well as 2MASS $J$-band and *Spitzer* Ch1 and Ch2. The results are listed in Table \[modelparam\] and also shown in the right panels of Figures \[contrifunc1700\]–\[contrifunc950\]. [We stress that the approach adopted here, although based on state-of-the-art atmosphere models, offers only a limited tool for identifying the specific pressure levels where modulations are introduced: In our approach of modifying the temperature but not the cloud opacity, the results strictly apply only to changes in atmospheric temperature [*alone*]{}. However, our model does make clear the atmospheric region which the spectra at a given wavelength are most sensitive to. Changes to the cloud opacity at or above the pressure levels we identify for each observations are likely to substantially alter the emergent flux whereas changes to the underlying cloud opacity (below the 80% contribution levels) are less likely to be as significant.]{} [A good example of how high-altitude clouds or haze layer can dramatically modulate the pressure levels probed in the infrared is provided by *Cassini* observations of hot spots in Jupiter’s equatorial regions. @choi2013 presents multi-band *Cassini* imaging monitoring of the hot spots and plumes that are seen to co-evolve near the jovian equator. The hot spots are interpreted as cloud clearing in the ammonia cloud deck, which may be rapidly obscured or revealed by high-altitude clouds. In this case, for example, the infrared observations probe deep in the atmosphere when cloud opacity is [*absent*]{}, but are limited to the top of the atmosphere when opacity is introduced by high-level clouds.]{} Heterogeneous Upper-atmosphere in All Objects --------------------------------------------- All six sources in our sample exhibit flux variations in both *Spitzer* channels during all 8 visits, regardless of spectral type and rotation period. We found that the derived characteristic model pressures above which 80% of the fluxes emerge from at Ch1 and Ch2 are less than 2 bars for the two L5 dwarfs and less than 1 bar for the T dwarfs, indicating heterogeneity in the upper atmospheres of all our targets. Interestingly, the relative flux contributions (Figures \[contrifunc1700\]–\[contrifunc950\]) show that in the model atmosphere of a T2 dwarf most of *Spitzer* Ch1 flux is from lower pressure levels than the Ch2 flux, while the case is reversed in the L5 and T6 models. For 2M2228 (T6), shown in top left panel of Figure \[contrifunc950\], flux between 1.1 and 1.7 $\mu$m emerges from a wide range of pressures levels. For example, the narrow $J$-band flux is mostly from $\sim$ 7.5 bars, while the flux of the $1.35-1.43~\mu$m water band primarily comes from $\sim$ 2 bar. In contrast, the L5 and T2 dwarfs emit most of the flux between 1.1 and 1.7 $\mu$m from a smaller range of pressures, which are around 6.6-4.3 bars for the L5 targets and 8.1-4.1 bars for the T2 targets. Observed Phase Shifts --------------------- Between *Spitzer* Ch1 and Ch2 light curves, we have found no detectable phase shifts from analyses using two different methods. According to the relative flux contributions derived from the models (Figures \[contrifunc1700\]–\[contrifunc950\]), most of the observed flux in both *Spitzer* channels comes from a relatively narrow range of pressure levels in the upper atmospheres, e.g., between 1 and 2 bars in Ch1 for the L5 model (Figure \[contrifunc1700\]), and between 0 and 0.3 bars in Ch2 for the T2 model (Figure \[contrifunc1400\]). For all four targets that have simultaneous *HST* and *Spitzer* observations, we have detected phase shifts between $J$-band and Ch1/Ch2 light curves (Table \[shiftcross\]). The rotation rates of our objects range from 1.37 hr (2M2228) to 4.2 hr (2M1821). With the small sample size, the phase shifts display no obvious correlation with rotation periods. For example, SIMP0136 (T2) and 2M1507 (L5) have similar rotation periods, but their phase shifts are very different. SIMP0136 (T2) exhibits $\sim 30^\circ$ phase shift, while the phase shifts of the mid-L and late-T dwarfs are all centered around 180$^\circ$. This might hint that L/T transition objects have peculiarities in their atmospheric properties with respect to regular L and T dwarfs, resulting in different phase shifts between the *HST* and *Spitzer* light curves. Correlation Between Pressure Levels and Phase Shifts ---------------------------------------------------- Simultaneous observations in multiple bandpasses probes different pressure levels in the atmosphere, providing vital information on the atmospheric properties in the vertical direction. With the *Storms* data set, we measured phase shifts between light curves of four narrow bands between 1.1 and 1.7 $\mu$m and two *Spitzer* broad bands centered at 3.6 and 4.5 $\mu$m. Here we explore the correlation between phase shifts of the six bandpasses and the characteristic model pressure levels from which most flux originates for those bandpasses. @buenzli2012 first detected pressure-dependent phase shifts in 2M2228, with lower pressure levels showing larger phase shifts. In Figure \[shftpressure2228\], we examine again the phase shifts as a function of model pressures for 2M2228 observed two years later. We find that the phase shifts measured among *HST* narrow bandpasses in this work are consistent with those reported by @buenzli2012. Our measured phase shifts between the $J$- and $H$-band light curves are $-5^\circ \pm 2^\circ$ and $-8^\circ \pm 2^\circ$ for two visits, respectively (Figure \[narrow2228\]), while @buenzli2012 found it to be $15^\circ \pm 2^\circ$. The sine wave fits to the $1.35-1.43~\mu$m (water) and $1.62-1.69~\mu$m (water and methane) bandpasses are not as good as $J$- and $H$-bands, and the phase shifts of those two bandpasses with respect to the narrow $J$-band are generally close to 180$^\circ$, consistent within errors with the results of @buenzli2012. [(Note that a 180$^\circ$ phase shift could be equivalent to an anti-correlation of fluxes at different wavelengths with no phase shift.)]{} The main difference between the results of @buenzli2012 and this work lies in the phase shift between the narrow $J$-band and the *Spitzer* channels, which @buenzli2012 measured to be $118^\circ \pm 7^\circ$, compared to our measurement of around 160$^\circ$. Our Fourier-based phase shift measurements between different wavelength bands are separated into two distinctive groups in terms of the pressure levels probed by the bandpasses. For each target (left panels of Figures \[shftpressure2228\]–\[shftpressure1821\]), the light curves probing deeper ($\gtrsim$ 4 bars) in the atmosphere are generally in phase, while the light curves probing the upper atmospheres ($\lesssim$ 4 bars) display similar phase offsets. For the L and the L/T dwarfs, the two groups of light curves appear to probe pressure levels separated approximately by the radiative-convective boundary calculated from the model atmospheres. For the T6 object, the radiative-convective boundary is deeper in the atmosphere than where our available wavelength bands can probe. Compared with the locations of the condensate clouds in the models (right panels of Figures \[shftpressure2228\]–\[shftpressure1821\]), the higher pressure region ($\gtrsim$ 4 bars) probed by our observations generally coincides with the dense part of the cloud layers. Variable cloud thickness could explain the observed flux variations in the light curves that probe higher pressure levels in the atmosphere. On the other hand, the pressure levels that the *Spitzer* channels probe have very low condensate density. Along with the phase differences, this might indicate that different sources of modulation are in play in the two regions. The transition between the two groups of light curves is very abrupt in pressure, indicating a break or discontinuity between the two pressure regions. A separate patchy cloud deck at the lower pressure levels could cause the flux variations probed by the *Spitzer* bands. SUMMARY ======= We monitored the light curves of two mid-L dwarfs, three L/T transition dwarfs, and one late-T dwarf over the course of 20 months. We cover at least 24 rotations in Spitzer observations for all six targets and sample at least six rotations with time-resolved *HST* spectroscopy for four objects. The key results of our study are: - All six targets are variable and exhibit light curve evolution over timescales of their rotation periods, ranging from 1.4 h to 13 h. For each object, we find variations in every wavelength band observed, demonstrating that the photospheres of all six objects are heterogeneous. - For three objects, we accurately determine rotation periods. For the other three, we adopt estimates of rotation periods from @metchev2015 and find those values consistent with much longer observations in this work. - We use state-of-the-art radiative transfer and atmospheric chemistry models to determine the flux contribution of each pressure layer to the spectral bands studied. Our observations probe model pressure levels between $\sim$8.1 to $\sim$0.2 bars, using light curves obtained in 7 different bandpasses for four objects. - We use two different methods to assess the phase shifts between the spectral bands studied for our objects. No phase shift is found in any of six objects between *Spitzer* Ch1 and Ch2 light curves. Both channels probes a narrow range of layers high in the atmospheres ($\lesssim$ 3 bars for the L dwarfs and $\lesssim$ 1 bar for the T dwarfs). - We detect phase shifts between *HST* $J$-band and *Spitzer* light curves for all four objects simultaneously observed by the two observatories. From the limited sample, the phase shift between the *HST* and *Spitzer* data does not show correlation with rotation period. - The *HST* $J$-band and Spitzer light curves of SIMP0136 (T2) shows a small phase shift ($\sim 30^{\circ}$), while for the other three targets such phase shifts are close to $180^{\circ}$. This might indicate that the L/T transition objects have peculiar atmospheric properties, compared to regular L and T dwarfs. More simultaneous multi-band observations are needed to check whether such different behavior in phase shifts are common for L-T transition dwarfs. - For 2M2228 (T6), phase shifts among narrow *HST* bands and *Spitzer* Ch2 persist between two sets of observations separated by two years, equivalent of thousands of rotations. The phase shifts among *HST* narrow bandpasses are generally consistent with that reported by @buenzli2012, while the phase shift between the narrow $J$-band and *Spitzer* Ch2 is different from that measured in 2012. No *HST* narrow-band phase shifts are found for 2M1507 (L5), 2M1821 (L5), and SIMP0136 (T2). - For the four sources with *HST* and *Spitzer* light curves, we identify a clear difference between higher pressures ($\gtrsim$ 4 bars) and lower pressures ($\lesssim$ 4 bars), visible in differences in the light curve shape and Fourier-based phase shifts. The pressure range that separates the two groups of light curves appears to be close to the estimate radiative/convective boundary for the mid-L and the L/T dwarfs, but for the T6 dwarf it appears to be significantly lower than the radiative/convective boundary. - We attribute the modulations introduced in the deeper atmosphere to cloud thickness variations occurring at or near the densest parts of the condensate clouds probes. In contrast, the *Spitzer* bands probe pressures where the @saumon2008 cloud models predict only low condensate volume mixing ratios, which are unlikely to account for the variations observed. The two groups of light curves and the abrupt transition between the phases of the two pressure regions may indicate that another heterogeneous cloud layer at much lower pressures could be responsible. Our results show evidence for a possible two-component vertical cloud structure, but not for more components, providing new insights and constraints for vertical cloud models. This work is part of the Spitzer Cycle-9 Exploration Program, Extrasolar Storms. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. Support for *HST* GO programs 13176 was provided of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. We acknowledge the outstanding help of Patricia Royle (STScI) and the Spitzer Science Center staff, especially Nancy Silbermann, for coordinating the HST and Spitzer observations. This research has benefitted from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at <http://pono.ucsd.edu/~adam/browndwarfs/spexprism> and the M, L, T, and Y dwarf compendium housed at DwarfArchives.org. -------------- ------- ------------------------------ --------------------------- --------------------------- --------------------------- Target Visit $\Delta T_{\rm{max}}$ \[hr\] $\Delta\phi$ ($^{\circ}$) $\Delta T_{\rm{min}}[hr]$ $\Delta\phi$ ($^{\circ}$) **SIMP0136** 1 -0.06 $\pm$ 0.07 -9.1 $\pm$ 11.0 0.04 $\pm$ 0.09 5.3 $\pm$ 12.7 2 -0.06 $\pm$ 0.11 -0.6 $\pm$ 15.8 0.07 $\pm$ 0.08 11.1 $\pm$ 13.0 3 0.09 $\pm$ 0.04 13.5 $\pm$ 6.4 -0.08 $\pm$ 0.07 -11.6 $\pm$ 10.7 4 0.03 $\pm$ 0.03 4.3 $\pm$ 4.9 -0.15 $\pm$ 0.18 -22.1 $\pm$ 26.4 6 0.02 $\pm$ 0.04 2.7 $\pm$ 6.3 -0.06 $\pm$ 0.10 -8.2 $\pm$ 14.5 7 -0.09 $\pm$ 0.13 -13.6 $\pm$ 19.8 -0.21 $\pm$ 0.11 -31.2 $\pm$ 16.9 **2M1324** 1 -0.59 $\pm$ 0.16 -16.1 $\pm$ 4.1 0.14 $\pm$ 0.12 3.7 $\pm$ 3.3 4 -0.21 $\pm$ 0.17 -5.8 $\pm$ 4.6 0.25 $\pm$ 0.17 6.8 $\pm$ 4.5 5 -0.01 $\pm$ 0.04 -0.2 $\pm$ 1.2 -0.14 $\pm$ 0.17 -3.8 $\pm$ 4.7 **2M1507** 3 -0.09 $\pm$ 0.11 -13.4 $\pm$ 15.8 0.08 $\pm$ 0.11 11.0 $\pm$ 16.2 **2M1821** 2 -0.01 $\pm$ 0.20 -1.1 $\pm$ 17.7 0.20 $\pm$ 0.17 18.4 $\pm$ 16.7 4 0.013 $\pm$ 0.23 1.2 $\pm$ 21.0 0.05 $\pm$ 0.18 4.6 $\pm$ 16.4 6 0.18 $\pm$ 0.12 15.8 $\pm$ 10.8 0.43 $\pm$ 0.20 38.3 $\pm$ 18.2 8 0.35 $\pm$ 0.15 31.3 $\pm$ 13.9 0.04 $\pm$ 0.14 3.7 $\pm$ 13.0 **2M2139** 2 0.21 $\pm$ 0.08 9.8 $\pm$ 4.0 -0.02 $\pm$ 0.08 -1.0 $\pm$ 3.9 3 -0.17 $\pm$ 0.05 7.9 $\pm$ 2.3 0.19 $\pm$ 0.09 9.0 $\pm$ 4.2 4 -0.07 $\pm$ 0.05 -3.3 $\pm$ 2.2 0.35 $\pm$ 0.13 16.3 $\pm$ 6.3 5 0.02 $\pm$ 0.04 1.1 $\pm$ 2.1 0.35 $\pm$ 0.13 16.5 $\pm$ 6.3 7 0.43 $\pm$ 0.15 20.4 $\pm$ 7.0 -0.08 $\pm$ 0.04 -3.6 $\pm$ 1.7 -------------- ------- ------------------------------ --------------------------- --------------------------- --------------------------- -------------- ---------- ------- ----------------------- ---------------------------------------- ---------------------------------------- -- Target $P$ (hr) Visit $P_{\rm{Visit}}$ (hr) $\Delta\phi_{\rm{R2-R3}}$ ($^{\circ}$) $\Delta\phi_{\rm{R3-R4}}$ ($^{\circ}$) **SIMP0136** 2.414 1 2.628 -0.4 $\pm$ 4.8 0.1 $\pm$ 4.6 2 2.623 7.3 $\pm$ 7.1 -7.5 $\pm$ 6.6 3 2.617 -5.7 $\pm$ 4.0 -0.3 $\pm$ 4.0 6 2.611 10.7 $\pm$ 6.1 -14.8 $\pm$ 7.2 7 2.531 12.8 $\pm$ 6.8 -16.9 $\pm$ 7.0 **2M1324** 13.0 1 12.339 13.8 $\pm$ 27.5 -11.8 $\pm$ 27.5 4 12.346 -2.6 $\pm$ 1.6 3.8 $\pm$ 1.8 **2M1821** 4.2 1 4.177 -1.8 $\pm$ 9.5 -5.9 $\pm$ 9.1 4 4.262 -8.8 $\pm$ 5.6 1.5 $\pm$ 6.7 5 4.289 4.1 $\pm$ 5.8 -3.9 $\pm$ 5.4 8 4.439 -2.9 $\pm$ 6.0 -8.5 $\pm$ 6.5 **2M2139** 7.614 5 7.790 -1.0 $\pm$ 9.0 0.2 $\pm$ 2.2 6 7.775 -16.3 $\pm$ 21.5 18.1 $\pm$ 22.3 7 7.654 -2.8 $\pm$ 1.6 3.2 $\pm$ 1.7 **2M2228** 1.369 3 1.516 8.6 $\pm$ 10.6 -5.6 $\pm$ 9.2 -------------- ---------- ------- ----------------------- ---------------------------------------- ---------------------------------------- --  \[fig1schedule\]                     

--- abstract: 'We examine the transition of a particle across the singularity of the compactified Milne (CM) space. Quantization of the phase space of a particle and testing the quantum stability of its dynamics are consistent to one another. One type of transition of a quantum particle is described by a quantum state that is continuous at the singularity. It indicates the existence of a deterministic link between the propagation of a particle before and after crossing the singularity. Regularization of the CM space leads to the dynamics similar to the dynamics in the de Sitter space. The CM space is a promising model to describe the cosmological singularity deserving further investigation by making use of strings and membranes.' author: - | Przemys[ł]{}aw Ma[ł]{}kiewicz$^\dag$ and W[ł]{}odzimierz Piechocki$^\ddag$\ Department of Theoretical Physics\ Sołtan Institute for Nuclear Studies,\ Hoża 69, 00-681 Warszawa, Poland;\ $^\dag$pmalk@fuw.edu.pl, $^\ddag$piech@fuw.edu.pl title: Probing the cosmological singularity with a particle --- Introduction ============ Presently available cosmological data indicate that known forms of energy and matter comprise only $4\%$ of the makeup of the Universe. The remaining $96\%$ is unknown, called ‘dark’, but its existence is needed to explain the evolution of the Universe [@Spergel:2003cb; @Bahcall:1999xn]. The dark matter, DM, contributes $22\%$ of the mean density. It is introduced to explain the observed dynamics of galaxies and clusters of galaxies. The dark energy, DE, comprises $74\%$ of the density and is responsible for the observed accelerating expansion. These data mean that we know almost nothing about the dominant components of the Universe! Understanding the nature and the abundance of the DE and DM within the standard model of cosmology has difficulties [@NGT; @GDS]. These difficulties have led many physicists to seek anthropic explanations which, unfortunately, have little predictive power. An alternative model has been proposed by Steinhardt and Turok (ST) [@Steinhardt:2001vw; @Steinhardt:2001st; @Steinhardt:2004gk]. It is based on the idea of a cyclic evolution, CE, of the Universe. The ST model has been inspired by string/M theories [@Khoury:2001bz]. In its simplest version it assumes that the spacetime can be modelled by the higher dimensional compactified Milne, CM, space. The attraction of the ST model is that it potentially provides a complete scenario of the evolution of the universe, one in which the DE and DM play a key role in both the past and the future. The ST model *requires* DE for its consistency, whereas in the standard model, DE is introduced in a totally *ad hoc* manner. Demerits of the ST model are extensively discussed in [@Linde:2002ws]. Response to the criticisms of [@Linde:2002ws] can be found in [@NGT]. The mathematical structure and self-consistency of the ST model has yet not been fully tested and understood. Such task presents a serious mathematical challenge. It is the subject of our research programme. The CE model has in each of its cycles a quantum phase including the cosmological singularity, CS. The CS plays key role because it joins each two consecutive classical phases. Understanding the nature of the CS has primary importance for the CE model. Each CS consists of contraction and expansion phases. A physically correct model of the CS, within the framework of string/M theory, should be able to describe propagation of a p-brane, i.e. an elementary object like a particle, string and membrane, from the pre-singularity to post-singularity epoch. This is the most elementary, and fundamental, criterion that should be satisfied. It presents a new criterion for testing the CE model. Hitherto, most research has focussed on the evolution of scalar perturbations through the CS. Successful quantization of the dynamics of p-brane will mean that the CM space is a promising candidate to model the evolution of the Universe at the cosmological singularity. Thus, it could be further used in advanced numerical calculations to explain the data of observational cosmology. Failure in quantization may mean that the CS should be modelled by a spacetime more sophisticated than the CM space. Preliminary insight into the problem has already been achieved by studying classical and quantum dynamics of a test particle in the two-dimensional CM space [@Malkiewicz:2005ii]. The present paper is a continuation of [@Malkiewicz:2005ii] and it addresses the two issues: the Cauchy problem at the CS and the stability problem in the propagation of a particle across the CS. Both issues concern the nature of the CS. In Sec. II we define and make comparison of the two models of the universe: the CM space and the regularized CM space. The classical dynamics of a particle in both spaces is presented in Sec. III. The quantization of the phase space of a particle is carried out in Sec. IV. In Sec. V we examine the stability problem of particle’s dynamics both at classical and quantum levels. We summarize our results, conclude and suggest next steps in Sec. VI. Spacetimes ========== The CM space ------------ For completeness, we recall the definition of the CM space used in [@Malkiewicz:2005ii]. It can be specified by the following isometric embedding of the 2d CM space into the 3d Minkowski space $$\label{emb} y^0(t,\theta) = t\sqrt{1+r^2},~~~~y^1(t,\theta) = rt\sin(\theta/r),~~~~y^2(t,\theta) = rt\cos(\theta/r),$$ where $ (t,\theta)\in {\mathbb R}^1 \times {\mathbb S}^1 $ and $ 0<r \in{\mathbb R}^1 $ is a constant labelling compactifications . One has $$\label{stoz} \frac{r^2}{1+r^2}(y^0)^2 - (y^1)^2- (y^2)^2 =0.$$ Eq. (\[stoz\]) presents two cones with a common vertex at $\:(y^0,y^1,y^2)= (0,0,0)$. The induced metric on (\[stoz\]) reads $$\label{line1} ds^2 = - dt^2 +t^2 d\theta^2 .$$ Generalization of the 2d CM space to the Nd spacetime has the form $$\label{line2} ds^2 = -dt^2 +dx^k dx_k + t^2 d\theta^2,$$ where $t,x^k \in \mathbb{R}^1,~\theta\in \mathbb{S}^1~(k= 1,\ldots, N-2)$. One term in the metric (\[line2\]) disappears/appears at $t=0$, thus the CM space may be used to model the big-crunch/big-bang type singularity [@Khoury:2001bz]. In what follows we restrict our considerations to the 2d CM space. Later, we make comments concerning generalizations. It is clear that the CM space is locally isometric to the Minkowski space at each point except the vertex $t=0$. The CM space is not a manifold, but an orbifold due to this vertex. The Riemann tensor components vanish for $t\neq 0$ and cannot be defined at $t=0$, since one dimension disappears/appears there. There is a space-like singularity at $t=0$ of removable type because any time-like geodesic with $t<0$ can be extended to some time-like geodesic with $t>0$. However, such an extension cannot be unique due to the Cauchy problem for the geodesic equation at the vertex (compact dimension shrinks away and reappears at $t=0$). The RCM space ------------- Since trajectory of a *test* particle coincides (by definition) with time-like geodesic, there is no obstacle for the test particle to reach and leave the CS. However, the Cauchy problem for a geodesic equation at the CS is not well defined. As the result, a test particle ‘does not know where to go’ at the singularity. Thus, the singularity acts as ‘generator’ of uncertainty in the propagation of a test particle from the pre-singularity to post-singularity era. In the present paper we propose to solve this problem by replacement of a test particle by a *physical* one. The test and physical particles differ in a number of ways. For instance, physical particle’s own gravitational field effects its motion [@Poisson:2004gg] and may modify the singularity of the CM space. We assume that these effects may be modelled by replacing the CM space by a regularized compactified Milne, RCM, space in such a way that the big-crunch/big-bang type singularity of the CM space is replaced by the big-bounce type singularity. In the RCM space the Cauchy problem does not occur because compact space dimension does not contract to a point, but to some ‘small’ value. As the result the propagation of a particle is uniquely defined in the entire spacetime. Particle’s propagation in the RCM space is similar to the corresponding one in the de Sitter space [@WP; @Piechocki:2003hh]. We define the RCM space by the following embedding into the 3d Minkowski space $$\label{rem} y^0(t,\theta) = t\sqrt{1+r^2},~~~y^1(t,\theta) = r\sqrt{t^2+\epsilon^2}\sin(\theta/r),~~~y^2(t,\theta) = r\sqrt{t^2+ \epsilon^2}\cos(\theta/r),$$ and we have the relation $$\label{conr} \frac{r^2}{1+r^2}(y^0)^2 - (y^1)^2- (y^2)^2 =-\epsilon^2r^2 .$$ The induced metric on the RCM space reads $$\label{line3} ds^2_\epsilon = - (1+\frac{r^2\epsilon^2}{t^2+\epsilon^2})\;dt^2 +(t^2+\epsilon^2)\; d\theta^2 ,$$ where $\epsilon\in {\mathbb R}$ is a small number. It is clear that now the space dimension $\theta$ does not shrink to zero at $t=0$. The scalar curvature has the form $$\label{sca} \mathcal{R}_\epsilon= \frac{2\epsilon^2(1+r^2)}{(\epsilon^2(1+r^2)+t^2)^2}$$ and the Einstein tensor corresponding to the metric (\[line3\]) is zero, thus (\[line3\]) defines some vacuum solution to the 2d Einstein equation. It is evident that at $t\neq 0$ we have $$\label{comp} \lim_{\epsilon\to 0} ds^2_\epsilon = - dt^2 +t^2d\theta^2~~~~~\mathrm{and}~~~~~\lim_{\epsilon\to 0} \mathcal{R}_\epsilon = 0.$$ It is obvious that (\[conr\]) turns into (\[stoz\]) as $\epsilon\rightarrow 0$. {width="55.00000%"} Figure 1 presents the RCM and CM spaces embedded into the 3d Minkowski space. We can see that the big-crunch/big-bang singularity of the CM space is represented in the RCM by the big-bounce type singularity. Classical dynamics ================== An action integral, $\mathcal{A}$, describing a relativistic test particle of mass $m$ in a gravitational field $g_{kl},~(k,l=0,1)$ may be defined by $$\label{action} \mathcal{A}=\int d\tau\: L(\tau),~~~~ L(\tau):=\frac{m}{2}\:(\frac{\dot{x}^k \dot{x}^l}{e} g_{kl}-e),$$ where $\dot{x}^k :=dx^k/d\tau,~\tau$ is an evolution parameter, $e(\tau)$ denotes the ‘einbein’ on the world-line, $x^0$ and $x^1$ are time and space coordinates, respectively. In case of the CM and RCM spaces the Lagrangian $L_\epsilon $ reads $$\label{lag} L_\epsilon(\tau)= \frac{m}{2e}\:\big((t^2+\epsilon^2) \dot{\theta}^2 - (1+\frac{r^2\epsilon^2}{t^2+\epsilon^2})\dot{t}^2 -e^2\big),$$ where $\epsilon =0$ corresponds to the CM space. The action (\[action\]) is invariant under reparametrization with respect to $\tau$. This gauge symmetry leads to the constraint $$\label{con} \Phi_\epsilon := p_k p_l \;g^{kl} + m^2 = \frac{p^2_\theta}{(t^2 + \epsilon^2)}- \frac{p^2_t}{1+\frac{r^2\epsilon^2}{t^2+\epsilon^2}} + m^2 =0,$$ where $p_t := \partial L_\epsilon/\partial\dot{t}\:$ and $p_\theta :=\partial L_\epsilon/\partial\dot{\theta}\:$ are canonical momenta, and where $g^{kl}$ denotes an inverse of the metric $g_{kl}$ defined by the line element (\[line3\]) (case $\epsilon =0$ corresponds to the CM space). Variational principle applied to (\[action\]) gives equations of motion of a particle $$\label{eq} \frac{d}{d\tau} p_\theta = 0, ~~~\frac{d}{d\tau}p_t - \frac{\partial L}{\partial t} = 0,~~~\frac{\partial L}{\partial e} = 0.$$ Since during evolution of the system $p_\theta$ is conserved, due to (\[eq\]), we can analyze the behaviour of $p_t$ by making use of the constraint (\[con\]). In case of the CM space $(\epsilon =0)$, for $p_\theta\neq 0$ there must be $p_t \rightarrow\infty$ as $t\rightarrow 0$. This problem cannot be avoided by different choice of coordinates[^1]. It is connected with the vanishing/appearance of the space dimension $\theta$ at $t=0$. Another interpretation of this problem is that different geodesics cross each other with the relative speed reaching the speed of light as they approach the singularity at $t=0$. The dynamics of a physical particle in the RCM space $(\epsilon \neq 0)$ does not suffer from such a problem, since for $p_\theta\neq 0$ the momentum component $p_t$ does not need to ‘blow up’ to satisfy (\[con\]). Geodesics in CM and RCM spaces ------------------------------ It was found in [@Malkiewicz:2005ii] an analytic general solution to (\[eq\]), for $\epsilon =0$, in the form $$\label{sol1} \theta(t)= \theta_0 - \sinh^{-1} \Big(\frac{p_\theta}{mt}\Big),$$ where $(p_\theta, \theta_0) \in {\mathbb R}^1\times{\mathbb S}^1$. It is clear that geodesics (\[sol1\]) ‘blow up’ at $t=0$, which is visualized in Fig. 2. For $\epsilon\neq 0$, Eqs. (\[eq\]) read $$\label{eq2} \frac{m(t^2+\epsilon^2)\dot{\theta}}{e}=p_\theta=const,~~~~ e^2=(1+\frac{r^2\epsilon^2}{t^2+\epsilon^2})\dot{t}^2-(t^2+ \epsilon^2)\dot{\theta}^2$$ and $$\label{eq3} \big(1+\frac{r^2\epsilon^2}{t^2+\epsilon^2}\big)\ddot{t}- \big(1+\frac{r^2\epsilon^2}{t^2+\epsilon^2}\big) \big(\frac{\dot{e}}{e}\big)\dot{t}- \frac{r^2\epsilon^2t} {(t^2+\epsilon^2)^2}\dot{t}+\dot{\theta}^2t=0.$$ From (\[eq2\]) and (\[eq3\]) we get $$\label{sol2} \Big(\frac{d\theta}{dt}\Big)^2=\frac{p_\theta^2(1+\frac{r^2\epsilon^2}{{t}^2+\epsilon^2})} {m^2(t^2+\epsilon^2)^2+p_\theta^2(t^2+\epsilon^2)},$$ where $p_\theta \in {\mathbb R}^1$. General solution to (\[sol2\]) reads $$\label{sol3} \theta (t) = \theta_0 ~+ ~p_\theta \int_{-\infty}^{t} \,d{\tau}\sqrt{\frac{1+\frac{r^2\epsilon^2}{{\tau}^2+\epsilon^2}}{m^2({\tau}^2+\epsilon^2)^2+ p_\theta^2({\tau}^2+\epsilon^2)}}$$ where $\theta_0\in {\mathbb S}^1$. The integral in (\[sol3\]) cannot be calculated analytically. Numerical solution of (\[sol2\]) is presented in Fig. 2. {width="56.00000%" height="42.00000%"} {width="\textwidth"} {width="56.00000%" height="42.00000%"} Figure 2 shows that a geodesic in the RCM space is bounded and continuous in the neighborhood of the singularity. In contrary, a geodesic in the CM space (drawing by making use of (\[sol1\])) blows up as $t\rightarrow\pm 0$. Phase space and basic observables --------------------------------- We define a phase space to be the set of independent parameters (variables) defining all particle geodesics. Thus the pase space, $\Gamma$, for (\[sol3\]) reads $$\label{phase1} \Gamma := \{(\sigma,p_\sigma)\;|\; \sigma\in {\mathbb R}^1\; mod\;2\pi r,\; p_\sigma \in {\mathbb R}^1\}= {\mathbb S}^1 \times {\mathbb R}^1.$$ The Cauchy problem at the singularity results from the vanishing/appearance of the space dimension $\theta$ at $t=0$. It is fairly probable that [*any*]{} simple regularization of the singularity of the CM space that prevents such collapse will lead to the cylindrical phase space (\[phase1\]). In [@Malkiewicz:2005ii] we have analyzed four types of propagations of a particle in the CM space. Now we can see that the regularization prefers the propagation in the CM space of the de Sitter type (see, Sec. III D of [@Malkiewicz:2005ii]), because only in this case the phase space topology has the form (\[phase1\]). Now, let us identify the *basic* canonical functions on the phase space, i.e. observables that can be used to define any *composite* observable of the underlying classical system. In case a phase space includes a variable with non-trivial topology, i.e. different from ${\mathbb R}^1$, it is a serious problem. However, it has been solved in two (equivalent) ways not long ago. In what follows we use the method used in the group theoretical quantization (see, [@Kastrup:2005xb] and references therein). In the next section we explain relation with another method. A natural choice [@Kastrup:2005xb] of the basic functions on (\[phase1\]) is $$\label{bf} S:=\sin(\sigma/r),~~C:=\cos(\sigma/r),~~P:=r p_\sigma .$$ The basic observables $S$ and $C$ are smooth single-valued functions on ${\mathbb S}^1$ (contrary to $\sigma$). The observables (\[bf\]) satisfy the Euclidean algebra $e(2)$ on $\Gamma$ $$\label{n1} \{S,C\}= 0,~~\{P,S\}=C,~~\{P,C\}=-S,$$ where $$\label{pb} \{\cdot,\cdot\} := \frac{\partial~\cdot}{\partial p_\sigma}\frac{\partial~\cdot}{\partial\sigma} - \frac{\partial~\cdot}{\partial \sigma}\frac{\partial~\cdot}{\partial p_\sigma}.$$ It is shown in [@Kastrup:2005xb] that the Euclidean group $E(2)$ can be used as the canonical group [@CJ] of the phase space $\Gamma$. Quantization of phase space =========================== By quantization we mean finding an irreducible unitary representation of the symmetry group of the phase space of the underlying classical system. The group $E(2)$ has the following irreducible unitary representation [@Kastrup:2005xb] $$\label{ua} [U(\alpha)\psi](\beta):= \psi[(\beta - \alpha)\;mod\;2\pi],~~~~\mathrm{for\; rotations}~~~~z\rightarrow e^{i\alpha}z,$$ $$\label{ut} [U(t)\psi](\beta):= [\exp{-i(a\cos\beta + b\sin\beta)}]\psi(\beta),~~~~ \mathrm{for\;translations}~~~~ z\rightarrow z+t,$$ where $z=|z|\;e^{i\beta},~~t=a+bi$, and where $\psi\in L^2({\mathbb S}^1)$. Making use of the Stone theorem, we can find an (essentially) self-adjoint representation of the algebra (\[n1\]). One has $$\label{algebra} [\hat{C},\hat{S}]=0,~~[\hat{P},\hat{S}]=-i\hat{C}, ~~[\hat{P},\hat{C}]=i\hat{S},$$ where $$\label{mom} \hat{P}\varphi(\beta):= -i\frac{\partial}{\partial\beta}\varphi(\beta),~~~~ \hat{S}\varphi(\beta):=\sin\beta\;\varphi(\beta),~~~~ \hat{C}\varphi(\beta):=\cos\beta\;\varphi(\beta).$$ The domain, $\Omega_\lambda$, of operators $\hat{P}, \hat{S}, \hat {C}$ reads $$\label{domain} \Omega_\lambda := \{\varphi\in L^2({\mathbb S}^1)~|~\varphi\in C^{\infty}[0,2\pi],\;\varphi^{(n)} (2\pi) = e^{i\lambda}\varphi ^{(n)}(0),~~n=0,1,2,\dots\},$$ where $ 0\leq\lambda < 2\pi $ labels various representations of $e(2)$ algebra. The space $\Omega_\lambda$ is dense in $L^2({\mathbb S}^1)$ so the unbounded operator $\hat{P}$ is well defined. As the operators $\hat{S}$ and $\hat{C}$ are bounded on the entire $L^2({\mathbb S}^1)$, the space $\Omega_\lambda$ is a common invariant domain for all operators and their products. In [@Malkiewicz:2005ii] we have found that the representation of the algebra specific to the case considered there in Sec. III D, has the form $$\label{quant3} [\hat{\alpha},\hat{U}]= \hat{U},$$ with $$\label{quant} \hat{\alpha}\varphi(\beta):= -i\frac{d}{d\beta}\varphi(\beta),~~~~~~ \hat{U}\varphi(\beta) :=e^{i\beta}\varphi(\beta),$$ where $0\leq\beta<2\pi$ and $\varphi\in \Omega_\lambda$. However, both representations, (\[mom\]) and (\[quant\]), are in fact the same owing to $$\label{same} e^{i\beta} = \cos\beta + i \sin\beta,~~~~~[\cos\beta, \sin\beta]=0 .$$ The space $\Omega_\lambda$, where $0\leq\lambda < 2\pi$, may be spanned by the set of orthonormal eigenfunctions of the operator $\hat{\alpha}$ $$\label{quant4} f_{k,\lambda}(\beta):= (2\pi)^{-1/2}\exp{i\beta (k+\lambda/2\pi}),~~~~~k=0,\pm 1,\pm 2,\ldots$$ However, the functions (\[quant4\]) are [*continuous*]{} on ${\mathbb S}^1$ only in the case when $\lambda =0$. Thus, the requirement of the continuity removes the ambiguity of quantization. Stability of System =================== To examine the stability problem of our system, we use the Hamiltonian formulation of the dynamics of a particle. Classical level --------------- By stability of the dynamics of a *classical* particle we mean such an evolution of a particle that can be described by the canonical variables which are bounded and continuous functions. Direct application of the results of [@Turok:2004gb] gives the following expression for the extended Hamiltonian [@PAM; @MHT], $H_\epsilon$, of a particle $$\label{ham1} H_\epsilon = \frac{1}{2}\; C_\epsilon \;\Phi_\epsilon ,$$ where $C_\epsilon$ is an arbitrary function of an evolution parameter $\tau$, and where $\Phi_\epsilon$ is the first-class constraint defined by (\[con\]). The equations of motion for canonical variables $(t,\theta;p_t,p_\theta)$ read $$\label{e1} \frac{dt}{d\tau}=\{t,H_\epsilon\}= -C_\epsilon(\tau)\frac{p_t (t^2 +\epsilon^2)} {t^2 +\epsilon^2+r^2 \epsilon^2},$$ $$\label{e2} \frac{d\theta}{d\tau}=\{\theta,H_\epsilon\}=C_\epsilon (\tau)\frac{p_\theta}{t^2 +\epsilon^2},$$ $$\label{e3} \frac{dp_t}{d\tau}=\{p_t,H_\epsilon\}= C_\epsilon(\tau)\Big(\frac{t p_\theta^2}{(t^2 +\epsilon^2)^2} +\frac{t p^2_t\epsilon^2 r^2}{(t^2+\epsilon^2+r^2\epsilon^2)^2} \Big),$$ $$\label{e4} \frac{dp_\theta}{d\tau}=\{p_\theta,H_\epsilon\}= 0,$$ where $$\{\cdot,\cdot\}=\frac{\partial\cdot}{\partial t}\frac{\partial\cdot} {\partial p_t} - \frac{\partial\cdot}{\partial p_t}\frac{\partial\cdot} {\partial t} + \frac{\partial\cdot}{\partial \theta}\frac{\partial\cdot} {\partial p_\theta} - \frac{\partial\cdot}{\partial p_\theta}\frac{\partial\cdot} {\partial \theta} \nonumber$$ To solve (\[e1\])-(\[e4\]), we use the gauge $\tau = t$. In this gauge (\[e1\]) leads to $$\label{e5} C_\epsilon(t)= -\frac{t^2+\epsilon^2+r^2\epsilon^2}{p_t(t^2+\epsilon^2)}.$$ Insertion of (\[e5\]) into (\[e3\]) and taking into account (\[con\]) gives $$\label{e6} \frac{d}{dt}p_t^2 = - \frac{2tp_t^2r^2\epsilon^2}{(t^2+\epsilon^2)(t^2+\epsilon^2+r^2\epsilon^2)} - \frac{2tp_{\theta}^2(t^2+\epsilon^2+r^2\epsilon^2)} {(t^2+\epsilon^2)^3}.$$ Solution to (\[e6\]) reads $$\label{e7} p_t^2 =c_1\Big( \frac{c_1p_{\theta}^2}{t^2+\epsilon^2}+c_2\Big)\; \frac{t^2+\epsilon^2+r^2\epsilon^2}{t^2+\epsilon^2},$$ where $p_\theta$ does not depend on time due to (\[e4\]). This result is consistent with the constraint (\[con\]) if we put $c_1=1$ and $c_2=m^2$. Insertion of (\[e7\]) into (\[e5\]) yields an explicit expression for $C_\epsilon(t)$. Next, insertion of so obtained $C_\epsilon(t)$ into (\[e2\]) gives (\[sol2\]) with the solution (\[sol3\]). Thus, we have found complete solutions[^2] to the equations (\[e1\])-(\[e4\]). It results from the functional form of solutions that for $\epsilon\neq 0$ the propagation of canonical variables is regular, i.e. has no singularities for any value of time. One may also verify that the constraint equation (\[con\]), with $p_t$ determined by (\[e7\]), is satisfied for each value of time either. Since $C_\epsilon(t)$, determined by (\[e5\]) is bounded, the Hamiltonian $H_\epsilon$ defined by (\[ham1\]) is weakly zero due to the constraint (\[con\]). Now, let us analyze the case $\epsilon =0$, which corresponds to the evolution of a particle in the CM space. The solutions of (\[e1\])-(\[e4\]), in the gauge $\tau =t$, are the following $$\label{ee1} C_0(t) = -1/p_t ,$$ $$\label{ee2} p_t^2 = p_\theta^2 /t^2 +m^2$$ $$\label{ee3} \theta(t) = - sinh^{-1} \Big(p_\theta /mt\Big)+ const ,$$ (where $p_\theta = const$), in agreement with (\[sol1\]). The constraint equation (\[con\]) reads $$\label{ee4} \Phi_0 = p_\theta^2 /t^2 - p_t^2 + m^2 = 0.$$ It results from (\[ee2\]) and (\[ee3\]) that only for $p_\theta =0$ the propagation of a particle in the CM space is regular for any value of time. The Hamiltonian (\[ham1\]) is also regular and is weakly equal to zero. Quite different situation occurs in the case $p_\theta\neq 0$. The equations (\[ee2\])-(\[ee4\]) and the Hamiltonian are singular at $t=0$. Thus the dynamics of a particle is unstable[^3]. Quantum level ------------- By stability of dynamics of a *quantum* particle we mean the boundedness from below of its quantum Hamiltonian. To construct the quantum Hamiltonian of a particle we use the following mapping (see, e.g. [@Ryan:2004vv]) $$\label{mapp} p_k p_l g^{kl} \longrightarrow \Box := (-g)^{-1/2}\partial_k [(-g)^{1/2} g^{kl} \partial_l ],$$ where $g:=det [g_{kl}]$ and $\partial_k := \partial/\partial x^k$. The Laplace-Beltrami operator, $\Box$, is invariant under the change of spacetime coordinates and it leads to Hamiltonians that give results consistent with experiments [@Ryan:2004vv], and which has been used in theoretical cosmology (see, [@Turok:2004gb] and references therein). In the case of the CM space the Hamiltonian, for $t<0$ or $t>0$, reads $$\label{nh} \hat{H} = \Box + m^2 = \frac{\partial}{\partial t^2} + \frac{1}{t}\frac{\partial}{\partial t} - \frac{1}{t^2}\frac{\partial^2} {\partial \theta^2} + m^2 .$$ The operator $\hat{H}$ was obtained by making use of (\[mapp\]), (\[line1\]) and (\[ham1\]) in the gauge $C_\epsilon =2$ (for $\epsilon =0$). In this gauge[^4] the Hamiltonian equals the first class constraint (\[ee4\]). Thus the Dirac quantization scheme [@PAM; @MHT] leads to the equation $$\label{hcn} \hat{H} \psi(\theta,t)= 0.$$ The space of solutions to (\[hcn\]) defines the domain of boundedness of $\hat{H}$ from below (and from above). Let us find the non-zero solutions of (\[hcn\]). Separating the variables $$\label{sep} \psi(\theta,t):= A(\theta)\;B(t)$$ leads to the equations $$\label{eqth} d^2 A/d\theta^2 + \rho^2 A = 0,~~~~\rho\in {\mathbb R}$$ and $$\label{eqt} \frac{d^2 B}{d t^2}+ \frac{1}{t} \:\frac{dB}{dt} + \frac{m^2 t^2 +\rho ^2}{t^2}\;B = 0,~~~~t\neq 0,$$ where $\rho$ is a constant of separation. Two independent continuous solutions on ${\mathbb S}^1$ read $$\label{solth} A_1(\rho,\theta)= a_1 \cos(\rho\theta),~~~~A_2(\rho,\theta)= a_2 \sin(\rho\theta),~~~~~~~a_1, a_2 \in {\mathbb R}.$$ Two independent solutions on $ {\mathbb R}$ (for $t<0$ or $t>0$) have the form [@Arfken:2005; @SWM] $$\label{solt} B_1(\rho,t)= b_1 \Re J(i\rho,mt),~~~~B_2(\rho,t)= b_2 \Re Y(i\rho,mt),~~~~~~~b_1, b_2 \in {\mathbb C},$$ where $\Re J$ and $\Re Y$ are the real parts of Bessel’s and Neumann’s functions, respectively. Since $\rho\in{\mathbb R}$, the number of independent solutions is: $2 \times 2 \times \infty$ ( for $t<0$ and $t>0$). At this stage we define the scalar product on the space of solutions (\[solth\]) and (\[solt\]) as follows $$\label{scalar} <\psi_1|\psi_2> := \int_{\widetilde{\Gamma}} d \mu \;\overline{\psi}_1 \;\psi_2,~~~~~~d\mu :=\sqrt{-g}\; d\theta \;dt = |t|\; d\theta \;dt,$$ where $\widetilde{\Gamma}:= [-T,0[ \times {\mathbb S}^1$ (with $T >0$) in the pre-singulaity epoch, and $\widetilde{\Gamma}:= ]0,T] \times {\mathbb S}^1$ in the post-singularity epoch. We assume that the CM space can be used to model the universe only during its quantum phase, which lasts the period $[-T, T$\]. Now we construct an orthonormal basis, in the left neighborhood of the cosmological singularity, out of the solutions (\[solth\]) and (\[solt\]). One can verify that the solutions (\[solth\]) are orthonormal and continuous on ${\mathbb S}^1$ if $\;a_1 = \pi^{-1/2}= a_2\;$ and $\;r\rho = 0,\pm 1,\pm 2,\ldots$. (This set of functions coincides with the basis (\[quant4\]) that spans the subspace $\Omega_\lambda$ if we replace $k$ by $r\rho$.) Some effort is needed to construct the set of orthonormal functions out of $\Re J(i\rho,mt)$ and $\Re Y(i\rho,mt)$. First, one may verify that these functions are square-integrable on the interval $[-T,T]$. This is due to the choice of the measure in the scalar product (\[scalar\]), which leads to the boundedness of the corresponding integrants. Second, having normalizable set of four independent functions, for each $\rho$, we can turn it into an orthonormal set by making use of the Gram-Schmidt procedure (see, e.g. [@Arfken:2005]). Our orthonormal and countable set of functions may be used to define the span $\mathcal{F}$. The completion of $\mathcal{F}$ in the norm induced by the scalar product (\[scalar\]) defines the Hilbert spaces $L^2(\widetilde{\Gamma} \times {\mathbb S}^1,d\mu)$. It is clear that the same procedure applies to the right neighborhood of the singularity. Finally, we can prove that the Hamiltonian (\[nh\]) is self-adjoint on $L^2(\widetilde{\Gamma} \times {\mathbb S}^1,d\mu)$. The proof is immediate if we rewrite (\[hcn\]) in the form $$\label{newf} \Box \;\psi = - m^2\;\psi .$$ It is evident that on the orthonormal basis that we have constructed above the operator $\Box$ is an identity operator multiplied by a real constant $-m^2$. The operator $\Box$ is bounded since $$\label{bound} \|\Box \| := \sup_{\|\psi\| =1} \|\Box\;\psi\| = \sup_{\|\psi\| =1} \| -m^2 \; \psi\| = m^2 < \infty ,$$ where $\|\psi\|:=\sqrt{<\psi|\psi>}$. The operator $\Box$ is also symmetric, because $m$ is a [*real*]{} constant. Since $\Box$ is bounded and symmetric, it is a self-adjoint operator (see, e.g. [@MRS]). Clearly, the self-adjointness of the Hamiltonian (\[nh\]) results from the self-adjointness of $\Box$. We have constructed the two Hilbert spaces: one for the pre-singularity epoch, $\mathcal{H}^{(-)}$, and another one to describe the post-singularity epoch, $\mathcal{H}^{(+)}$. Next problem is to ‘glue’ them into a single Hilbert space, $\mathcal{H}=L^2([-T,T] \times {\mathbb S}^1,d\mu)$, that is needed to describe the entire quantum phase. From the mathematical point of view the gluing seems to be problematic because the Cauchy problem for the equation (\[hcn\]) is not well defined[^5] at $t=0$, and because we have assumed that $t\neq 0$ in the process of separation of variables to get Eqs. (\[eqth\]) and (\[eqt\]). However, arguing based on the physics of the problem enables the gluing. First of all we have already agreed that a *classical* test particle is able to go across the singularity (see, subsection II B). One can also verify that the probability density $$\label{amp} P(\theta,t):= \sqrt{-g}\;|\psi(\theta,t)|^2 = |t|\;|\psi(\theta,t)|^2$$ is bounded and continuous in the domain $\;[-T,T] \times {\mathbb S}^1$. Figures 3 and 4 illustrate the behavior of $P(\theta,t)$ for two examples of gluing the solutions having $\rho =0$. The cases with $\rho \neq 0$ have similar properties. Thus, the assumption that the gluing is possible is justified. However one can glue the two Hilbert spaces in more than one way, as it was done in the quantization of the phase space in our previous paper [@Malkiewicz:2005ii]. In what follows we present two cases, which are radically different. ### Deterministic propagation Among all solutions (\[solt\]) there is one, corresponding to $\rho =0$, that attracts an attention [@SWM]. It reads $$\label{n1B} B_1(0,mt)= b_1\;\Re J(0,mt),~~~~~~b_1\in {\mathbb R},$$ and has the following power series expansion close to $t=0$ $$\label{psn} B_1(0,x)/b_1 = 1- \frac{x^2}{4}+\frac{x^4}{64} - \frac{x^6}{2304} + \mathcal{O}[x^8] .$$ It is visualized in Fig. 5a. The solution (\[n1B\]) is smooth at the singularity, in spite of the fact that (\[eqt\]) is singular at $t=0$. It defines a solution to (\[hcn\]) that does not depend on $\theta$, since the non-zero solution (\[solth\]) with $\rho=0$ is just a constant. Thus, it is unsensitive to the problem that one cannot choose a common coordinate system for both $t<0$ and $t>0$. The solution $B_1$ (and the trivial solution $B_0 := 0$) can be used to construct a one-dimensional Hilbert space $\mathcal{H}=L^2([-T,T] \times {\mathbb S}^1,d\mu)$. The scalar product is defined by (\[scalar\]) with $\widetilde{\Gamma}$ replaced by $\Gamma := [-T,T] \times {\mathbb S}^1$. It is obvious that the Hamiltonian is self-adjoint on $\mathcal{H}$. The solution (\[n1B\]) is [*continuous*]{} at the singularity. It describes an unambiguous propagation of a quantum particle. Thus, we call it a [*deterministic*]{} propagation. It is similar to the propagation of a particle in the RCM space considered in the next subsection. Since (\[eqt\]) is a second order differential equation, it should have two independent solutions. However, the second solution cannot be continuous at $t=0$. One may argue as follows: The solution (\[n1B\]) may be obtained by ignoring the restriction $t \neq 0$ and solving (\[eqt\]) with the following initial conditions $$\label{incon} B(0,0)=1,~~~~~~dB(0,0)/dt = 0.$$ Equations (\[eqt\]) and (\[incon\]) are consistent, because the middle term of the r.h.s. of (\[eqt\]) may be equal to zero due to (\[incon\]) so the resulting equation would be non-singular at $t=0$. Another initial condition of the form $\;B(0,0)= const\;$ and $\;dB(0,0)/dt = 0\;$ would be linearly dependent on (\[incon\]). Thus, it could not lead to the solution which would be continuous at $t=0$ and linearly independent on (\[n1B\]). This qualitative reasoning can be replaced by a rigorous derivation using the power series expansion method [@Arfken:2005]. Applying this method one obtains that near the singularity $t=0$ the solution to (\[eqt\]) behaves like $\;t^\omega\;$ and that the corresponding indicial equation reads $$\label{ind} \omega^2 = -\rho^2 .$$ Thus, for $\rho\neq 0$ the two solutions behave like $\;t^{\pm i \rho}\;$, i.e. are bounded but not continuous (see, Eq. (\[solt\])). For $\rho =0$ the indicial equation has only one solution $\;\omega =0\;$ which leads to an analytic solution to (\[eqt\]) defined by (\[n1B\]). In such a case, it results from the method of solving the singular linear second order equations [@Arfken:2005], the second solution to (\[eqt\]) may behave like $\;\ln |t|\;$. In fact it reads [@SWM] $$\label{n2B} B_2(0,mt)= b_2\;\Re Y(0,mt),~~~~~~b_2\in {\mathbb R},$$ and is visualized in Fig. 5b. It cannot be called a deterministic propagation due to the discontinuity at the singularity $t=0$. ### Indeterministic propagation All solutions (\[solt\]), except (\[n1B\]), are discontinuous at $t=0$. This property is connected with the singularity of (\[eqt\]) at $t=0$. It is clear that due to such an obstacle the identification of corresponding solutions on both sides of the singularity is impossible. However there are two natural constructions of a Hilbert space out of $\mathcal{H}^{(-)}$ and $\mathcal{H}^{(+)}$ which one can apply:\ [*(a) Tensor product of Hilbert spaces*]{}\ The Hilbert space is defined in a standard way [@EP] as $\mathcal{H}:= \mathcal{H}^{(-)}\otimes\mathcal{H}^{(+)}$ and it consists of functions of the form $$\label{tp} f(t_1,\theta_1;t_2,\theta_2) \equiv (f^{(-)}\otimes f^{(+)})(t_1,\theta_1;t_2,\theta_2) := f^{(-)}(t_1,\theta_1)\;f^{(+)}(t_2,\theta_2) ,$$ where $f^{(-)}\in \mathcal{H}^{(-)}$ and $f^{(+)}\in \mathcal{H}^{(+)}$. The scalar product reads $$\label{sten} <f\;|\;g>:= <f^{(-)}|\;g^{(-)}>\;<f^{(+)}|\;g^{(+)}> ,$$ where $$\label{stm} <f^{(-)}|\;g^{(-)}>:=\int_{-T}^0 dt_1 \int_0^{2\pi}d \theta_1 \;|t_1|\; f^{(-)}(t_1,\theta_1)\; g^{(-)}(t_1,\theta_1)$$ and $$\label{stp} <f^{(+)}|\;g^{(+)}>:=\int_0^{T} dt_2 \int_0^{2\pi}d \theta_2 \;|t_2|\; f^{(+)}(t_2,\theta_2)\; g^{(+)}(t_2,\theta_2) .$$ The action of the Hamiltonian is defined by $$\label{mamt} \hat{H} \big(f^{(-)}\otimes f^{(+)}\big):= \big(\hat{H} f^{(-)}\big) \otimes f^{(+)} + f^{(-)}\otimes \big(\hat{H} f^{(+)}\big).$$ The Hamiltonian is clearly self-adjoint on $\mathcal{H}$. The quantum system described in this way appears to consist of two independent parts. In fact it describes the same quantum particle but in two subsequent time intervals separated by the singularity at $t=0$.\ [*(b) Direct sum of Hilbert spaces*]{}\ Another standard way [@EP] of defining the Hilbert space is $\mathcal{H}:= \mathcal{H}^{(-)}\bigoplus\mathcal{H}^{(+)}$. The scalar product reads $$\label{dssc} <f_1|f_2>:= <f_1^{(-)}|f_2^{(-)}> + <f_1^{(+)}|f_2^{(+)}> ,$$ where $$\label{dsf} f_k := (f_k^{(-)},f_k^{(+)}) \in \mathcal{H}^{(-)}\times\mathcal{H}^{(+)},~~~~~~k=1,2,$$ and where $f_k^{(-)}$ and $f_k^{(+)}$ are two completely independent solutions in the pre-singularity and post-singularity epochs, respectively. (The r.h.s of (\[dssc\]) is defined by (\[stm\]) and (\[stp\]).) The Hamiltonian action on $\mathcal{H}$ reads $$\label{hamds} \mathcal{H}\ni (f^{(-)},f^{(+)})\longrightarrow \hat{H} (f^{(-)},f^{(+)}):= (\hat{H} f^{(-)},\hat{H} f^{(+)}) \in \mathcal{H}.$$ It is obvious that $\hat{H}$ is self adjoint on $\mathcal{H}$. By the construction, the space $\mathcal{H}^{(-)}\bigoplus\mathcal{H}^{(+)}$ includes vectors like $(f^{(-)},0)$ and $(0,f^{(+)})$, which give non-vanishing contribution to (\[dssc\]) (but yield zero in case (\[sten\])). The former state describes the annihilation of a particle at $t=0$. The latter corresponds to the creation of a particle at the singularity. These type of states do not describe the propagation of a particle [*across*]{} the singularity. The annihilation/creation of a massive particle would change the background. Such events should be eliminated from our model because we consider a [*test*]{} particle which, by definition, cannot modify the background spacetime. Since $\mathcal{H}^{(-)}$ and $\mathcal{H}^{(+)}$, being vector spaces, must include the zero solutions, the Hilbert space $\mathcal{H}^{(-)}\bigoplus\mathcal{H}^{(+)}$ cannot model the quantum phase of our system. Regularization -------------- In the RCM space the quantum Hamiltonian, $\hat{H}_\epsilon$, for any $t \in [-T,T]$, reads (we use the gauge $C_\epsilon =2$) $$\hat{H_\epsilon}= \frac{\sqrt{t^2 + \epsilon^2 + \epsilon^2 r^2}}{t^2 +\epsilon^2}\; \frac{\partial^2}{\partial\theta^2} - \frac{t^2 +\epsilon^2}{\sqrt{t^2 + \epsilon^2 + \epsilon^2 r^2}}\;\frac{\partial^2}{\partial t^2} - \nonumber$$ $$\label{rh} \frac{t(t^2 + \epsilon^2 + 2 \epsilon^2 r^2)}{(t^2 + \epsilon^2 + \epsilon^2 r^2)^{3/2}}\;\frac{\partial}{\partial t} - \sqrt{t^2 + \epsilon^2 + \epsilon^2 r^2}\; m^2$$ Since the Hamiltonian is equal to the first-class constraint, the physical states are solutions to the equation $$\label{rh1} \hat{H_\epsilon}\Psi=0 .$$ As in the case of $\epsilon =0$, the space of solutions to (\[rh1\]) defines the domain of boundedness of $\hat{H_\epsilon}$. Substitution $\Psi(\theta,t)=A(\theta)B(t)$ into (\[rh1\]) yields $$\label{eqthr} \frac{d^2A}{d\theta^2}+\rho^2A=0,~~~~\rho\in {\mathbb R},$$ and $$\label{eqtr} \frac{(t^2+\epsilon^2)^2}{t^2+\epsilon^2+r^2\epsilon^2}\;\frac{d^2B}{dt^2}+ \frac{t(t^2+\epsilon^2)(t^2+\epsilon^2+2r^2\epsilon^2)} {(t^2+\epsilon^2+2r^2\epsilon^2)^2}\;\frac{dB}{dt}+m^2(t^2+\epsilon^2+\rho^2)B=0.$$ The equation (\[eqthr\]) looks the same as in the non-regularized case (\[eqth\]) so the two independent solutions on ${\mathbb S}^1$ read $$\label{solthr} A_1(\rho,\theta)= \pi^{-1/2} \cos(\rho\theta),~~~~~A_2(\rho,\theta)= \pi^{-1/2} \sin(\rho\theta),$$ where $r\rho = 0,\pm 1,\pm 2,\ldots$ (orthogonality and continuity conditions). Equation (\[eqtr\]) is non-singular in $ [-T,T] $ so for each $\rho$ it has two independent solutions which are bounded and smooth in the entire interval. One may represent these solutions by a symmetric and an anti-symmetric functions. We do not try to find the analytic solutions to (\[eqtr\]). What we really need to know are general properties of them. In what follows we further analyze only numerical solutions to (\[eqtr\]) by making use of [@SWM]. Since in the regularized case the solutions are continuous in the entire interval $[-T,T]$, the problem of gluing the solutions (the main problem in case $\epsilon = 0$) does not occur at all. Thus, the construction of the Hilbert space, $\mathcal{H}_\epsilon$, by using the space of solutions to (\[eqthr\]) and (\[eqtr\]) is straightforward. The construction of the basis in $\mathcal{H}_\epsilon$ may be done by analogy to the construction of the basis in $\mathcal{H}^{(-)}$ (described in the subsection B). The only difference is that now $t \in [-T,T]$ and instead of (\[solt\]) we use the solutions to (\[eqtr\]). Let us denote them by $B_i(\rho,mt)$, where $i=s$ and $i=a$ stand for symmetric and anti-symmetric solutions, respectively. Figures 6 and 7 present two examples of solutions to (\[eqtr\]) for $\rho =1$ and the corresponding probability densities. We can see that $P(\theta,t)$ is a bounded and continuous function on $[-T,T]\times {\mathbb S}^1$, as in the case of $\epsilon =0$ (cp with Figs. 3 and 4). We define the scalar product as follows $$\label{sc} <\psi_1|\psi_2> := \int_\Gamma d \mu \;\overline{\psi}_1 \;\psi_2,~~~~~~d\mu :=\sqrt{-g}\; d\theta \;dt = \sqrt{t^2+\epsilon^2+r^2\epsilon^2}\; d\theta \;dt,$$ where ${\Gamma}:= [-T,T] \times {\mathbb S}^1$, and where an explicit form of $d\mu$ is found by making use of (\[line3\]). It is evident that $\hat{H}_\epsilon$ is self-adjoint on $\mathcal{H}_\epsilon$. The main difference between the deterministic case with $\epsilon =0$ and the present case $\epsilon >0$ is that in the former case the Hilbert space is one dimensional ($\rho =0$), whereas in the latter case it is $2\times 2 \times \infty$ dimensional ($r\rho \in {\mathbb Z}$). Summary and conclusions ======================= The Cauchy problem at the cosmological singularity of the geodesic equations may be ‘resolved’ by the regularization, which replaces the double conical vertex of the CM space by a space with the vertex of the big-bounce type, i.e. with non-vanishing space dimension at the singularity. We have presented a specific example of such regularization of the CM space. Both classical and quantum dynamics of a particle in the regularized CM space are deterministic and stable. We have examined these aspects of the dynamics at the phase space and Hamiltonian levels. The classical and quantum dynamics of a particle in the regularized CM space is similar to the dynamics in the de Sitter space [@WP; @Piechocki:2003hh]. We are conscious that our regularization of the singularity is rather [*ad hoc*]{}. Our arguing (presented at the beginning of Sec. IIB) that taking into account the interaction of a physical particle with the singularity may lead effectively to changing of the latter into a big-bounce type singularity should be replaced by analyzes. However, examination of this problem is beyond the scope of the present paper, but will be considered elsewhere. The classical dynamics in the CM space is unstable (apart from the one class of geodesics). However, the quantum dynamics is well defined. The Cauchy problem of the geodesics is not an obstacle to the quantization. The examination of the quantum stability has revealed surprising result that in one case a quantum particle propagates deterministically in the sense that it can be described by a quantum state that is continuous at the singularity. This case is very interesting as it says that there can exist deterministic link between the data of the pre-singularity and post-singularity epochs. All other states have discontinuity at the singularity of the CM space, but they can be used successfully to construct a Hilbert space. This way we have proved the stability of the dynamics of a [*quantum*]{} particle. At the quantum level the stability condition requiring the boundedness from below of the Hamiltonian operator means the imposition of the first-class constraint onto the space of quantum states to get the space of [*physical*]{} quantum states. The resulting equation depends on all spacetime coordinates. In the pre-singularity and post-singularity epochs the CM space is locally isometric to the Minkowski space [@Malkiewicz:2005ii]. Owing to this isometry, the stability condition is in fact the Klein-Gordon, KG, equation. The space of solutions to the KG equation in these two epochs and the corresponding Hilbert space are fortunately non-trivial ones, otherwise our quantum theory of a particle would be empty. Quantization of the phase space carried out in Sec. IV (and in our previous paper [@Malkiewicz:2005ii]), corresponds to some extent to the method of quantization in which one first solves constraints at the classical level and then quantize the resulting theory. Quantization that we call here examination of the stability at the quantum level, is effectively the method in which we impose the constraint, but at the quantum level. The results we have obtained within both methods of quantization are consistent. It means that the quantum theory of a particle in the compactified Milne space does exist. The CM space seems to model the cosmological singularity in a satisfactory way[^6]. It turns out that the time-like geodesics of our CM space may have interpretation in terms of cosmological solutions of some sophisticated higher dimensional field theories [@Russo:2004am; @Bergshoeff:2005cp; @Bergshoeff:2005bt]. We have already discussed some aspects of this connection in our previous paper (see Sec. 5 of [@Malkiewicz:2005ii]). Presently, we can say that in one case (see Sec. 4 of [@Russo:2004am] and Sec. V.B.1 of this paper) this analogy extends to the quantum level: transition of a particle through the cosmological singularity in both models is mathematically well defined. In both cases the operator constraint is used to select quantum physical states. Elaboration of this analogy needs an extension of our results to the Misner space (that consists of the Milne and Rindler spaces) because it is the spacetime used in [@Russo:2004am]. Another subtlety is connected with the fact that we carry out analysis in the compactified space, whereas the authors of [@Russo:2004am] use the covering space. There exists another model to describe the evolution of the universe based on string/M theory. It is called the pre-big-bang model [@Gasperini:2002bn]. However, the ST model is more self-consistent and complete. Other sophisticated model called loop quantum cosmology, LQC, is based on non-perturbative formulation of quantum gravity called the loop quantum gravity, LQG [@TT; @CR]. It is claimed that the CS is resolved in this approach [@Bojowald:2006da]. However, this issue seems to be still open due to the assumptions made in the process of truncating the infinite number degrees of freedom of the LQG to the finite number used in the LQC [@Brunnemann:2005in; @Brunnemann:2005ip]. This model has also problems in obtaining an unique semi-classical approximations [@Nicolai:2005mc], which are required to link the quantum phase with the nearby classical phase in the evolution of the Universe. For response to [@Nicolai:2005mc] we recommend [@Thiemann:2006cf]. Quantization of dynamics of [*extended*]{} objects in the CM space is our next step. There exist promising results on propagation of a string and membrane [@Turok:2004gb; @Niz:2006ef]. 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[^2]: This way we have also verified, in the gauge $\tau =t$, the equivalence between our Hamiltonian and the Lagrangian formulations of the dynamics of a particle. [^3]: The division of the set of geodesics into regular and unstable depends on the choice of coordinates, but it always includes the unstable ones. [^4]: In the preceding subsection, concerning the classical dynamics, the choice of gauge was different. But since the theory we use is gauge invariant, the different choice of the gauge does not effect physical results. [^5]: Except one case discussed later. [^6]: Our result should be further confirmed by the examination of the dynamics of a particle in a higher dimensional CM space.

--- abstract: '**Abstract** Electronic transport through a quantum dot chain embodied in an Aharonov-Bohm interferometer is theoretically investigated. In such a system, it is found that only for the configurations with the same-numbered quantum dots side-coupled to the quantum dots in the arms of the interferometer, some molecular states of the quantum dot chain decouple from the leads. Namely, in the absence of magnetic flux all odd molecular states decouple from the leads, but all even molecular states decouple from the leads when an appropriate magnetic flux is introduced. Interestingly, the antiresonance position in the electron transport spectrum is independent of the change of the decoupled molecular states. In addition, when considering the many-body effect within the second-order approximation, we show that the emergence of decoupling gives rise to the apparent destruction of electron-hole symmetry. By adjusting the magnetic flux through either subring, some molecular states decouple from one lead but still couple to the other, and then some new antiresonances occur.' author: - 'Yu Han$^{a,b}$' - 'Weijiang Gong$^{a}$' - 'Haina Wu$^{a}$' - 'Guozhu Wei$^{a,c}$' title: 'Decoupling and antiresonance in electronic transport through a quantum dot chain embodied in an Aharonov-Bohm interferometer' --- Introduction ============ During the past years, electronic transport through quantum-dot(QD) systems has been extensively studied both experimentally and theoretically. The atom-like characteristics of a QD, such as the discrete electron levels and strong electron correlation, manifest themselves by the experimental observations of Coulomb blockade[@Thomas; @Meirav; @Sakaki; @Zhang], conductance oscillation[@Kool], and Kondo effect[@Mahalu; @Gores; @Cronenwett; @Heary] in electronic transport through a QD. Therefore, a single QD is usually called an artificial atom, and a mutually coupled multi-QD system can be regarded as an artificial molecule. Thanks to the progress of nanotechnology, it now becomes possible to fabricate a variety of coupled QD structures with sizes to be smaller than the electron coherence[@Xie; @Shailos]. In comparison with a single QD, coupled QD systems possess higher freedom in implementing some functions of quantum devices, such as the QD cellular automata[@Loss1] and solid-state quantum computation[@Loss2; @Klein]. Motivated by an attempt to find some interesting electron transport properties, recently many experimental and theoretical works have become increasingly concerned about the electronic transport through various multi-QD systems[@Tarucha; @Yu; @Vidan; @Waugh; @Amlani]. According to the previous researches, the peaks of the linear conductance spectra of coupled-QD systems reflect the eigenenergies of the corresponding coupled QDs. On the contrary, the zero point of the conductance, called antiresonance, originates from the destructive quantum interference among electron waves passing through different transmission paths. With respect to the coupled-QD structures, the typical ones are the structures of the so-called T-shaped QDs [@Iye; @Sato; @Ihn; @Santos; @Tor; @Gong1]and the parallel QDs [@Konign; @Orellanan; @Zhun; @Sunn]( i.e., the QDs embodied in the Aharonov-Bohm (AB) interferometer). A unique property of electron transport through the T-shaped QD systems is that the antiresonance points coincide with the eigenenergies of the side-coupled QDs, which has also been observed experimentally[@Iye; @Sato]. On the other hand, the parallel-coupled QD systems, which offer two channels for the electron tunneling, have also attracted much attention. In such structures, with the adjustment of magnetic flux the AB effect has been observed. Meanwhile, the appropriate couplings between the molecular states of the coupled QDs and leads can be efficiently adjusted, which gives rise to the tunable Fano effect[@Orellanan; @Zhun]. Moreover, under the condition of an appropriate external field, some molecular states can decouple completely from the leads, which is referred to as the formation of bound states in continuum in some literature[@Orellana1]. According to the previous researches, the existence of decoupling plays a nontrivial role in the quantum interference of QD structures, especially, it changes the property of the quantum interference[@Bao]. Therefore, it is still desirable to clarify the decoupling in electronic transport through some coupled-QD structures. Since the development of nanotechnology, it is feasible to fabricate the coupled QDs, in particular the QD chain, in the current experiment[@Berry; @Martin]. Thereby we are now theoretically concerned with the electron transport properties of the this structure, by considering it embodied in the AB interferometer. As a result, we find that for the structures with the same-numbered QDs side-coupled to the QDs in the two arms of the interferometer, some molecular states of the QD chain decouple from the leads, and which molecular states decouple from the leads is determined by the adjustment of magnetic flux. Besides, in the case of the many-body effect being considered, the existence of decoupling gives rise to the destruction of electron-hole symmetry. model\[theory\] =============== The coupled-QD structure we consider is illustrated in Fig.\[structure\](a). The Hamiltonian that describes the electronic motion in such a structure reads $H=H_{C}+H_{D}+H_{T}$, in which $H_C$ is the Hamiltonian for the noninteracting electrons in the two leads, $H_D$ describes the electron in the QD chain, and $H_{T}$ denotes the electron tunneling between the leads and QDs. They take the forms as follows. $$\begin{aligned} H_{C}&&=\underset{\sigma k\alpha\in L,R}{\sum }\varepsilon _{k\alpha}c_{k\alpha\sigma}^\dag c_{k\alpha\sigma},\notag\\ H_{D}&&=\sum_{\sigma,m=1}^{{\scriptscriptstyle N}}\varepsilon _{m}d_{m\sigma}^\dag d_{m\sigma}+\sum_m U_m n_{m\uparrow}n_{m\downarrow}\notag\\ &&+\sum_{\sigma,m=1}^{N-1}(t_md^\dag_{m+1\sigma}d_{m\sigma} +{\mathrm {H.c.}}),\notag\\ H_{T} &&=\underset{k\alpha\sigma}{\sum }( V_{\alpha j}d_{j\sigma}^\dag c_{k\alpha\sigma}+V_{\alpha j+1}d_{j+1\sigma}^\dag c_{k\alpha\sigma}+{\mathrm {H.c.}}),\notag\\\end{aligned}$$ where $c_{k\alpha\sigma}^\dag$ $( c_{k\alpha\sigma})$ is an operator to create (annihilate) an electron of the continuous state $|k,\sigma\rangle$ in lead-$\alpha$ with $\sigma$ being the spin index, and $\varepsilon _{k\alpha}$ is the corresponding single-particle energy. $d^{\dag}_{m\sigma}$ $(d_{m\sigma})$ is the creation (annihilation) operator of electron in QD-$m$, $\varepsilon_m$ denotes the electron level in the corresponding QD, $t_m$ is the interdot hopping coefficient, and $U_m$ represents the intradot Coulomb repulsion. $n_{m\sigma}=d_{m\sigma}^\dag d_{m\sigma}$ is the electron number operator in QD-$m$. We assume that only one level is relevant in each QD and the value of $\varepsilon _{m}$ is independent of $m$, i.e, $\varepsilon _{m}=\varepsilon_0$. In the expression of $H_{T}$, the sequence numbers of the two QDs in the interferometer arms are taken as $j$ and $j+1$, and $V_{\alpha j}$ and $V_{\alpha j+1}$ with $\alpha=L, R$ denotes the QD-lead coupling coefficients. We adopt a symmetric QD-lead coupling configuration which gives that $V_{Lj}=Ve^{i\phi_L/2}$, $V_{Lj+1}=Ve^{-i\phi_L/2}$, $V_{Rj}=Ve^{-i\phi_R/2}$, and $V_{Rj+1}=Ve^{i\phi_R/2}$ with $V$ being the QD-lead coupling strength. The phase shift $\phi_\alpha$ is associated with the magnetic flux $\Phi_\alpha$ threading the system by a relation $\phi_\alpha=2\pi\Phi_\alpha/\Phi_{0}$, in which $\Phi_{0}=h/e$ is the flux quantum. To study the electronic transport properties of such a structure, the linear conductance at zero temperature is obtained by the Landauer-Büttiker formula $$\mathcal {G}=\frac{e^{2}}{h}\sum_\sigma T_\sigma(\omega)|_{\omega=\varepsilon_F}.\label{conductance}$$ $T(\omega)$ is the transmission function, in terms of Green function which takes the form as[@Meir1; @Jauho] $$T_\sigma(\omega)=\mathrm {Tr}[\Gamma^LG^r_\sigma(\omega)\Gamma^RG^a_\sigma(\omega)],\label{transmission}$$ where $\Gamma^L$ is a $2\times 2$ matrix, describing the strength of the coupling between lead-L and the QDs in the interferometer arms. It is defined as $[\Gamma^{L}]_{ll'}=2\pi V_{{{\scriptscriptstyle L}}l}V^*_{{{\scriptscriptstyle L}}l'}\rho_{{\scriptscriptstyle L}}(\omega)$ ( $l,l'=[j,j+1]$). We will ignore the $\omega$-dependence of $\Gamma^{L}_{ll'}$ since the electron density of states in lead-L, $\rho_{{\scriptscriptstyle L}}(\omega)$, can be usually viewed as a constant. By the same token, we can define $[\Gamma^R]_{ll'}$. In fact, one can readily show that $[\Gamma^L]_{ll}=[\Gamma^R]_{ll}$ in the case of identical QD-lead coupling, hence we take $\Gamma=[\Gamma^L]_{ll}=[\Gamma^R]_{ll}$ to denote the QD-lead coupling function. In Eq. (\[transmission\]) the retarded and advanced Green functions in Fourier space are involved. They are defined as follows: $G_{ll',\sigma}^r(t)=-i\theta(t)\langle\{d_{l\sigma}(t),d_{l'\sigma}^\dag\}\rangle$ and $G_{ll',\sigma}^a(t)=i\theta(-t)\langle\{d_{l\sigma}(t),d_{l'\sigma}^\dag\}\rangle$, where $\theta(x)$ is the step function. The Fourier transforms of the Green functions can be performed via $G_{ll',\sigma}^{r(a)}(\omega)=\int^{\infty}_{-\infty} G_{ll',\sigma}^{r(a)}(t)e^{i\omega t}dt$. These Green functions can be solved by means of the equation-of-motion method[@Liu; @Liu2]. By a straightforward derivation, we obtain the retarded Green functions which are written in a matrix form as $$\begin{aligned} G^r_\sigma(\omega)=\left[\begin{array}{cc} g_{j\sigma}(z)^{-1} & -t_j+i\Gamma_{j,j+1}\\ -t^*_j+i\Gamma_{j+1,j}& g_{j+1\sigma}(z)^{-1}\\ \end{array}\right]^{-1}\ \label{green},\end{aligned}$$ with $z=\omega+i0^+$ and $\Gamma_{ll'}={1 \over 2}([\Gamma^L]_{ll'}+[\Gamma^R]_{ll'})$. $g_{l\sigma}(z)=[(z-\varepsilon_{l})S_{l\sigma}-\Sigma_{l\sigma}+i\Gamma_{ll}]^{-1}$, is the zero-order Green function of the QD-$l$ unperturbed by QD-$l'$, in which the selfenergies $$\begin{aligned} &&\Sigma_{j\sigma}=\frac{t_{j-1}^2}{(z-\varepsilon_{j-1})S_{j-1\sigma} -\frac{t^2_{j-2}}{(z-\varepsilon_{j-2})S_{j-2\sigma}-\ddots\frac{t_{2}^2}{(z-\varepsilon_{2})S_{2\sigma} -\frac{t^2_{1}}{(z-\varepsilon_{1})S_{1\sigma}}}}}\notag\\ &&\Sigma_{j+1\sigma}=\frac{t_{j+1}^2}{(z-\varepsilon_{j+1})S_{j+1\sigma} -\frac{t^2_{j+2}}{(z-\varepsilon_{j+2})S_{j+2\sigma}-\ddots\frac{t_{N-1}^2}{(z-\varepsilon_{N-1})S_{N-1\sigma} -\frac{t^2_{N}}{(z-\varepsilon_{N})S_{N\sigma}}}}}\notag\end{aligned}$$ account for the laterally coupling of the QDs to QD-$j$ and QD-$j+1$, respectively[@Liu]. The quantity $S_{m\sigma}=\frac{z-\varepsilon_{m}-U_m}{z-\varepsilon_{m}-U_m+U_m\langle n_{m\bar{\sigma}}\rangle}$ ($m\in[1,N]$) is the contribution of the intradot Coulomb interaction up to the second-order approximation[@Gong1]. In addition, the advanced Green function can be readily obtained via a relation $G^a_\sigma(\omega)=[G^r_\sigma(\omega)]^\dag$. It is easy to understand that in the noninteracting case, the linear conductance spectrum of the coupled QD structure reflects the eigenenergy spectrum of the “molecule" made up of the coupled QDs. In other words, each resonant peak in the conductance spectrum represents an eigenenergy of the total QD molecule, rather than the levels of the individual QDs. Therefore, it is necessary to transform the Hamiltonian into the molecular orbital representation of the QD chain. We now introduce the electron creation(annihilation) operators corresponding to the molecular orbits, i.e., $f_{m\sigma}^\dag\; (f_{m\sigma})$. By the diagonalization of the single-particle Hamiltonian of the QDs, we find the relation between the molecular and atomic representations (here each QD is regarded as an “atom"). It is expressed as $[\bm{f}_\sigma^\dag]=[\bm{\eta}][\bm{d}_\sigma^\dag]$. The $N\times N$ transfer matrix $[\bm{\eta}]$ consists of the eigenvectors of the QD Hamiltonian. In the molecular orbital representation, the single-particle Hamiltonian takes the form: $H=\underset{k\sigma\alpha\in L,R}{\sum }\varepsilon _{\alpha k}c_{\alpha k\sigma }^\dag c_{\alpha k\sigma}+\sum_{m=1, \sigma}e _{m}f_{m\sigma}^\dag f_{m\sigma}+\underset{\alpha k\sigma}{\sum } v_{\alpha m}f_{m\sigma}^\dag c_{\alpha k\sigma}+{\mathrm {h.c.}}$, in which $e_m$ is the eigenenergy of the coupled QDs. The coupling between the molecular state $|m\sigma\rangle$ and the state $|k,\sigma\rangle$ in lead-$\alpha$ can be expressed as $$v_{\alpha m}=V_{\alpha j}[\bm\eta]^\dag_{jm}+V_{\alpha j+1}[\bm\eta]^\dag_{j+1,m}.\label{gamma}$$ In the case of symmetric QD-lead coupling, the above relation can be rewritten as $v_{\alpha m}=V([\bm\eta]^\dag_{jm}+[\bm\eta]^\dag_{j+1,m}e^{i\phi_\alpha})$. Fig.\[structure\](b) shows the illustration of the QD structure in the molecular orbital representation. We here define $\gamma^\alpha_{mm}=2\pi v_{\alpha m} v^*_{\alpha m}\rho_\alpha(\omega)$ which denotes the strength of the coupling between the molecular state $|m\sigma\rangle$ and the leads. Numerical results and discussions \[result2\] ============================================= With the theory in the above section, we can perform the numerical calculation to investigate the linear conductance spectrum of this varietal parallel double-QD structures, namely, to calculate the conductance as a function of the incident electron energy. Prior to the calculation, we need to introduce a parameter $t_0$ as the unit of energy. We choose the parameter values $t_{m}=\Gamma=t_{0}$ for the QDs to carry out the numerical calculation. And $\varepsilon_{0}$, the QD level, can be shifted with respect to the Fermi level by the adjustment of gate voltage experimentally. Typically, the case of $\phi_L=\phi_R=\phi$ are first considered. Fig.\[QD2\] shows the linear conductance spectra ($\cal{G}$ versus $\varepsilon_0$) for several structures with the QD number $N=2$ to $4$. It is obvious that the 2-QD structure just corresponds to the parallel double QDs with interdot coupling mentioned in some previous works[@Orellanan; @Zhun]. Its conductance spectrum presents a Breit-Wigner lineshape in the absence of magnetic flux, as shown in Fig.2(a). Such a result can be analyzed in the molecular orbital representation. Here the $[\bm\eta]$ matrix, takes a form as $[\bm\eta]={1 \over\sqrt{2}}\left[\begin{array}{cc} -1& 1\\ 1& 1\\ \end{array}\right]$, presenting the relation between the molecular and ‘atomic’ representations. Then with the help of Eq. (\[gamma\]) one can find that here the bonding state completely decouples from the leads and only the antibonding state couples to the leads, which leads to the appearance of the Breit-Wigner lineshape in the conductance spectrum. On the other hand, when introducing the magnetic flux with $\phi=\pi$, we can see that the decoupled molecular state is changed as the antibonding state, as exhibited by the dashed line in Fig.\[QD2\](a). In such a case, only the bonding state couples to the leads and the conductance profile also shows a Breit-Wigner lineshape. In Fig.\[QD2\](b) the conductance curves as a function of gate voltage are shown for the 3-QD structure. Obviously, there exist three conductance peaks in the conductance profiles and no decoupled molecular state appears. We can clarify this result by calculating $v_{\alpha m}=V([\bm\eta]^\dag_{jm}+[\bm\eta]^\dag_{j+1,m}e^{i\phi})$. Via such a relation, one can conclude that $v_{\alpha m}$ is impossible to be equal to zero in this structure regardless of the adjustment of magnetic flux. Thus one can not find the decoupled molecular states, the state-lead coupling may be relatively weak, though. Just as shown in Fig.\[QD2\](b), in the absence of magnetic flux the distinct difference of the couplings between the molecular states and leads offer the ‘more’ and ‘less’ resonant channels for the quantum interference. Then the Fano effect occurs and the conductance profile presents an asymmetric lineshape. In addition, the Fano lineshape in the conductance spectrum is reversed by tuning the magnetic flux to $\phi=\pi$, due to the modulation of magnetic flux on $v_{\alpha m}$. When the QD number increases to $N=4$, there will be two configurations corresponding to this structure, i.e, the cases of $j=1$ and $j=2$. As a consequence, the conductance spectra of the two structures remarkably differ from each other. With respect to the configuration of $j=1$, as shown in Fig.\[QD2\](c), the electron transport properties presented by the conductance spectra are similar to those in the case of the 3-QD structure, and there is also no existence of decoupled molecular states. However, for the case of $j=2$, as shown in Fig.\[QD2\](d), it is clear that in the absence of magnetic flux, there are two conductance peaks in the conductance spectrum, which means that the decoupling phenomenon comes into being. Alternatively, in the case of $\phi=\pi$, there also exist two peaks in the conductance profile. But the conductance peaks in the two cases of $\phi=0$ and $\pi$ do not coincide with one another. We can therefore find that in this structure, when $\phi=n\pi$ the decoupling phenomena will come about, and the adjustment of magnetic flux can effectively change the appearance of decoupled molecular states. By a further calculation and focusing on the conductance spectra, we can understand that in the case of $\phi=2n\pi$, the odd (first and third) molecular states of the coupled QDs decouple from the leads; In contrast, the even (second and fourth) molecular states of the QDs will decouple from the leads if $\phi=(2n-1)\pi$. Additionally, in Fig.\[QD2\](d) it shows that the conductance always encounters its zero when the level of the QDs is the same as the Fermi level of the system, which is irrelevant to the tuning of magnetic flux from $\phi=0$ to $\pi$. In order to obtain a clear physics picture about decoupling, we analyze this problem in the molecular orbital representation. By solving the $[\bm\eta]$ matrix and borrowing the relation $v_{\alpha m}=V_{\alpha j}[\bm\eta]^\dag_{jm}+V_{\alpha j+1}[\bm\eta]^\dag_{j+1,m}$, it is easy to find that in the case of zero magnetic flux, $v_{\alpha 1}$ and $v_{\alpha 3}$ are always equal to zero, which brings out the completely decoupling of the odd molecular states from the leads. Unlike this case, when $\phi=\pi$ the values of $v_{\alpha 2}$ and $v_{\alpha 4}$ are fixed at zero, and such a result leads to the even molecular states to decouple from the leads. However, the underlying physics responsible for antiresonance is desirable to clarify. We then analyze the electron transmission by the representation transformation. We take the case of $\phi=0$ as an example, where only two molecular states $|2\sigma\rangle$ and $|4\sigma\rangle$ couple to the leads due to decoupling. Accordingly, $|2\sigma\rangle$ and $|4\sigma\rangle$ might be called as well the bonding and antibonding states. As is known, the molecular orbits of coupled double QD structures, e.g, the well-known T-shaped QDs, are regarded as the bonding and antibonding states. Therefore, by employing the representation transformation $[\bm{a}_\sigma^\dag]=[\bm{\beta}][\bm{f}_\sigma^\dag]$, such a configuration can be changed into the T-shaped double-QD system (see Fig.\[structure\](c)) of the Hamiltonian ${\cal H}=\underset{k\sigma\alpha\in L,R}{\sum }\varepsilon _{\alpha k}c_{\alpha k\sigma }^\dag c_{\alpha k\sigma}+\sum^2_{\sigma, n=1}E_{n}a_{n\sigma}^\dag a_{n\sigma} +\tau_1 a_{2\sigma}^\dag a_{1\sigma}+\underset{\alpha k\sigma}{\sum } w_{\alpha1}a_{1\sigma}^\dag c_{\alpha k\sigma}+h.c.$. By a further derivation, the relations between the structure parameters of the two QD configurations can be obtained with $E_1=\varepsilon_0+t_0$, $E_2=\varepsilon_0$, $\tau_1=t_0$, and $w_{\alpha1}=V_{\alpha1}$ respectively with $[\bm\beta]={1\over\sqrt{2\sqrt{5}}}\left[\begin{array}{ccc} -\sqrt{\sqrt{5}-1} & \sqrt{\sqrt{5}+1}\\ \sqrt{\sqrt{5}+1}& \sqrt{\sqrt{5}-1} \\ \end{array}\right].$ The 4-QD structure is then transformed into the T-shaped double QDs with $\varepsilon_0$ being the level of dangling QD. Just as discussed in the previous works[@Gong1], in the T-shaped QDs antiresonance always occurs when the dangling QD level is aligned with the Fermi level of the system, one can then understand that in this 4-QD system, the antiresonant point in the conductance spectrum is consistent with $\varepsilon_0=\varepsilon_F=0$. When paying attention to the $[\bm\eta]^\dag$ matrix, one will see that $[\bm\eta]^\dag_{12}=[\bm\eta]^\dag_{43}$, $[\bm\eta]^\dag_{22}=-[\bm\eta]^\dag_{33}$, $[\bm\eta]^\dag_{32}=[\bm\eta]^\dag_{23}$, and $[\bm\eta]^\dag_{42}=-[\bm\eta]^\dag_{13}$ for the 4-QD structure. As a result, such relations give rise to $v_{\alpha 2}|_{\phi=0}=v_{\alpha 3}|_{\phi=\pi}$ and $v_{\alpha 4}|_{\phi=0}=v_{\alpha 1}|_{\phi=\pi}$. So, when $\phi=\pi$ the magnetic flux reverses the lineshape of the conductance spectrum in the case of $\phi=0$. Based on these properties, we can realize that the quantum interference in the $\phi=\pi$ case is similar to that in the case of $\phi=0$. Therefore, the antiresonant point in the conductance spectra is independent of the adjustment of magnetic flux. Similar to the analysis above, we can expect that in the structure of with $t_m=t_0$ and $\varepsilon_m=\varepsilon_0$, when $N$ is even and $j=\frac{N}{2}$ there must be the appearance of decoupled molecular states. To be concrete, in the case of $\phi=2n\pi$, the odd (first and third) molecular states of the coupled QDs decouple from the leads, whereas the even (second and fourth) molecular states of the QDs will decouple from the leads if $\phi=(2n-1)\pi$. This expectation can be confirmed because of $[\bm\eta]^\dag_{jm}=-[\bm\eta]^\dag_{j+1,m} (m\in \text{odd})$ and $ [\bm\eta]^\dag_{jm}=[\bm\eta]^\dag_{j+1,m} (m\in \text{even})$ for the QDs. The numerical results in Fig.\[6-8\], describing the conductances of 6-QD and 8-QD structures, can support our conclusion. Besides, with the help of the representation transformation the antiresonance positions can be clarified by transforming these structure into the T-shaped QD systems. Fig.\[manybody1\] shows the calculated conductance spectra of the double-QD structure by incorporating the many-body effect to the second order and considering the uniform on-site energies of all the QDs $U_m=U=2t_0$ as well as $4t_0$, respectively. It is seen that the conductance spectra herein split into two groups due to the Coulomb repulsion. But in each group the present decoupling phenomenon and antiresonance are similar to those in the noninteracting case. In addition, for each case ($U=2t_0$ or $4t_0$) between the two separated groups a conductance zero emerges in the conductance spectra. According to the previous discussions, when the many-body terms are considered within the second-order approximation, a pseudo antiresonance, resulting from the electron-hole symmetry, should occur at the position of $\varepsilon_0=-\frac{U}{2}$ where $\langle n_{m\sigma}\rangle={1\over 2}$ [@Gong1; @Liu; @Liu2]. However, with respect to such a situation, unlike the conventional electron-hole symmetry, the position of such a conductance zero departs from $\varepsilon_0=-\frac{U}{2}$ remarkably. Taking the structure of double QDs as an example, in the absence of magnetic flux, this conductance zero appears at the position of $\varepsilon_0=-{3\over 2}t_0$ when $U=2t_0$, but it presents itself at the point of $\varepsilon_0=-3t_0$ in the case of $U=4t_0$, as shown in Fig.\[manybody1\](a) and (c). Meanwhile, we can obtain mathematically that around the point of $\varepsilon_0=-\frac{U}{2}$, the average electron occupation number $\langle n_{m\sigma}\rangle$ is not equal to ${1\over 2}$ any more. Hence, the presence of decoupled molecular state destroys the electron-hole symmetry. All these numerical results can be explained as follows. For such a double-QD structure, one can understand that when the many-body terms are considered within the second-order approximation, the molecular levels are given by $\varepsilon_0-t_0, \varepsilon_0+t_0, \varepsilon_0-t_0+U$, and $\varepsilon_0+t_0+U$, respectively. So, when $\varepsilon_0=-{U\over 2}$, they distribute symmetrically about the Fermi level of the system, which results in $\langle n_{m\sigma}\rangle={1\over 2}$ and ${\cal G} =0$. But, since the unique QD-lead coupling manner of our model, in the absence of magnetic flux the bonding states ( with the levels $\varepsilon_0-t_0$ and $\varepsilon_0-t_0+U$) completely decouple from the leads and only the other two states ( $\varepsilon_0+t_0$ and $\varepsilon_0+t_0+U$ ) provide the channels for the electron transport. Obviously, under the condition of $\varepsilon_0=-{U\over 2}$ the levels $\varepsilon_0+t_0$ and $\varepsilon_0+t_0+U$ do not distribute symmetrically about the Fermi level of the system, thus the electron-hole symmetry is broken, which shows itself as the result of $\langle n_{m\sigma}\rangle\neq{1\over 2}$. Alternatively, in the case of $\phi=\pi$, only the bonding states couple to the leads, which also destroys the electron-hole symmetry, corresponding to the results in Fig.\[manybody1\](b) and (d). The case of 4-QD structure, as shown Fig.\[manybody2\], can also be clarified based on such an approach. Next we turn to focus on the situation of $\phi_\alpha=n\pi$ and $\phi_{\alpha'}\neq n\pi$. By virtue of Eq.(\[gamma\]) we can expect that for such a case, some molecular states of the QDs will decouple from lead-$\alpha$ but they still couple to lead-$\alpha'$. In Fig. \[phiLR\] it shows the research on the electron transport within the double-QD and four-QD $j=2$ structures. For the double-QD structure, it is obvious that in the case of $\phi_L=0$ and $\phi_R=0.5\pi$ both the bonding and antibonding states couple to lead-R but the bonding state decouples from lead-L. The corresponding numerical result is shown in Fig.\[phiLR\](a). Clearly, due to the decoupling of the bonding state from lead-L, antiresonance occurs at the position of $\varepsilon_0=t_0$. This means that in this case antiresonance will come about when the level of such a decoupled molecular state is consistent with the Fermi level, corresponding to the discussions in the previous works[@Liu2]. In the case of $\phi_L=0$ and $\phi_R=\pi$, the bonding state decouples from lead-$L$ with the antibonding state decoupled from lead-R. So herein there is no channel for the electron tunneling and the conductance is always equal to zero, despite the shift of QD level, corresponding to the dotted line in Fig.\[phiLR\](a). On the other hand, by fixing $\phi_R=\pi$ and increasing $\phi_L$ to $0.5\pi$, one will see that the decoupling of antibonding state from lead-R results in the antiresonance at the point of $\varepsilon_0=-t_0$. With regard to the 4-QD $j=2$ structure, the decoupling-induced antiresonance is also remarkable in the case of $\phi_\alpha=n\pi$ and $\phi_{\alpha'}\neq n\pi$. As shown in Fig.\[phiLR\](b) there are two kinds of antiresonant points in such a case: One originates from the quantum interference between the coupled molecular states, and the other is caused by the decoupling of some molecular states. With respect to the many-body effect, as shown in Fig. \[phiLR\](c), apart from the appearance of two groups in the conductance spectra, the electron-hole symmetry remains, which is due to that there is no completely decoupling of the molecular states from the leads in such a case. In Fig.\[infinite\] the linear conductances of the semi-infinite and infinite QD chains are presented as a function of gate voltage. As shown in Fig.\[infinite\], regardless of the semi-infinite or infinite QD chains, no conductance peak is consistent with any eigenlevel since the molecular states of the QDs become a continuum in such a case. In the case of continuum, although some molecular states decouple from the leads, it can not affect the electron transport for reasons that the electron transmission paths can not be differentiated. Thus no antiresonance appears in the conductance spectra. However, when investigating the influence of the difference between $\phi_L$ and $\phi_R$ on the electron transport, we find that in the situation of $\phi_L=0$ and $\phi_R=0.5\pi$ the conductance of the infinite QD chain encounters its zero at the down side of energy band, as shown in Fig.\[infinite\](d). Such a result can be explained as follows. The coupling of a semi-infinite QDs to QD-$l$ indeed brings out the self-energy $\Sigma_{l\sigma}=\frac{1}{2}(-\varepsilon_0-i\sqrt{4t_0^2-\varepsilon_0^2})$. Then at both sides of the energy band (i.e., $\varepsilon_0=\pm2t_0$), the structure is just transformed into a new double-QD configuration with $\varepsilon_1=\varepsilon_2={\varepsilon_0\over 2}=t_0$. As a result, in the case of $\varepsilon_0=2t_0$ the bonding state of such a new double-QD structure, which has decoupled from lead-L since $\phi_L=0$ and $\phi_R=0.5\pi$ ( as discussed in the previous paragraph ), is aligned with the Fermi level, thereupon, the electron tunneling here presents antiresonance. Alternatively, for the case of $\phi_L=0.5\pi$ and $\phi_R=\pi$ a similar reason gives rise to antiresonance at the position of $\varepsilon_0=-2t_0$. In addition, it is apparent that in the case of $\phi_L=0$ and $\phi_R=\pi$ the conductance is always fixed at zero. This is because that in such a situation any molecular state coupled to lead-$\alpha$ is inevitable to decouple from lead-$\alpha'$, though the molecular states of the QDs is continuum. Thus, there is still no channel for the electron transport. summary ======= With the help of nonequilibrium Green function technique, the electron transport through a QD chain embodied in an AB interferometer has been theoretically investigated. It has been found that for the configurations with the same-numbered QDs coupled to the QDs in the interferometer arms, in the case of $\phi=2n\pi$ all odd molecular states of the QDs decouple from the leads, but all even molecular states decouple from the leads when the magnetic flux phase factor is equal to $(2n-1)\pi$. With the increase of magnetic flux from $(2n-1)\pi$ to $2n\pi$, the antiresonance position in the electron transport spectrum is independent of the change of the decoupled molecular states. By representation transformation, these results are analyzed in detail and the quantum interference in these structures are therefore clarified. When the many-body effect is considered up to the second-order approximation, we showed that the emergence of decoupling gives rise to the apparent destruction of electron-hole symmetry. Finally, the cases of different magnetic fluxes through the two subrings were studied, it showed that via the adjustment of the magnetic flux through either subring, some molecular states would decouple from one lead but still couple to the other, which cause the occurrence of new antiresonances. At last, we would like to point out that the theoretical model in the present work can also be regarded as a double-QD AB interferometer with some impurities side-coupled to the QDs in its arms[@Liu2; @EPL], thus the calculated results can mimic the influence of impurity states on the electronic transport behaviors in such a structure. 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--- bibliography: - 'bibkumwed.bib' --- Introduction {#introduction .unnumbered} ============ Let $C$ be a smooth genus 2 curve. Let $Pic^1(C)$ be the Picard variety that parametrizes all degree 1 line bundles on $C$ and $\Theta$ the canonical theta divisor made up set-theoretically of line bundles $L$ s.t. $h^0(C,L)\neq 0$. We will denote ${\mathcal{SU}_C(2)}$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. The description of this moduli space dates back almost fourty years ago to the paper [@rana:cra]. Ramanan and Narashiman proved that ${\mathcal{SU}_C(2)}$ is isomorphic to the linear system $|2\Theta|\cong {\mathbb{P}}^3$ on $Pic^1(C)$ and that the semi-stable locus is exactly the Kummer quartic surface image of the Jacobian $Jac(C)$ via the Kummer map. In this paper we look at ${\mathcal{SU}_C(2)}$ in a different frame. Let $\omega$ be the canonical bundle on $C$, we consider the space ${\mathbb{P}}Ext^1(\omega, \omega^{-1})=: {{\mathbb{P}}^4_{\omega}}= |\omega^3|^*$. This space parametrizes extension classes $(e)$ of $\omega$ by $\omega^{-1}$. $$0 {\longrightarrow}\omega {\longrightarrow}E_e {\longrightarrow}\omega^{-1} {\longrightarrow}0\ \ \ \ \ \ \ (e)$$ Therefore there exists a classifying map $$\varphi:{{\mathbb{P}}^4_{\omega}}\dashrightarrow {\mathcal{SU}_C(2)}$$ that associates the vector bundle $E_e$ to the extension class $(e)$. Bertram showed in [@ab:rk2] that $\varphi$ is given by the quadrics in the ideal $\mathcal{I}_C(2)$ of the curve, that is naturally embedded as a sextic in ${{\mathbb{P}}^4_{\omega}}$. In the first part of the paper we describe the fibers of the map $\varphi$. Let $E\in{\mathcal{SU}_C(2)}$ and $C_E$ the closure $\overline{\varphi^{-1}(E)}$ of the fiber of $E$, then the principal results of section 1 are the following. Let $E$ be a stable vector bundle, then $dim C_E=1$ and $C_E$ is a smooth conic. Let $E$ be a strictly semi-stable vector bundle, then $C_E$ is singular. If $E\cong L\oplus L^{-1}$ for $L\in JC[2]/\mathcal{O}_C$ then $C_E$ is a double line. Moreover we show that the fiber over the $S$-equivalence class of the bundle ${\mathcal{O}}_C \oplus {\mathcal{O}}_C$ is a cone $S\in {{\mathbb{P}}^4_{\omega}}$ over a twisted cubic curve. In Section 2 we blow up ${{\mathbb{P}}^4_{\omega}}$ along the surface $S$ and $|2\Theta|$ at the origin of the Kummer surface $K^0$ that represents the semi-stable boundary. Let $Bl_S({{\mathbb{P}}^4_{\omega}})$ be the blow up of ${{\mathbb{P}}^4_{\omega}}$ and ${\mathbb{P}}^3_0$ the blow-up of $|2\Theta|$, we describe the induced map $\tilde{\varphi}:Bl_S({\mathbb{P}}^4){\longrightarrow}{\mathbb{P}}^3_{{\mathcal{O}}}$ and we prove that the restriction of $\tilde{\varphi}$ to the exceptional divisors is a conic bundle. The main theorem of Section 2 is in fact the following. The morphism $$\tilde{\varphi}:Bl_S({\mathbb{P}}^4){\longrightarrow}{\mathbb{P}}^3_{{\mathcal{O}}}$$ is a conic bundle whose discriminant locus is the blow-up at the origin of the Kummer surface $K^0$. Moreover we construct a rank 2 vector bundle $\mathcal{A}$ on ${\mathbb{P}}^2$, such that ${\mathbb{P}}(\mathcal{A}\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}) \cong Bl_S({\mathbb{P}}^4)$. This leads us to prove that our conic bundle can be seen as a section $$\tilde{\varphi}\in \mathrm{Hom}({\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}\oplus{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1),Sym^2 (\mathcal{A}^*\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}) \otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1)).$$ This vector space has dimension 16 thus we have a moduli map that associates to a smooth genus 2 curve the conic bundle induced by its classifying map. $$\begin{aligned} \Xi:\{ \mathrm{smooth\ genus\ 2\ curves} \} & {\longrightarrow}& {\mathbb{P}}^{15}={\mathbb{P}}B;\\ C & \mapsto & \tilde{\varphi}_C.\end{aligned}$$ In Section 3 we study the stability and the deformations of the bundle $\mathcal{A}$. By applying the Hoppe criterion (Prop. \[kit\]), we get the following Theorem. The vector bundle ${\mathcal{A}}$ on ${{\mathbb{P}}^2_{\omega}}$ is stable. Finally, via a few cohomology calculations we find the dimension of $Ext^1(\mathcal{A},\mathcal{A})$. The space of deformations of ${\mathcal{A}}$ has dimension $dim(Ext^1({\mathcal{A}},{\mathcal{A}}))=5$. *Acknowledgments:* I would like to thank my Phd advisor Christian Pauly for his suggestions, Chiara Brambilla for a useful advice and Yves Laszlo for posing the problem of the deformations of the vector bundle $\mathcal{A}$. The classifying map =================== Preliminaries on extension classes ---------------------------------- Let $C$ be a smooth genus 2 curve and $\lambda$ the hyperelliptic involution on $C$; we will denote $\mathcal{W}$ the set of the Weierstrass points of $C$. Let also $Pic^d(C)$ be the Picard variety parametrizing degree $d$ line bundles over $C$ and $Jac(C)=Pic^0(C)$ the Jacobian variety of $C$. We will denote $K^0$ the Kummer surface obtained as quotient of $Jac(C)$ by $\pm Id$ and $K^1$ the quotient of $Pic^1(C)$ by the involution $\tau:\xi\mapsto \omega \otimes \xi^{-1}$. Moreover we remark that the 16 theta characteristics are the fixed points of the involution $\tau$. Let $\Theta\subset Pic^1(C)$ be the Riemann theta divisor. It is isomorphic to the curve $C$ via the Abel-Jacobi embedding $$\begin{aligned} \label{eq:aj} Aj: C & \hookrightarrow & Pic^1(C), \\ p & \mapsto & {\mathcal{O}}_C(p). \nonumber\end{aligned}$$ Let ${\mathcal{SU}_C(2)}$ be the moduli space of semi-stable rank two vector bundles on $C$ with trivial determinant. It is isomorphic to ${\mathbb{P}}^3\cong |2\Theta|$, the isomorphism being given by the map [@bo:fib1] $$\begin{aligned} \label{tetta} \theta: {\mathcal{SU}_C(2)}& {\longrightarrow}& |2\Theta|,\\ E & \mapsto & \theta(E);\nonumber\end{aligned}$$ where $$\theta(E):=\{L \in Pic^1(C)| h^0(C,E\otimes L)\neq 0\}.$$ With its natural scheme structure, $\theta(E)$ is in fact linearly equivalent to $2\Theta$. The Kummer surface $K^0$ is embedded in $|2\Theta|$ and points in $K^0$ correspond to bundles $E$ whose S-equivalence class $[E]$ contains a decomposable bundle of the form $M\oplus M^{-1}$, for $M\in Jac(C)$. Furthermore on the semistable boundary the morphism $\theta$ restricts to the Kummer map. Let $\omega$ be the canonical line bundle on $C$. We introduce the 4-dimensional projective space $${\mathbb{P}}^4_{\omega}:={\mathbb{P}}Ext^1(\omega,\omega^{-1})=|\omega^3|^*.$$ A point $e \in {{\mathbb{P}}^4_{\omega}}$ corresponds to an isomorphism class of extensions $$0{\longrightarrow}\omega^{-1} {\longrightarrow}E_e {\longrightarrow}\omega{\longrightarrow}0.\ \ \ \ \ \ (e)$$ We denote by $\varphi$ the rational classifying map $$\begin{aligned} \varphi:{{\mathbb{P}}^4_{\omega}}& \dashrightarrow & |2\Theta|\\ e & \mapsto & \textrm{S-equivalence class of } E_e.\end{aligned}$$ Let $\mathcal{I}_C$ be the ideal sheaf of the curve $C\subset {{\mathbb{P}}^4_{\omega}}$, Bertram ([@ab:rk2], Theorem 2) showed that there is an isomorphism (induced via pull-back by $\varphi$) $$H^0({\mathcal{SU}_C(2)},\mathcal{O}(2\Theta))\cong H^0({{\mathbb{P}}^4_{\omega}},\mathcal{I}_C\otimes\mathcal{O}(2)).$$ Therefore the classifying map $\varphi$ is the rational map given by the full linear system of quadrics contained in the ideal of $C \subset {{\mathbb{P}}^4_{\omega}}$. In fact the locus of non semi-stable extensions is exactly represented by $C$, as the next lemma shows. \[lem:ber\][@ab:rk2] Let $(e)$ be an extension class in ${{\mathbb{P}}^4_{\omega}}$ and $Sec(C)$ the secant variety of $C\subset {{\mathbb{P}}^4_{\omega}}$, then the vector bundle $E_e$ is not semistable if and only if $e \in C$ and it is not stable if and only if $e \in Sec(C)$. \[re:marco\] One can say even more. In fact, given $x,y\in C$ the secant line $\overline{xy}$ is the fiber of $\varphi$ over the S-equivalence class of $\omega(-x-y)\oplus\omega^{-1}(x+y)$.\ This implies directly the following Corollary. The image of the secant variety Sec(C) by the classifying map $\varphi$ is the Kummer surface $K^0\subset |2\Theta|$. The hyperelliptic involution $\lambda$ acts on the canonical line bundle over $C$ and on its spaces of sections. A straightforward Riemann-Roch computation shows that $h^0(C,\omega^3)^*=5$. Let $\pi: C\rightarrow {\mathbb{P}}^1$ be the hyperelliptic map. There is a canonical linearization for the action of $\lambda$ on $\omega$ that comes from the fact that $\omega=\pi^*\mathcal{O}_{{\mathbb{P}}^1}(1)$. In fact, by Kempf’s Theorem ([@dn:pfv], Théorème 2.3), a line bundle on $C$ descends to ${\mathbb{P}}^1$ if and only if the involution acts trivially on the fibers over Weierstrass points. Thus we choose the linearization $\delta:\lambda^*\omega\stackrel{\sim}{\rightarrow} \omega$ that induces the identity on the fibers over Weierstrass points. This means that $$Tr(\lambda: L_{w_i}\rightarrow L_{w_i})=1,$$ for every Weierstrass point $w_i$. Moreover we have that $d\lambda_{w_i}=-1$, which implies, via the Atiyah-Bott-Lefschetz fixed point formula ([@gh:pag], p.421), that $$h^0(C,\omega^3)_+ - h^0(C,\omega^3)_-=3.$$ Since $ h^0(C,\omega^3)_+ + h^0(C,\omega^3)_-=5$, this means that $h^0(C,\omega^3)_+=4$ and $h^0(C,\omega^3)_-=1$ and we can see that $$H^0(C,\omega^3)_-=\sum_{i=1}^6 w_i.$$ Furthermore, we have $$E_{\lambda(e)}=\lambda^*E_e$$ thus the points of ${{\mathbb{P}}^3_{\omega+}}:= {\mathbb{P}}H^0(C,\omega^3)_+^*$ represent involution invariant extension classes. We have studied this classes in [@bol:wed]; in this paper our aim is to describe more precisely the classifying map. The fibers of the classifying map --------------------------------- Let $p\in {\mathbb{P}}^3\cong |2\Theta|$ be a general point, then the fiber $\varphi^{-1}(p)$ is the intersection of 3 quadrics $$Q_1\cap Q_2 \cap Q_3 = \varphi^{-1}(p).$$ If $dim (\varphi^{-1}(p))=1$ then the fiber $\varphi^{-1}(p)$ is a degree 8 curve. Since $C\subset \varphi^{-1}(p)$ and $deg(C)=6$, the residual curve $\varphi^{-1}(p)- C$ is a conic.\ We will often denote $V$ the space of global sections $H^0(C,\omega).$ We have the following equality $$H^0(C,\omega^3)_+=Sym^3V$$ and we denote $X\subset {\mathbb{P}}(Sym^3 V^*)$ the twisted cubic curve image of ${\mathbb{P}}V \cong {\mathbb{P}}^1$ via the cubic Veronese morphism. Let’s consider now the linear subspace $<D>$ of ${{\mathbb{P}}^4_{\omega}}$ generated by the points of a divisor $D\in |\omega^2|$. Since $D\in |\omega^2|$, we can write it down as $$D= a + b + \lambda(a) + \lambda (b)$$ for $a,b \in C$. Furthermore we remark that the annihilator of $<D>$ is $H^0(C,\omega^3(-a-b-\lambda(a)-\lambda(b)))$, that has dimension equal to 2. This means that the linear envelop $<D>$ is a ${\mathbb{P}}^2$, and we shall denote it as ${\mathbb{P}}^2_{ab}$. [@ln85]\[carol\] Let $c,d \in C$ and $e \in {{\mathbb{P}}^4_{\omega}}$ an extension $$0{\longrightarrow}\omega^{-1}\stackrel{i_e}{{\longrightarrow}} E_e \stackrel{\pi_e}{{\longrightarrow}}\omega{\longrightarrow}0.$$ Then $e \in {\mathbb{P}}^2_{cd}$ if and only if it exists a section $\beta \in H^0(C,Hom(\omega^{-1},E))$ s.t. $$Zeros(\pi_e \circ \beta)=c+d+ \lambda(c) + \lambda (d).$$ We will denote $[{\mathcal{O}}_C \oplus {\mathcal{O}}_C]$ the $S$-equivalence class of the rank 2 bundle ${\mathcal{O}}_C \oplus {\mathcal{O}}_C$. \[pr:giu\] Let $$|\omega^3|^*={{\mathbb{P}}^4_{\omega}}\stackrel{\varphi}{\dashrightarrow}{\mathbb{P}}^3=|2\Theta|$$ be the classifying map. Then the closure of $\varphi^{-1}([{\mathcal{O}}_C \oplus {\mathcal{O}}_C])$ is the cone $S$ over a twisted cubic curve $X\subset {{\mathbb{P}}^3_{\omega+}}$. *Proof:* The vertex of $S$ is the point $x={\mathbb{P}}H^0(C,\omega^3)^*_-\in {{\mathbb{P}}^4_{\omega}}$, that is the projectivized anti-invariant eigen-space. This means that every line contained in $S$ is a secant of $C$ invariant under the involution of ${{\mathbb{P}}^4_{\omega}}$. Such a secant line can be written as $\overline{p\lambda(p)}$. The image of such a secant line via $\varphi$ is the origin, hence $S \subset \varphi^{-1}([{\mathcal{O}}_C \oplus {\mathcal{O}}_C])$. In order to prove the opposite inclusion we remark that a vector bundle $E$ contained in the $S$-equivalence class of the origin satisfies the following exact sequence $$0{\longrightarrow}{\mathcal{O}}_C\stackrel{\nu_E}{{\longrightarrow}} E {\longrightarrow}{\mathcal{O}}_C {\longrightarrow}0.$$ This implies that the trivial bundle is a sub-bundle of $E$. Let us consider the morphism $\varsigma$, composition of $\nu_E$ and $\pi_E$: $$\varsigma: {\mathcal{O}}_C \stackrel{\nu_E}{{\longrightarrow}} E \stackrel{\pi_E}{{\longrightarrow}} \omega.$$ Then $\delta \in H^0(C,\omega)=Hom ({\mathcal{O}}_C,\omega)$ and we obtain the following diagram. $$\begin{array}{ccccccccc} & & & & {\mathcal{O}}_C & & & &\\ & & & & \downarrow &\stackrel{\searrow}{\varsigma} & & &\\ 0 & {\rightarrow}& \omega^{-1} & {\rightarrow}& E & {\rightarrow}& \omega & {\rightarrow}& 0.\\ \end{array}$$ This means that the morphism $\varsigma$ is a section of $H^0(C,\omega)$ and its divisor is of type $a + \lambda(a)$ for $a \in C$. Because of Proposition \[carol\] this implies that the extension classes contained in the fiber $\varphi^{-1}([{\mathcal{O}}_C \oplus {\mathcal{O}}_C])$ belong to an invariant secant line, therefore $\varphi^{-1}([{\mathcal{O}}_C \oplus {\mathcal{O}}_C])\subset S \square$\ We will see that the map $\varphi$ defines a conic bundle on $|2\Theta|- [{\mathcal{O}}_C \oplus {\mathcal{O}}_C]$. We will now describe the fibers of $\varphi$ on the open set complementary to the origin.\ Let $E\in{\mathcal{SU}_C(2)}$ we will call $C_E$ the closure $\overline{\varphi^{-1}(E)}$ of the fiber of $E$. \[yo\] Let $E$ be a stable vector bundle, then $dim C_E=1$ and $C_E$ is a smooth conic. *Proof:* Let $E$ be a stable bundle and $e$ the following equivalence class of extensions $$\label{eq:fly} 0{\longrightarrow}\omega^{-1}{\longrightarrow}E_e\cong E {\longrightarrow}\omega {\longrightarrow}0$$ For a general bundle $E$, $C_E$ has dimension equal to 1. In fact for a genus 2 curve the Riemann-Roch theorem gives $\chi(E\omega)=4+2(-1)=2$. Moreover, by Serre duality, we have $h^1(E\omega)=h^0(E^*)$ and $h^0(E^*)=h^0(Hom(E,\mathcal{O}_C))=0$ because $E$ is stable.\ We define the following map $$\begin{aligned} j:\varphi^{-1}(E) & {\longrightarrow}& {\mathbb{P}}H^0(C,E\omega)={\mathbb{P}}^1,\\ e & \mapsto & j(e),\end{aligned}$$ that sends the extension class $e\in C_E$ on the point of ${\mathbb{P}}H^0(C,E\omega)$ corresponding to the first morphism of the exact sequence \[eq:fly\]. This map has degree 1 and it is not defined on the points of $C_E\cap C$.\ Furthermore we remark that the projection from ${\mathcal{SU}_C(2)}$ with centre $[{\mathcal{O}}_C \oplus {\mathcal{O}}_C]$ can be described in the following way. $$\begin{aligned} \Delta:{\mathcal{SU}_C(2)}& \dashrightarrow & {\mathbb{P}}^2 = |\omega^2|,\\ E & \mapsto & D(E).\\\end{aligned}$$ Here by $D(E)$ we mean the divisor on $C$ with the following support. $$Supp(D(E))=\{p\in C| h^0(E\otimes \mathcal{O}_C(p))\neq 0\}.$$ In fact the projection from ${\mathcal{O}}$ is exactly the restriction to $C$, embedded in $Pic^1(C)$ via the Abel-Jacobi map of equation \[eq:aj\], of the map $\theta$ from equation \[tetta\]. Now we consider the determinant map $$\begin{aligned} \bigwedge^2 H^0(E\omega) & {\longrightarrow}& H^0(\omega^2),\\ s\wedge t & \mapsto & Zero(s\wedge t).\\\end{aligned}$$ Let $p\in C$ be a point of the curve, if $p\in$ Zero$(s \wedge t)$ then there exists a non zero section $s_p\in H^0(C,E\omega (-p))$. Hence $h^0(C,E\omega(-p))\neq 0$. Moreover, if we make the hyperelliptic involution $\lambda$ act on $E\omega(-p)$ we find that $h^0(C,E\omega(-p))=h^0(C,E\otimes {\mathcal{O}}_C(p))\neq 0$. This implies that the zero divisor of $s\wedge t$ is $D(E)$. Now $D(E)$ has degree 4 and for every $p\in D(E)$ it exists a section $s_p$. We remark then that the morphism $j$ is surjective on the open set ${\mathbb{P}}H^0(C,E\omega)/ \{s_p|p\in D(E) \}$, that means that it is dominant.\ In order to end the proof we need 3 technical lemmas. \[shit\] Let $D\in |\omega^2|$ be the divisor $a+ b + \lambda(a) + lambda (b)$. The image of $$\varphi_{|{\mathbb{P}}^2_{ab}}:{\mathbb{P}}^2_{ab} \dashrightarrow {\mathbb{P}}^3$$ is the fiber of $\Delta$ over $D$, i.e. the line passing by $[{\mathcal{O}}_C\oplus {\mathcal{O}}_C]$ and the point corresponding to $D$ in $|\omega^2|$. *Proof:* We remark first that $D=C\cap {\mathbb{P}}^2_{ab}$, because $H^0(C,\omega^3(-a-b-\lambda(a)-\lambda(b)- c))=H^0(C,\omega - c)=1$ for every $c\in C$. This implies that the restriction $$\varphi_{|{\mathbb{P}}^2_{ab}}:{\mathbb{P}}^2_{ab} \dashrightarrow {\mathbb{P}}^3$$ is given by quadrics passing by the four points of $D$. The space of quadrics on ${\mathbb{P}}^2$ has dimension 5 and we impose 4 independent linear conditions. So the image of ${\mathbb{P}}^2_{ab}$ via $\varphi_{|{\mathbb{P}}^2_{ab}}$ is a ${\mathbb{P}}^1\subset {\mathbb{P}}^3$. Let $e$ be an extension class in ${\mathbb{P}}^2_{ab}$ and $E_e$ its image via $\varphi$ in ${\mathcal{SU}_C(2)}$. Now, by Proposition \[carol\], the extension $e$ belongs to ${\mathbb{P}}^2_{ab}$ if and only if it exists a section $\alpha \in H^0(C,Hom(\omega^{-1},E))$ s.t., using the notation of the following diagram, we have Zeros$(\pi_e \circ \alpha)=D$. $$\begin{array}{ccccccccc} & & & & \omega^{-1} & & & &\\ & & & & \alpha \downarrow & \searrow & & & \\ 0 & {\rightarrow}& \omega^{-1} & \stackrel{i_e}{{\rightarrow}} & E_e & \stackrel{\pi_e}{{\rightarrow}} & \omega & {\rightarrow}0. \\ \end{array}$$ This implies that $\alpha$ and $i_e$ are 2 independent sections of $E\omega$ and Zeros$(i_e\wedge \alpha)= D$. $\square$\ \[anto\] Let $E\in {\mathcal{SU}_C(2)}$, then we have the equality $$C_E\cap C= D(E).$$ *Proof:* Let $c,d \in C$ and let us suppose that $D(E)=c+d+\lambda(c)+\lambda(d)$. We remind that ${\mathbb{P}}^2_{cd}$ is the plane s.t. $c+d+\lambda(c)+\lambda(d)\subset {\mathbb{P}}^2_{cd}$. By Lemma \[shit\] the fiber $C_E$ is a conic contained in ${\mathbb{P}}^2_{cd}$ and passing by the 4 points of $D(E)$. Since $D(E)=C\cap {\mathbb{P}}^2_{cd}$ we obtain the equality $C_E\cap C= D(E)$.$\square$ \[ss\] Let $E\in {\mathcal{SU}_C(2)}$ be a stable bundle, then we have a decomposition $$H^0(C,E\omega)=H^0(C,E\omega)_+\oplus H^0(C,E\omega)_-$$ of $H^0(C,E\omega)$ in two eigen-spaces of dimension 1. *Proof:* We will use the Atiyah-Bott-Lefschetz formula, so we must choose a linearization $$\nu:\lambda^*E \stackrel{\sim}{{\longrightarrow}} E$$ for the action of $\lambda$ on $E$ and watch how it acts on the fibers over the points of $\mathcal{W}$. The bundle $E$ is an extension $$0{\longrightarrow}\omega^{-1} {\longrightarrow}E {\longrightarrow}\omega {\longrightarrow}0$$ so a linearization on $E$ is defined once one chooses two linearizations on $\omega $ and $\omega^{-1}$. We have already chosen $$\delta:\lambda^*\omega {\longrightarrow}\omega$$ that acts trivially on the fibers over the points of $\mathcal{W}$. For the line bundle $\omega^{-1}$ we have two different choices: the linearization that acts trivially on the fibers over the points of $\mathcal{W}$ and its inverse. Let $x\in \mathcal{W}$, then we can decompose $$\label{deco} E_x=\omega_x \oplus \omega^{-1}_x.$$ If we choose the first linearization on $\omega^{-1}$ then, by Kempf Lemma, the vector bundle $E$ would be the pull-back of a bundle $F$ over ${\mathbb{P}}^1$ defined by the exact sequence $$\label{pale} 0{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}(-1) {\longrightarrow}F {\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}(1) {\longrightarrow}0.$$ We remark that the only vector bundle on ${\mathbb{P}}^1$ that verifies the exact sequence (\[pale\]) is ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(1)$ and it is not semi-stable. Then the choice of $\delta$ as a linearization for $\omega$ forces us to choose the linearization $$\tilde{\delta}:\lambda^*\omega^{-1} \stackrel{\sim}{{\longrightarrow}} \omega^{-1}$$ that induces $-Id$ on the fibers over the points of $\mathcal{W}$. We recall that $h^0(C,E\omega)=2$. Thanks to the decomposition (\[deco\]) the trace of the linearization $\delta \oplus \tilde{\delta}$ is zero. Then by the Atiyah-Bott-Lefschetz formula we have $$h^0(C,E\omega)_+-h^0(C,E\omega)_-=0,$$ that means $$h^0(C,E\omega)_+=h^0(C,E\omega)_-=1.$$ $\square$ *Continuation of the proof of Theorem \[yo\]:* Thanks to Lemmas \[shit\] and \[anto\] we can extend the morphism $j$ to the closure $C_E=\overline{\varphi^{-1}(E)}$. We send every point $p\in D(E)$ on the section $s_p$ that vanishes in $p$. We will denote this morphism $$\tilde{j}:C_E {\longrightarrow}{\mathbb{P}}^1.$$ The conic $C_E$ is either smooth, or the union of two disjoint lines, or a double line. The morphism $\tilde{j}$ is surjective on ${\mathbb{P}}^1$ and its degree is 1. Then it exists a morphism $\tau$ s.t. $ \tilde{j}\circ \tau = Id_{{\mathbb{P}}^1}$. This means that we have an isomorphism between ${\mathbb{P}}^1={\mathbb{P}}H^0(C,E\omega)$ and a component of $C_E$. The morphism $\tau$ is equivariant under the action of $\lambda$ and by Lemma \[ss\] the space $H^0(C,E\omega)$ has two eigen-spaces. This means that the component that is image of $\tau$ must cut ${{\mathbb{P}}^3_{\omega+}}$ in two different points, that means that $C_E$ is a smooth conic. $\square$ \[hl2\] Let $E$ be a strictly semi-stable vector bundle, then $C_E$ is singular. If $E\cong L\oplus L^{-1}$ for $L\in JC[2]/\mathcal{O}_C$ then $C_E$ is a double line. *Proof:* If the bundle $E$ is strictly semi-stable we know, by Lemma \[lem:ber\] and Remark \[re:marco\] that the fiber consists of two lines so it is either a rank 2 or a rank 1 conic. Moreover the fibers over the points of $JC[2]/\mathcal{O}_C$ are the double lines $\overline{w_iw_j}$, for $w_i,w_j$ two different Weierstrass points. These are all the $\lambda$-invariant couples of points, that means that the 15 2-torsion points are the rank 1 locus. $\square$\ The conic bundle ================ In the following, we will often denote ${\mathbb{P}}^2_{\omega}$ the linear system $|\omega^2|$. We will now define a rank 2 projective bundle $\mathcal{E}$ on ${{\mathbb{P}}^2_{\omega}}$ strictly connected to the classifying map.\ Since the rational map $$\Delta: {\mathcal{SU}_C(2)}\dashrightarrow {{\mathbb{P}}^2_{\omega}}$$ is surjective, every point of ${{\mathbb{P}}^2_{\omega}}$ can be represented by a divisor $\Delta(E)$ for a semi-stable bundle $E$ on $C$. We start by constructing a rank 2 vector bundle $\mathcal{A}$ on ${{\mathbb{P}}^2_{\omega}}$. Let us first define the fiber $\mathcal{A}_E$ over the point $\Delta(E)$: we want that $\mathcal{A}_E\subset H^0(C,\omega^3)^*_+$ and that its dual is the cokernel of the natural multiplication map $$\label{wl} 0 {\longrightarrow}H^0(C,\omega) \stackrel{+\Delta(E)}{{\longrightarrow}} H^0(C,\omega^3)_+ {\longrightarrow}\mathcal{A}_E^*{\longrightarrow}0.$$ Furthermore we can generalize the sequence \[wl\] to an exact sequence (in fact a global version of the one just defined) of vector bundles on ${{\mathbb{P}}^2_{\omega}}$. In order to do this we define a new rank 2 vector bundle $\mathcal{G}$ on ${\mathbb{P}}^2$. We remark in fact that it exists a natural morphism of vector bundles $$\nu:{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1) {\longrightarrow}H^0(C,\omega^2)\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}$$ that sends the fiber over one point $p$ on the line in $H^0(C,\omega^2)$ whose projectivized is $p$ . Now we twist by $H^0(C,\omega)$ the morphism $\nu$: we get the morphism $$Id_{H^0(C,\omega)}\otimes \nu=:\nu^{\prime} :H^0(C,\omega) \otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1) {\longrightarrow}H^0(C,\omega) \otimes H^0(C,\omega^2)\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}.$$ Moreover it exists a natural multiplication morphism $$\mu:H^0(C,\omega) \otimes H^0(C,\omega^2)\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} {\longrightarrow}H^0(C,\omega^3)_+\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}.$$ We define $$\mathcal{G}:=H^0(C,\omega) \otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1)$$ and the injective morphism $$\alpha:= \mu \circ \nu^{\prime}: \mathcal{G} {\longrightarrow}H^0(C,\omega^3)_+\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}.$$ The bundle $\mathcal{G}$ is in fact a sub-bundle of ${\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}\otimes H^0(C,\omega^3)_+$: the fiber over every $\Delta(E)\in {{\mathbb{P}}^2_{\omega}}$ is composed by the divisors of $H^0(C,\omega^3)_+$ of the form $\Delta (E) + \delta$, with $\delta \in H^0(C,\omega)$. We define $$\mathcal{A}^*:=coker (\alpha: \mathcal{G} {\longrightarrow}H^0(C,\omega^3)_+\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}})$$ and we get the following exact sequence of bundles on ${{\mathbb{P}}^2_{\omega}}$. $$\label{ff} 0{\longrightarrow}\mathcal{G} {\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}\otimes H^0(C,\omega^3)_+ {\longrightarrow}\mathcal{A^*}{\longrightarrow}0.$$ We define the vector bundle $\mathcal{E}$ on ${{\mathbb{P}}^2_{\omega}}$ as $$\mathcal{E}:=\mathcal{A}\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}.$$ The situation is resumed in the following diagram. $$\begin{array}{ccc} {\mathbb{P}}\mathcal{E} & \hookrightarrow & |\omega^3|^* \times {{\mathbb{P}}^2_{\omega}}\\ &&\\ \stackrel{|}{\downarrow} & &\stackrel{|}{\downarrow} \\ &&\\ {\mathbb{P}}\mathcal{A} & \hookrightarrow & {\mathbb{P}}Sym^3 V^* \times {{\mathbb{P}}^2_{\omega}}\\ &&\\ & \searrow & \downarrow \\ &&\\ & & {{\mathbb{P}}^2_{\omega}}\\ \end{array}$$ Let $y \in {{\mathbb{P}}^2_{\omega}}$. Then we can identify $y$ and a divisor $a+b$ of degree 2 on $X$, with $a,b\in X$. The fiber ${\mathbb{P}}\mathcal{A}_y$ is the secant line $\overline{ab}$ to $X\subset {\mathbb{P}}Sym^3 V^*$ and ${\mathbb{P}}\mathcal{E}_y$ is the plane $<\overline{ab},x> \subset |\omega^3|^*$.\ Let $$pr_x: {{\mathbb{P}}^4_{\omega}}\dashrightarrow {{\mathbb{P}}^3_{\omega+}}$$ be the projection with centre $x$. We have a commutative diagram of rational maps $$\begin{array}{ccc} {{\mathbb{P}}^4_{\omega}}& \stackrel{\varphi}{\dashrightarrow} & |\mathcal{I}_C(2)|^*={\mathbb{P}}^3 \\ \stackrel{|}{\downarrow} pr_x & & \stackrel{|}{\downarrow} \Delta \\ {{\mathbb{P}}^3_{\omega+}}& \stackrel{\phi}{\dashrightarrow} & {{\mathbb{P}}^2_{\omega}}\cong |\mathcal{I}_X(2)|^*, \end{array}$$ where $\phi$ is defined as follows. Given a $t\in {\mathbb{P}}^3 \setminus X$, let $l_t$ be the only secant line to $X$ passing by $t$. The application $\phi$ sends $t$ on the pencil ${\mathbb{P}}^1\subset|\mathcal{I}_X(2)|$ given by the quadrics vanishing on the union $X\cup l_t$. *Proof:* First we show that there exists a unique secant line $l$ to $X$ that passes by $t$. Projecting $X$ with centre $t$ we remark that the image is a plane cubic, that has one knot by the genus formula. This implies that there exists a unique secant line to $X$ passing by $t$. Moreover the projection $pr_x$ induces an isomorphism $$pr_x^*|\mathcal{I}_X(2)|\cong |I_S(2)|.$$ Since $\varphi^{-1}({\mathcal{O}})=S$ (Prop. \[pr:giu\]) and $\Delta$ is the projection with centre $[{\mathcal{O}}\oplus {\mathcal{O}}]$, the diagram commutes. $\square$ The map $\phi$ can be defined in a different way. Since\ ${{\mathbb{P}}^2_{\omega}}\cong {\mathbb{P}}Sym^2 H^0(C, \omega)$, $\phi$ is the map that sends $t\in {{\mathbb{P}}^3_{\omega+}}\setminus X$ on the pair of points of $X$ cut out by the unique secant line $l_t$ to $X$ passing by $t$. Let $Bl_X{{\mathbb{P}}^3_{\omega+}}$ be the blow-up of ${{\mathbb{P}}^3_{\omega+}}$ along the twisted cubic and $$\mu:Bl_X{{\mathbb{P}}^3_{\omega+}}{\longrightarrow}{{\mathbb{P}}^3_{\omega+}}$$ the projection on ${{\mathbb{P}}^3_{\omega+}}$. Since $X$ is scheme-theoretically defined by the 3-dimensional space of quadrics $\mathcal{I}_X(2)$ it exists a morphism $\tilde{\phi}$ that makes the following diagram commute. $$\begin{array}{ccc} Bl_X{{\mathbb{P}}^3_{\omega+}}& & \\ \mu\downarrow & \stackrel{\tilde{\phi}}{\searrow} & \\ X\subset{{\mathbb{P}}^3_{\omega+}}& \stackrel{\phi}{\dashrightarrow} & {{\mathbb{P}}^2_{\omega}}\\ \end{array}$$ Hence the morphism $\tilde{\phi}$ defines a ${\mathbb{P}}^1$-fibration on ${{\mathbb{P}}^2_{\omega}}$. Futhermore the exceptional divisor $E\subset Bl_X{{\mathbb{P}}^3_{\omega+}}$ is the projective bundle ${\mathbb{P}}(N_{X|{{\mathbb{P}}^3_{\omega+}}})$ of the normal bundle of $X\subset{{\mathbb{P}}^3_{\omega+}}$. We have an isomorphism $$N_{X|{\mathbb{P}}^3}\cong {\mathcal{O}}_{{\mathbb{P}}^1}(5) \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(5).$$ *Proof:* Let $$\begin{aligned} i:{\mathbb{P}}V & {\longrightarrow}& {{\mathbb{P}}^3_{\omega+}};\\ {[u:v]} & \mapsto & [u^3:u^2v:vu^2:v^3];\end{aligned}$$ be the Veronese embedding. We have the following exact sequence. $$\label{edy} 0{\longrightarrow}T_X {\longrightarrow}i^*T_{{{\mathbb{P}}^3_{\omega+}}} {\longrightarrow}N_{X|{\mathbb{P}}^3}{\longrightarrow}0.$$ Since $X\cong{\mathbb{P}}^1$ we have $T_X\cong {\mathcal{O}}_{{\mathbb{P}}^1}(2)$. Then we pull-back via $i^*$ the Euler exact sequence and we get $$0{{\longrightarrow}} {\mathcal{O}}_{{\mathbb{P}}^1} \stackrel{k}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}(3)^{\oplus 4}\stackrel{h}{{\longrightarrow}} i^* T_{{{\mathbb{P}}^3_{\omega+}}}{\longrightarrow}0.$$ Let $l$ be a local section of ${\mathcal{O}}_{{\mathbb{P}}^1}$ and $u,v$ the coordinates on ${\mathbb{P}}^1$, then we can write the morphism $k$ down in the following way $$\begin{aligned} k:{\mathcal{O}}_{{\mathbb{P}}^1} & {\longrightarrow}& {\mathcal{O}}_{{\mathbb{P}}^1}(3)^{\oplus 4};\\ l & \mapsto & (u^3l,u^2vl,uv^2l,v^3l).\end{aligned}$$ We denote $X,Y,Z,T$ the coordinates on ${\mathcal{O}}_{{\mathbb{P}}^1}(3)^{\oplus 4}$. Moreover the morphism $h$ is given by the equations of the line image of ${\mathcal{O}}_{{\mathbb{P}}^1}$ in ${\mathcal{O}}_{{\mathbb{P}}^1}(3)^{\oplus 4}$. Therefore we have $$\begin{aligned} h:{\mathcal{O}}_{{\mathbb{P}}^1}(3)^{\oplus 4} & {\longrightarrow}& i^*T_{{{\mathbb{P}}^3_{\omega+}}};\\ (X,Y,Z,T) & \mapsto & (vX-uY,vY-uZ,vZ-uT).\end{aligned}$$ Hence we have $i^*T_{{{\mathbb{P}}^3_{\omega+}}}\cong {\mathcal{O}}_{{\mathbb{P}}^1}(4)^{\oplus 3}$ and we can rewrite the exact sequence (\[edy\]) in the following way $$0 {\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1} (2)\cong T_X \stackrel{di}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^1}(4)^{\oplus 3}\cong i^*T_{{{\mathbb{P}}^3_{\omega+}}} {\longrightarrow}N_{X|{\mathbb{P}}^3}{\longrightarrow}0,$$ where $di$ is the differential of $i$. On the affine open set $\{v\neq 0\}$ the morphism $di$ is defined by the following equations $$\begin{aligned} di:{\mathcal{O}}_{{\mathbb{P}}^1} (2) & {\longrightarrow}& {\mathcal{O}}_{{\mathbb{P}}^1}(4)^{\oplus 3};\\ l & \mapsto & (3lu^2,2luv,lv).\end{aligned}$$ Let $(C,D,F)$ be the coordinates on ${\mathcal{O}}_{{\mathbb{P}}^1}(4)^{\oplus 3}$, then the equations of the line image of ${\mathcal{O}}_{{\mathbb{P}}^1}(2)$ in ${\mathcal{O}}_{{\mathbb{P}}^1}(4)^{\oplus 3}$ are $$(vA-uB,vB-uC);$$ this implies that $N_{X|{\mathbb{P}}^3}\cong {\mathcal{O}}_{{\mathbb{P}}^1}(5) \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(5).\square$ Since ${\mathcal{O}}_{{\mathbb{P}}^1}(5) \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(5)\cong ({\mathcal{O}}_{{\mathbb{P}}^1} \oplus {\mathcal{O}}_{{\mathbb{P}}^1}) \otimes {\mathcal{O}}_{{\mathbb{P}}^1}(5)$, we have $${\mathbb{P}}(N_{X|{\mathbb{P}}^3})\cong X \times {\mathbb{P}}^1 \cong {\mathbb{P}}^1 \times {\mathbb{P}}^1.$$ We will denote $$\sigma: {\mathbb{P}}\mathcal{A} {\longrightarrow}{{\mathbb{P}}^2_{\omega}}$$ the projection of the projective bundle and $E$ the exceptional divisor in $Bl_X{{\mathbb{P}}^3_{\omega+}}$. \[pr:a\] There exists an isomorphism of projective bundles on ${{\mathbb{P}}^2_{\omega}}$ $$\begin{array}{ccc} {\mathbb{P}}\mathcal{A} & \stackrel{\sim}{{\longrightarrow}} & Bl_X{{\mathbb{P}}^3_{\omega+}}\\ \sigma \searrow & & \swarrow_{\tilde{\phi}} \\ & {{\mathbb{P}}^2_{\omega}}& \\ \end{array}$$ *Proof:* Let $x\in {{\mathbb{P}}^2_{\omega}}$, $a+b$ the divisor on $X$ that corresponds to $x$ and $t\in {\mathbb{P}}\mathcal{A}_x$. We recall that $\mathcal{A}$ is a sub-bundle of ${\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}\otimes H^0(\omega^3)_+^*$ and the projectivization gives us an embedding $$\begin{aligned} k:{\mathbb{P}}\mathcal{A} & \hookrightarrow & {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^3_{\omega+}}.\\\end{aligned}$$ Moreover we have $$k({\mathbb{P}}\mathcal{A}_x)=\overline{ab}\subset {{\mathbb{P}}^3_{\omega+}}.$$ We will often consider ${\mathbb{P}}\mathcal{A}$ as a sub-variety of ${{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^3_{\omega+}}$. If $t\in {{\mathbb{P}}^3_{\omega+}}\setminus X$ then $\overline{ab}$ is the only secant line to $X$ passing by $t$.\ We define a morphism $$\begin{aligned} \varpi:= \tilde{\phi} \times \mu : Bl_X{{\mathbb{P}}^3_{\omega+}}& {\longrightarrow}& {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^3_{\omega+}}.\\\end{aligned}$$ The morphism $\varpi$ has a birational inverse $$\varpi^{-1}:{\mathbb{P}}\mathcal{A}\subset {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^3_{\omega+}}\dashrightarrow Bl_X{{\mathbb{P}}^3_{\omega+}}$$ defined as follows. We define $\varpi^{-1}$ on the open set of ${\mathbb{P}}\mathcal{A}$ given by the couples $(x,t)$ s.t. $x=a+b$ is a point of ${{\mathbb{P}}^2_{\omega}}$ s.t. $a\neq b$ and $t\in \{{\mathbb{P}}\mathcal{A}_x\setminus X\}$. We send $(x,t)\in{\mathbb{P}}\mathcal{A}$ on $\mu^{-1}(t)\in Bl_X{{\mathbb{P}}^3_{\omega+}}$. Then by Zariski’s main theorem the morphism $\varpi$ induces an isomorphism between $Bl_X{{\mathbb{P}}^3_{\omega+}}$ and ${\mathbb{P}}\mathcal{A}$. $\square$\ Let $$\beta: {\mathbb{P}}(\mathcal{A}\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}) \dashrightarrow {\mathbb{P}}\mathcal{A}$$ be the natural projection and $\eta$ the composed map $$\eta: {\mathbb{P}}\mathcal{E} \stackrel{\beta}{\dashrightarrow} {\mathbb{P}}\mathcal{A} \stackrel{\sigma}{{\longrightarrow}} {{\mathbb{P}}^2_{\omega}}.$$ Since the map $\varphi$ is given by the quadrics in the ideal of $C\subset {{\mathbb{P}}^4_{\omega}}$, there exists a morphism $$\overline{\varphi}:Bl_C{{\mathbb{P}}^4_{\omega}}{\longrightarrow}|2\Theta|$$ that makes the following diagram commute. $$\begin{array}{ccc} Bl_C{{\mathbb{P}}^4_{\omega}}& & \\ &&\\ \downarrow & \searrow^{\overline{\varphi}} & \\ &&\\ {{\mathbb{P}}^4_{\omega}}& \stackrel{\varphi}{\dashrightarrow} & |2\Theta|\\ \end{array}$$ We will denote ${\mathbb{P}}^3_{{\mathcal{O}}}$ the blow-up of $|2\Theta|={\mathbb{P}}^3$ at the point $[{\mathcal{O}}\oplus {\mathcal{O}}]$ and $pr_0 $ the morphism that resolves the projection $\Delta$ with centre $[{\mathcal{O}}\oplus {\mathcal{O}}]$ and that makes the following diagram commute. $$\begin{array}{ccc} {\mathbb{P}}^3_{{\mathcal{O}}} & & \\ &&\\ \downarrow & \searrow^{pr_0} & \\ &&\\ {\mathbb{P}}^3 & \stackrel{\Delta}{\dashrightarrow} & {{\mathbb{P}}^2_{\omega}}\\ \end{array}$$ The morphism $$pr_0:{\mathbb{P}}_{{\mathcal{O}}}^3{\longrightarrow}{{\mathbb{P}}^2_{\omega}}$$ defines a ${\mathbb{P}}^1$-fibration on ${{\mathbb{P}}^2_{\omega}}$ hence ${\mathbb{P}}_{{\mathcal{O}}}^3\cong {\mathbb{P}}M$ for some rank 2 vector bundle $M$ on ${{\mathbb{P}}^2_{\omega}}$. We denote $F= {\mathbb{P}}\mathrm{T}_{{\mathcal{O}}}{\mathbb{P}}^3 \cong {{\mathbb{P}}^2_{\omega}}$ the exceptional divisor over the origin in ${\mathbb{P}}^3_{{\mathcal{O}}}$ . The vector bundle $M$ is defined up to a line bundle $L$, because, as projective varieties ${\mathbb{P}}M \cong {\mathbb{P}}(M\otimes L)$, so we choose $M$ once and for all as the vector bundle s.t. $${\mathcal{O}}_{{\mathbb{P}}M}(1) = {\mathcal{O}}_{{\mathbb{P}}_{{\mathcal{O}}}^3}(F).$$ We have the equality $$M={\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} \oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1).$$ *Proof:* We have $M^*=pr_{0*}{\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(F)$ and we consider the restriction exact sequence $$\label{ja} 0{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}{\longrightarrow}{\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(F) {\longrightarrow}{\mathcal{O}}_F(F) {\longrightarrow}0.$$ We push down via $pr_{0}$ the exact sequence (\[ja\]) and we get $$0{\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} {\longrightarrow}M^* {\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1){\longrightarrow}0.$$ This means that $M^*$ determines an extension class $(e)$ in $Ext^1({\mathcal{O}}(-1),{\mathcal{O}})$. We remark that $$Ext^1({\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1),{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}})=H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1))=\{0\}$$ thus $M^*$ is the trivial extension, i.e. $$M^*={\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} \oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} (-1).$$ $\square$ By Proposition \[pr:giu\], we have $\overline{\varphi^{-1}({\mathcal{O}})}=S$, so there exists a morphism $$\tilde{\varphi}: Bl_S {{\mathbb{P}}^4_{\omega}}{\longrightarrow}{\mathbb{P}}^3_{{\mathcal{O}}}$$ that makes the following diagram commute. $$\begin{array}{ccc} Bl_S{{\mathbb{P}}^4_{\omega}}& & \\ &&\\ \downarrow & \searrow^{\tilde{\varphi}} &\\ &&\\ Bl_C {{\mathbb{P}}^4_{\omega}}& & {\mathbb{P}}^3_{{\mathcal{O}}} \\ &&\\ \downarrow & \searrow^{\overline{\varphi}} & \downarrow \\ &&\\ {{\mathbb{P}}^4_{\omega}}& \stackrel{\varphi}{\dashrightarrow} & |2\Theta|\\ \end{array}$$ We will denote $\varrho$ the composed map $$\varrho:Bl_S{{\mathbb{P}}^4_{\omega}}\stackrel{\tilde{\varphi}}{{\longrightarrow}} {\mathbb{P}}^3_{{\mathcal{O}}} \stackrel{pr_0}{{\longrightarrow}} {{\mathbb{P}}^2_{\omega}}$$ and $\pi$ the projection $$\pi:Bl_S{{\mathbb{P}}^4_{\omega}}{\longrightarrow}{{\mathbb{P}}^4_{\omega}}.$$ Let $S\subset {{\mathbb{P}}^4_{\omega}}$ be the cone over the twisted cubic $X$ of Proposition \[pr:giu\]. Then there exists an isomorphism of projective bundles on ${{\mathbb{P}}^2_{\omega}}$ $$\begin{array}{ccc} {\mathbb{P}}\mathcal{E} & \stackrel{\sim}{{\longrightarrow}} & Bl_S{{\mathbb{P}}^4_{\omega}}\\ &&\\ \eta \searrow & & \swarrow\varrho\\ &&\\ & {{\mathbb{P}}^2_{\omega}}. & \\ \end{array}$$ *Proof:* Let $x\in {{\mathbb{P}}^2_{\omega}}$, $a+b$ the divisor on $X$ corresponding to $x$ and $s\in {\mathbb{P}}\mathcal{E}_x$. We recall that $\mathcal{E}$ is a sub-bundle of ${\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}\otimes H^0(\omega^3)^*$, thus we have an embedding $$\begin{aligned} j:{\mathbb{P}}\mathcal{E} & \hookrightarrow & {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^4_{\omega}}.\\\end{aligned}$$ Moreover we have $$j({\mathbb{P}}\mathcal{E}_x)=x \times \langle a + b + \lambda(a) + \lambda (b) \rangle \subset {{\mathbb{P}}^4_{\omega}}.$$ We will often consider ${\mathbb{P}}\mathcal{E}$ as a sub-variety of ${{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^4_{\omega}}$. We also remark that $$j({\mathbb{P}}\mathcal{E}_x)\cap {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^3_{\omega+}}= k({\mathbb{P}}\mathcal{A}_x)=x \times \overline{ab}.$$ We define a morphism $$\begin{aligned} \varpi^{\prime}:=(\varrho,\pi): Bl_S{{\mathbb{P}}^4_{\omega}}& {\longrightarrow}& {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^4_{\omega}}.\\\end{aligned}$$ The morphism $\varpi^{\prime}$ has a birational inverse $$\varpi^{\prime -1}: {\mathbb{P}}\mathcal{E}\subset {{\mathbb{P}}^2_{\omega}}\times {{\mathbb{P}}^4_{\omega}}\dashrightarrow Bl_S{{\mathbb{P}}^4_{\omega}}$$ defined as follows. We define $\varpi^{\prime -1}$ on the open set of ${\mathbb{P}}\mathcal{E}$ given by the couples $(x,s)$ s.t. $x=a+b$ is a point of ${{\mathbb{P}}^2_{\omega}}$ s.t. $a\neq b$ and $s\in \{{\mathbb{P}}\mathcal{E}_x\setminus S\}$. The pair $(x,s)$ is sent on $\pi^{-1}(s)\in Bl_S{{\mathbb{P}}^4_{\omega}}$. Then, by Zariski’s main theorem $\varpi^{\prime}$ induces an isomorphism between $Bl_S{{\mathbb{P}}^4_{\omega}}$ and ${\mathbb{P}}\mathcal{E}.\square$\ We recall that we denoted $E$ the exceptional divisor of $Bl_X{{\mathbb{P}}^3_{\omega+}}$. \[hate\] The restricted map $$\tilde{\phi}_{|E}:E {\longrightarrow}{{\mathbb{P}}^2_{\omega}}$$ is a morphism of degree 2 ramified along the conic that is the image of ${\mathbb{P}}V$ via the quadratic Veronese embedding $Ver_2$. *Proof:* We recall that $E\cong X\times {\mathbb{P}}^1$. Moreover we remark that $\tilde{\phi}_{|E}$ is given by the differential of $\varphi$ and we have $$\tilde{\phi}_{|{\{a\}}\times {\mathbb{P}}^1}: {\mathbb{P}}(N_{X|{\mathbb{P}}^3,a}) {{\longrightarrow}} {{\mathbb{P}}^2_{\omega}}.$$ Let $a,b\in X$, then we have $$\tilde{\phi} ( \overline{ab}-\{a,b\})= a+b \in |\omega^2|.$$ Let $v_b\in \mathrm{T}_a{\mathbb{P}}^3$ the tangent vector to ${\mathbb{P}}^3$ with direction $\overline{ab}$. Then, since the line $\overline{ab}$ is contracted to a point, we have $$\tilde{\phi}(v_b)=a+b \in |\omega^2|.$$ Furthermore, every normal vector $v\in {\mathbb{P}}^1 \cong {\mathbb{P}}(N_{X|{\mathbb{P}}^3,a})$ is of type $v_b$ for a point $b\in X$. This implies that $$\tilde{\phi}(\{a\} \times {\mathbb{P}}^1) =D_a:=\{a+b|b\in X\}.$$ The line $D_a\subset {{\mathbb{P}}^2_{\omega}}$ is the tangent line at the point $2a$ to the conic in ${{\mathbb{P}}^2_{\omega}}$ obtained as the image of the Veronese morphism $$\begin{aligned} Ver_2: {\mathbb{P}}V & {\longrightarrow}& |{\mathcal{O}}_{{\mathbb{P}}^1}(2)|^*={{\mathbb{P}}^2_{\omega}};\\ p & \mapsto & 2p.\end{aligned}$$ Let $a+b$ be again the divisor on $X$ corresponding to $x \in {{\mathbb{P}}^2_{\omega}}$, then the fiber $\tilde{\phi}^{-1}(x)$ in $X \times {\mathbb{P}}^1\cong {\mathbb{P}}(N_{X|{\mathbb{P}}^3})$ is composed by two points $\{(a,\alpha),(b,\beta)\}$ if $x$ is not contained in the conic. We also remark that if $x$ is a point contained in the conic the fiber is just one point. This defines a degree 2 covering ramified along the conic.$\square$\ We will denote $\tilde{E}$ the exceptional divisor of $Bl_{S}{{\mathbb{P}}^4_{\omega}}\cong {\mathbb{P}}\mathcal{E}$. We have $$\tilde{E}\cong\overline{\beta^{-1}(E)}.$$ \[hl\] The restricted map $$\tilde{\varphi}_{|\tilde{E}}:\tilde{E} {\longrightarrow}E \stackrel{2:1}{{\longrightarrow}} F \cong {{\mathbb{P}}^2_{\omega}}$$ defines a conic bundle *Proof:* The situation is the following $$\begin{array}{cccc} \tilde{E} \subset & {\mathbb{P}}(\mathcal{A} \oplus {\mathcal{O}}) = Bl_{S}{{\mathbb{P}}^4_{\omega}}& &\\ \stackrel{|}{\downarrow}\beta & &\searrow \tilde{\varphi} &\\ &&&\\ E \subset & {\mathbb{P}}\mathcal{A} = Bl_X {{\mathbb{P}}^3_{\omega+}}& \longrightarrow & {\mathbb{P}}^3_{{\mathcal{O}}}\\ &&&\\ \sigma \downarrow & & &\\ &&&\\ {{\mathbb{P}}^2_{\omega}}& & & \\ \end{array}$$ Let us consider the composed map $$\eta_{|{\mathbb{P}}\tilde{E}}:= \sigma \circ \beta_{|{\mathbb{P}}\tilde{E}}: {\mathbb{P}}\tilde{E} {\longrightarrow}{\mathbb{P}}^2.$$ The fiber of $\eta$ over a point $x\in {\mathbb{P}}^2$ is a rank two conic if $x\not \in Ver_2({\mathbb{P}}^1)$ and a double line if $x\in Ver_2({\mathbb{P}}^1). \square$\ Proposition \[hl\] and Theorem \[hl2\] imply the following theorem. \[yooo\] The morphism $$\tilde{\varphi}:Bl_S({\mathbb{P}}^4){\longrightarrow}{\mathbb{P}}^3_{{\mathcal{O}}}$$ is a conic bundle whose discriminant locus is the blow-up at the origin of the Kummer surface $K^0$. Moreover we remark that the conic $Ver_2({\mathbb{P}}^1)$ is the tangent cone at the origin of the Kummer surface. We recall that $Pic({\mathbb{P}}\mathcal{A}) \cong \mathbb{Z}^2$, notably $$Pic({\mathbb{P}}\mathcal{A}) = {\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(1) \mathbb{Z} \times \sigma^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1) \mathbb{Z}.$$ We also recall that the map $$\begin{aligned} \mu:{\mathbb{P}}\mathcal{A} & {\longrightarrow}& {{\mathbb{P}}^3_{\omega+}}= {\mathbb{P}}Sym^3 V;\\ {\mathbb{P}}\mathcal{A}_x & \mapsto & \overline{ab};\\\end{aligned}$$ that sends the fiber over $x$ on the secant line $\overline{ab}$ to $X$ is the projection of the blow-up of ${{\mathbb{P}}^3_{\omega+}}$ along $X$, hence $\mu^{-1}(X)=E$. We have $$\mu^*{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1) = {\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(1).$$ *Proof:* Our aim is to determine two integers $l,k\in \mathbb{Z}$ s.t. $$\mu^*{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1)={\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(l)\otimes \sigma^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(k).$$ Since the map $\mu$ is the projection of the blow-up of ${{\mathbb{P}}^3_{\omega+}}$ along $X$, we have $l=1$. Now we need to determine $k$. We have $$H^0({{\mathbb{P}}^3_{\omega+}},\mu^*{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1))=H^0({{\mathbb{P}}^3_{\omega+}},\mu_*\mu^*{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1)),$$ and by projection formula this is equal to $H^0({{\mathbb{P}}^3_{\omega+}},{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1)\otimes \mu_*{\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}).$ Since the fibers of $\mu$ are connected $$\mu_*{\mathcal{O}}_{{\mathbb{P}}\mathcal{A}} = {\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}.$$ Therefore $$H^0({\mathbb{P}}\mathcal{A},\mu^*{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1))=H^0({{\mathbb{P}}^3_{\omega+}},{\mathcal{O}}_{{{\mathbb{P}}^3_{\omega+}}}(1))=Sym^3V.$$ By taking the cohomology of the exact sequence (\[ff\]) we get an isomorphism $$\label{fly} H^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^*) \cong Sym^3 V.$$ In order to determine $k$ we compute $H^0({\mathbb{P}}\mathcal{A},{\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(1)\otimes \sigma^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(k))$. By the projection formula we get $$\label{dp} H^0({\mathbb{P}}\mathcal{A},{\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(1)\otimes \sigma^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(k))=H^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^*\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(k)).$$ We twist the exact sequence (\[ff\]) by the line bundle ${\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(k)$ and we take the cohomology of the obtained sequence. This leads us to conclude that the equality $$H^0({{\mathbb{P}}^2_{\omega}},\mathcal{A}^*\otimes{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(k))= Sym^3V$$ is possible only for $k=0. \square$\ Now we compute the class of the exceptional divisor $E$ in the Picard group of ${\mathbb{P}}\mathcal{A}$. We have an isomorphism in $Pic({\mathbb{P}}\mathcal{A})$ $${\mathcal{O}}(E)_{{\mathbb{P}}\mathcal{A}} \cong {\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(2) \otimes \sigma^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1).$$ *Proof:* First we compute the first factor. We recall that the restricted map $${\tilde{\phi}}:E{\longrightarrow}{{\mathbb{P}}^2_{\omega}}$$ defines a degree 2 morphism ramified along a smooth conic (Thm. \[hate\]). If $x\in {{\mathbb{P}}^2_{\omega}}$ then the intersection $E_x = \tilde{E} \cap {\mathbb{P}}\mathcal{A}_x$ is made up of two points. This implies that $${\mathcal{O}}(E)_{{\mathbb{P}}\mathcal{A}} \cong {\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(2) \otimes \sigma^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(m),$$ for some $m\in \mathbb{Z}$. We have the following equalities. $$H^0({\mathbb{P}}\mathcal{A}, {\mathcal{O}}(1))= H^0({{\mathbb{P}}^2_{\omega}}, \sigma_* {\mathcal{O}}_{{\mathbb{P}}\mathcal{A}} (1)) = H^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^*).$$ Let $a,b\in X$ and let $x\in{{\mathbb{P}}^2_{\omega}}$ be as usual the point corresponding to the divisor $a+b$. Let us consider a smooth quadric $Q\subset |\mathcal{I}_X(2)|.$ Then we have $$\mu^{-1}(Q)=E + R_Q \subset |{\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(2)|,$$ where we denote by $R_Q\subset{\mathbb{P}}\mathcal{A}$ the residual divisor. Moreover, we have either $\overline{ab}\subset Q$, or $\overline{ab} \cap Q = \{a,b\}$. We define $$\mathcal{C}_Q:=\{a+b=x\in {\mathbb{P}}^2|\overline{ab}\subset Q\}$$ and $R_Q=\sigma^{-1}(C_Q)$. It is well known that, since $Q$ is smooth, we have an isomorphism $Q \cong {\mathbb{P}}^1 \times {\mathbb{P}}^1$ via the Segre embedding and that $X \subset Q$ can be seen as the zero locus of a bihomogeneus polynomial of degree $(1,2)$. This means that we have an embedding $$\begin{aligned} X={\mathbb{P}}^1 & \hookrightarrow & Q = {\mathbb{P}}^1 \times {\mathbb{P}}^1;\\ {[u:v]} & \mapsto & ([u^2,v^2],[u,v]).\\\end{aligned}$$ This implies that, if we choose a $p\in {\mathbb{P}}^1$ and we let $t$ vary in the other ${\mathbb{P}}^1$, the lines of the ruling $\{t\}\times p$ intersect $X$ in two points. The lines of the other ruling of $Q$ intersect $X$ in just one point. Let $\alpha$ be the following morphism $$\begin{aligned} \alpha:{\mathbb{P}}^1 & {\longrightarrow}& {\mathbb{P}}^1;\\ {[u:v]}& \mapsto & [u^2:v^2].\end{aligned}$$ Then we have a linear embedding $$H^0(X,{\mathcal{O}}(1)) \stackrel{\alpha^*}{{\longrightarrow}} H^0(X,{\mathcal{O}}(2))$$ and the line ${\mathbb{P}}(\alpha^*(H^0(X,{\mathcal{O}}(1))))$ is $\mathcal{C}_Q$. This implies that $${\mathcal{O}}_{{\mathbb{P}}\mathcal{A}} (R)= \sigma^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1).$$ and thus $${\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(E)={\mathcal{O}}_{{\mathbb{P}}\mathcal{A}}(2)\otimes \sigma^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1).$$ $\square$ The morphism $\tilde{\varphi}$ then makes the following diagram of projective bundles on ${\mathbb{P}}^2$ commute. $$\begin{array}{ccc} Bl_S {{\mathbb{P}}^4_{\omega}}\cong {\mathbb{P}}(\mathcal{A} \oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}) & \stackrel{\tilde{\varphi}}{{\longrightarrow}} & {\mathbb{P}}[{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} \oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1)]={\mathbb{P}}^3_{{\mathcal{O}}}\\ &&\\ \varrho\searrow & & \swarrow pr_0\\ &&\\ & {{\mathbb{P}}^2_{\omega}}& \\ \end{array}$$ The pull back $\tilde{\varphi}^*$ induces a homomorphism $$\label{wft} \tilde{\varphi}^*: Pic({\mathbb{P}}^3_{{\mathcal{O}}}) {\longrightarrow}Pic({\mathbb{P}}(\mathcal{A}\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}})),$$ and both Picard groups are isomorphic to $\mathbb{Z}\times \mathbb{Z}$. The homomorphism of equation (\[wft\]) is the following. $$\begin{aligned} \mathbb{Z}\times \mathbb{Z} & {\longrightarrow}& \mathbb{Z}\times \mathbb{Z}\\ (a,b) & \mapsto & (2a,b-a).\end{aligned}$$ *Proof:* We have the following equalities: $$\begin{aligned} Pic({\mathbb{P}}^3_{{\mathcal{O}}})= {\mathbb{Z}}{\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1) \times {\mathbb{Z}}pr_0^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1);\\ Pic({\mathbb{P}}(\mathcal{A}\oplus {\mathcal{O}}))= {\mathbb{Z}}{\mathcal{O}}_{{\mathbb{P}}(\mathcal{A}\oplus {\mathcal{O}})}(1) \times {\mathbb{Z}}\eta^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1).\\\end{aligned}$$ Moreover, since $pr_0\circ \tilde{\varphi} = \eta$, we have $$\tilde{\varphi}(0,b) = (0,b).$$ We recall that we chose $F$ in such a way that ${\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1)={\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(F)$, this means that we have $$\tilde{\varphi}^* {\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1) = \tilde{\varphi}^* {\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(E)={\mathcal{O}}_{{\mathbb{P}}\mathcal{E}}(\tilde{\varphi}^{-1}(F)) = {\mathcal{O}}_{{\mathbb{P}}\mathcal{E}}(\beta^{-1}(E))= {\mathcal{O}}_{{\mathbb{P}}\mathcal{E}}(2) \otimes \eta^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1).$$ This implies $$(a,0)\mapsto (2a,-a).$$ $\square$\ As a consequence of the last proposition, we have $$\label{side} \tilde{\varphi}^* {\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1) = {\mathcal{O}}_{{\mathbb{P}}\mathcal{E}}(2) \otimes \eta^*{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1).$$ Furthermore $\tilde{\varphi}$ induces a natural morphism $$\label{stro} {\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1) {\longrightarrow}\tilde{\varphi}_* \tilde{\varphi}^* {\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1).$$ By applying $pr_{0*}$ to the morphism (\[stro\]) and using the equality (\[side\]), we obtain a morphism of sheaves on ${{\mathbb{P}}^2_{\omega}}$ $$\tilde{\varphi}^*: pr_{0*}{\mathcal{O}}_{{\mathbb{P}}^3_{{\mathcal{O}}}}(1)= {\mathcal{O}}\oplus {\mathcal{O}}(-1)=M^* {\longrightarrow}\eta_*({\mathcal{O}}_{{\mathbb{P}}\mathcal{E}}(2) \otimes \eta^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1)).$$ By the projection formula, $$\eta_*({\mathcal{O}}_{{\mathbb{P}}\mathcal{E}}(2) \otimes \eta^* {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1))\cong Sym^2 (\mathcal{A}^*\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}) \otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1).$$ We showed that ${\mathbb{P}}M^*$ defines a ${\mathbb{P}}^1$-bundle on ${{\mathbb{P}}^2_{\omega}}$. Let $y\in {{\mathbb{P}}^2_{\omega}}$ and let $c,d$ be the points of $X$ s.t. $y$ is the divisor $c+d$ on $X$. We recall that $C$ is a degree 2 covering of $X\cong {\mathbb{P}}V$. Then ${\mathbb{P}}(\mathcal{A}\oplus {\mathcal{O}})_y\subset {{\mathbb{P}}^4_{\omega}}$ is the ${\mathbb{P}}^2$ generated by the two pairs of points of $C$ whose images in $X$ are respectively $c$ and $d$. The fiber ${\mathbb{P}}M^*_y$ is in fact the pencil of conics in ${\mathbb{P}}\mathcal{E}_y$ that pass by these four points. \[db\] We have $$\begin{aligned} h^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^*(-1))=0;\\ h^1({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^*(-1))=0.\end{aligned}$$ *Proof:* We twist the exact sequence (\[ff\]) by the vector bundle ${\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1)$ and we find $$0 {\longrightarrow}V \otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-2) {\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1) \otimes Sym^3V {\longrightarrow}\mathcal{A}^*(-1){\longrightarrow}0.$$ By taking cohomology, we get $$h^0({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}}(-1))=h^1({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}}(-2))=0.$$ This gives us the first equality. Then, we have $$h^1({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}}(-1))=0$$ and by duality $$H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-2))\cong H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-1),$$ hence $h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-2))=0$. This implies the second equality.$\square$ We have $$h^0({{\mathbb{P}}^2_{\omega}},Sym^2 \mathcal{A}^*)=10.$$ *Proof:* By twisting the exact sequence (\[ff\]) by $\mathcal{A}^*$ we obtain the following exact sequence. $$0{\longrightarrow}\mathcal{A}^*(-1) \otimes V {\longrightarrow}\mathcal{A}^* \otimes Sym^3 V {\longrightarrow}\mathcal{A}^{*2} {\longrightarrow}0.$$ This implies that we have $$H^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^{*2})\cong H^0( {{\mathbb{P}}^2_{\omega}},\mathcal{A}^*)\otimes Sym^3V)= Sym^3V\otimes Sym^3V,$$ and therefore $h^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^{*2})=16$. Moreover we remark that $$h^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^{*2})= h^0({{\mathbb{P}}^2_{\omega}}, \wedge^2 \mathcal{A}^*) \oplus h^0({{\mathbb{P}}^2_{\omega}},Sym^2 \mathcal{A}^*).$$ By taking determinants in the exact sequence (\[ff\]) we get that $\bigwedge^2 \mathcal{A}^*={\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(2)$ and $h^0({{\mathbb{P}}^2_{\omega}}, \bigwedge^2 \mathcal{A}^*)=h^0({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}}(2))=6$. This implies directly the lemma.$\square$ We have $$h^0({{\mathbb{P}}^2_{\omega}},Sym^2A^*(-1))=1.$$ *Proof:* We twist the exact sequence (\[ff\]) by the line bundle $\mathcal{A}^*(-1)$ and we get the following exact sequence. $$0{\longrightarrow}\mathcal{A}^*(-2) \otimes V {\longrightarrow}\mathcal{A}^*(-1) \otimes Sym^3 V {\longrightarrow}\mathcal{A}^{*2}(-1) {\longrightarrow}0.$$ By passing to cohomology we have that $$h^1({{\mathbb{P}}^2_{\omega}},\mathcal{A}^*(-1))\otimes Sym^3V=h^0({{\mathbb{P}}^2_{\omega}},\mathcal{A}^*(-1))\otimes Sym^3V=0.$$ Thanks to Lemma \[db\] we have then $$\label{dak} H^0({{\mathbb{P}}^2_{\omega}}, \mathcal{A}^{*2}(-1))\cong H^1({{\mathbb{P}}^2_{\omega}},\mathcal{A}^*(-2))\otimes V.$$ We twist the exact sequence (\[ff\]) by the line bundle ${\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-2)$ and we get $$0 {\longrightarrow}V \otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-3) {\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-2) \otimes Sym^3V {\longrightarrow}\mathcal{A}^*(-2){\longrightarrow}0.$$ Moreover we remark that $h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-2)=0$ and $h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-3))=1$. By duality $h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-2))=h^0({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-1))=0$, then $$h^1({{\mathbb{P}}^2_{\omega}},\mathcal{A}^*(-2))=h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-3))\otimes V=2.$$ The equality (\[dak\]) implies that $h^0({{\mathbb{P}}^2_{\omega}},\mathcal{A}^{*2}(-1)=4$. The vector bundle $\mathcal{A}^{2*}(-1)$ decomposes as $ Sym^2 \mathcal{A}^*(-1) \oplus \bigwedge^2 \mathcal{A}^*(-1)$. Since $$h^0({{\mathbb{P}}^2_{\omega}},\bigwedge^2 \mathcal{A}^*(-1))=h^0({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1))=3$$ we have that $$h^0({{\mathbb{P}}^2_{\omega}},Sym^2 \mathcal{A}^*(-1))=4-3=1.$$ $\square$ We have $$dim (\mathrm{Hom}({\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}\oplus{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1),Sym^2 (\mathcal{A}^*\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}})\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1)))=16.$$ *Proof:* We will denote $$B:=\mathrm{Hom}({\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}\oplus{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1),Sym^2 (\mathcal{A}^*\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}})\otimes {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1)).$$ We have $$B= H^0({{\mathbb{P}}^2_{\omega}},Sym^2\mathcal{A}^*\oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}\oplus \mathcal{A}^* \oplus Sym^2 \mathcal{A}^*(-1) \oplus {\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(-1) \oplus \mathcal{A}^*(-1)).$$ Summing the dimensions of the direct summands we have $$dim B=10+1+4+1=16.$$ $\square$ The projective plane ${{\mathbb{P}}^2_{\omega}}={\mathbb{P}}Sym^2V$ does not depend on the curve $C$, because $V$ is an abstract vector space. Since it is defined by the exact sequence $\ref{ff}$, even $\mathcal{A}$ does not depend on the curve $C$. This means that to a given a genus 2 curve $C$ we can associate a conic bundle over ${\mathbb{P}}^2$, notably the bundle defined by the section $$\tilde{\varphi}_C\in {\mathbb{P}}B.$$ Its discriminant locus is the blow-up (at the origin) of the Kummmer surface $K^0=Jac(C)/\pm Id$. In this way we build a moduli map $$\begin{aligned} \Xi:\{ \mathrm{smooth\ genus\ 2\ curves} \} & {\longrightarrow}& {\mathbb{P}}^{15}={\mathbb{P}}B;\\ C & \mapsto & \tilde{\varphi}_C.\end{aligned}$$ \[klu\] Let $Y$ a smooth projective variety, $$f:G{\longrightarrow}Y$$ a conic bundle, then there exists a rank 3 vector bundle $F$ on Y, a line bundle $L$ and one section $q\in H^0(Y,Sym^2 H\otimes L^k)$ for some integer $k$, s.t. G is the zero scheme of $q$ in the projective bundle ${\mathbb{P}}F$. *Proof:* The proof follows that of Proposition 1.2 of [@bo:ji]. The assertion is equivalent to the existence of a line bundle $N$ on $G$ inducing the sheaf ${\mathcal{O}}_{G_s}(1)$ on every fiber $G_s$. More precisely we will have $H=f_*N$. If we consider the sheaf of differentials of maximum degree $\omega_G$ then by the adjunction formula we have $\omega_{G|G_s}\cong \omega_{G_s}$ and $\omega_{G_s}\cong {\mathcal{O}}_{G_s}(-1)$. Then we take $N =\omega_G^{-1}.\square$ Let $$\tilde{\varphi}:Bl_S{{\mathbb{P}}^4_{\omega}}{\longrightarrow}{\mathbb{P}}^3_{{\mathcal{O}}}$$ be the conic bundle of Theorem \[yooo\]. Then $Bl_S{{\mathbb{P}}^4_{\omega}}$ is a divisor in the total space of the bundle ${\mathbb{P}}(pr_0^*\mathcal{E})$ on ${\mathbb{P}}^3_{{\mathcal{O}}}$. *Proof:* Let $y\in{\mathbb{P}}^3_{{\mathcal{O}}}$, then we have $$\tilde{\varphi}^{-1}(y)\subset {\mathbb{P}}(\mathcal{E}_{pr_0(y)}).$$ Then the rank 3 vector bundle associated to $\tilde{\varphi}$ is $pr_0^*\mathcal{E}.\square$ Moreover $Bl_S{{\mathbb{P}}^4_{\omega}}= {\mathbb{P}}\mathcal{E}$. This means we have the following diagram, where the lower square is a fiber product.                                 In this diagram $Z$ is the embedding of $Bl_S{{\mathbb{P}}^4_{\omega}}$ in ${\mathbb{P}}(pr_0^*\mathcal{E})$ induced by the universal property of fiber product. Stability and deformations of $\mathcal{A}$. ============================================ Stability --------- In this section we will go through the question of the stability of ${\mathcal{A}}$ and we will calculate its space of deformations. Let $H$ be the hyperplane class that generates $Pic({\mathbb{P}}^2)$. If we take the cohomology of the exact sequence \[ff\] (and of its dual sequence) and we compute the Chern polynomials we find that $$\begin{aligned} c_1({\mathcal{A}})=-2H,\\ c_2({\mathcal{A}})=3H^2,\\ c_1({\mathcal{A}}^*)=2H,\\ c_2({\mathcal{A}}^*)=3H^2.\end{aligned}$$ Then we have that the slope $\mu({\mathcal{A}}^*)=1$. If we twist by ${\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1)$ the exact sequence \[ff\] that defines ${\mathcal{A}}^*$ we get $$\label{stai} 0{\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}} \otimes H^0(C,\omega) {\longrightarrow}{\mathcal{O}}_{{{\mathbb{P}}^2_{\omega}}}(1) \otimes H^0(C,\omega^3)_+ {\longrightarrow}{\mathcal{A}}^*(1) {\longrightarrow}0.$$ According to [@dk] a bundle on ${\mathbb{P}}^n$ that has a linear resolution like ${\mathcal{A}}^*(1)$ a *Steiner bundle*. Let $K$ be a complex projective manifold with $Pic(K)\cong {\mathbb{Z}}$ and $E$ a vector bundle of rank $r$ on $K$. Then the bundle $E$ on $K$ is called *normalized* if $c_1(E)\in \{-r+1,\dots,-1,0\}$, i.e. if $-1<\mu(E)\leq 0$. We denote by $E_{norm}$ the unique twist of $E$ that is normalized. The following criterion for the stability of vector bundles on $K$ is a consequence of the definition. **(Hoppe)**[@hl]\[kit\] Let $V$ be a vector bundle on a projective manifold $K$ with $Pic(K)\cong {\mathbb{Z}}$. If $H^0(X,\wedge^q V_{norm})=0$ for any $1\leq q \leq rkV-1$, then $V$ is stable. The vector bundle ${\mathcal{A}}$ on ${{\mathbb{P}}^2_{\omega}}$ is stable. *Proof:*We remark that $Pic({{\mathbb{P}}^2_{\omega}})\cong {\mathbb{Z}}$ and that $c_1({\mathcal{A}}^*(-1))=0$, this means that ${\mathcal{A}}^*_{norm}={\mathcal{A}}^*(-1)$. By taking the cohomology of the exact sequence \[stai\] we remark that $H^0({\mathcal{A}}^*(-1))=0$. By proposition \[kit\] then ${\mathcal{A}}^*$ is stable and this in turn implies that ${\mathcal{A}}$ is stable. $\square$ Deformations of ${\mathcal{A}}$ ------------------------------- Before computing directly the dimension of the space of deformations of ${\mathcal{A}}$, i.e. $dim (Ext^1({\mathcal{A}},{\mathcal{A}}))$, we need some technical lemmas. \[black\] We have $$\begin{aligned} h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*)=0,\\ h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*)=0.\end{aligned}$$ *Proof:* We take the cohomology of the exact sequence \[ff\] and we get $$0 {\rightarrow}H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*) {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-1) {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}}) {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*) {\rightarrow}0.$$ By duality we have $H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-1)\cong H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-2))$ and $H^2({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}})\cong H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(-3))$ and both spaces are zero dimensional. This implies our statement. $\square $\ Furthermore, we recall that $H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*)\cong H^0({{\mathbb{P}}^2_{\omega}},\omega^3)_+$. \[yl\] We have $$\begin{aligned} h^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))=10,\\ h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))=0,\\ h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))=0.\end{aligned}$$ *Proof:* We take the cohomology of the exact sequence \[stai\] and we get $$0{\rightarrow}H^0(C,\omega) {\rightarrow}H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1))\otimes H^0(C,\omega^3)_+ {\rightarrow}H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1)) {\rightarrow}H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}){\rightarrow}$$ $${\rightarrow}H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1)) {\rightarrow}H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1)) {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}}) {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1)) {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1)) {\rightarrow}0.$$ Since $h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{O}})=0$, we have $h^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))=10$. Moreover, by duality we have also $$h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1))=h^2({{\mathbb{P}}^2_{\omega}}, {\mathcal{O}})= h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{O}}(1))=0.$$ Hence $h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))= h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))=0. \square$\ We are ready to state the main result of this section. The space of deformations of ${\mathcal{A}}$ has dimension equal to 5. *Proof:* We twist by ${\mathcal{A}}^*$ the dual of the exact sequence \[ff\] and we obtain the following. $$\label{ev} 0 {\longrightarrow}{\mathcal{A}}^*\otimes {\mathcal{A}}{\longrightarrow}{\mathcal{A}}^* \otimes H^0(C,\omega^3)_+^* {\longrightarrow}{\mathcal{A}}^*(1) \otimes H^0(C,\omega)^* {\longrightarrow}0.$$ By taking cohomology we get the following long exact sequence $$0 {\rightarrow}H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}}) {\longrightarrow}H^0({{\mathbb{P}}^2_{\omega}}, {\mathcal{A}}^*)\otimes H^0(C,\omega^3)_+^* {\longrightarrow}H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))\otimes H^0(C,\omega)^* {\rightarrow}$$ $${\rightarrow}H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}}) {\rightarrow}H^1({{\mathbb{P}}^2_{\omega}}, {\mathcal{A}}^*)\otimes H^0(C,\omega^3)_+^* {\rightarrow}H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))\otimes H^0(C,\omega)^* {\rightarrow}$$ $${\rightarrow}H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}}) {\rightarrow}{\rightarrow}H^2({{\mathbb{P}}^2_{\omega}}, {\mathcal{A}}^*)\otimes H^0(C,\omega^3)_+^* {\rightarrow}H^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))\otimes H^0(C,\omega)^* {\rightarrow}0.$$ Lemma \[yl\] and Lemma \[black\] imply that $h^2({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}})=0$. Moreover we have that $dim H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*)\otimes H^0(C,\omega^3)_+^*=16$ and $dim H^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*(1))\otimes H^0(C,\omega)^*=20$. This in turn implies that $$h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}}) - h^0({{\mathbb{P}}^2_{\omega}},A^*\otimes {\mathcal{A}})=20-16=4.$$ Now $H^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}})\cong Ext^1({\mathcal{A}},{\mathcal{A}})$ and, since ${\mathcal{A}}$ is stable $$dim Hom({\mathcal{A}},{\mathcal{A}})=h^0({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}})=1.$$ This means that $h^1({{\mathbb{P}}^2_{\omega}},{\mathcal{A}}^*\otimes {\mathcal{A}})=5.\square$ Michele Bolognesi\ Dipartimento di Matematica Università di Pavia\ Via Ferrata 1\ 27100 Pavia\ Italy\ E-mail: michele.bolognesi@unipv.it

--- abstract: 'We study the effects of SO(10) D-terms on the allowed parameter space (APS) in models with $t - b - \tau$ and $b - \tau$ Yukawa unifiction. The former is allowed only for moderate values of the D-term, if very precise ($\le$ 5%) unification is required. Next we constrain the parameter space by looking for different dangerous directions where the scalar potential may be unbounded from below (UFB1 and UFB3). The common trilinear coupling $A_0$ plays a significant role in constraing the APS. For very precise $t - b - \tau$ Yukawa unification, $-\MSX \lapp A_0 \lapp \MSX$ can be probed at the LHC, where $\MSX$ is the common soft breaking mass for the sfermions. Moreover, an interesting mass hierarchy with very heavy sfermions but light gauginos, which is strongly disfavoured in models without D-terms, becomes fairly common in the presence of the D-terms. The APS exhibits interesting characteristics if $\MSX$ is not the same as the soft breaking mass $\MTN$ for the Higgs sector. In $b - \tau$ unification models with D-terms, the APS consistent with Yukawa unification and radiative electroweak symmetry breaking, increases as the UFB1 constraint becomes weaker. However for $A_0 \lapp 0$, a stronger UFB3 condition still puts, for a given $\MSX$, a stringent upper bound on the common gaugino mass ($\MHF$) and a lower bound on $\MSX$ for a given $\MHF$. The effects of sign of $\mu$ on Yukawa unification and UFB constraints are also discussed.' author: - | [*Amitava Datta*]{} [^1]   and [*Abhijit Samanta*]{} [^2]\ Department of Physics, Jadavpur University, Kolkata - 700 032, India title: ' Effects of the SO(10) D-Term on Yukawa Unification and Unstable Minima of the Supersymmetric Scalar Potential' --- PACS no: 12.60.Jv, 14.80.Ly, 14.80.Cp Introduction ============ It is quite possible that the Standard Model (SM), is not the ultimate theory of nature, as is hinted by a number of theoretical shortcomings. One of the most popular choices for physics beyond SM is supersymmetry (SUSY) [@susy-review]. However, the experimental requirement that SUSY must be a broken symmetry introduces a plethora of new soft breaking parameters. There are important constraints on this large parameter space from the negative results of the sparticle searches at colliders like LEP[@lepbound] and Tevatron[@tevatronbound]. In addition there are important theoretical constraints which are often introduced for aesthetic reasons. From practical point of view, however, the most important effect of such constraints is to reduce the number of free parameters. For example, the assumption that the soft breaking terms arise as a result of gravitational interactions leads to the popular minimal supergravity (mSUGRA) model with five free parameters only, defined at a high energy scale where SUSY is broken. They are the common scalar mass ($m_0$), the common gaugino mass ($\MHF$), the common trilinear coupling ($A_0$), the ratio of vacuum expectation values of two Higgs field ($\tan\beta$) and the sign of $\mu$, the higgsino mass parameter. In this paper we shall restrict ourselves to variations of this basic framework. A very useful way to further constrain the allowed parameter space (APS) of softly broken SUSY models is to consider the dangerous directions of the scalar potential, where the potential may be unbounded from below (UFB) or develop a charge and/or color breaking (CCB) minima [@oldufb]. Different directions are chosen by giving vacuum expectation value (VEV) to one or more coloured and / or charged scalar fields, while the VEVs of the other scalars are taken to be zero. In a very interesting paper which revived interest in UFB and CCB constraints, Casas [*et al*]{} [@casas] investigated the effects of such constraints on SUSY models. Though their formulae are fairly model-independent, they had carried out the numerical analysis within the framework of mSUGRA for moderate values of $\tan\beta$ only, when one can ignore the effects of b and $\tau$ Yukawa couplings in the relevant renormalization group equations (RGEs). Their main result was that a certain UFB constraint known as UFB3 with VEVs given in the direction of the slepton fields puts the tightest bound on the SUSY parameter space that they considered (see eq. (93) of [@casas] and the discussions that follow). In an earlier paper [@paper1], we had extended and complemented the work of [@casas] by looking at the APS subject to such ‘potential constraints’ for large values of $\tan\beta$, motivated by partial $b$-$\tau$ [@b-tau; @partial] or full $t$-$b$-$\tau$ Yukawa unification [@sotenbr]. Such unifications are natural consequences of an underlying Grand Unified Theory (GUT). We considered a popular model in which the GUT group SO(10) breaks directly into the SM gauge group SU(3) $\times$ SU(2) $\times$ U(1). All matter fields belonging to a particular generation is contained in a 16 dimensional representation of SO(10). With a minimal Higgs field content (one [**10**]{}-plet containing both the Higgs doublets required to give masses to u and d type quarks) all three Yukawa couplings related to the third generation fermions must unify at the GUT scale. If one assumes more than one [**10**]{}-plet, at least the bottom and the tau Yukawa couplings should unify. In [@paper1] we assumed a common soft breaking (SB) mass ($\MSX$) at $M_G$ for all sfermions of a given generation. Similarly a common mass parameter ($\MTN$) was chosen for both the Higgs fields. We then studied the stability of the potential for two sets of boundary conditions: i) the mSUGRA motivated universal scenario ($\MSX = \MTN$), and ii) a nonuniversal scenario ($\MSX \neq \MTN$). The second condition is motivated by the fact that a common scalar mass at the Planck scale, generated, by the SUGRA mechanism, may lead to nonuniversal scalar masses at $M_G$ due to different running of $m_{10}$ and $m_{16}$, as they belong to different GUT multiplets [@running]. In this paper we shall extend the work of [@paper1] by considering the APS due to Yukawa unification and UFB constraints in the presence of SO(10) breaking D-terms. The group $SO(10)$ contains $SU(5)\times U(1)_X$ as a subgroup. It is well known that the breaking of SO(10) to the lower rank SM group may introduce nonzero D-terms at the GUT scale[@d-term]. We further assume that the D-terms are linked to the breaking of $U(1)_X$ only. It should be noted that if one assumes the existence of additional $U(1)$’s at high energies, it is quite natural to assume that the D-term contributions to scalar masses are non-zero[@d-term]. The only uncertainty lies in the magnitude of the D-terms which may or may not be significant. The squark - slepton and Higgs soft breaking masses in this case can be parametrized as $m_{\tilde Q}^2 = m_{\tilde E}^2 = m_{\tilde U}^2 = m_{16}^2 + m_D^2$\ $m_{\tilde D}^2 =m_{\tilde L}^2 =m_{16}^2 -3 m_D^2$\ $m_{H_{d,u}}^2 =m_{10}^2 \pm 2 m_D^2$\ where $\tilde Q$ and $\tilde L$ are SU(2) doublets of squarks and sleptons, $\tilde E$, $\tilde U$ and $\tilde D$ are SU(2) singlet charged sleptons, up and down type squarks respectively. The unknown parameter $m_D^2$ (the D-term) can be of either sign. The mass differences arise because of the differences in the $U(1)$ quantum numbers of the sparticles concerned. As can be readily seen from the above formula for $m_D^2>$ 0, the left handed sleptons and right handed down type squarks (belonging to the $\bar 5$ representation of $SU(5)$), are lighter than the members of the $10$ plet of $SU(5)$. In recent times the phenomenology of the D-terms has attained wide attention[@tata; @pheno-d]. D-terms acquire particular significance in the context of $t-b-\tau$ unification as has already been noted in the literature [@tata]. A new result of this work is that while moderate values of D-terms facilitate very accurate unification, high values of this parameter spoil it. The UFB and CCB constraints depend crucially on the particle spectra at the properly chosen scale where the true minimum and the dangerous minimum can be reliably evaluated from the tree level potential ($V_{tree}$) [@casas; @gamberini]. Such spectra, in turn, depend on the boundary conditions at the GUT scale. The SO(10) breaking D-terms alter the sparticle spectra at the GUT scale and may affect the stability of the potential. In this paper we focus our attention on the impact of such D-terms on the APS restricted by Yukawa unification and the stability of the potential in both universal and nonuniversal scenarios. Throughout the paper we ignore the possibility that nonrenormalizable effective operators may stabilise the potential [@nonrenorm]. The dangerous minima that we encounter in our analysis typically occur at scales $\lapp 10^8$ GeV where the effects induced by the nonrenormalizable operators, which in principle can be significant in the vicinity of the GUT scale, are not likely to be very serious. It has been pointed out in the literature that the standard vacuum, though metastable, may have a lifetime longer than the age of the universe [@claudson], while the true vacuum is indeed charge and colour breaking. If this be the case, the theory seems to be acceptable in spite of the existence of the unacceptable UFB minima that we have analysed. However, the life-time calculation, which is relatively straightforward for a single scalar field, is much more uncertain in theories where the potential is a function of many scalar fields. Thus it is difficult to judge the reliability of these calculations. Moreover, the constraints obtained by us does not loose their significance even if the false vacuum idea happens to be the correct theory. If these constraints are violated by future expeimental data then that would automatically lead to the startling conclusion that we are living in a false vacuum and charge and colour symmetry may eventually breakdown. It has been known for quite some time that while $\mu > 0$(in our sign convention which is opposite to that of Haber and Kane [@susy-review]) is required by Yukawa unification, the opposite sign is preferred by the data on the branching ratio of b $\rightarrow$ s $\gamma$ and that on $g_\mu$ - 2 ( see [@ferrandis; @referee] for some of the recent analyses and references to the earlier works). It has been shown in [@ferrandis] and also in the first paper of [@referee] that in a narrow region of the parameter space there is no conflict between data and Yukawa unification. We have analysed the parameter space found in [@ferrandis] in the light of the stability of the vacuum and the results are given in the next section (see Table 1 in particular). The above conflict may also be resolved by introducing non-universal gaugino masses ( see Chattopadhaya and Nath in [@referee]). In section 2 we discuss the effects of $\tan\beta$, $m_D$ and sign of $\mu$ on Yukawa unification and stability of the potential. In subsection 2.2 and 2.3 we study the APS for both $t-b-\tau$ and $b-\tau$ unification in conjunction with the UFB constraints. In the last section we summarise and conclude. Results ======= General Discussions ------------------- The methodology of finding the spectra is the same as in [@paper1], which is based on the computer program ISASUGRA, a part of the ISAJET package, vesion 7.48[@ISAJET]. The parameters $\mu$ and $B$ are fixed by radiative electroweak symmetry breaking (REWSB) [@rewsb] at a scale $M_S = \sqrt{m_{\tilde t_L} m_{\tilde t_R}}$. We further require that the lightest neutralino ($\tilde \chi^0$) be the lightest supersymmetric particle (LSP). The above two constraints will also be used to obtain the allowed parameter space (APS) although their use may not be mentioned explicitly everywhere. We then fix $\tan\beta$ to its lowest value required by Yukawa unification. Next we check the experimental constraints on sparticle masses. Finally we impose the UFB constraints. Before discussing the basic reasons of how Yukawa unification plays a significant role in restricting the APS, we will review the different uncertainties of Yukawa unification. The effectiveness of Yukawa unification as a restrictor of the APS diminishes, as expected, as the accuracy with which we require the unification to hold good is relaxed. There are several reasons why the unification may not be exact. First, there may be threshold corrections [@threshold], both at the SUSY breaking scale (due to nondegeneracy of the sparticles) and at $M_G$, of which no exact estimate exist. Secondly, we have used two-loop RGEs for the evolutions of gauge couplings as well as Yukawa couplings and one loop RGEs for the soft breaking parameters, but higher order loop corrections may be important at a few percent level at higher energy scales. Finally the success of the unification program is also dependent on the choice of $\alpha_s(M_Z)$ which is not known as precisely as $\alpha_1$ or $\alpha_2$. To take into account such uncertainties, one relaxes the Yukawa unification condition to a finite amount (5%, 10% or 20%) which should indirectly take care of the above caveats. The demand of very accurate Yukawa coupling unification at $M_G$ puts severe constraint on $\tan\beta$ restricting it to very large values only. The accuracy of the $t-b-\tau$ unification is usually relaxed since there are more elements of uncertainty, [*e.g.*]{}, the choice of the Higgs sector. To quantify this accuracy, one can define three variables $r_{b\tau}, r_{tb}$ and $r_{t\tau}$ where generically $r_{xy}=Max(Y_x/Y_y,Y_y/Y_x)$. For example, to check whether the couplings unify, one should select only those points in the parameter space where, $Max(r_{b\tau},r_{tb},r_{t\tau})<1.10$ (for 10% $t-b-\tau$ unification) and $r_{b\tau}<1.05$ (for 5% $b-\tau$ unification). Now we will focus on the basic reasons which lead to upper and lower bounds on the APS in the $\MSX-\MHF$ plane, if partial ($b-\tau$) or full ($t-b-\tau$) unification is required. It is wellknown that for precise Yukawa unification one should have $\mu >0$. The partial Yukawa unification can be accommodated at relatively low values of $\tan\beta$ when the phenomenologically interesting small $\MSX,\MHF$ region of the parameter space is allowed (viz. for $\MSX,\MHF\sim$ 200 GeV, the required minimum value of tan$\beta\sim 30,$ and for $\MSX,\MHF\sim$ 800 GeV, ${(\tan\beta)}_{\rm min}\sim$ 41). On the otherhand $\tan\beta$ cannot be arbitrarily increased due to the REWSB. This basic trend, which often makes the two constraints incompatible, remains unaltered irrespective of the choice of the other parameters. The constraints due to Yukawa unification and REWSB are relatively weak for large negative values of $A_0$ and becomes stronger as this parameter is algebraically increased [^3]. On the other hand, the UFB constraints are very potent for large negative values of $A_0$. The expanded APS allowed by Yukawa unification, is eaten up by the UFB constraints. In this sense the two sets of constraints are complementary[@paper1]. $Y_t$ varies relatively slowly with respect to tan$\beta$ compared to $Y_{\tau}$ and $Y_b$. For very accurate (5 %) $t-b-\tau$ unification, we, therefore, need high values of $\tan\beta\sim 47 - 51$. In this case the low $\MSX - \MHF$ region is excluded by the REWSB condition, leading to lower bounds much stronger than the experimental ones and the resulting APS is restricted to phenomenologically uninteresting high $\MSX,\MHF$ region. For example, with $\tan\beta$ =49.5 the lowest allowed values are $\MSX=$600GeV, $\MHF=1000$GeV leading to rather heavy sparticles. In the presence of D-terms a larger APS is obtained even if very accurate full unification is required[@tata]. A new finding of this paper is that though moderate values of $m_D$ leads to better Yukawa unification, somewhat larger values spoil it. Although the D-terms do not affect the evolution of the Yukawa couplings directly through the RGEs, they change the initial conditions through SUSY radiative corrections to $m_b(m_Z)$ [@pierce]. This is illustrated in figs. \[yu0\]—\[yu3\], where approximate unification is studied for three different values of $m_D$. The choice of other SUSY parameters for these figures are as follows: $m_{10}=m_{16}$ = 1500GeV,  $m_{1/2}$ = 500GeV,  tan$\beta$=48.5, $A_0$ = 0 and $\mu >0$. From fig \[yu0\] ($m_D=0$), we see that the accuracy of unification is rather modest ($\sim 15\%$). As $m_D$ is further increased to $m_{16}/5$ (fig.\[yu5\]), the $\tilde b\tilde g$ loop corrections (see eq. (8) of [@pierce]) to $m_b(m_Z)$ increases and leads to better unification. However, if we increase $m_D$ further to $m_{16}/3$, the accuracy of unification deteriorates ( fig. \[yu3\]) since $m_b(m_Z)$ suffers a correction which is too large. We have checked that this feature holds for a wide choice of SUSY parameters. Quite often the APS expanded due to the presence of D-terms is significantly reduced by the UFB constraints. As discussed in our earlier work[@paper1], the variation of $\MHU$ and $\MHD$, the soft breaking masses of the two Higgs bosons, with respect to the common trilinear coupling $A_0$ is of crucial importance in understanding this. Here we extend the discussion for non-zero values of the D-term, $m_D =$ $\MSX/5$ and $\MSX/3$. The effects are illustrated in fig. \[mh2md\]. As we increase the magnitude of the D-term, the UFB3 becomes more potent though UFB1 looses its restrictive power for a fixed value of $\tan\beta$. To clarify this result we examine two important expressions of Casas [@casas]. The first one is ++2\^2 2 |B|, \[ufbone\] which is known as the UFB1 condition and should be satisfied at any scale $\hat{Q}>M_S$, in particular at the unification scale $\hat{Q}=M_G$. The second one is the UFB3 constraint, V\_[UFB3]{}=\[+m\_[L\_[i]{}]{}\^[2]{}\]ł|H\_u|\^2+\[m\_[L\_j]{}\^[2]{} +m\_[E\_[j]{}]{}\^[2]{}+m\_[L\_[i]{}]{}\^[2]{}\]ł|H\_u|- , \[ufbthree\] where $g^\prime$ and $g$ are normalised gauge couplings of $U(1)$ and $SU(2)$ respectively, $\lambda_{E_j}$ is a Yukawa coupling and $i,j$ are generation indices. We find that larger $m_D$ drives $\MHU$ to more negative values, while $\MHD$ is driven to positive values (see fig. \[mh2md\]). In addition, it follows from REWSB condition that as the difference $\MHD - \MHU$ increases, the higgsino mass parameter $\mu$ increases. As a result the UFB1 constraint becomes weaker for large $m_D$ values (see eq. \[ufbone\]). On the otherhand at the GUT scale, $m_{L_i}^2$ becomes smaller for larger $m_D$. From eq. \[ufbthree\] it can be concluded that the parameter space where $\MHU + m_{L_i}^2$ is negative increases and the model is more succeptible to the UFB3 codition. These effects will be reflected in $b-\tau$ unification as well. For precise ($\leq$ 5%) $t-b-\tau$ unification the required $\tan\beta$ is very high ($\sim 49$) and the allowed $\MSX$ values are large. Here the magnitude of $\mu$ as determined by the REWSB becomes very low even for moderate values of the D-term ($m_D \approx \MSX/5$). Consequently UFB1 still disallows a significant part of the enlarged APS obtained with introduction of the D-term. However, the effectiveness of the UFB1 constraint depends crucially on $A_0$. We see that $A_0 \lapp \MSX$ is ruled out by UFB1 if the D-term is zero. In presence of the D-terms the UFB1 constraints become weaker but still have some restrictive power for $A_0 \lapp 0$. Moreover UFB3 becomes weaker for large $\MSX$ in general. Now we shall discuss the impact of the sign of $\mu$ on both UFB constraints and Yukawa unification. Yukawa unification generally favours $\mu > 0$. The sign of $\mu$ affects unification through loop corrections[@pierce] to the bottom Yukawa coupling, which are incorporated at the weak scale. These corrections lower the bottom Yukawa coupling significantly, consequently the GUT scale value bcomes very low, which tends to spoil Yukawa unification. Baer[@ferrandis] showed that for $\mu < 0$ full unification with low accuracy ($\sim 30\%$) is possible. This is interesting since approximate unification then becomes consistent with the constraints from $b \rightarrow s \gamma$ and $g - 2$ of the muon. It was shown in [@ferrandis] that the Yukawa unified APS favours $A_0 \approx -2\MSX$ and $\MTN\approx\sqrt{2}\MSX$ (see fig. 1 of [@ferrandis]). We have extended the analysis of ref[@paper1] for $\mu < 0$ and have found that the UFB1 condition looses it effectiveness for $\mu <0$. The bottom Yukawa coupling affects the value of $\MHD$ through renormalization group (RG) running and cannot make $\MHD$ large negative as in the $\mu >0$ case. This is why UFB1 is weakened (see eq. \[ufbone\]). We have studied the APS obtained in [@ferrandis] and found that UFB1 can disallow certain negative values of $m_D^2$ depending on the magnitudes of $\MSX$ and $\MHF$. If $\MSX, \MHF$ are increased, relatively small negative values of $m_D^2$ make the potential unstable under UFB1 condition. Some representative regions of APS are shown in Table 1. In obtaining Table 1, $A_0$, $\MTN$ and $\tan\beta$ are varied within the ranges indicated by ref[@ferrandis]. -------- -------- ----------------------- -- -- -- $\MSX$ $\MHF$ allowed $m_D^2$ (GeV) (GeV) (GeV$^2$) 600 300 $\gapp -(\MSX/4.3)^2$ 1000 300 $\gapp -(\MSX/4.5)^2$ 1000 500 $\gapp -(\MSX/5.0)^2$ -------- -------- ----------------------- -- -- -- : *[Representative D-terms allowed by UFB1 for $\mu <0$. ]{}* We have also checked that precise $b-\tau$ Yukawa unification is not possible for $\mu <0$ except for very low $\tan\beta(\sim 1)$. As $\tan\beta$ is increased, $Y_\tau$ at $M_G$ increases rapidly compared to $Y_b$. This is clear from $\wt t\wt\chi^+$ loop correction (see eqn. 15 of [@pierce]). $t$-$b$-$\tau$ Unification -------------------------- Through out this section we shall restrict ourselves to unification within 5%. For the sake of completeness and systematic analysis, we start our discussion for large negative values of $A_0$ (say, $A_0 = -2\MSX$), though it is not interesting from the point of view of collider searches. We first consider moderate values of the D - term ( $m_D = \MSX/5$). Though at low values of $\tan\beta$ the large negative values of $A_0$ are favoured by REWSB (see, the following section on $b-\tau$ unifiaction), they are strongly disfavoured at large tan$\beta$ ($\sim$ 49) which is required by full unification. Only a narrow band of $\MHF$ is allowed. However, the APS corresponds to rather heavy sparticles (e.g., $\MSX(\MHF) \gapp$ 1100(1300)GeV) which are of little interest even for SUSY searches at the LHC. Non-universality affects the APS marginally; no significant change can be obtained. Thus no squarks - gluino signal is expected at LHC for $A_0 \lapp -2\MSX$ irrespective of the boundary conditions (universal or non-universal) on the scalar masses. Over a small region of the APS somewhat lighter sleptons ($\MSL \sim$ 1000GeV) are still permitted. Moreover, the tiny APS allowed by the unification criterion is ruled out by the UFB1 condition. For $A_0 \gapp 2\MSX$, the APS is qualitatively the same as that for $A_0 \lapp -2\MSX$, with the only difference that the UFB constraint does not play any role. Relatively large APSs with phenomenologically interesting sparticle masses open up for $-\MSX \lapp A_0 \lapp \MSX$, which is favourable for both Yukawa unification and REWSB. The common feature of the APS is that gluino masses almost as low as the current experimental lower bound with much heavier squark and slepton masses($\gapp$ 1TeV) can be obtained irrespective of universality or non-universality of scalar masses. It should be stressed that this mass pattern cannot be accommodated without the D-terms. In the presence of D-term this mass hierarchy becomes a distinct possibility. In fig. \[tbtaum0p1md5a0\_1\] we present the $\MSX - \MHF$ plane for $A_0 = -\MSX$ in the universal model. A large APS is obtained by the unification criterion alone. For each $\MSX$ there are lower and upper bounds on $\MHF$. For $\MSX < 1200$GeV relatively low values of $\MHF$ are excluded by REWSB while very high values are excluded by the requirement that the neutralino be the LSP. The value of $\MSX$ can be as low as 700 GeV, which corresponds $\MHF \geq$ 1100 GeV, yielding $\MGL \geq$ 2422 GeV, $\MSQ \approx$ 2200 GeV, $\MSL \approx$ 829 GeV. For $\MSX \gapp 1200$GeV, low values of $\MHF$ are quite common. Scanning over the APS we find that the lowest allowed gluino mass is just above the experimental lower bound. Corresponding to this gluino mass the minimum sfermion masses are $\MSQ \approx$ 1200 GeV, $\MSL \approx$ 1200 GeV. As $A_0$ is further increased algebraically from $-\MSX$, the APS slightly decreases due to unification and REWSB constraints. For $A_0 = 0$, we obtain an upper limit $\MSX \leq$ 2400 GeV. However, the lower limits on $\MSX$ is relaxed by $\sim$ 200 GeV in comparison to the $A_0 = -\MSX$ case. As we further increase the value of $A_0$ to $A_0 = \MSX$, the APS is almost the same as that for $A_0 = -\MSX$. This trend is observed in all cases irrespective of universality or non-universality of the scalar masses and even for $b-\tau$ unification. We find that as the absolute value of $A_0$ increases, Yukawa unification is less restricted, while REWSB is somewhat disfavoured. When both act in combination, we get a relatively large APS for $\left |A_0\right | = \MSX$ and a somewhat smaller one for $A_0 = 0$. As the potential constraints are switched on for $A_0 = -\MSX$, an interesting upper bound on $\MHF$ for each given $\MSX$ is imposed by the UFB1 constraint (fig. \[tbtaum0p1md5a0\_1\]). As a result practically over the entire APS, the gauginos are required to be significantly lighter than the sfermions. Moreover, the allowed gaugino masses are accessible to searches at the LHC. We next focus on the impact of a particular type of non-universality ($\MTN <\MSX$) for the negative $A_0$ scenario. The shape of the APS is affected appreciably. As $\MTN$ decreases, $Y_b$ gets larger SUSY threshold corrections than $Y_{\tau}$ and $Y_t$; this disfavours Yukawa unification. On the other hand $\MHU$ and $\MHD$ becomes more negative for even smaller values of $\MSX$ and $\MHF$, which disfavors REWSB. The overall APS is somewhat smaller compared to the universal case, which is illustrated in fig. \[tbtaum0p.8md5a0\_1\] for $\MTN = .8\MSX$ (compare with fig. \[tbtaum0p1md5a0\_1\]). The UFB1 constraint still imposes an upper bound on the gaugino mass for a given $\MSX$ as in the universal case. As a result the gauginos are within the striking range of LHC practically over the entire APS. We also note from fig. \[tbtaum0p.8md5a0\_1\] that $\MSX \gapp 1600$ GeV over the entire APS. For a different pattern of non-universality ($\MTN > \MSX$), Yukawa unification alone narrows down the APS considerably. However, it is seen that regions with simultaneously low values of $\MSX$ and $\MHF$ are permitted. This happens in this specific nonuniversal scenario only. On the otherhand, the parameter space with large $\MSX$ and small $\MHF$, preferred by the earlier scenarios, is disfavoured. With $\MTN = 1.2\MSX$ (fig. \[tbtaum0p1.2md5a0\_1\]), it is found that $\MHF\gapp$ 300 GeV. The unification allowed parameter space, however, is very sensitive to the UFB conditions which practically rules out the entire APS for negative $A_0$. No major change is noted in the APS for $A_0 = 0$ and $A_0 = \MSX$ apart from the fact that the UFB constraints get weaker. We now discuss the impact of larger D-terms on the parameter space. For example, with $m_D = \MSX/3$, the APS reduces drastically in the universal as well as non-universal scenario with $\MTN < \MSX$, irrespective of $A_0$. This is illustrated in fig. \[tbtaum0p1md3a0\_1\]. and is in complete agreement with our qualitative discussion in the earlier section. Only in the specific nonuniversal scenario with $\MTN > \MSX$, slightly larger $m_D$ is preferred. However, $m_D$ cannot be increased arbitrarily. For $\MTN=1.2\MSX$, the APS begins to shrink again for $m_D \gapp \MSX/3$ and we find no allowed point for $m_D = \MSX/2$. As $\MTN$ is increased further, Yukawa unification occurs in a narrower APS. This, nevertheless, is a phenomenologically interesting region where lower $\MSX-\MHF$ values can be accommodated. For example, $\MSX(\MHF) =$400(300)GeV is allowed with $\MTN=1.5\MSX, m_D=\MSX/3$, $A_0$=0 and tan$\beta \sim$ 51, leading to $\MGL$ = 742GeV, $\MSQ \approx$ 700 GeV, $\MSL \approx$ 400GeV and $m_{\tilde \tau_1}$ = 274GeV. However, we cannot increase $\MTN$ arbitrarily either, the APS reduces drastically for $\MTN \gapp 1.5 \MSX$ irrespective of the value of $m_D$. This trend qualitatively remains the same even if $A_0$ is changed. This effect can be seen in $b-\tau$ unification as well. $b$-$\tau$ Yukawa unification ----------------------------- In our earlier work[@paper1] without D-terms, we had shown that the APS is strongly restricted due to Yukawa unification and UFB constraints. The minimum value of tan$\beta$ required for unification is $\approx$ 30. If D-terms are included, Yukawa unification and REWSB occur over a larger region of the parameter space. This is primarily due to two reasons: i) Yukawa unification can now be accommodated for lower values of tan$\beta$ ($\sim 20$) and ii) REWSB is allowed at somewhat higher values of $\tan\beta$ than the values permitted in $m_D = 0$ case. This reduces the conflict between Yukawa unification and REWSB. As a result $\MHF$ almost as low as that allowed by the LEP bound on the chargino mass is permitted over a wide range of $\MSX$. In some cases the upper bound on $\MSX$ for a given $\MHF$ is also relaxed. Similarly for a fixed $\MSX$, the upper bound on $\MHF$ is sometimes relaxed by few hundred GeVs. Through out this work we require this partial unification to an accuracy of $<$ 5%. Now we will focus our attention on large negative values of $A_0$ ($A_0 = -2\MSX$) with $m_D = \MSX/5$ in the universal scenario. The unification allowed APS, as shown in fig. \[btaum0p1md5a0\_2\], expands compared to the $m_D$ = 0 scenario (compare with fig. 6 of [@paper1]). Moreover, the phenomenologically interesting scenario with light gauginos but very heavy sleptons and squarks beyond the reach of LHC, which was rather disfavoured without the D-terms (see [@paper1]), is now viable. Without the D-term, the APS was severely restricted by the UFB conditions for large negative values of $A_0$. As discussed earlier, inclusion of the D-term increases the value of $\mu$. As a result UFB1 looses its constraining power; lower values of $\MHF$ are allowed for large $\MSX$ by UFB1. On the other hand as the value of D-term increases, UFB3 becomes more powerful and the upper bounds on $\MHF$ for relatively low values of $\MSX$ get stronger (for $\MSX$ = 600(1000)GeV, $\MHF <$ 300(600)GeV). For $A_0 > 0$, the APS again expands. However, the UFB constraints are found to be progressively weaker as $A_0$ is increased from $A_0 = -2\MSX$. We next consider the non-unversal scenerio $\MTN \ne \MSX$. If we take $\MTN < \MSX$ (say, $\MTN=.6\MSX$) and $m_D = \MSX/5 $, the unification allowed parameter space for $A_0 = -2\MSX$, as shown in fig. \[btaum0p.6md5a0\_2\], is more or less the same as in the universal scenario. The entire APS is, however, ruled out due to a very powerful constraint obtained from the UFB3 condition. This conclusion obviously holds for larger values of $m_D$. For $\MTN > \MSX$ and large negative $A_0$ ($A_0 = - 2\MSX$), the unification allowed APS (fig. \[btaum0p1.2md5a0\_2\]) is smaller compared to that in the universal case (fig. \[btaum0p1md5a0\_2\]). The same trend was also observed with $m_D=0$[@paper1]. The APS, however, is significantly larger than that for $m_D = 0$. For a given $\MHF$ ($\MSX$) the upper-bound on $\MSX$ ($\MHF$) gets weaker for non-zero D-terms. Relatively light gluinos consistent with current bounds are allowed over a larger region of the parameter space. The UFB constraints restrict the APS further and put rather strong bounds on $\MHF$ and $\MSX$. A large fraction of this restricted APS is accessible to tests at LHC energies. The usual reduction of the APS due to unification constraints as $A_0$ is increased from $A_0 = - 2\MSX$ also holds in this nonuniversal scenario. If we increase $m_D$ further, the APS due to Yukawa unification reduces for reasons already discussed. The UFB1 constraint also gets weaker. On the other hand the UFB3 constraints become rather potent. For example, i) with $A_0 = - 2\MSX$ and $m_D \gapp \MSX/3$ the entire APS for $\MTN = \MSX$ or $\MTN < \MSX$ is ruled out. ii) $A_0 = -\MSX$ and $m_D \gapp \MSX/2$ the entire APS corresponding to $\MTN = \MSX$ or $\MTN < \MSX$ is ruled out. On the other hand, for $\MTN > \MSX$ the APS further reduces as $m_D$ is increased. Conclusion ========== For moderate values of the D-terms $(m_D \approx \MSX/5)$, the APS expands in general compared to the $m_D=0$ case for both $b-\tau$ and $t-b-\tau$ Yukawa unification. A large fraction of the enlarged APS is, however, reduced by the requirement of vacuum stability and the predictive power is not lost altogether. D-terms with much larger magnitudes, however, are not favourable for unification. In the $t-b-\tau$ Yukawa unified model (accuracy $\leq 5\%$), a band of very low gaugino mass close to the current experimental lower limit is a common feature in the presence of D-terms. For a given $\MSX$ there is an upper bound on $\MHF$ from unification and stability of the potential constraints. This happens for $-\MSX\lapp A_0\lapp\MSX$. Outside this range of $A_0$, the APS is very small with sparticle masses of the first two generations well above 1 TeV. In $b-\tau$ unification, UFB3 strongly restricts the APS while UFB1 becomes less potent in the presence of D-terms. [***Acknowledgements***]{}: The work of AD was supported by DST, India (Project No. SP/S2/k01/97) and BRNS, India (Project No. 37/4/97 - R & D II/474). AS acknowledges CSIR, India, for his research fellowship. [99]{} For reviews see, [*e.g.*]{}, H.P. Nilles, Phys. Rep. [**110**]{}, 1 (1984); P. Nath, R. Arnowitt and A. Chamseddine, Applied N=1 Supergravity, ICTP series in Theoretical Physics, vol. I, World Scientific (1984); H. Haber and G.L. Kane, Phys. Rep. [**117**]{}, 75 (1985). See, ALEPH Coll., A. Heister , (B544,73,2002); OPAL Coll. G. 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[^1]: Electronic address: adatta@juphys.ernet.in [^2]: Electronic address: abhijit@juphys.ernet.in [^3]: This is due to the fact that unification holds at relatively lower values of $\tan\beta$ as one goes to larger negative values of $A_0$. There is, therefore, more room for increasing $\tan\beta$, if required, without violating REWSB condition. This point was not ellaborated in our earlier work[@paper1].

--- author: - 'Shigefumi [Naka]{}, Haruki [Toyoda]{} and Aiko [Kimishima]{},' title: 'Q-Deformed Bi-Local Fields' --- Introduction ============ The aim of non-local field theories proposed originally by Yukawa[@non-local] was twofold: firstly, characteristic properties of elementary particles such as mass spectrum of hadrons are to be derived from their extended structure. Secondly, those fields are expected to save the divergence difficulty, which is inherent in the local field theories with local interactions. The bi-local field theory[@bi-local] was the first attempt by Yukawa in this line of thought. As for the bi-local fields, the first aim has been studied by many authors in the context of effective relativistic two-particle systems of quark and anti-quark bound systems. In particular, the two-particle systems bounded by a relativistic harmonic oscillator potential were desirable to get a linear mass square spectrum associated with the Regge behavior in their scattering amplitude[@Barger-Cline][@Regge]. On the other hand, the second aim was rather unsuccessful mainly because the bi-local fields are reduced to superposition of infinite local fields with different masses, although some attempts claimed that the second order self-energy becomes convergent associated with the direction of center of mass momenta. In addition to this, the problem of unitarily of scattering matrix and that of the causality are also serious problems for those fields, since the bi-local system allows time-like relative motions in general. Usually, such a degree of freedom is frozen by an additional subsidiary condition[@Takabayasi], which is not always successful, however, for interacting cases. This situation may be different from that of string models which are characterized by the Virasoro condition associated with the parameterization invariance in such an extended model; nevertheless the study of bi-local field theories does not come to end, since a small change in models, sometimes, will cause a significant change in their physical properties. Under these backgrounds, the purpose of this paper is to study the q-deformation[@Macfarlane][@Wess][@Sogami][@Quantum-Groups] of a bi-local system characterized by the relativistic harmonic oscillator potential, since the q-deformation is well defined for harmonic oscillator systems. In a previous paper[@5-dimension], we have studied a q-deformed 5-dimensional spacetime such that the extra dimension generates a harmonic oscillator type of potential for particles embedded in that spacetime. Then, the propagator of particles in that spacetime acquires significant convergent property by requiring that the 4-dimensional spacetime variables and the extra dimensional one are mixed by the deformation. It is, then, happen that the spacetime variables become non-commutative between 4-dimensional components and the fifth one. We can expect the same situation in the bi-local system if we carry out the deformation of the relative variables in the bi-local system on a parallel with the extra dimension in the q-deformed 5-dimensional spacetime. In the next section, we try to formulate the bi-local system with q-deformed relative coordinates. In that case, we have to define the deformation carefully to keep the covariance, since the q-deformation of harmonic oscillator is ambiguous other than one dimensional space. In §3, the interaction of the bi-local field is discussed within the level of calculating Feynman diagrams. Some scattering amplitudes between the bi-local system and external scalar fields is also studied paying attention to their Regge behavior. We also calculate a self-energy type of diagram to study the convergence of the model to the second order. §4. is devoted to the summary and discussions; and in Appendix A, we also give the discussions on the representation of q-deformation in N-dimensional space. Formulation =========== The classical action of equal mass two-particle system that leads to the standard set of bi-local field equations is simply given by $$S=\sum_{i=1}^2\frac{1}{2}\int d\tau\left\{\frac{\dot{x}^{(i)2}}{e_i}+e_i\left(m^2+V(\bar{x}) \right) \right\}, ~(\bar{x}=x^{(1)}-x^{(2)}) \label{BL action}$$ where $e_i(\tau),(i=1,2)$ are gauge variables, einbein’s, that assure the reparametrization invariance with respect to $\tau$. Namely, it is assumed that the $e_i$ transforms $$e_i(\tau)=\left(\frac{df(\tau')}{d\tau'}\right)^{-1}e_i(\tau'), \label{gauge transformation}$$ according as $\tau=f(\tau')$. By definition, the momentum conjugate to $x^{(i)}$ is given by $p^{(i)}=\frac{\delta S}{\delta x^{(i)}}=\frac{1}{e_i}\dot{x}^{(i)}$, and so, the variation of $S$ with respect to $e_i$ gives rise to the constraints [^1] $$\frac{\delta S}{\delta e_i}=-\left\{p^{(i)2}-\left( m^2+V(\bar{x}) \right) \right\} =0,~(i=1,2). \label{BLC}$$ The sum of these constraints becomes $$\sum_{i=1}^2(p^{(i)2}-m^2)-2V(\bar{x})=\frac{1}{2}P^2+2(\bar{p}^2-m^2)-2V(\bar{x})=0, \label{BLC-1}$$ where $P=p^{(1)}+p^{(2)}$ and $\bar{p}=\frac{1}{2}(p^{(1)}-p^{(2)})$. On the other hand, the subtraction between two constraints leads to the constraint $$P\cdot\bar{p}=0 , \label{BLC-2}$$ which can eliminate the degree of freedom of the relative time $\bar{x}^0$ at the rest frame $P=(P^0, {\boldsymbol{0}})$. In practice, however, compatibility between the constraints (\[BLC-1\]) and (\[BLC-2\]) dependents on the functional form of $V(\bar{x})$. A simple way to remove this problem is to make the substitutions $\bar{x}^\mu \rightarrow \bar{x}_\perp^\mu$ and $\bar{p}^\mu \rightarrow \bar{p}_\perp^\mu$, where $\bar{x}_\perp =(\eta_{\mu\nu}-\frac{P_\mu P_\nu}{P^2})\bar{x}^\nu$ and so on. As for the covariant harmonic oscillator potential $V(\bar{x})=-\kappa^2 \bar{x}^2$, there is another way, which is similar to the Gupta-Bleuler formalism in Q.E.D.. In this case, if we define the oscillator variables $(a^\dagger,a)$ in q-number theory by $$\bar{x}=\frac{1}{\sqrt{2\kappa}}(a^\dagger + a),~~~~ \bar{p}=i\sqrt{\frac{\kappa}{2}}(a^\dagger - a) ,$$ then Eq(\[BLC-1\]) and Eq.(\[BLC-2\]) can be understood as the wave equation for the bi-local system and its subsidiary condition such that $$\begin{aligned} & \left(\alpha^\prime P^2 + \frac{1}{2}\{ a^\dagger_\mu,a^\mu \}-\omega \right)|\Phi \rangle =0, \label{wave-eq} & \\ & P\cdot a|\Phi \rangle = 0, & \label{subsidiary}\end{aligned}$$ where $\alpha^\prime=\frac{1}{8\kappa}$ and $\omega=\frac{m^2}{2\kappa}$. The compatibility between Eqs.(\[wave-eq\]) and (\[subsidiary\]) can be verified easily; and further, the Eq.(\[BLC-2\]) holds in the sense of the expectation value $\langle\Phi|P\cdot\bar{p}|\Phi\rangle = 0$. Now, let us consider the q-deformation, the mapping $(a_\mu,a_\mu^\dagger)\rightarrow (A_\mu,A_\mu^\dagger)$, of this bi-local system. In our standpoint, the deformation is a way getting a new wave equation without changing the geometrical meaning of particle’s cordinates $x^{(i)},^(i=1,2)$. As shown in Appendix A, however, the way of q-deformation is not unique; and we follow the case (iii) in Appendix A defined by the mapping $$\begin{aligned} A_\mu=a_\mu\sqrt{\frac{[N]_q}{N_\mu}},~~A_\mu^\dagger=\sqrt{\frac{[N]_q}{N_\mu}}a_\mu^\dagger ~, \label{mapping} \\ (N=-a_\mu^\dagger a^\mu;~N_0=-a_0^\dagger a_0,N_i=a_i^\dagger a_i). \nonumber\end{aligned}$$ Here $[N]_q$ is the q-deformed number operator defined in Eq.(\[\[N\]\]) with $$\alpha=1~~~{\rm and}~~~\beta=\beta_0-\alpha_0^\prime P^2~~,(\alpha_0^\prime >0). \label{beta}$$ We note that the $\beta$ in Eq.(\[\[N\]\]) may be a q-number variable commuting with oscillator variables. In the bi-local system, Eq.(\[beta\]) gives a possible form of q-number $\beta$, which does not break the Lorentz covariance and the transnational invariance of the resultant wave equation. Then, the q-deformed counterpart of Eq.(\[wave-eq\]) becomes $$\left(\alpha^\prime P^2 + \frac{1}{2}\{ A^\dagger_\mu,A^\mu \}(P^2,N)-\omega \right)|\Phi \rangle = 0, \label{q-wave-eq}$$ where $$\frac{1}{2}\{ A^\dagger_\mu,A^\mu \}(P^2,N)=-2\frac{\sinh\{(N+\beta_0-\alpha_0^\prime P^2 +\frac{1}{2})\log q\}}{\sinh(\frac{1}{2}\log q)} \label{q-mass}$$ The subsidiary condition (\[subsidiary\]) is again compatible with the q-deformed wave equation (\[q-wave-eq\]). Further, if we take the limit $q\rightarrow 1$, then Eq.(\[q-wave-eq\]) will be reduced to Eq.(\[wave-eq\]) provided that $\alpha_0^\prime=\frac{3}{4}\alpha^\prime$ and $\beta_0=\frac{3}{4}\omega-4$. It should also be noted that the resultant wave equation is covariant under the Lorentz transformation though $(A_\mu,A_\mu^\dagger)$ do not transform as four vectors. Now, the mass square of the q-deformed bi-local system is determined by solving $m_n^2=-\frac{1}{2\alpha^\prime}\{A,A^\dagger\}(m_n^2)+\frac{\omega}{\alpha^\prime}$ for each $n(=0,1,2,\cdots)$, the eigenvalue of $N$. The $m_n^2$ can be solved uniquely under the sign of $\alpha_0^\prime$ in Eq.(\[beta\]) so that there exist no spacelike solutions as in Fig.1. It should also be noted that the operator acting on the physical state $|\Phi \rangle$ in Eq.(\[q-wave-eq\]) behaves as $(\cdots)=\sum_{k\geq 1}c_k(n)(P^2-m_n^2)^k$ with $c_1(n)=\frac{\partial}{\partial P^2}(\cdots)|_{P^2=m_n^2}\neq 0$ near the on mass-shell points $P^2=m_n^2$; in other words, those points are the first order zero of the free propagator [^2] . The wave equation Eq.(\[q-wave-eq\]), thus, solved generally in the following form: ![The line $y=x$ vs. $y=\frac{1}{2\alpha^\prime}\{A^\dagger,A \}(x)-\frac{\omega}{\alpha^\prime}$ with $n=0,1,2,\cdots$; the mass-square eigenvalues are given as x-coordinates of the intersections.[]{data-label="fig:1"}](mass-2.eps){width="5cm" height="3cm"} $$|\Phi \rangle =\sum_{n=0}^\infty \int\frac{d^3 p}{\sqrt{(2\pi)^2 2p_n^0}}\left(e^{-ip_n\cdot x}|\Phi_n\rangle c_n + h.c. \right),$$ where $N|\Phi_n\rangle=n|\Phi_n\rangle,~P\cdot a|\Phi_n\rangle=0$ and $p_n=(\sqrt{{{\boldsymbol{p}}}^2+m_n^2},{{\boldsymbol{p}}}),~(n=0,1,2,\cdots)$. It should also be stressed that the wave equation does not allow [^3] of a complex $P^2$ in addition to a space-like $P^2$. This means that the free bi-local field is similar to local free fields in such a sense that its spacetime development can be determined by the Cauchy data without confliction with the causality. On the other hand, the Feynman propagator $$G(P^2,N)=\left(P^2+\frac{1}{2\alpha^\prime}\{A,A^\dagger\}(P^2,N)-\frac{\omega}{\alpha^\prime}+i\epsilon \right)^{-1} \label{propagator-1}$$ decreases exponentially according as $N,|P^2| \rightarrow \infty$. As will be shown in the next section, this enables us to get a finite vacuum loop amplitude in contrast to local field theories. Interaction of the bi-local field ================================= As discussed in the previous section, we understand the q-deformation as a way getting a new dynamical system by the mapping $(a_\mu, a_\mu^\dagger)\rightarrow (A_\mu, A_\mu^\dagger)$. In other words, the physical meaning of the variables $X^\mu$ and $\bar{x}^\mu$ is not changed, but a modification is down only for the wave equation of free bi-local field. Then, the vertex function $|V \rangle$ corresponding to Fig.\[fig:2\] should be determined by the conditions[@Goto-Naka] $$\begin{aligned} ~[ x^{(1)}(b) - x^{(2)}(a) ] |V \rangle & = & 0 \nonumber \\ ~[ p^{(1)}(b) + p^{(2)}(a) ] |V \rangle & = & 0 \label{3-vertex-1}\\ (a,b,c~~{\rm cyclic})~~~~~~ && \nonumber\end{aligned}$$ ![The dashed line denotes the local scalar field corresponding to the ground state of $b$.[]{data-label="fig:3"}](3-vertex.eps){width="5cm" height="4cm"} ![The dashed line denotes the local scalar field corresponding to the ground state of $b$.[]{data-label="fig:3"}](vertex-2.eps){width="5cm" height="4cm"} with the subsidiary conditions $P(i)\cdot a(i)|V\rangle =0,(i=a,b,c)$. The explicit form of the vertex operator can be obtained by a standard way. In what follows, however, for the sake of simplicity, we consider a simpler case such that the particle b in Fig.\[fig:3\] is the ground state of the system, which can be identified with an external scalar field. Then the vertex conditions (\[3-vertex-1\]) become $$\begin{aligned} ~[ x^{(1)}(a) - x^{(2)}(c) ] |V \rangle & = & [ x^{(2)}(a) - x^{(1)}(c) ] |V \rangle = 0 \nonumber \\ ~[ p^{(1)}(a) + p^{(2)}(c) ] |V \rangle & = & [ p^{(2)}(a) + p^{(1)}(c)+p(b) ] |V \rangle = 0 , \label{3-vertex-2}\end{aligned}$$ where $p(b)$ is the momentum of the external scalar field. The conditions (\[3-vertex-2\]) with $P(i)\cdot a(i)|V\rangle =0,(i=a,c)$ can be solved easily; and, in $p$-representation for center of mass variables, we obtain $$\begin{aligned} |V\rangle = &g& \delta^{(4)}(P(a)+P(b)+P(c)) \\ \nonumber & \times & e^{-\frac{i}{2\sqrt{2\kappa}}(a(a)_\perp^\dagger-a(c)_\perp^\dagger)\cdot P(b)+a(a)_\perp^\dagger\cdot a(c)_\perp^\dagger}|0\rangle , \label{vertex-1}\end{aligned}$$ where $g$ is the coupling constant. The $|0\rangle=|0_a\rangle\otimes |0_c\rangle$ is the product of the ground states defined by $a(i)_\mu|0_i\rangle=0_i$ and $\langle 0_i|0_i \rangle =1~(i=a,c)$. Further, the projection of $a(i)_\mu$ to its physical components is written as $a_\perp(i)_\mu=O_{\mu\nu}(i)a^\nu(i)$ with $O_{\mu\nu}(i)=\eta_{\mu\nu}-P_\mu(i)P_\nu(i)/P^2(i)~(i=a,c)$. If we remark that $$\begin{aligned} && \langle \phi_a\{a(a)\}|\otimes \langle \phi_c\{a(c)\} | e^{-\frac{i}{2\sqrt{2\kappa}}(a(a)_\perp^\dagger-a(c)_\perp^\dagger)\cdot P(b)+a(a)_\perp^\dagger\cdot a(c)_\perp^\dagger}|0\rangle \nonumber \\ &=& \langle \phi_c\{a(c)\}|e^{\frac{i}{2\sqrt{2\kappa}}a(c)_\perp^\dagger\cdot P(b)} \langle 0_a|e^{a(a)_\perp\cdot a(c)_\perp^\dagger}|0_c \rangle e^{\frac{i}{2\sqrt{2\kappa}}a(a)_\perp\cdot P(b)}|\phi_a\{a(a)^\dagger\}\rangle \nonumber \\ &=& \langle \phi_c\{a(c)\}|:e^{\frac{i}{2\sqrt{2\kappa}}(a(c)^\dagger+a(c))_\perp\cdot P(b)}:|\phi_a\{a(c)^\dagger\} \rangle , \end{aligned}$$ we obtain another expression of the vertex operator $$\tilde{V}(c,b,a)=g\delta^{(4)}(P(a)+P(b)+P(c)):e^{\frac{i}{2}\bar{x}^{(1)}(c)_\perp\cdot P(b)}: , \label{vertex-2}$$ in which the operators $(a(a),a(a)^\dagger)$ are identified as $(a(c),a(c)^\dagger)$. In this case, we may confine our discussion to the physical space constructed out of $a_\perp^\dagger$ and $|0\rangle$; and then, we can write the propagator (\[propagator-1\]) in the following form: $$G(P^2,N_\perp) = \int_C \frac{d\zeta}{2\pi i\zeta} \sum_{n=0}^\infty \zeta^{(N_\perp - n)} G(P^2,n), \label{propagator-2}$$ where $N_\perp=-a_\perp^\dagger\cdot a_\perp$. The integral of the complex variable $\zeta$ is taken along a closed contour $C$ surrounding $\zeta=0$; for the later use, we set $C$ as a circle such as $|\zeta|<1$. Using the above propagator and the vertex operator (\[vertex-1\]) or (\[vertex-2\]), first, let us calculate the scattering amplitude corresponding to Fig.[\[fig:4\]]{}. For convenience, we put $a$ and $a'$ in Fig.\[fig:4\] as the coherent states $|z_a\rangle=e^{-a(a)_\perp^\dagger\cdot z_a}|0\rangle$ and $|z_{a'}\rangle=e^{-a(a')_\perp^\dagger\cdot z_{a'}}|0\rangle$, respectively. ![A second order scattering amplitude of bi-local system by two external fields with $s=(P(a)+P(b))^2$ and $t=(P(a)-P(a'))^2$.[]{data-label="fig:4"}](scatt.eps){width="4.5cm" height="5cm"} Then, we have the second order scattering amplitude $$\begin{aligned} A^{(2)} &=& \langle z_{a'}|\tilde{V}(a',b',c)G(s,N(c)_\perp)\tilde{V}(c,b,a)|z_a\rangle \nonumber \\ &=& g^2\delta^4(P(a')+P(b')-P(a)-P(b))A^{(2)}_s, \label{scattering-1}\end{aligned}$$ where $$\begin{aligned} A^{(2)}_s &=& e^{-\frac{i}{2\sqrt{2\kappa}}(z_{\perp a'}^*\cdot P(b')-z_{\perp a}\cdot P(b))}\sum_{n=0}^\infty G(s,n)\times \nonumber \\ & \times & \frac{[(z_{a'}^*+\frac{i}{2\sqrt{2\kappa}}P(a'))\cdot O(c)\cdot (z_a-\frac{i}{2\sqrt{2\kappa}}P(a))]^n}{n!}. \label{scattering-2}\end{aligned}$$ The summation with respect to $n$ in Eq.(\[scattering-2\]) is convergent for a fixed $s$, since $G(s,k)\sim -\sinh(\frac{1}{2}\log q)e^{-(n+\beta_0-\alpha'_0 s+\frac{1}{2})\log q}$ for $(n+\beta_0-\alpha'_0 s+\frac{1}{2})\log q \gg 1$. In particular, for the simpler case $z_a=z_{a'}=0$ corresponding to Fig.\[fig:5\], the amplitude takes a simple form $$A^{(2)}_s =\sum_{n=0}^\infty G(s,n)\frac{[\frac{1}{8\kappa}(\frac{2m_0^2-t)}{2}-\frac{s}{2})]^n}{n!}. \label{scattering-3}$$ Then, one can see that in the limit $s \rightarrow \infty$ while $t$ fixed, the $n$-th term in the right-hand side decreases rapidly as $\sim (-\frac{\alpha'}{2}s)^ne^{-\alpha'_0 s\log q}$, since $G(s,n)\sim q^n\sinh(\frac{1}{2}\log q)e^{-(\alpha'_0 s-\beta_0-\frac{1}{2})\log q}$ for $\alpha'_0 s\log q \gg 1$. On the other hand, in the limit $t \rightarrow \infty$ while $s$ fixed, we can write $$A^{(2)}_s \sim \alpha'\sum_{n=0}^\infty \frac{1}{i}\int_0^\infty d\tau e^{i\tau \alpha'G(s,n)^{-1}} \frac{1}{n!}\left(-\frac{1}{2}\alpha' t\right)^n . \label{scattering-3}$$ I should be notice that $G(s,n)$ decreases rapidly according as $|n-\alpha_0(s)|$ increase, where $\alpha_0(s)=\alpha'_0 s-\beta_0-\frac{1}{2}$. Hence, if we approximate $\alpha'G(s,n)^{-1} \simeq \alpha(s)-\dot{\alpha}(n-\alpha_0(s))$ with $\alpha(s)=\alpha' s-\omega,~\alpha_0(s)=\alpha'_0 s-\beta_0 -\frac{1}{2}$ and $\dot{\alpha}=1/\sinh(\frac{1}{2}\log q)$, then summation with respect to $k$ in Eq.(\[scattering-3\]) will give the factor $\exp(-\frac{1}{2}\alpha' t e^{-i\tau\dot{\alpha}})$. Then, evaluating the integral with respect to $\tau$ by the method of steepest descent, we obtain $$A^{(2)}_s \sim F(s)\times (\alpha^\prime t)^{\alpha_1(s)} ,$$ where $F(s)$ is a function of $s$, and $\alpha_1(s)=(\frac{\alpha^\prime}{\dot{\alpha}}+\alpha^\prime_0)s-(\frac{\omega}{\dot{\alpha}}+\beta_0+\frac{1}{2})$. This means that the $t$-channel amplitude $A^{(2)}_t$ in Fig.\[fig:6\] obtained by interchanging $s$ and $t$ from $A^{(2)}_s$ shows the Regge behavior for a large $s$ while $t$ fixed. ![The t-channel amplitude obtained by the interchange $s \leftrightarrow t$ from Fig.\[fig:6\].[]{data-label="fig:6"}](s-chan.eps){width="4cm" height="4cm"} ![The t-channel amplitude obtained by the interchange $s \leftrightarrow t$ from Fig.\[fig:6\].[]{data-label="fig:6"}](t-chan.eps){width="5cm" height="4cm"} As the final of this section, let us consider the loop diagram Fig.\[fig:7\] corresponding to the self-energy $\delta m^2$ for the ground state, which can be written as ![The self-nenergy diagram for the ground state.[]{data-label="fig:7"}](loop.eps){width="5cm" height="3cm"} $$\begin{aligned} \delta m^2 &\sim g^2\int d^4p Tr\left[G((p-k)^2,N_\perp(p-k)):e^{-\frac{i}{2}\bar{x}\cdot O(p-k)\cdot k}:G(p^2,N_\perp(p)):e^{\frac{i}{2}\bar{x}\cdot O(p)\cdot k}: \right] \nonumber \\ &= \delta m^2(|p|\lnsim |k|) + \delta m^2(|p|\gg |k|) . \label{self-energy}\end{aligned}$$ The first term of the r.h.s in Eq.(\[self-energy\]) will be finite; and the second term can be roughly evaluated as $$\begin{aligned} \delta m^2(|p| \gg |k|) &\sim g^2\int d^4p \int_C \frac{d\zeta}{2\pi i\zeta} \int_C \frac{d\zeta'}{2\pi i\zeta'} \sum_{n=0}^\infty \sum_{n'=0}^\infty \times \nonumber \\ &Tr_{phys} \left[\zeta^{N_\perp(p)}:e^{-\frac{i}{2}\bar{x}\cdot O(p)\cdot k}:\zeta'^{N_\perp(p)}:e^{\frac{i}{2}\bar{x}\cdot O(p)\cdot k}: \right]\zeta^{-n}G(p^2,n)\zeta'^{-n'}G(p^2,n') , \label{self-energy-2}\end{aligned}$$ where $Tr_{phys}[\cdots]$ means the tracein the physical subspace defined by Eq.(\[subsidiary\]). Since the operators in Eq.(\[self-energy-2\]) are constructed out of $(a_\perp,a_\perp^\dagger)$, the trace can be calculated by using the coherent state[@Coherent] $|z\rangle =e^{-z\cdot a^\dagger}|0\rangle$ as follows $$\begin{aligned} Tr_{phys}[\cdots] &= \int\left(\prod_{\mu=0}^3\frac{d^2z^\mu}{\pi}\right) e^{\bar{z}^*\cdot z}\langle \bar{z}|\cdots|z\rangle |_{z_\parallel=0} \nonumber \\ &= \frac{1}{(1-\zeta\zeta')^4}\exp \left[ -\frac{m_0^2}{2p^2}\frac{2\zeta\zeta'-(\zeta + \zeta')}{1-\zeta\zeta'}\right] \simeq \frac{1}{(1-\zeta\zeta')^4}, \label{trace}\end{aligned}$$ where $\bar{z}=(-z^0,z^1,z^2,z^3)$ and $z_\parallel=\frac{P(P\cdot z)}{P^2}$. As a function of $\zeta (\zeta')$, the last form of Eq.(\[trace\]) is singular at $\zeta=\zeta'^{-1} (\zeta'=\zeta^{-1})$, which is, however, located in outside of the counter $C$. Further, since $G(p^2,n)$ has no pole of imaginary $p^0$, we can evaluate the integral with respect to $p$ in Eq.(\[self-energy-2\]) by means of analytic continuation $p=(i\bar{p}^0,\bar{p}^i)$. Thus, approximating $G(-\bar{p}^2,n) \simeq -\alpha'\sinh(\frac{1}{2}\log q)e^{-(n+\beta_0+\alpha_0'\bar{p}^2+\frac{1}{2})\log q}$ for large $\bar{p}$, the summation with respect to $n,n'$ gives rise to $[\alpha'\sinh(\frac{1}{2}\log q)e^{-(\beta_0+\alpha_0'\bar{p}^2+\frac{1}{2})\log q}]^2(1-\zeta q^{-1})(1-\zeta'q^{-1})$. Then we can carried out the $p$-integral in Eq.(\[self-energy-2\]) so that $$\begin{aligned} \delta m^2(|p| \gg |k|) & \sim i \left(\frac{g\pi\alpha'\sinh(\frac{1}{2}\log q)}{2\alpha_0'\log q}e^{-(\beta_0+\frac{1}{2})\log q}\right)^2 \nonumber \\ & \int_C\frac{d\zeta}{2\pi i\zeta} \int_C\frac{d\zeta'}{2\pi i\zeta'}\frac{1}{(1-\zeta q^{-1})(1-\zeta' q^{-1})(1-\zeta\zeta')^4}. \label{self-energy-3} \end{aligned}$$ The integral with respect to $\zeta$ and $\zeta'$ comes to be $1$ provided that the radius of the contour $C$ is smaller than $q$; therefore, the self energy in our model is be convergent [^4] . Summary and discussions ======================= The relativistic two-particle system, the bi-local system, bounded by 4-dimensional harmonic oscillator potential yields a successful description of two-body meson like states. The mass square spectrum, then, arises from the excitation of relative variables, which are independent of center of mass variables. In this paper, we have tried to construct a q-deformed bi-local system in such a way that the center of mass momenta are included in the deformation parameters of relative variables. In other words, the q-deformed relative variables become functions of the center of mass momenta and of original relative variables, which are independent of center of mass variables. Then, the q-deformed relative coordinates become non-commutative, while the center of mass coordinates remain as commutative variables. As a result of this deformation, the mass square operator gives rise to a non-linear mass square spectrum. The way of deformation is not unique, and we have defined it from a heuristic point of view such that the mass square operator becomes a Lorentz scalar in spite of non-covariant property of oscillator variables. In the q-deformed bi-local system, the wave function of the system acquires new aspects such that the propagator of the system dumps rapidly according as $|P^2|$ or $N$ tends to infinity. With that in mind, further, we have studied the interaction of the bi-local system with external scalar fields, which are identified with the ground state of the system. To this end, we have defined the three vertexes among two bi-local systems and one scalar field; and then, we have verified the flowing: First, the second order t-channel scattering amplitude shows the Regge behavior in the limit $t \rightarrow \infty$ while $s$ fixed. Secondly, we have calculated a second-order self-energy diagram, and it is shown that the self energy diagram of the bi-local field comes to be convergent due to the characteristic property of the propagator. The convergence problem, however, is still subtle, since it is necessary to choose a suitable contour in the integral representation of the propagator. In order to fix the field theoretical properties in the q-deformed bi-local system, it will be necessary to study the higher order diagrams. Further, for interacting case, the problem of causality is remained as an open question, since the field equation contains higher order derivatives with respect to time parameter. In addition to those, the way of q-deformation (\[mapping\]) for 4-dimensional oscillator variables is not sufficient because of its non covariant property. Thus, it should also be required to study the guiding principle defining the deformation from various points of view. These are interesting and important subjects for a future study. Acknowledgements {#acknowledgements .unnumbered} ================ The authors wish to express their thanks to the members of their laboratory for discussions and encouragement. Representation of a q-oscillator ================================ We here review the representation of the q-oscillator variables defined by $$[A,A^\dagger]_q \equiv A A^\dagger -q^\alpha A^\dagger A =q^{-\alpha ( N + \beta) }, \label{q-commutator}$$ where $\alpha,\beta$ and $q (\geq 1)$ are real parameters. The $N=a^\dagger a$ is the number operator for the ordinary oscillator variables satisfying $[a,a^\dagger]=1$. The explicit mapping between $(a,a^\dagger)$ and $(A,A^\dagger)$ is given by $$A = a \sqrt{\frac{[N]_q}{N}},~~A^\dagger = \sqrt{\frac{[N]_q}{N}}a^\dagger ,$$ where $N=a^\dagger a$, and $[N]_q$ is a function of $N$ determined in what follows. Now, remembering $Na=a(N-1)$ and $Na^\dagger=a^\dagger(N+1)$, one can verify that $$A A^\dagger = [N+1]_q,~~A^\dagger A = [N]_q .$$ Then Eq.(\[q-commutator\]) can be reduced to $$[N+1]_q - q^\alpha [N]_q = q^{-\alpha (N +\beta) } . \label{recurrence}$$ The recurrence equation (\[recurrence\]) can be solved easily in the following form: $$[N]_q = q^{-\alpha\beta } \frac{q^{\alpha N} - q^{-\alpha N}}{q^\alpha - q^{-\alpha}} + q^{\alpha N}[0]_q ,$$ where $[0]_q$ is an arbitrary initial term for $N=0$. We, here, choose $$[0]_q= \frac{q^{\alpha\beta}-q^{-\alpha\beta}}{q^\alpha - q^{-\alpha}}$$ ; then, $[N]_q$ has the simple form $$[N]_q = \frac{q^{\alpha(N+\beta)}-q^{-\alpha(N+\beta)}}{q^\alpha -q^{-\alpha}} \label{[N]}$$ After this q-deformation, the hamiltonian type of combination $\frac{1}{2}\{ a^\dagger ,a \}+\beta$ of the ordinary oscillators should be replaced with $\frac{1}{2} \{ A^\dagger ,A \}$, to which we have the expression $$\begin{aligned} \frac{1}{2} \{ A^\dagger,A \} &=& \frac{1}{2}\left( [N]_q + [N+1]_q \right) \nonumber \\ &=& \frac{1}{2}\frac{\sinh \left[\alpha(N + \beta +\frac{1}{2})\log q \right]}{\sinh(\frac{1}{2}\alpha\log q)}. \label{q-hamiltonian}\end{aligned}$$ Indeed, one can verify that the $\frac{1}{2} \{ A,A^\dagger \}$ becomes $\frac{1}{2}\{ a^\dagger ,a \}+\beta$ according as $q \rightarrow 1$. This implies that the $\beta$ plays the role of an additional factor to the zero point energy of the four-dimensional oscillator for $q=1$; on the other hand, the $\alpha$ can be absorbed in the definition of $q$ by the substitution $q^\alpha \rightarrow q$. The q-deformation can be extended to N-dimensional oscillator variables, though there arise several problems. Let us consider the D-dimensional oscillator variables defined by $[a_i,a_j^\dagger]=\delta_{ij},[a_i,a_j]=[a\i^\dagger,a_j^\dagger]=0~(i,j=1,2,\cdots,D)$. We have to remarked that there is no mapping $A_i(a,a^\dagger)$ satisfying $$[A_i,A_j^\dagger]_q=\delta_{ij}f(N)~~{\rm and}~~[A_i,A_j]_q=[A_i^\dagger,A_j^\dagger]_q=0,$$ where $N=\sum_ia_i^\dagger a_i$. Indeed, since $A_j[A_i,A_j^\dagger]_q=q^{2\alpha}[A_i,A_j^\dagger]_q A_j+q^\alpha [A_i,f(N)]$ for $i\neq j$, we have $[A_i,f(N)]=0$, which holds only when $A_i$ is a function of $N$; that is, a function without vector indices $\{i\}$. The followings, however, may be available mappings:\ Case (i) $A_i=a_i\sqrt{\frac{[N_i]_q}{N_i}}$\ In this case, we have the simple q-deformed algebra $$[A_i,A_j^\dagger]_q =\delta_{ij}q^{-\alpha ( N_i + \beta) }, \label{q-commutator-2}$$ which spoils, however, $U(D)$ symmetry even for $\sum_i\{A_i,A_i^\dagger\}$. Case (ii) $A_i=a_i\sqrt{\frac{[N]_q}{N}}$\ This is a covariant mapping which keeps the $U(D)$ vector property of $A_i$. The algebra is, however, not simple so that $$[A_i,A_j^\dagger]_q=\frac{[N+1]_q}{N+1}\delta_{ij}+\left(\frac{[N+1]_q}{[N]_q}\frac{N}{N+1}-q^\alpha \right)A_j^\dagger A_i$$ In this case, the hamiltonian type of combination becomes $$\frac{1}{2}\sum_i\{ A_i^\dagger,A_i \} = \frac{1}{2}\frac{\sinh \left[\alpha(N + \beta +\frac{1}{2})\log q \right]}{\sinh(\frac{1}{2}\alpha\log q)}+\frac{1}{2}\frac{D+1}{N+1}\frac{\sinh \left[\alpha(N + \beta +1)\log q \right]}{\sinh(\alpha\log q)}.$$ Case (iii) $A_i = a_i\sqrt{\frac{[N]_q}{N_i}}$\ The mapping looks like to break $U(D)$ symmetry of the q-deformed algebra, but we have $$[A_i,A_j^\dagger]_q = q^{-\alpha(N+\beta)}A_i[N]_q^{-1}A_j^\dagger \label{commutator-3}$$ and the simple hamiltonian type of combination $$\frac{1}{2}\sum_i\{ A_i^\dagger,A_i \}=\frac{D}{2}\frac{\sinh \left[\alpha(N + \beta +\frac{1}{2})\log q \right]}{\sinh(\frac{1}{2}\alpha\log q)}. \label{q-hamiltonian-2}$$ The Eq.(\[q-hamiltonian-2\]) is the direct extension of Eq.(\[q-hamiltonian\]), though $\{A_i\}$ do not carry D-dimensional vector property from $\{a_i\}$. [99]{} H. Yukawa, Phys. Rev. [**77**]{} (1950), 219; Y. Katayama and H. Yukawa, Prog. Theor. Phys. Suppl. No.41 (1968), 1. See also, S. Naka, [*On the Study in Japan of Theory of Elementary Particle Extended in Space-Time*]{}, Proceedings of the International Symposium on Extended Objects and Bound Systems, ed. O. Hara, S. Ishida and S. Naka (World Scientific, 1992). H. Yukawa, Phys. Rev. [**91**]{} (1953), 415,416. See also the papers cited in the following review article: T. Gotō, S. Naka and K. Kamimura, Prog. Theor. Phys. Suppl. No.67 (1979), 69. See for example,\ V. D. Barger and D. B. Cline, [*Phenomenological Theories of High Energy Scattering*]{} (W. A. Benjamin, Inc. 1969), p41. L. Van Hove, Phys. Letters [**24B**]{} (1967), 183.\ M. Bandoo, T. Inoue, Y. Takada and S.Tanaka, Prog. Theor. Phys. [**38**]{} (1967), 715.\ S. Ishida, J. Otokozawa, Prog. Theor. Phys. [**47**]{} (1974), 2117. T. Takabayasi, Nuovo Cimento [**33**]{} (1964), 668. A.J. Macfarlane, J. of Phys. [**A 22**]{} (1989), 4581. M. Fichtmuller, A. Lorec, J. Wess, Z. Phys. [**C 71**]{} (1996), 533.\ B.L. Cerchiai, R. Hinterding, J. Madore, J. Wess, Eur. Phys. J. C(1999) DOI10.1007/s100 I.S. Sogami, K. Koizumi and R.M. Mir-Kasimov, Prog. Theor. Phys. [**110**]{} (2003), 819. As for a review article:\ M. Chaichian and A. Demichev, [*Introduction to Quantum Groups*]{}, World Scientific, 1996. S. Naka and H. Toyoda, Prog. Theor. Phys. [**109**]{} (2003), 103. T. Gotō and S. Naka, Prog. Theor. Phys. [**51**]{} (1974), 299. T. Gotō and T. Obara, Prog. Theor. Phys. [**49**]{} (1973), 322.\ [^1]: If we eliminate $e_i$ by using these constraints, the action of the bi-local system can be rewritten as ${\displaystyle S=\sum_i\int d\tau \sqrt{(m^2+V(\bar{x}))\dot{x}^{(i)2}} }$   provided that   $m^2+V(\bar{x})\neq 0$. [^2]: Indeed, $ c_1(n) = \alpha^\prime +2\alpha_0^\prime\ln q \frac{\cosh[(n+\beta_0-\alpha_0^\prime m_n^2+\frac{1}{2})\ln q}{\sinh(\frac{1}{2}\ln q)} \neq 0 $    for $\alpha',\alpha_0'>0 $. [^3]: The eigenvalues of $P^2$ are obtained as the solutions of the equation $z=A\sinh(B-Cz)+\omega$, where $z=\alpha'P^2,A=2/\sinh(\frac{1}{2}\log q),B=(N+\beta_0+\frac{1}{2})\log q$, and $C=\omega(\alpha_0'/\alpha')\log q$. Writing $z=x+iy$ for a complex $z$, the equation can be decomposed into $$x=A\sinh(B-Cx)\cos y+\omega~~~{\rm and}~~~y=-A\cosh(B-Cx)\sin y ~.$$ The second equation is satisfied only for $y=0$ provided $A>0$; thus, there are no complex eigenvalues for $P^2$. [^4]: If we do not ignore $m_0^2/p^2$ term in the exponential in Eq.(\[trace\]), then the substitution $$1\Big/(1-\zeta q^{-1})(1-\zeta' q^{-1})(1-\zeta\zeta')^4 \rightarrow \exp \left[ -8\sqrt{-(\alpha_0'm_0^2\log q)\frac{2\zeta\zeta'-(\zeta + \zeta')}{1-\zeta\zeta'}}\right] \Big/ (1-\zeta q^{-1})(1-\zeta' q^{-1})(1-\zeta\zeta')^4$$ must be done. In this case, the integral in Eq.(\[self-energy-3\]) is still convergent.

--- abstract: 'We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.' address: - 'School of Math. Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.' - 'School of Math. Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.' - 'Institute of Mathematics Hebrew Univ. of Jerusalem, Jerusalem 91904, Israel' author: - 'Jon. Aaronson, Zemer Kosloff $\&$ Benjamin Weiss' title: Symmetric Birkhoff sums in infinite ergodic theory --- [^1] §0 Introduction {#introduction .unnumbered} =============== Pointwise ergodicity {#pointwise-ergodicity .unnumbered} -------------------- Pointwise ergodicity for infinite measure preserving transformations fails. Let $\xbmt$ be a conservative, ergodic , measure preserving transformation with $m(X)=\infty$ then (see [@IET]) for any sequence of constants $a_n>0$ $$\begin{aligned} \tag*{\Football}&\text{either}\ \varliminf_{n\to\infty}\frac{S_n(1_A)}{a_n}=0\ \ \text{a.e.}\ \ \forall\ A\in\B,\ m(A)<\infty\\ & \text{or}\ \exists\ n_k\to\infty\ \text{so that}\ \ \frac{S_{n_k}(1_A)}{a_{n_k}}\xyr[k\to\infty]{}\infty\ \ \ \forall\ A\in\B,\ m(A)>0.\end{aligned}$$ This is for the one-sided Birkhoff sums $S_n(1_A)(x):=\sum_{k=0}^{n-1}1_A(T^kx)$. For an invertible $\xbmt$, and the one-sided Birkhoff sums replaced by the two-sided Birkhoff sums $$\Si_n(1_A)(x):=\sum_{|k|\le n}1_A(T^kx),$$ the   analogue may fail. Infinite measure examples were given in [@MS] with constants $a_n>0$ so that $$\begin{aligned} \tag*{{\Large\Pointinghand}}\Si_n(1_A)\ \asymp\ a_n\ \text{a.e.} \ \forall\ \ A\in\B,\ 0<m(A)<\infty\end{aligned}$$ where for eventually positive sequences $a_n,\ b_n$, $a_n\ \asymp\ b_n$ means existence of $M>1$ so that $M^{-1}<\tfrac{a_n}{b_n}<M\ \forall\ n$ large. An early hint of this possibility can be found in [@DE] where an example $\xbmt$ is given for which the forward sums are not comparable to the backward sums, namely: for $A\in\B,\ 0<m(A)<\infty$ and a.e. $x\in X$, $$\begin{aligned} \tag*{\rightthumbsdown} \varliminf_{n\to\infty}\frac{S^{(T^{-1})}_n(1_A)(x)}{S^{(T)}_n(1_A)(x)}=0\ \ \&\ \ \varlimsup_{n\to\infty}\frac{S^{(T^{-1})}_n(1_A)(x)}{S^{(T)}_n(1_A)(x)}=\infty.\end{aligned}$$ Indeed, in view of ,      is a consequence of , which is in turn satisfied by the [@DE] example (see theorem 3 below). Our main result (theorem 2 in §2) is that can never be upgraded to the convergence: $$\begin{aligned} \tag*{{\leftthumbsup}}\frac{\Si_n(1_A)}{a_n}\ \xyr[n\to\infty]{}m(A)\ \ \ \forall\ \ A\in\B,\ 0<m(A)<\infty.\end{aligned}$$ For larger groups the situation is different. Example of actions of large groups satisfying analogues for compactly supported, integrable functions are given in Theorem 1.1 of [@GN]. In this context (example 5.1 in §5) we show that certain infinite $\Bbb Z^2$ actions satisfy the analogue for all integrable functions. We also discuss the various possibilities for infinite ergodic $\Bbb Z^2$ actions in terms of the recurrence of the generators. We also consider among two natural classes of infinite, ergodic examples: cutting and stacking constructions with bounded cutting numbers, which all satisfy (theorem 3 in §3), and transformations admitting [generalized recurrent events]{} where is characterized by a “trimmed sum" type small tail condition of the first return time functions to generalized recurrent events (theorem 5 in §4). (see §1) is necessary for . In theorem 4, we show ([*inter alia*]{}) that the return sequences of boundedly rational ergodic transformations admitting generalized recurrent events have a certain [extended regular variation]{} property. This is also needed for the proof of theorem 5 and is contrast to the rank one, boundedly rational ergodic transformations considered. The ratio ergodic theorem holds for $\Bbb Z^d$ actions (see [@EH] for $d=1$ and [@H] for $d\ge 2$) and shows that for an ergodic $\Bbb Z^d$ action, if any of these statements holds for some $A\in\B,\ 0<m(A)<\infty$, then it holds $\forall\ \ A\in\B,\ 0<m(A)<\infty$. Thus the properties are invariant under similarity (see [@IET]). §1 Preliminaries {#preliminaries .unnumbered} ================ Bounded rational ergodicity {#bounded-rational-ergodicity .unnumbered} --------------------------- As in [@BRE], the conservative, ergodic, measure preserving transformation $\xbmt$ is called [*boundedly rationally ergodic*]{} ([BRE]{}) if $\exists\ A\in\B,\ 0<m(A)<\infty$ so that $$\begin{aligned} &\tag{a}S_n(1_A)(x)\le Ma_n(A)\ \text{a.e. on}\ A\ \forall\ n\ge 1\\ & \text{where}\ a_n(A):=\sum_{k=0}^{n-1}\frac{m(A\cap T^{-k}A)}{m(A)^2}.\end{aligned}$$ In this case ([@BRE]), $\xbmt$ is [*weakly rationally ergodic*]{} ([WRE]{}), that is, writing $a_n(T):=a_n(A)$ (where $A$ is as in (a)), there is a dense hereditary ring $$R(T)\subset\mathcal F:=\{F\in\B:\ m(F)<\infty\}$$ (including all sets satisfying (a)) so that $$\sum_{k=0}^{n-1}m(F\cap T^{-k}G)\sim m(F)m(G)a_n(T)\ \forall\ F,\ G\in R(T)$$ and in particular, $$a_n(F)\sim a_n(T)\ \forall\ F\in R(T),\ m(F)>0.$$ For invertible transformations, the one sided properties ([RE]{} $\&$ [BRE]{}) are equivalent to their 2-sided analogues: $\xbmt$ is: if $\exists\ A\in\B,\ 0<m(A)<\infty$ so that $$\begin{aligned} &\tag{a'}\Si_n(1_A)(x)\le M\o a_n(A)\ \text{a.e. on}\ A\ \forall\ n\ge 1\\ & \text{where}\ \o a_n(A):=\sum_{k=-(n-1)}^{n-1}\frac{m(A\cap T^{k}A)}{m(A)^2}\sim 2a_n(A);\end{aligned}$$ and , if there is a dense hereditary ring $$\o R(T)\subset\mathcal F:=\{F\in\B:\ m(F)<\infty\}$$ (including all sets satisfying (a’)) so that $$\o a_n(F)\sim 2a_n(T)\ \forall\ F\in \o R(T),\ m(F)>0.$$ In case $T$ is weakly rationally ergodic, $\exists\ \u\b(T)\in [0,1],\ \a(T),\ \b(T)\in [1,\infty]$ so that a.e., $\forall\ f\in\ L^1(m)_+$: $$\begin{aligned} &\varlimsup_{n\to\infty}\frac1{a_n(T)}S_n(f)\ =\ \a\int_Xfdm\ \\ & \varlimsup_{n\to\infty}\frac1{2a_n(T)}\Si_n(f)\ =\ \b\int_Xfdm\\ & \varliminf_{n\to\infty}\frac1{2a_n(T)}\Si_n(f)\ =\ \u\b\int_Xfdm\end{aligned}$$ and $T$ is boundedly rationally ergodic if and only if $\a(T)<\infty$. Since bounded rational ergodicity of an invertible $\xbmt$ implies that of $T^{-1}$ we have that $\a(T)<\infty\ \Lra\ \a(T^{-1})<\infty$.   Let $\xbmt$ be an invertible, conservative, ergodic , measure preserving transformation. If   $T$ satisfies wrt some sequence of normalizing constants, then $T$ is boundedly rationally ergodic, (hence weakly rationally ergodic). If   $T$ is boundedly rationally ergodic, then $$\begin{aligned} &\tag*{{$\clubsuit$}}\b(T)\le \a(T)=\a(T^{-1})\le 2\b(T)\ \&\\ & \tag*{\Stopsign} \ \ \u\b(T)\ \le\ \frac{\a(T)}2.\end{aligned}$$      Suppose that $$\Si_n(1_A)\ \asymp\ \ a(n)\ \text{for some and hence all}\ A\in\B,\ 0<m(A)<\infty.$$ Fix $A\in\B,\ 0<m(A)<\infty$. By Egorov’s theorem $\exists\ M>1,\ N\in\Bbb N\ \&\ B\in\B(A),\ m(B)>0$ so that $$\frac{\Si_n(1_A)(x)}{a(n)}<M\ \ \forall\ x\in B,\ n\ge N.$$ On the other hand, $\exists\ \e>0$ so that $$\varliminf_{n\to\infty}\frac{\Si_n(1_B)}{a(n)}\ge 4\e\ \ \text{a.e.}$$ whence, by Fatou’s lemma $$a_n(B)\ge\frac13\int_B\Si_n(1_B)dm\ge\e a(n)\ \forall\ n\ \ \text{large}.$$ To see bounded rational ergodicity, for $n\ge 1$ large and $x\in B$, $$S_n(1_B)(x)\le \Si_n(1_A)(x)\le Ma(n)\le\frac{M}\e\cdot a_n(B).\ \ \ \CheckedBox\text{(i)}$$ It suffices to show that $\a(T)\ge\a(T^{-1})$. Fix $A\in\mathcal F_+\ \&\ \e>0$. By Egorov, $\exists\ B\in\B_+\cap A\ \&\ N_0\ge 1$ so that $$S_n^+(1_B)(x)<(\a+\e)a_n(T)m(B)\ \forall\ x\in B,\ n\ge N_0.$$ For $n\ge 1\ \&\ x\in B$, let $$K_n(x):=\max\,\{0\le k\le n:\ T^{-k}(x)\in B\};$$ then $K_n\xyr[n\to\infty]{}\infty$ a.e. $\&$ whenever $K_n(x)\ge N_0$, $$\begin{aligned} S_n^-(1_B)&=S^-_{K_n}(1_B)=S_{K_n}(1_B)\circ T^{-K_n}\\ &\lesssim (\a+\e)a(K_n)m(B)\\ & \le (\a+\e)a_n(T)m(B).\ \CheckedBox\ (\clubsuit)\end{aligned}$$    Using Egorov’s theorem we can, given $\e>0$, find $B\in\mathcal F,\ m(B)>0$ and $N_\e\ge 1$ so that $$S_n^{(T^{\pm 1})}(1_B)\ \le\ (\a+\e)a_n(T)\ \ \text{a.e. on}\ X\ \ \forall \ n\ge N_\e.$$ It follows that for $n\ge N_\e$, $$\begin{aligned} \frac1{2a_n(T)}\Si_n(1_B)& =\frac1{2a_n(T)}S_n^{(T)}(1_B) +\frac1{2a_n(T)}S_n^{(T^{-1})}(1_B)\\ &\le \frac1{2a_n(T)}S_n^{(T)}(1_B) +\frac{\a+\e}2\end{aligned}$$ whence $$\varliminf_{n\to\infty}\frac1{2a_n(T)}\Si_n(1_B)\le \frac{\a+\e}2+\varliminf_{n\to\infty}\frac1{2a_n(T)}S_n^{(T)}(1_B)=\frac{\a+\e}2.\ \CheckedBox$$ §2 No absolutely normalized convergence of two-sided Birkhoff sums. {#no-absolutely-normalized-convergence-of-two-sided-birkhoff-sums. .unnumbered} ===================================================================   Let $\xbmt$ be an infinite, invertible, conservative, ergodic , measure preserving transformation, then fails.    Suppose otherwise, that $$\begin{aligned} \tag*{{\leftthumbsup}}\frac{\Si_n(1_A)}{2a(n)}\ \xyr[n\to\infty]{}m(A)\ \ \ \forall\ \ A\in\B,\ 0<m(A)<\infty\end{aligned}$$ By proposition 1(i), $T$ is boundedly rationally ergodic, hence weakly rationally ergodic. We claim first that $a(n)\sim a_n(T)$. To see this, let $A\in R(T)$ and let $B\in\B(A),\ m(B)>0$ so that $$\frac{\Si_n(1_A)}{2a(n)}\ \xyr[n\to\infty]{}m(A)\ \ \text{uniformly on}\ B.$$ It follows that $$\sum_{k=0}^{n-1}m(B\cap T^{-k}A)\sim\frac12\int_B\Si_n(1_A)dm\sim m(A)m(B)a(n).$$ On the other hand, since $A\in R(T)$, $$\sum_{k=0}^{n-1}m(B\cap T^{-k}A)\sim m(A)m(B)a_n(T)$$ showing that indeed $a(n)\sim a_n(T)$. We claim next that $\a(T)=2$. Indeed by , $\a\ge 2$ and by [[$\clubsuit$]{}]{}, $\a\le 2$. Thus $$\varlimsup_{n\to\infty}\frac1{2a(n)}S_n(1_F)=m(F)\ \text{a.e.}\ \ \forall\ F\in\B.$$ The rest of this proof is on a “single orbit" which we proceed to specify. Fix $A\in\mathcal F_+$. By Egorov, $\exists\ B\in\B(A),\ m(B)>\frac34m(A)$ so that $$\sup_{N\ge n}\frac1{2a(N)}S_N(1_A),\ \frac1{2a(n)}\Si_n(1_A)\xyr[n\to\infty]{}\ m(A)\ \text{uniformly on}\ B.$$ Call a point $x\in B$ [*admissible*]{} if $$\begin{aligned} &\tag*{A(i)}\frac{S_N(1_B)(x)}{S_N(1_A)(x)}\ \xyr[n\to\infty]{}\ \frac{m(B)}{m(A)};\\ &\tag*{A(ii)} \frac1{2a(n)}\Si_n(1_B)(x)\xyr[n\to\infty]{}\ m(B)\\ &\tag*{A(iii)}\sup_{N\ge n}\frac1{2a(N)}S_N(1_B)\xyr[n\to\infty]{}\ m(B),\\end{aligned}$$ and $\exists$ $K\subset\Bbb N$, an [*$x$-admissible subsequence*]{} in the sense that $$\begin{aligned} &\tag*{A(iv)} T^nx\in B\ \forall\ n\in K\ \ \&\ \\ &\tag*{A(v)} \frac1{2a(n)}S_n(1_B)(x)\xyr[n\to\infty,\ n\in K]{}\ m(B).\end{aligned}$$ An [*admissible pair*]{} is $(x,K)\in B\x 2^\Bbb N$ where $x$ is an [admissible point]{} and $K$ is an [$x$-admissible subsequence]{}. Note that if $(x,K)$ is an admissible pair, then by A(iv) and A(i), $$\frac1{2a(n)}S_n(1_A)(x)\xyr[n\to\infty,\ n\in K]{}\ m(A).$$   Almost every $x\in B$ is admissible. By , $\a(T)=2$ and the ratio theorem, almost every $x\in B$ satisfies A(i), A(ii) $\&$ A(iii). Also by $\a(T)=2$, for a.e. $x\in B,\ \exists\ K\subset\Bbb N$ satisfying A(v). We claim that if $K:=\{k_n:\ n\ge 1\},\ k_n\uparrow$, then $K':=\{k'_n:\ n\ge 1\}$ where $k'_n:=\max\{j\le k_n:\ T^jx\in B\}$ is $x$-admissible. Evidently $K'$ is infinite and satisfies A(iv). To check A(v): $$2a(k_n)\ge 2a(k_n')\overset{\text{\tiny A(iii)}}{\text{\Large$\gtrsim$}} S_{k_n'}(1_B)(x)=S_{k_n}(1_B)(x)\overset{\text{\tiny A(v)}}{\text{\Large$\sim$}}2a(k_n).\ \ \CheckedBox\text{A(v)}$$    If $x\in B,\ K\subset\Bbb N$ and $\{J_n:\ n\in K\}$ satisfy $$\begin{aligned} &\frac1{2a(n)}S_n(1_A)(x)\xyr[n\to\infty,\ n\in K]{}\ m(A); \\ &\ n\ge J_n\xyr[n\to\infty,\ n\in K]{}\ \infty;\\ &\ \varliminf_{n\to\infty,\ n\in K}\frac{a(J_n)}{a(n)}=:\rho>0,\end{aligned}$$ then $$\frac1{2a(J_n)}S_{J_n}(1_A)(x)\xyr[n\to\infty,\ n\in K]{}\ 2m(A).$$ $$\begin{aligned} \frac1{2a(J_n)}S^{-}_{J_n}(1_A)(x) &\lesssim \frac1{2\rho a(n)}S^{-}_{n}(1_A)(x)\ \ \text{as}\ n\to\infty,\ n\in K;\\ &= \frac1{\rho}\left(\frac1{2a(n)}\Si_{n}(1_A)(x)-\frac1{2 a(n)}S_{n}(1_A)(x)\right)\\ &\xyr[n\to\infty,\ n\in K]{}\ 0\ \ \ \ \because\ \ x\in B.\end{aligned}$$ $$\begin{aligned} \therefore\ \ \frac1{2a(J_n)}S_{J_n}(1_A)(x) &= \frac1{2a(J_n)}\Si_{J_n}(1_A)(x)- \frac1{2a(J_n)}S^{-}_{J_n}(1_A)(x)\\ &\xyr[n\to\infty,\ n\in K]{}\ m(A).\ \ \CheckedBox\end{aligned}$$    Let $(x,K)\in B\x 2^\Bbb N$ be an admissible pair, then $$\begin{aligned} \frac1{12}\le \varliminf_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\ \ \&\ \ \ \ \varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\le\frac14.\end{aligned}$$   We show first that $$\begin{aligned} \tag{a}\varliminf_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\ge\frac1{12}.\end{aligned}$$    Define $$J_i:=\min\,\{\ell\ge \frac{in}9:\ T^\ell x\in B\}\wedge\frac{i(n+1)}9;\ \ \ (0\le i\le 8),$$ then $$\begin{aligned} 2m(B)a(n)&\lesssim\ S_n(1_B)(x)\ \ \text{as}\ n\to\infty,\ n\in K\\ &=\sum_{i=0}^8S_{\frac{n}9}(1_B)(T^{\frac{in}9}x)\\ &= \sum_{i=0}^8S_{\frac{(i+1)n}9-J_i}(1_B)(T^{J_i}x)\\ &\le \sum_{i=0}^8S_{\frac{n}9}(1_B)(T^{J_i}x)\\ &\le \sum_{i=0}^8\|S_{\frac{n}9}(1_B)\|_{L^\infty(B)}\\ &\lesssim 18m(A)a(\frac{n}9)\ \ \text{as}\ n\to\infty.\end{aligned}$$ Thus $$\varliminf_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\ge\frac{2m(B)}{18m(A)}>\frac1{12}.\ \ \ \CheckedBox{\rm(a)}$$ Next, we show: $$\begin{aligned} \tag{b}\varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{n}3)}{a(n)}\le\frac12.\end{aligned}$$ By (a), $\{\frac{n}3:\ n\in K\}$ satisfies the preconditions of lemma 0 and so $$\begin{aligned} \frac1{2a(\tfrac{n}3)}S_{\frac{n}3}(1_A)(x)\ \xyr[n\to\infty,\ n\in K]{}\ m(A).\end{aligned}$$ For $n\in K$, let $$J_n:=\max\,\{j\le\frac{n}3:\ T^jx\in B\}.$$ We claim that $a(J_n)\sim a(\tfrac{n}3)$ as $n\to\infty,\ n\in K$ since: $$\begin{aligned} 2a(\tfrac{n}3)m(B)&\ge 2a(J_n)m(B)\\ &\underset{n\to\infty}{\text{\Large$\gtrsim$}} S_{J_n}(1_B)(x)\ \ \because\ x\in B;\\ &=S_{\frac{n}3}(1_B)(x)\\ &\underset{n\to\infty,\ n\in K}{\text{\Large$\sim$}}2a(\tfrac{n}3)m(B).\end{aligned}$$ Finally $$\begin{aligned} 2m(A)a(n)&\sim\ \Si_n(1_{A})(T^{J_n}x)\ \ \because\ x\in B;\\ &=\sum_{k=-n+J_n}^{n+J_n}1_A(T^kx)\\ &\ge \Si_{J_n}(1_{A})(T^{J_n}x)+\Si_{J_n}(1_A)(T^nx)\\ &\sim 4m(A)a(J_n)\ \ \ \ \because\ T^{J_n}x,\ T^nx\in B;\\ &\sim 4m(A)a(\tfrac{n}3).\ \ \CheckedBox{\rm (b)}\end{aligned}$$ Next, we iterate (b): $$\begin{aligned} \tag{c}\varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\le\frac14.\end{aligned}$$  Let $L_n:=\min\,\{J\ge\frac{n}3:\ T^Jx\in B\}$. We claim that $$\begin{aligned} \tag*{\text{\Large\Lightning}}\exists\ N\ge 1\ \ \text{so that}\ L:=\{L_n:\ n\in K\ n\ge N\}\ \ \text{ is $x$-admissible}.\end{aligned}$$ Firstly, for $n\in K,\ T^nx\in B$ whence $L_n\le n$. Since $$\varliminf_{n\to\infty,\ n\in K}\frac{a(L_n)}{a(n)}\ge \varliminf_{n\to\infty,\ n\in K}\frac{a(\frac{n}3)}{a(n)}\ \ \overset{\text{\tiny (a)}}{\text{\Large$\ge$}}\ \ \frac1{12}$$ we have by lemma 1 that $$\frac1{2a(L_n)}S_{L_n}(1_B)(x)\xyr[n\to\infty,\ n\in K]{}\ m(B)$$ and $L:=\{L_n:\ n\in K\}$ is $x$-admissible.  Applying (a) to $L=\{L_n:\ n\in K\}$, we obtain $$\begin{aligned} \varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{L_n}3)}{a(L_n)}\le \frac12.\end{aligned}$$ To obtain (c) from this, it suffices to show that $$\begin{aligned} \tag*{\Biohazard} a(L_n)\underset{n\to\infty,\ n\in K}{\text{\Large $\sim$}}\ a(\frac{n}3)\end{aligned}$$ By lemma 1, $$\frac1{2a(\frac{n}3)}S_{\frac{n}3}(1_B)(x)\xyr[n\to\infty,\ n\in K]{}\ m(B)$$ whence, as $n\to\infty,\ n\in K$ $$\begin{aligned} 2a(\frac{n}3)m(B)&\ \sim\ S_{\frac{n}3}(1_B)(x)\\ &\sim S_{L_n}(1_B)(x)\\ &\sim 2a(L_n)m(B)\\ &\ge 2a(\frac{n}3)m(B).\ \CheckedBox\text{\ \Biohazard\ \&\ {\rm (c)}}\end{aligned}$$ This completes the proof of lemma 2.    $$\begin{aligned} \tag*{\bf Lemma 3:}\ \text{If}\ &x\in B_0,\ \l,\ \rho\in (0,1)\ \&\ \varliminf_{n\to\infty,\ n\in K}\frac{a(\l n)}{a(n)}\ge\rho,\ \ \text{then}\\ & \varlimsup_{n\to\infty,\ n\in K}\frac{a((1-\l) n)}{a(n)}\le\ 1-\rho. \end{aligned}$$ Firstly we claim that as $n\to\infty,\ n\in K$, $$\begin{aligned} \tag*{\Coffeecup}S^{-}_{\l n}(1_A)(T^nx)\sim 2m(A)a(\l n). \end{aligned}$$ To see , note that as $n\to\infty,\ n\in K$, $$S^{-}_n(1_A)(T^nx)=S_n(1_A)(x)\sim 2m(A)a(n).$$ Since $T^nx\in B_1$, we have that $$\begin{aligned} \frac1{2a(n)}S_n(1_A)(T^nx)&=\frac1{2a(n)}\Si_n(1_A)(T^nx)- \frac1{2a(n)}S^{-}_n(1_A)(T^nx)o(a(n))\\ & \xyr[n\to\infty,\ n\in K]{}\ 0 \end{aligned}$$ whence also $$\frac1{2a(\l n)}S_{\l n}(1_A)(T^nx)\ \xyr[n\to\infty,\ n\in K]{}\ 0$$ and $$\begin{aligned} \frac1{2a(\l n)}S^{-}_{\l n}(1_A)(T^nx)&=\frac1{2a(\l n)}\Si_{\l n}(1_A)(T^nx)-\frac1{2a(\l n)}S^{+}_{\l n}(1_A)(T^nx)\\ & \xyr[n\to\infty,\ n\in K]{}\ m(A).\ \CheckedBox\text{\Coffeecup} \end{aligned}$$ To prove the lemma, note that, as $n\to\infty,\ n\in K$, $$\begin{aligned} 2m(A)a((1-\l)n)&\sim S_{(1-\l)n}(1_A)(x)\\ &= S_{n}(1_A)(x)-S^{-}_{\l n}(1_A)(T^nx)\\ &\sim 2m(A)(a(n)-a(\l n))\\ &\lesssim 2(1-\rho)m(A)a(n). \ \CheckedBox \end{aligned}$$ Fix an admissible pair $(x,K)\in B\x 2^\Bbb N$, then $$\varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\le\frac14\ \ \&\ \ \ \varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{8n}9)}{a(n)}\le\frac{11}{12}.$$ For $n\in K$, let $$J=J_n(x):=\min\,\{j\ge \frac{n}9:\ T^jx\in B\}.$$ We claim that $J\le\frac89$; else, as $n\to\infty,\ n\in K$: $$\begin{aligned} 2a(n)m(B)&\lesssim S_n(1_B)(x)\\ &= S_J(1_B)(x)+S_{(n-J)\vee 0}(1_B)(T^Jx)\\ &= S_{\frac{n}9}(1_B)(x)+S_{(n-J)\vee 0}(1_B)(T^Jx) \\ &\le S_{\frac{n}9}(1_B)(x)+S_{\frac{n}9}(1_B)(T^Jx)\ \text{assuming}\ J>\frac89;\\ &\lesssim 4m(A)a(\frac{n}9) \end{aligned}$$ whence $$\frac{a(\frac{n}9)}{a(n)}\ \ \ \underset{n\to\infty,\ n\in K}{\text{\Large$\gtrsim$}}\ \ \ \frac{2m(B)}{4m(A)}=\frac38$$ and $$\frac38\le \varlimsup_{n\to\infty,\ n\in K}\frac{a(\frac{n}9)}{a(n)}\le\frac14.\ \ \ \ \XBox$$ This contradiction shows that indeed $J\le\frac89$. Finally, since $\frac{n}9\le J\le \frac{8n}9$: $[J-\frac{8n}9,J+\frac{8n}9]\supset [0,n]$ and as $n\to\infty,\ n\in K$, $$\begin{aligned} 2a(\frac{8n}9)m(A)&\sim \Si_{\frac{n}9}(1_A)(T^Jx)\\ &\ge S_n(1_A)(x)\\ &\sim 2a(n)m(A) \end{aligned}$$ whence $$\frac{11}{12}\ \underset{n\to\infty,\ n\in K}{\text{\Large$\gtrsim$}}\ \frac{a(\frac{8n}9)}{a(n)}\ \underset{n\to\infty,\ n\in K}{\text{\Large$\gtrsim$}}\ \ 1.\ \XBox$$ This last contradiction contradicts .  Remark on quantitative estimates {#remark-on-quantitative-estimates .unnumbered} -------------------------------- The proof of theorem 2 can be adapted to show that $\exists\ \D>0$ so that for any $\xbmt$ satisfying , we have $$\begin{aligned} \b(T)-\underline{\b}(T)\ \ge\ \D. \end{aligned}$$ The question of estimating the best $\D>0$ arises. For the examples appearing in this paper, $\D\ge\frac12$. §3 Rank one towers {#rank-one-towers .unnumbered} =================== These are [CEMPT]{}s constructed by cutting and stacking as in [@FR-Book], [@FR-AMM], [@CH], Ch. 7 of [@N2]. Let $c_n\in \Bbb N,\ c_n\ge 2\ \ (n\ge 1)$ and let $S_{n,k}\ge 0,\ \ (n\ge 1,\ 1\le k\le c_n)$. The [*rank one transformation*]{} with [*construction data*]{} $$\{(c_n;S_{n,1},\dots,S_{n,c_n}):\ n\ge 1\}$$ is an invertible piecewise translation of the interval $J_T=(0,S_T)$ where $$S_T:=1+\sum_{n\ge 1}\frac1{C_n}\sum_{k=1}^{c_n}S_{n,k}\le \infty\ \text{with}\ C_n:=c_1\cdots c_n.$$ This is defined as the limit of a nested sequence of Rokhlin towers $(\tau_n)_{n\ge 1}$ of intervals where $\tau_1=[0,1]$ and $\tau_{n+1}$ is constructed from $\tau_n$ by cutting $\tau_n$ into $c_n$ columns,putting $S_{n,k}$ spacer intervals above the $k^{\text{th}}$ column ($1\le k\le c_n$);and stacking. The transformation $T$ constructed, being an invertible, piecewise translation of $J_T$, preserves Lebesgue measure. It is conservative and ergodic.     Let $\xbmt$ be the conservative, ergodic measure preserving transformation with construction data $$\{(c_n;S_{n,1},\dots,S_{n,c_n}):\ n\ge 1\}.$$ If $\sup_{n\ge 1}c_n<\infty$, then $T$ satisfies .  Let $c_n\le J\ \ (n\ge 1)$ and let $q_n$ be the height of $\tau_n$ ($n\ge 1$). For $x\in I:=[0,1]$ and $n\ge 1$ we have $$C_n\le\Si_{q_n}(1_I)(x)\le 2C_{n}.$$ Define $a(n)$ by $$a(n):=C_\nu\ \ \text{for}\ \ q_\nu\le n<q_{\nu+1},$$ then, for $q_\nu\le n<q_{\nu+1}$ $$a(n)=C_\nu\le\Si_{q_\nu}(1_I)(x)\le \Si_{n}(1_I)(x)\le\Si_{q_{\nu+1}}(1_I)(x)\le 2C_{\nu+1}\le 2Ja(n).$$ Finally $\varlimsup_{n\to\infty}\frac{\Si_n(1_A)}{a(n)}\ \&\ \varliminf_{n\to\infty}\frac{\Si_n(1_A)}{a(n)}$ are $T$-invariant whence constant by ergodicity and we have .   Remark {#remark .unnumbered} ------   Bounded rational ergodicity was established in [@DGPS]. §4 Weakly pointwise dual ergodic transformations {#weakly-pointwise-dual-ergodic-transformations .unnumbered} ================================================ As in [@AZ], the conservative ergodic measure preserving transformation  $(X,\B,m,T)$ is called [*weakly pointwise dual ergodic*]{} if $\exists\ a(n)>0, n\ge 1,$ such that for each $f\in L^1_+(m)$, $$\begin{aligned} & \frac1{a(n)}\sum_{k=0}^{n-1}\T^k f \xrightarrow[n\to\infty]{m}\ \ \int_Xfdm\ \ \&\\ & \varlimsup_{n\to\infty}\frac1{a(n)}\sum_{k=0}^{n-1}\T^k f=\int_Xfdm\ \ \text{a.e.}\ .\end{aligned}$$ This property entails [WRE]{} and the return sequence $a_n(T)\sim a(n)$. Our next result shows that the return sequence of bounded rationally ergodic, weakly pointwise dual ergodic transformation must be large. This is in contrast with the rank one transformations considered in theorem 3 whose return sequences can grow arbitrarily slowly. Let  $\xbmt$ be weakly pointwise dual ergodic with return sequence $a(n)=a_n(T)$. If $\a(T)<\infty$ then $\exists\ M>1$ and $N:\Bbb N\to\Bbb N$ so that $$\begin{aligned} \tag{\dsjuridical}a(pn)=M^{\pm 1}pa(n)\ \forall\ p>1,\ n\ge N(p).\end{aligned}$$ Remark {#remark-1 .unnumbered} ------   The property () is called [extended regular variation]{} ([ER]{}) with Karamata indices $1$ in [@BGT]. Fix $\Om\in\B,\ m(\Om)=1$ a limited set in the sense of [@AZ], that is satisfying $$\left\|\frac1{a(n)}\sum_{k=0}^{n-1}\T^k 1_\Om\right\|_{L^\infty(\Om)}\xrightarrow[n\to\infty]{}\ 1.$$ WLOG, $a(n)=\sum_{k=0}^nu_k$ where $u_n:=m(\Om\cap T^{-n}\Om)$. By lemma 4.1 in [@AZ], $$\begin{aligned} &\tag{a}\frac{\widehat{U}_s}{u(s)}\xrightarrow[s\to 0+]{m}1\ \ \&\\ &\tag{b} \ \frac1{u(s)}\|\widehat{U}_s\|_{L^\infty(\Om)}\xrightarrow[s\to 0+]{}1\end{aligned}$$ where $\widehat{U}_s:=\sum_{n\ge 0}e^{-sn}\widehat{T}^n1_{\Om}$ and $u(s):=\sum_{n\ge 0}e^{-sn}u_n$. For $s>0$, set $$U_s:=\sum_{n\ge 0}e^{-sn}1_{\Om}\circ T^n,$$ then $$\int_{\Om} U_sdm=\sum_{n\ge 0}u_ne^{-sn}=:u(s).$$ We claim first that for $p\in\Bbb N$ $$\begin{aligned} \tag{\dsliterary} \int_\Om U_s^pdm\ \sim\ \ p!\prod_{k=1}^pu(ks)\ \ \text{as}\ s\to 0+.\end{aligned}$$ Firstly, by convexity, $$\begin{aligned} \tag{\dsaeronautical}\int_\Om U_s^pdm\ \ge\ \(\int_\Om U_sdm\)^p=u(s)^p\ \forall\ p\ge 1,\ s>0.\end{aligned}$$ Next, for $p\in\Bbb N$ fixed, $$\begin{aligned} \tag{\dsarchitectural} U_s^p=&p!V(p,s)+E(p,s)U_s^{p-1}\ \text{where}\\ & \ V(p,s)=\sum_{0\le n_1\le\dots\le n_p}e^{-s(n_1+\dots+n_p)}\prod_{k=1}^p1_{\Om}\circ T^{n_k} \\&\text{and}\ \ |E(p,s)|\le M_p\ \forall\ s>0\ \text{where $M_p$ is constant.}.\end{aligned}$$ Thus Now $$\begin{aligned} V(p,s)&=\sum_{n=0}^\infty e^{-ns}1_{\Om}\circ T^n\sum_{n\le n_2\le\dots\le n_p}e^{-s(n_2+\dots+n_p)}\prod_{k=2}^p1_{\Om}\circ T^{n_k}\\ &\overset{n_k=n+\nu_k}{\text{\Large$=$}}\ \sum_{n=0}^\infty e^{-ns}1_{\Om}\circ T^n\sum_{0\le \nu_2\le\dots\le \nu_p}e^{-s((p-1)n+\nu_2+\dots+\nu_p)}\prod_{k=2}^p1_{\Om}\circ T^{n+\nu_k}\\ & =\sum_{n=0}^\infty e^{-nps}1_{\Om}\circ T^n\sum_{0\le \nu_2\le\dots\le \nu_p}e^{-s(\nu_2+\dots+\nu_p)}\prod_{k=2}^p1_{\Om}\circ T^{n+\nu_k}\\ &= \sum_{n=0}^\infty e^{-nps}1_{\Om}\circ T^nV(p-1,s)\circ T^n.\end{aligned}$$ whence $$\begin{aligned} \int_{\Om} V(p,s)dm&=\int_{\Om} V(p-1,s)\widehat{U}_{ps}dm \\ &\overset{\text{\tiny (b)}}{\underset{s\to 0+}{\text{\Large $\lesssim$}}}\ u(ps)\int_{\Om} V(p-1,s)dm.\end{aligned}$$ Thus $$\begin{aligned} \tag{\dsagricultural} \int_{\Om} V(p,s)dm\underset{s\to 0+}{\text{\Large $\lesssim$}}\ \prod_{k=1}^pu(ks).\end{aligned}$$ So far, by (), () and (), we have $$\begin{aligned} \tag{\dsmathematical} u(s)^p\ \asymp\ \int_\Om U_s^pdm\ =\ p!\int_\Om V(p,s)dm+O(u(s)^{p-1}). \end{aligned}$$ Thus, to finish the proof of (), it suffices to show that $$\begin{aligned} \tag{\dsheraldical}\int_{\Om} V(p,s)dm\underset{s\to 0+}{\text{\Large $\gtrsim$}}\ u(ps)\int_{\Om} V(p-1,s)dm.\end{aligned}$$ To this end, using () and (), we see that $$\int_{\Om} U_s^{2p}dm=O((\int_{\Om} U_s^{p}dm)^2)\ \text{as}\ s\to 0+$$ because $$\int_{\Om} U_s^{2p}dm\asymp u(s)^{2p}=(\int_\Om U_sdm)^p\cdot(\int_\Om U_sdm)^p\overset{\text{\tiny convexity}}{\text{\Large $\le$}}(\int_\Om U_s^pdm)^2.$$ We’ll need to know that $$\begin{aligned} \tag{\dschemical} \int_{A}U_s^pdm\underset{s\to 0+}{\text{\Large $\sim$}}\ \ m(A)\int_\Om U_s^pdm\ \forall\ A\in\B(\Om). \end{aligned}$$ Let $$\Phi_s:=\frac{U_s}{\int_\Om U_sdm},$$ then $\int_\Om\Phi_sdm=1\ \&\ \sup_{s>0}\int_\Om\Phi_s^2dm<\infty$. Thus $\{\Phi_s:\ s>0\}$ is weakly sequentially compact in $L^2(\Om)$ and for (), it suffices to show that $$\Phi_s\xrightarrow[s\to 0]{}\ 1\ \ \text{weakly in}\ \ L^2(\Om).$$ To see this note that $$\begin{aligned} \tag{\ddag} e^{-s\v}U_s\circ T_\Om=U_s-1 \ \text{on}\ \Om\end{aligned}$$ where $T_\Om$ is the [*induced transformation*]{} on $\Om$ defined by $T_\Om x:=T^{\v(x)}x$ where $\v(x):=\min\,\{n\ge 1:\ T^nx\in\Om\}$ (aka the [*first return time*]{} function). As is well known, $(\Om,\B(\Om),m_\Om,T_\Om)$ is an ergodic probability preserving transformation where $m_\Om(A):=m(A|\Om)$ . It follows from () that $$e^{-sp\v}U_s^p\circ T_\Om=(U_s-1_\Om)^p=U_s^p+\sum_{k=0}^{p-1}\tbinom{p}k(-1)^kU_s^k=U_s^p+\mathcal E_{p,s}U_s^{p-1}$$ where $|\mathcal E_{p,s}|\le 2^p$. Thus $$|U_s^p\circ T_\Om-U_s^p|\le (1-e^{-sp\v})U^p_s\circ T_\Om+2^pU_s^{p-1}$$ and $$\begin{aligned} \|U_s^p\circ T_\Om-U_s^p]|_{L^2(\Om)}& \le \|1-e^{-sp\v}\|_{L^2(\Om)}\|U^p_s\circ T_\Om\|_{L^2(\Om)}+2^p\|U_s^{p-1}\|_{L^2(\Om)}\\ &= \|1-e^{-sp\v}\|_{L^2(\Om)}\|U^p_s\|_{L^2(\Om)}+2^p\|U_s^{p-1}\|_{L^2(\Om)}\\ &=o(\int_\Om U_s^pdm)\ \text{as\ $s\to 0$\ by (\dsmathematical)}\end{aligned}$$ whence $$\|\Phi_s\circ T_\Om-\Phi_s\|_{L^2(\Om)}\xrightarrow[s\to 0]{}\ 0.$$ Now suppose that $\Psi\in L^2(\Om),\ t_k\to 0$ so that $$\begin{aligned} \Phi_{t_k}\xrightarrow[N\to\infty]{}\ \Psi\ \ \text{weakly in}\ L^2(\Om),\end{aligned}$$ then (since $m_\Om\circ T_\Om^{-1}=m_\Om$ ) $$\begin{aligned} \Phi_{t_k}\circ T_\Om\xrightarrow[N\to\infty]{}\ \Psi\circ T_\Om\ \ \text{weakly in}\ L^2(\Om),\end{aligned}$$ and by $ \|U_s^p\circ T-U_s^p]|_{L^2(\Om)}=o(\int_\Om U_s^pdm)$, $$\begin{aligned} \Phi_{t_k}\circ T_\Om\xrightarrow[N\to\infty]{}\ \Psi\ \ \text{weakly in}\ L^2(\Om).\end{aligned}$$ It follows that $\Psi=\Psi\circ T_\Om$. By ergodicity, $\Psi\equiv\int_\Om\Psi dm=1$. So the only weak limit point of $\Phi_s$ as $s\to 0$ is the constant 1.   () Suppose that () fails and let $\e>0$ and let $s_j\to 0$ be sequence so that $$\begin{aligned} \int_{\Om} V(p,s_j)dm\ \ {\text{\Large $\lesssim$}}\ (1-2\e)u(ps_j)\int_{\Om} V(p-1,s_j)dm.\end{aligned}$$ By (a) and Egorov’s theorem, there is a subsequence $t_k\to 0$ and $A\in\B(\Om),\ m(A)>1-\e$ so that $\widehat{U}_{t_k}\sim u(t_k)$ as $k\to\infty$ uniformly on $A$, whence $$\begin{aligned} \int_{\Om} V(p,t_k)dm&\ge \int_A V(p-1,t_k)\widehat{U}_{pt_k} dm \\ &\text{\Large $\sim$}\ u(pt_k)\int_{A} V(p-1,t_k)dm \\ &\overset{\text{\tiny (\dschemical)}}{\text{\Large $\gtrsim$}}m(A)u(pt_k)\int_{\Om} V(p-1,t_k)dm \\ &> (1-\e)u(pt_k)\int_{\Om} V(p-1,t_k)dm.\ \ \ \CheckedBox\text{(\dsheraldical) $\&$ (\dsliterary)}\end{aligned}$$ Next, we claim that $\exists\ M>1\ \&\ \D:\Bbb N\to\Bbb R_+$ so that $$\begin{aligned} \tag{\dsmilitary}\frac1M<\frac{pu(ps)}{u(s)}<M\ \ \forall\ p\ge 1,\ \ 0<s<\D(p). \end{aligned}$$ of (): We now use the assumption $\a=\a(T)<\infty$. Since $u_s\underset{s\to 0+}{\text{\Large $\lesssim$}}\ \a u(s)$ a.e., by Egorov’s theorem, $\exists\ A\in \B(\Om)$ so that $u_s\underset{s\to 0+}{\text{\Large $\lesssim$}}\ \a u(s)$ uniformly on $A$. Using this and (), we have $$m(A)p!\prod_{k=1}^pu(ks)\underset{s\to 0+}{\text{\Large $\sim$}}\int_{A}u_s^pdm \underset{s\to 0+}{\text{\Large $\lesssim$}} m(A)\a^pu(s)^p.$$ Fixing $c>0$ so that $$(p!)^{\frac1p}\ge cp\ \forall\ p\ge 1,$$ it follows that $$cpu(ps)\le \left(p!\prod_{k=1}^pu(ks)\right)^{\frac1p} \underset{s\to 0+}{\text{\Large $\lesssim$}}\a u(s).$$ This proves (). Using (), we can now apply the de Haan-Stadtmüller theorem (theorem 1 in [@dH-S] and theorem 2.10.2 in [@BGT]) that $\exists\ I>1$ so that $$\begin{aligned} u(s)\ \ \ =\ \ \ I^{\pm 1} a(\frac1s)\end{aligned}$$ thus obtaining ().    Interarrival stochastic processes and generalized recurrent events {#interarrival-stochastic-processes-and-generalized-recurrent-events .unnumbered} ------------------------------------------------------------------ Let $\xbmt$ be a conservative, ergodic measure preserving transformation. The [*induced transformation*]{} on $\Om\in\mathcal F_+$ is the probability preserving transformation $$(\Om,\mathcal B(\Om) ,m_\Om,T_\Om)$$ where  $m_\Om:=m(\cdot\,|\Om)$;  $T_\Om:\Om\to\Om$ is the [*first return* ]{} or [*induced*]{} transformation defined by $T_\Om x:=T^{\v(x)}x$ where $\v=\v_\Om:\Om\to\Bbb N$ is the [*first return time*]{} function defined by $\v(x):=\min\,\{n\ge 1:\ T^nx\in\Om\}$. The (one-sided) [*interarrival (stochastic) process*]{} of $\Om$ is the stochastic process $(\v\circ T_\Om^n)_{n\ge 0}$ defined on $\Om$. It corresponds to a factor induced transformation on $\Om$ corresponding to the sub-invariant factor factor algebra $\B_0:=\s(\{T^{-n}\Om:\ n\ge 0\})$. As in [@AN], a stochastic process $(X_1,X_2,\dots)$ is [*continued fraction mixing*]{} if $\vartheta(1)<\infty\ \&\ \vartheta(n)\downarrow 0$ where $$\vartheta(n):=\sup\{|\tfrac{\Bbb P(A\cap B)}{\Bbb P(A)\Bbb P(B)}-1|:\ A\in\s_{1}^k,\ B\in\s_{k+n}^\infty,\ \Bbb P(A)\Bbb P(B)>0,\ k\ge 1\}.$$ Here, $\s_k^N$ denotes the $\s$-algebra generated by the random variables $\{X_j: k\le j<N+1\}$ for $k<N+1\le\infty$. Let $\xbmt$ be a conservative, ergodic measure preserving transformation. We’ll call $\Om\in\mathcal F_+$ a [*generalized recurrent event for $T$*]{} if its interarrival stochastic process $(\v_\Om\circ T^n)_{n\ge 0}$ is continued fraction mixing with coefficients satisfying $\sum_{n=1}^\infty\frac{\vartheta(n)}n<\infty$. Remarks {#remarks .unnumbered} ------- \(i) Any recurent event (as in 5.2 of [@IET]) has an independent, interarrival stochastic process whence is a generalized recurrent event. (ii) Examples are also obtained by noting that (as shown in [@GM]) any stationary stochastic process driven by a mixing Gibbs-Markov map and with observable measurable with respect to the Markov partition is continued fraction mixing with exponentially decaying coefficients. \(iii) By lemma 3.7.4 in [@IET], a transformation admitting a generalized recurrent event has a factor where the generalized recurrent event is a Darling-Kac set (and is hence pointwise dual ergodic). For $\Om\in\mathcal F_+$, set $$L(t)=L_\Om(t):=\int_\Om(\v_\Om\wedge t)dm_\Om,\ \ \ a(t):=\frac{t}{L(t)}\ \ \&\ \ b:=a^{-1}.$$ Let $\xbmt$ be a conservative, ergodic measure preserving transformation equipped with a generalized recurrent event $\Om\in\mathcal F_+$, then $\xbmt$ satisfies if and only if $$\begin{aligned} \tag*{\symqueen}\sum_{n=1}^\infty\left(\frac{m_\Om([\v_\Om\ge n])}{L_\Om(n)}\right)^2<\infty. \end{aligned}$$ In this case $$\begin{aligned} \tag*{\symking}\b(T)=1\ \&\ \u\b(T)=\frac12.\end{aligned}$$ Remark {#remark-2 .unnumbered} ------   As shown in [@AN] the condition    characterizes the “trimmed sum" convergence property: $$\frac1{b(n)}(\v_n-\max_{0\le k\le n-1}\v\circ T_\Om^k)\ \xyr[n\to\infty]{}\ 1\ \text{a.s.}$$ (see the earlier [@M1; @M2] for the independent case and [@DV] for the case of continued fraction partial quotients). Define $\v^{\pm}:\Om\to\Bbb N$ by $$\v^{\pm}(x):=\inf\,\{n\ge 1:\ T_\Om^{\pm n}(x)\in\Om\}\ \&\ \v^{\pm}_J:=\sum_{j=0}^{J-1}\v^{\pm}\circ T_\Om^{\pm j}$$ and define $$\v_n(x):=\begin{cases} &\v^{+}_n(x)\ \ \ \ \ \ \ \ \ n\ge 1,\\ & 0\ \ \ \ \ \ \ \ \ n=0,\\ & -\v_{-n}^{-}(\s^{-1}x)\ \ \ \ \ \ \ \ \ n\le -1.\end{cases}$$ It follows that $$\begin{aligned} \tag*{{\Large\dsrailways}}\Si_n(1_\Om)(x)=\#\{k\in\Bbb Z:\ |\v_k(x)|\le n\}. \end{aligned}$$ For $n\in\Bbb N,\ t>0$: $B_n(t)$ by $$B_n(t):=\bigcup_{k=2^n+1}^{2^{n+1}}[\v^{-}\circ T_\Om^{-k}>tb(2^n)]\cap [\v^{+}\circ T_\Om^{k}>tb(2^n)]$$ where $b=a^{-1},\ a(n)=a_n(T).$ Following the ideas in the proof of lemma 1.2 in [@AN], we claim that $$\begin{aligned} \tag*{\Leftscissors}P(B_n(t))\asymp 2^nP([\v^{+}>tb(2^n)])^2\end{aligned}$$  Evidently, $$\begin{aligned} P(B_n(t))&\le\sum_{k=2^n+1}^{2^{n+1}}P([\v^{-}\circ T_\Om^{-k}>tb(2^n)]\cap [\v^{+}\circ T_\Om^{k}>tb(2^n)])\\ &\le\sum_{k=2^n+1}^{2^{n+1}}(1+\vartheta(2^{n+1}))P([\v^{-}\circ T_\Om^{-k}>tb(2^n)])P( [\v^{+}\circ T_\Om^{k}>tb(2^n)])\\ &= (1+\vartheta(2^{n+1}))2^nP([\v^{+}>tb(2^n)])^2.\end{aligned}$$ For the other inequality, choose $\kappa\ge 1$ so that $\vartheta(\kappa)<\tfrac12\ \&$ $2^nP([\v^{\pm}>tb(2^n)])\le\tfrac12\ \forall\ n\ge\kappa$. Fix $n\gg \kappa\ \&\ 2^n< k\le 2^{n+1}$ and define $$A_k^{(n)}:=[\v^{+}\circ T_\Om^k\wedge\v^{-}\circ T_\Om^{-k}>tb(2^n)]\cap \bigcap_{2^n< j\le 2^{n+1},\ |j-k|\ge\kappa}[\v^{+}\circ T_\Om^j\wedge\v^{-}\circ T_\Om^{-j}\le tb(2^n)].$$ It follows that $$\begin{aligned} P(&A_k^{(n)})=\\ &P([\v^{+}\circ T_\Om^k\wedge\v^{-}\circ T_\Om^{-k}>tb(2^n)]\cap \bigcap_{2^n< j\le 2^{n+1},\ |j-k|\ge\kappa}[\v^{+}\circ T_\Om^j\wedge\v^{-}\circ T_\Om^{-j}\le tb(2^n)])\\ &\ge (1-\vartheta(\kappa))^3P([\v>tb(2^n])^2P(\bigcap_{2^n< j\le 2^{n+1}}[\v^{+}\circ T_\Om^j\wedge\v^{-}\circ T_\Om^{-j}\le tb(2^n)])^3\\ &=\tfrac18P([\v>tb(2^n])^2(1-P(\bigcap_{2^n< j\le 2^{n+1}}[\v^{+}\circ T_\Om^j\wedge\v^{-}\circ T_\Om^{-j}\le tb(2^n)]))^3\\ &\ge\tfrac18P([\v>tb(2^n])^2(1-2^nP([\v>tb(2^n)]))^3\\ &\ge \frac1{64}P([\v>tb(2^n])^2.\end{aligned}$$ Moreover $$\sum_{2^n< k\le 2^{n+1}}1_{A_k^{(n)}}\le\ (2\kappa+1) 1_{B_n(t)}$$ whence $$\begin{aligned} P(B_n(t)) &\ge\frac1{2\kappa+1}\sum_{2^n< k\le 2^{n+1}}P(A_k^{(n)})\\ &\ge \frac{2^n}{64(2\kappa+1)}P([\v>tb(2^n])^2.\ \ \CheckedBox\ \text{(\Leftscissors)}\end{aligned}$$ It follows from () and continued fraction mixing that $$P(B_n(t)\cap B_{n'}(t))\ \asymp\ P(B_n(t))P(B_{n'}(t))\ \ \text{for}\ n\ne n'\in \Bbb N.$$ The Borel Cantelli lemmas now ensure (as in [@AN] $\&$ [@M1; @M2]) that $$\begin{aligned} \tag*{\dstechnical}\sum_{n= 1}^\infty 1_{B_n(t)}=\infty\ \text{a.s.}& \iff\ \sum_{n= 1}^\infty 2^nP([\v^{+}>tb(2^n)])^2 =\infty\\ & \ \iff\ \sum_{n= 1}^\infty P([\v^{+}>tb(n)])^2 =\infty.\end{aligned}$$ If, in addition, $b$ is [*weakly regularly varying*]{} in the sense that $$\exists\ M>1\ \text{such that}\ A(2t)\le MA(t)\ \&\ 2A(t)\le A(Mt)\ \forall\ \text{large}\ t\in\Bbb R_+,$$ then the convergence of   for some $t>0$ implies its convergence for every $t>0$; a situation characterized by   (for more details, see [@AN]). To continue, we pass to the [*one-sided factor*]{} $$\pi:\xbmt\to (X_0,\B_0,m_0,T_0)$$ defined by $$\pi^{-1}\B_0=\mathcal F_\Om=\mathcal F_\Om:=\s(\{T^{-n}\Om:\ n\ge 0\}).$$ Fix $\Om_0\in\B_0,\ \pi^{-1}\Om_0=\Om$, then $\Om_0$ is a Darling Kac set for $T_0$. Suppose that  is satisfied then $L(n)$ is slowly varying (see [@BGT]) and by the asymptotic renewal equation (3.8.6 in [@IET]) $a_n(T)\propto\frac{n}{L(n)}$ is $1$-regularly varying and, in particular, weakly regularly varying. Moreover, $L(n\log\log n)\sim L(n)$ whence (see [@LB]) $\a(T)=1$. By proposition 1, $$\varliminf_{n\to\infty}\frac1{2a_n(T)}{\Si_n(1_A)}\le \frac{m(A)}2\ \text{a.e.}\ \forall\ A\in\B,\ 0<m(A)<\infty.$$ Next, $\sum_{k=1}^\infty 1_{B_k}<\infty$ a.s. and by theorem 1.1 in [@AN], $\exists\ \e:\Bbb N\x\Om\to\{-,+\}$ so that $$\frac{\v^{\e(n,x)}_n(x)}{b(n)}\xyr[n\to\infty]{}\ 1\ \text{a.s. where}\ b=a^{-1},\ a(n)=a_n(T).$$ In addition, it follows that a.s., $$\begin{aligned} &\frac1{a_n(T)}S_n^{(T^{\e(a(n),x)})}(1_\Om)(x)\ \ \xyr[n\to\infty]{}\ 1\ \text{a.s., whence}\\ & \varliminf_{n\to\infty}\frac1{2a_n(T)}{\Si_n(1_\Om)}\ge \frac12.\ \ \ \CheckedBox\text{{\Large\Pointinghand} $\&$ \symking}\end{aligned}$$ It follows from $\a(T)<\infty$ via theorem 4 that $b$ is weakly regularly varying. If   fails, then as above, for every $t>0$ $\sum_{k=1}^\infty 1_{B_k(t)}=\infty$ a.s. and a.e. $x\in\Om,\ \exists\ n_k\to\infty$ so that $$\v_{n_k}^{+}(x),\ \v_{n_k}^{-}(x)\ >tb(n_k).$$ Set $N_k:=\lfl tb(n_k)\rfl$, then $$S_{N_k}^{(T^{\pm 1})}(1_\Om)<n_k\sim a(\frac{N_k}t)\le \frac{MI^2}t a(N_k).$$ It follows that $$\begin{aligned} \varliminf_{n\to\infty}\frac1{2a(N)}{\Si_n(1_\Om)}(x)&\le \varlimsup_{k\to\infty}\frac1{2a(N_k)}{\Si_{N_k}(1_\Om)}(x)\\ &\le \varlimsup_{k\to\infty}\frac1{2a(N_k)}{S_{N_k}^{(T)}(1_\Om)}(x)+ \varlimsup_{k\to\infty}\frac1{2a(N_k)}{S_{N_k}^{(T^{- 1})}(1_\Om)}(x)\\ &\le \frac{MI^2}t\xyr[t\to\infty]{}\ 0.\ \ \CheckedBox\end{aligned}$$ §5 The multidimensional situation {#the-multidimensional-situation .unnumbered} ================================= Example 5.1 {#example-5.1 .unnumbered} ----------- Let $\xbm$ be $\Bbb R$ equipped with Borel sets and Lebesgue measure. Let $\a,\ \b\in\Bbb R$ be linearly independent over $\Bbb Q$ and define $$\tau =\tau^{(\a,\b)}:\Bbb Z^2\to\text{\tt MPT}\,\xbm$$ by $$\tau_{(k,\ell)}(x):=x+k\a+\ell\b.$$ Define $$\Xi_n^{(\tau)}(f):=\sum_{|k|,\ |\ell|\le N}f\circ \tau_{(k,\ell)}.$$ We claim that $$\begin{aligned} \tag{{\leftthumbsup}}\frac{\Xi_n^{(T)}(f)}{2N+1}\ \xyr[N\to\infty]{}\ R\int_Xfdm\ \ \text{a.e.}\ \forall\ f\in\ L^1(m)\end{aligned}$$ where $R:=\frac{\min\,\{|\a|,|\b|\}}{\max\,\{|\a|,|\b|\}}$. Here $R=\frac1{|\a|}\ \ \&\a\notin\Bbb Q$. We have that $W=[0,1)$ is a [*maximal wandering set*]{} for $\tau_{0,1}$ in the sense that $$X=\bigcupdot_{n\in\Bbb Z}\tau_{0,n}W,$$ whence, since $|\a|>1$, $\exists\ \kappa:\Bbb Z\x W\to\Bbb Z$ so that for $x\in W$, $$\frak n(x,W):=\{u\in\Bbb Z^2:\ \tau_u(x)\in W\}=\{(\ell,\kappa(\ell,x)):\ \ell\in\Bbb Z\}.$$ Here $|\kappa(\ell,x)|=|\ell\a|\pm 1\ \forall\ \ell\in\Bbb Z$, whence for $N\ge 1,\ x\in W$, $$\frak n(x,W)\cap[-N,N]^2=\{(\ell,\kappa(\ell,x)),\ell):\ \ell\in\Bbb Z,\ |\ell|,\ |\kappa(\ell,x)|\le N\}$$ and $$\Xi_N^{(\tau)}(1_W)(x)=\#\{(\kappa(\ell,x),\ell):\ \ell\in\Bbb Z,\ |\ell|\le N\}\sim\frac{ 2N}{|\a |}.$$ Next define $S:W\to W$ by $S(x):=\tau_{(1,\kappa(1,x))}$, then $$S(x)=x+\a\ \mod\ 1.$$ Thus $\tau$ is ergodic and for $f:X\to\Bbb R$, supported and continuous on $W$, we have on $W$: $$\begin{aligned} \frac{\Xi_n^{(T)}(f)}{2N+1}&=\frac1{2N+1}\sum_{|k|,\ |\ell|\le N}f\circ \tau_{(k,\ell)} \\ &\sim\frac1{2N}\sum_{|\ell|\le |\a|N}f\circ S^\ell\\ &\xyr[N\to\infty]{}\int_Wfdm_W=\frac1{|\a|}\int_Xfdm\ \ \text{uniformly on}\ W.\end{aligned}$$ The proposition follows from this via [@H].   It is not hard to show that the above action $T$ is uniquely ergodic in the sense that the only $T$-invariant Radon measures on $\Bbb R$ are multiples of $m$; and the convergence () is uniform on compact subsets for bounded continuous functions $f$. Example 5.2 {#example-5.2 .unnumbered} ----------- Let $f\in\mathcal P(\Bbb N)$, and let $\Om:=\Bbb N^\Bbb Z$ and $P=P_f\in\mathcal P(\Om)$ be product measure defined by $$P(\{\om\in\Om:\ \om_{k+i}=n_i\ \forall\ 1\le i\le N\})=\prod_{1\le i\le N}f_{n_i}\ \ \ \ \ (k\in\Bbb Z).$$ Let $$\xbm:=(\Om\x\Bbb Z,\B(\Om\x\Bbb Z),P_f\x\#),$$ let $\s:\Om\to\Om$ is the shift and $$\psi:\Bbb Z^2\to\text{\tt MPT}\,\xbm$$ by $$\psi_{1,0}(\om,n):=(\s\om,n+\om_0)\ \ \&\ \psi_{0,1}(\om,n):=(\om,n+1).$$ The action is ergodic since for $\om\in\Om$, $$\{\psi_{k,\ell}(\om,0):\ k,\ \ell\in\Bbb Z\}=\bigcup_{n\in\Bbb Z}\{\s^j(\om):\ j\in\Bbb Z\}\x\{n\}.$$ Moreover, writing $$s_k(\om):=\begin{cases} &\sum_{j=0}^{k-1}\om_j\ \ \ \ \ k\ge 1,\\ & 0\ \ \ \ \ \ \ \ \ \ k=0,\\ & -\sum_{j=1}^{|k|}\om_{-j}=-s_{-k}(\s^k\om)\ \ \ \ \ k\le -1,;\end{cases}$$ we have $$\begin{aligned} \sum_{|k|,\ |\ell|\le N}1_{\Om\x\{0\}}(\psi_{k,\ell}(\om,0))&=\sum_{|k|,\ |\ell|\le N}1_{\Om\x\{0\}}((\s^k\om,s_k(\om)-\ell))\\ &= \#\{k\in [-N,N]:\ |s_k(\om)|\le N\}.\end{aligned}$$ Let $u=(u_0,u_1,\dots)$ be the renewal sequence with lifetime distribution $f$ and let $a_u(n):=\sum_{k=1}^nu_k$. By and theorem 5, we have that the following conditions (on $f\in\mathcal P(\Bbb N)$) are equivalent: $$\begin{aligned} &\tag{\Pointinghand}\exists\ a_n>0\ \ \text{so that}\ \ \Xi^{(\psi)}(1_{\Om\x\{0\}})\ \asymp\ a_n \\ &\tag{\rightpointleft} \Xi^{(\psi)}(1_{\Om\x\{0\}})\ \asymp\ a_u(n);\\ &\tag{\symqueen}\sum_{n=1}^\infty\left(\frac{f([n,\infty))}{L_f(n)}\right)^2<\infty\ \text{where}\ L_f(n):=\sum_{k=1}^nf([k,\infty)).\end{aligned}$$ In this case () fails. The above examples show that a conservative, ergodic, infinite measure preserving $\Bbb Z^2$ action having a dissipative generator with a maximal wandering set of finite measure can satisfy (); satisfy () while not satisfying (), not satisfy (). It follows from theorem 2 that a a conservative, ergodic, infinite measure preserving $\Bbb Z^2$ action having a dissipative generator with a maximal wandering set of infinite measure cannot satisfy (), the other two possibilities being available. Question {#question .unnumbered} --------    There are conservative, ergodic, infinite measure preserving $\Bbb Z^2$ actions with both generators conservative. We do not know which of the above possibilities are available for such an action. [^1]: ©2013++

[**Conditioned limit theorems for products of positive random matrices** ]{} C. Pham $^($[^1]$^)$ $^($[^2]$^)$ **Abstract** Inspired by a recent paper of I. Grama, E. Le Page and M. Peigné, we consider a sequence $(g_n)_{n \geq 1}$ of i.i.d. random $d\times d$-matrices with non-negative entries and study the fluctuations of the process $(\log \vert g_n\cdots g_1\cdot x\vert )_{n \geq 1}$ for any non-zero vector $x$ in $\mathbb R^d$ with non-negative coordinates. Our method involves approximating this process by a martingale and studying harmonic functions for its restriction to the upper half line. Under certain conditions, the probability for this process to stay in the upper half real line up to time $n$ decreases as $c \over \sqrt n$ for some positive constant $c$. Keywords: exit time, Markov chains, product of random matrices. Introduction ============ Many limit theorems describe the asymptotic behaviour of random walks with i.i.d. increments, for instance the strong law of large numbers, the central limit theorem, the invariant principle... Besides, the fluctuations of these processes are well studied, for example the decay of the probability that they stay inside the half real line up to time $n$. The Wiener-Hopf factorization is usually used in this case but so far, it seems to be impossible to adapt in non-abelian context. Recently, much efforts are made to apply the results above for the logarithm of the norm of the product of i.i.d. random matrices since it behaves similarly to a sum of i.i.d. random variables. Many limit theorems arose for the last 60 years, initiated by Furstenberg-Kersten [@FK], Guivarc’h-Raugi [@GR], Le Page [@LePage82]... and recently Benoist-Quint [@BQ]. Let us mention also the works by Hennion [@H1] and Hennion-Herve [@HH] for matrices with positive entries. However, the studies on the subject of fluctuation was quite sparse a few years ago. Thanks to the approach of Denisov and Wachtel [@DW] for random walks in Euclidean spaces and motivated by branching processes, I. Grama, E. Le Page and M. Peigné recently progressed in [@GLP1] for invertible matrices. Here we propose to develop the same strategy for matrices with positive entries by using [@HH]. We endow $\mathbb R^d$ with the norm $\vert \cdot \vert $ defined by $\displaystyle \vert x\vert := \sum_{i=1}^d \vert x_i\vert $ for any column vector $x=(x_i)_{1\leq i \leq d}$. Let $\mathcal C$ be the cone of vectors in $\mathbb R^d$ with non-negative coordinates $$\mathcal C := \{x \in \mathbb R ^d: \forall 1 \le i \le d, x_i \ge 0 \}$$ and $\mathbb X$ be the limited cone defined by $$\mathbb X := \{ x \in \mathcal C, |x| =1 \}.$$ Let $S $ be the set of $d\times d$ matrices with non-negative entries such that each column contains at least one positive entry; its interior is $\mathring {S }:=\{g =(g(i, j))_{1\leq i, j \leq d}/ g(i, j) >0\} $. Endowed with the standard multiplication of matrices, the set $S$ is a semigroup and $ \mathring S$ is the ideal of $S$, more precisely, for any $g \in \mathring S$ and $h \in S$, it is evident that $gh \in \mathring S$. We consider the following actions: - the left linear action of $S$ on $\mathcal C$ defined by $(g,x) \mapsto gx$ for any $g \in S$ and $x \in \mathcal C$, - the left projective action of $S$ on $\mathbb X$ defined by $(g,x) \mapsto g \cdot x := \frac{gx}{|gx|}$ for any $g \in S$ and $x \in \mathbb X$. For any $g =(g(i, j))_{1 \le i, j \le d} \in S$, without confusion, let $$v(g) := \min_{1\leq j\leq d}\Bigl(\sum_{i=1}^d g(i, j)\Bigr)\quad {\rm and} \quad \vert g\vert :=\max_{1\leq j\leq d}\Bigl(\sum_{i=1}^d g(i, j)\Bigr) ,$$ then $\vert \cdot \vert$ is a norm on $S$ and for any $x \in \mathcal C$, $$\label{controlnormgx} 0< v(g)\ \vert x\vert \leq \vert gx\vert \leq \vert g\vert \ \vert x\vert.$$ We set $N(g):= \max \left({1\over v(g)}, \vert g\vert\right) $. On some probability space $(\Omega, \mathcal F, \mathbb P)$, we consider a sequence of i.i.d. $S$-valued matrices $ (g_n)_{n \geq 0}$ with the same distribution $\mu$ on $S$. Let $L_0= Id$ and $ L_n := g_n \ldots g_1$ for any $n \ge 0$. For any fixed $x \in \mathbb X$, we define the $\mathbb X$-valued Markov chain $(X_n^x)_{n \geq0}$ by setting $X_n^x := L_n \cdot x$ for any $n \ge 0$ (or simply $X_n$ if there is no confusion). We denote by $P$ the transition probability of $(X_n)_{n \ge 0}$, defined by: for any $ x \in \mathbb{X}$ and any bounded Borel function $\varphi : \mathbb X \to \mathbb{C}$, $$\begin{aligned} P\varphi ( x) := \int_{S} {\varphi ( g\cdot x) \mu (dg)} =\mathbb E[\varphi(L_1\cdot x)]. \end{aligned}$$ Hence, for any $n \ge 1$, $$P^n\varphi(x) =\mathbb E[\varphi(L_n\cdot x)].$$ We assume that with positive probability, after finitely many steps, the sequence $(L_n)_{n\ge 1}$ reaches $\mathring S$. In mathematical term, it is equivalent to writing as $$\begin{aligned} \mathbb P \left( \bigcup _{n \ge 1} [L_n \in \mathring S] \right) >0.\end{aligned}$$ On the product space $S \times \mathbb X$, we define the function $\rho$ by setting for any $(g,x) \in S \times \mathbb X $, $$\rho(g,x):=\log |gx|.$$ Notice that $gx = e^{\rho(g,x)} g \cdot x$; in other terms, the linear action of $S$ on $\mathcal C$ corresponds to the couple $(g \cdot x, \rho(g,x))$. This function $\rho$ satisfies the cocycle property $\rho(gh, x)= \rho(g, h\cdot x)+\rho(h, x)$ for any $g, h\in S$ and $x \in \mathbb X$ and implies the basic decomposition for any $x \in \mathbb X$, $$ \log \vert L_n x \vert = \sum_{k=1}^n \rho(g_k, X^x_{k-1}).$$ For any $a \in \mathbb R$ and $n \ge 1$, let $S_0 := a$ and $S_n = S_n (x,a) := a + \sum_{k=1}^n \rho(g_k, X_{k-1})$. Then the sequence $(X_n, S_n)_{n \geq0}$ is a Markov chain on $\mathbb X \times \mathbb R$ with transition probability $\widetilde P$ defined by: for any $(x, a) \in \mathbb X \times \mathbb R$ and any bounded Borel function $\psi : \mathbb X \times \mathbb R \to \mathbb{C}$, $$\widetilde P \psi ( x, a) = \int_{S} {\psi (g \cdot x, a+\rho( g, x)) \mu (dg)} .$$ For any $(x, a)\in \mathbb{X}\times \mathbb R$, we denote by $\mathbb{P}_{ x, a}$ the probability measure on $(\Omega, \mathcal F)$ conditioned to the event $ [X_{0}=x, S_0 = a]$ and by $\mathbb{E}_{x, a}$ the corresponding expectation; for the sake of brevity, by $\mathbb{P}_{ x}$ we denote $\mathbb P _{x,a}$ when $S_0 =0$ and by $\mathbb{E}_{x}$ the corresponding expectation. Hence for any $n \ge 1$, $$\begin{aligned} \label{eqn4} \widetilde P^n\psi(x, a) = \mathbb E[\psi(L_n\cdot x, a+\log \vert L_n x\vert)]= \mathbb E_{x,a} [\psi( X_n, S_n)]. \end{aligned}$$ Now we consider the restriction of $\widetilde P_+$ to $\mathbb X \times \mathbb R^+$ of $\widetilde P$ defined by: for any $(x,a) \in \mathbb X \times \mathbb R ^+$ and any bounded function $\psi : \mathbb X \times \mathbb R \to \mathbb C$, $$\widetilde P _+ \psi (x, a) = P (\psi {\bf 1}_{\mathbb X \times \mathbb R^+_\ast})(x,a).$$ Let us emphasize that $\widetilde P_+$ may not be a Markov kernel on $\mathbb X \times \mathbb R ^+$. Let $ \tau := \min \{ n \ge 1:\, {S_n} \leq 0\} $ be the first time the random process $(S_n)_{n \ge 1}$ becomes non-positive; for any $(x, a) \in \mathbb X \times \mathbb{R^+}$ and any bounded Borel function $\psi : \mathbb X \times \mathbb R \to \mathbb{C}$, $$\begin{aligned} \label{eqn5} \widetilde P_+\psi(x, a)=\mathbb E_{x, a}[\psi(X_1, S_1); \tau >1]= \mathbb E [\psi (g_1\cdot x, a+ \rho(g_1,x)); a+\rho(g_1,x) > 0].\end{aligned}$$ A positive $\widetilde P_+$-harmonic function $V$ is any function from $\mathbb X \times \mathbb R ^+$ to $\mathbb R ^+$ satisfying $ \widetilde P_+ V =V $. We extend $V$ by setting $V(x, a) =0$ for $(x, a) \in \mathbb X \times \mathbb R^-_\ast$. In other words, the function $V$ is $\widetilde P_+$-harmonic if and only if for any $x \in \mathbb X$ and $a\ge0$, $$\begin{aligned} \label{eqn7} V(x, a) = \mathbb E_{x,a} [V(X_1, S_1); \tau >1]. \end{aligned}$$ From Theorem II.1 in [@HH], under conditions P1-P3 introduced below, there exists a unique probability measure $\nu$ on $\mathbb X$ such that for any bounded Borel function $\varphi$ from $\mathbb X$ to $\mathbb R$, $$(\mu \ast \nu) (\varphi)= \int_{S} \int_{\mathbb X} \varphi(g \cdot x) \nu(dx) \mu(dg) = \int_{\mathbb X} \varphi(x) \nu(dx) = \nu (\varphi).$$ Such a measure is said to be $\mu$-invariant. Moreover, the upper Lyapunov exponent associated with $\mu$ is finite and is expressed by $$\begin{aligned} \label{eqn10} \gamma_\mu = \int_{S} \int_{\mathbb X} \rho(g,x) \nu (dx) \mu (dg) . \end{aligned}$$ Now we state the needed hypotheses for later work. [**HYPOTHESES**]{} [**P1**]{} [*There exists $\delta_0>0$ such that $\displaystyle \int_{S} N(g)^{ \delta_0} \mu(dg) <+\infty$.*]{} [**P2**]{} [*There exists no affine subspaces $A$ of $\R ^d$ such that $A \cap \mathcal C$ is non-empty and bounded and invariant under the action of all elements of the support of $\mu$.*]{} [**P3**]{} [*There exists $n_0 \geq 1$ such that $ \mu^{*n_0}(\mathring {S }) >0.$*]{} [**P4**]{} [*The upper Lyapunov exponent $\gamma_\mu$ is equal to $0$.*]{} [**P5**]{} [*There exists $\delta >0$ such that $\mu \{g \in S : \forall x \in \mathbb X, \log|gx| \ge \delta \} >0$.*]{} In this paper, we establish the asymptotic behaviour of $\mathbb P_{x,a} (\tau >n)$ by studying the $\widetilde P_+$-harmonic function $V$. More precisely, Proposition \[theo2\] concerns the existence of a $P^+$-harmonic function and its properties whereas Theorem \[theo3\] is about the limit behaviour of the exit time $\tau$. \[theo2\] Assume hypotheses P1-P5. 1. For any $x \in \mathbb X$ and $a \ge 0$, the sequence $\Bigl(\mathbb E _{x,a} [ S_n; \tau >n ] \Bigr) _{n \ge 0}$ converges to the function $ V(x, a):= a- \mathbb E_{x,a} M_\tau$. 2. For any $x \in \mathbb X$ the function $V(x, \cdot)$ is increasing on $\mathbb R^+ $. 3. There exists $c >0$ and $A >0$ such that for any $x \in \mathbb X$ and $a\ge 0$, $$\frac{1}{c} \vee (a-A) \leq V(x, a) \leq c(1+a).$$ 4. For any $x \in \mathbb X$, the function $V(x, .)$ satisfies $\displaystyle \lim_{a \to +\infty} \frac{V(x, a)}{a} =1$. 5. The function $V$ is $\widetilde P_+$-harmonic. The function $V$ contains information of the part of the trajectory which stays in $\mathbb R^+$ as stated in Theorem \[theo3\]. \[theo3\] Assume P1-P5. Then for any $x \in \mathbb X$ and $a \ge0$, $$\mathbb P _{x,a} (\tau >n) \sim \frac{2V(x, a)}{\sigma \sqrt{2 \pi n}} \,\,\mbox{as} \,\, n \to +\infty.$$ Moreover, there exists a constant $c$ such that for any $x \in \mathbb X$, $a\ge 0$ and $n \ge 1$, $$\sqrt n \mathbb P _{x,a} (\tau > n) \leq c V(x,a).$$ As a direct consequence, we prove that the sequence $(\frac{S_n}{\sigma \sqrt n})_{n \ge 1}$, conditioned to the event $\tau >n$, converges in distribution to the Rayleigh law as stated below. \[theo4\] Assume P1-P5. For any $x \in \mathbb X$, $a\ge 0$ and $t >0$, $$\lim_{n \to +\infty} \mathbb P _{x,a} \left( \frac{S_n}{ \sqrt n} \leq t \mid \tau >n\right) = 1- \exp \left(- \frac{t^2}{2\sigma ^2} \right).$$ In section 2, we approximate the chain $(S_n)_{n \ge 0}$ by a martigale and in section 3, we study the harmonic function $V$ and state the proof of Proposition \[theo2\]. We use the coupling argument to prove Theorem \[theo3\] and Theorem \[theo4\] in section 4. At last, in section 5 we check general conditions to apply an invariant principle stated in Theorem 2.1 in [@GLP1]. Throughout this paper, we denote the absolute constants such as $C, c,c_1, c_2, \ldots$ and the constants depending on their indices such as $c_\varepsilon, c_p, \ldots$. Notice that they are not always the same when used in different formulas. The integer part of a real constant $a$ is denoted by $[a]$. Approximation of the chain $(S_n)_{n \ge 0}$ ============================================ In this section, we discuss the spectral properties of $P$ and then utilise them to approximate the chain $(S_n)_{n \ge 0}$. Throughout this section, we assume that conditions P1-P4 hold true. Spectral properties of the operators $P$ and its Fourier transform ------------------------------------------------------------------ Following [@H2], we endow $\mathbb X$ with a bounded distance $d$ such that $g$ acts on $\mathbb X$ as a contraction with respect to $d$ for any $g \in S$. For any $x,y \in \mathbb X$, we write: $$\begin{array}{l} \displaystyle m\left( {x,y} \right) = \min_{1 \le i \le d} \left\{ {\displaystyle \frac{{{x_i}}}{{{y_i}}} | {y_{i } >0}} \right\} \end{array}$$ and it is clear that $ 0 \le m\left( {x,y} \right) \le 1$. For any $x,y \in \mathbb X$, let $ d\left( {x,y} \right): = \varphi \left( {m\left( {x,y} \right)m\left( {y,x} \right)} \right)$, where $\varphi$ is the one-to-one function defined for any $s \in [0,1]$ by $\varphi \left( s \right): = \displaystyle \frac{1 - s}{1 + s}$. Set $c\left( g \right): = \sup \left\{ {d\left( {g\cdot x,g\cdot y} \right),x,y \in \mathbb{X}} \right\}$ for $g \in S $; the proposition below gives some more properties of $d$ and $c(g)$. [@H2] \[propH\] The quantity $d$ is a distance on ${\mathbb{X}}$ satisfying the following properties: 1. $\sup \{ d(x,y ): x, y \in {\mathbb{X}} \} =1 $. 2. $|x-y| \le 2d(x,y)$ for any $x,y \in \mathbb X$. 3. $c(g) \le 1$ for any $g \in S$, and $c(g) < 1$ if and only if $g \in \mathring S$. 4. $d\left( {g\cdot x,g\cdot y} \right) \le c\left( g \right)d\left( {x,y} \right) \le c(g)$ for any and $x,y \in {\mathbb{X}}$. 5. $c\left( {gh} \right) \le c\left( g \right)c\left( {h} \right)$ for any $g, h \in S $. From now on, we consider a sequence $(g_n)_{n \ge 0}$ of i.i.d. $S$-valued random variables, we set $a_k := \rho (g_k, X_{k-1})$ for $k \ge 1$ and hence $S_n = a + \sum_{k=1}^n a_k$ for $n \ge 1$. In order to study the asymptotic behavior of the process $(S_n)_{n \geq 0}$, we need to consider the “Fourier transform” of the random variables $a_k$, under $\mathbb P_x, x \in \mathbb X$, similarly for classical random walks with independent increments on $\mathbb R$. Let $P_t$ be the family of “Fourier operators” defined for any $t \in \mathbb R$, $ x \in \mathbb{X}$ and any bounded Borel function $\varphi : \mathbb X \to \mathbb{C}$ by $$\begin{aligned} \label{eqn201} P_t\varphi ( x) := \int_{S} e^{it \rho(g, x)}\varphi ( g\cdot x) \mu (dg)= \mathbb E_x\left[ e^{it a_1} \varphi(X_1) \right] \end{aligned}$$ and for any $ n \geq 1$, $$\begin{aligned} \label{eqn8} P^n_t\varphi ( x)= \mathbb E[e^{it \log \vert L_n x\vert} \varphi(L_n\cdot x)]=\mathbb E_x [e^{it S_n} \varphi(X_n)]. \end{aligned}$$ Moreover, we can imply that $$\begin{aligned} \label{eqn1.131} P^mP_t^n\varphi(x)&=&\mathbb E \left[ e^{it\log\vert g_{m+n}\cdots g_{m+1}(L_m\cdot x)\vert}\varphi(L_{m+n}\cdot x)\right] \notag \\ &=& \mathbb E_x \left[ e^{it (a_{m+1}+\cdots +a_{m+n})}\varphi(X_{n+m}) \right] \end{aligned}$$ and when $\varphi = 1$, we obtain $$\begin{aligned} \label{eqn91} \mathbb E_x \left[ e^{it S_n}\right] = P^n_t1(x)\quad {\rm and} \quad \mathbb E_x \left[ e^{it (a_{m+1}+\cdots +a_{m+n})} \right] = P^mP_t^n 1(x). \end{aligned}$$ We consider the space $C(\mathbb X)$ of continuous functions from $\mathbb X$ to $\mathbb C$ endowed with the norm of uniform convergence $\vert.\vert_\infty$. Let $L$ be the subset of Lipschitz functions on $\mathbb X$ defined by $$L := \{\varphi \in C(\mathbb X): |\varphi|_L := |\varphi|_\infty + m(\varphi) <+\infty\},$$ where $m(\varphi):= \sup_{\stackrel{x, y \in \mathbb X}{x\neq y}} {\vert \varphi(x)-\varphi(y)\vert \over d(x, y)}$. The spaces $(C(\mathbb X), \vert \cdot\vert_\infty)$ and $(L, \vert \cdot\vert _L)$ are Banach spaces and the canonical injection from $L$ into $C(\mathbb X)$ is compact. The norm of a bounded operation $A$ from $L$ to $L$ is denoted by $|A|_{L \to L} := \sup_{\varphi \in L} |A \varphi|_L$. We denote $L'$ the topological dual of $L$ endowed with the norm $\vert\cdot \vert_{L'}$ corresponding to $\vert \cdot \vert _L$; notice that any probability measure $\nu$ on $\mathbb X$ belongs to $L'$. For further uses, we state here some helpful estimations. \[lemHH\] For $g \in S$, $x,y,z \in \mathbb X$ such that $d(x,y) <1$ and for any $t\in \mathbb R$, $$\begin{aligned} \label{lem2.21} |\rho(g,x)| \le 2 \log N(g), \end{aligned}$$ and $$\begin{aligned} \label{lem2.22} |e^{it\rho(g,y)} - e^{it\rho(g,z)} | \le \Bigl( 4 \min\{2|t| \log N(g), 1 \} + 2 C |t| \Bigr) d(y,z), \end{aligned}$$ where $C= \sup \{ \frac{1}{u} \log \frac{1}{1-u} : 0 < u \le \frac{1}{2} \} <+\infty$. [**Proof.**]{} For the first assertion, from (\[controlnormgx\]), we can imply that $\vert \log |gx| \vert \le \log N(g)$. For the second assertion, we refer to the proof the Theorem III.2 in [@HH]. Denote $\varepsilon(t) := \int_S \min \{ 2 |t| \log N(g) ,2\} \mu (dg)$. Notice that $\lim_{t \to 0} \varepsilon(t) =0$. [@HH] \[spectre\] Under hypotheses [**P1, P2, P3**]{} and [**P4**]{}, for any $t \in \mathbb R$, the operator $P_t$ acts on $L$ and satisfies the following properties: 1. Let $\Pi : L \to L$ be the rank one operator defined by $\Pi(\varphi)= \nu(\varphi ) 1$ for any function $\varphi \in L$, where $\nu$ is the unique $P$-invariant probability measure on $\mathbb X$ and $R:= P- \Pi$. The operator $R: L\to L$ satisfies $$\Pi R=R\Pi=0 ,$$ and its spectral radius is less than $1$; in other words, there exist constants $ C>0$ and $0 < \kappa < 1$ such that $\vert R ^n\vert_{L \to L}\leq C \kappa^n$ for any $n \geq 1$. 2. There exist $\epsilon>0$ and $ 0 \le r_\epsilon <1$ such that for any $t\in [-\epsilon, \epsilon]$, there exist a complex number $\lambda_t$ closed to $1$ with modulus less than or equal to $ 1$, a rank one operator $\Pi_t$ and an operator $R_t$ on $L$ with spectral radius less than or equal to $ r_\epsilon$ such that $$P_t= \lambda_t \Pi_t+R_t \quad {\it and} \quad \Pi_t R_t=R_t \Pi_t = 0.$$ Moreover, $C_P:= \displaystyle \sup_{\stackrel{-\epsilon\le t\leq \epsilon}{n \geq 0}}\vert P_t^n\vert _{L \to L}<+\infty$. 3. For any $p\geq 1$, $$\label{momentsoforderp} \sup_{n \geq 0}\sup_{x \in \mathbb X}\mathbb E_x \vert \rho(g_{n+1}, X_n)\vert ^p<+\infty.$$ [ **Proof.**]{} [**(a)**]{} We first check that $P_t$ acts on $(L, \vert\cdot \vert _L)$ for any $t \in \mathbb R$. On one hand, $|P _t \varphi|_\infty \leq |\varphi|_\infty $ for any $\varphi \in L$. On the other hand, by (\[lem2.22\]) for any $x, y \in \mathbb X$ such that $ x \ne y$, $$\begin{aligned} \frac {|P _t \varphi (x)- P _t \varphi (y)| }{d(x,y)} &\le& \int_{S} \left( \left| \frac {e^{it\rho(g,x)} - e^{it\rho(g,y)} } {d(x,y)} \right| |\varphi(g \cdot x)| + \left| \frac {\varphi(g \cdot x) -\varphi(g \cdot y)} {d(x,y)} \right|\right) \mu(dg) \\ &\le& |\varphi|_\infty (4 \varepsilon(t) + 2C |t| ) + \int_{S} \left( \frac{| \varphi(g \cdot x) -\varphi(g \cdot y)|}{d(g \cdot x, g \cdot y)} \frac{d(g \cdot x, g \cdot y)}{d(x,y)} \right) \mu(dg) ,\\ &\le& |\varphi|_\infty (4 \varepsilon(t) + 2C |t| ) + m(\varphi), \end{aligned}$$ which implies $m(P _t \varphi) \leq |\varphi|_\infty (4 \varepsilon(t) + 2C |t| ) + m(\varphi) < +\infty$. Therefore $P _t \varphi \in L$. Let $\Pi$ be the rank one projection on $L$ defined by $\Pi \varphi = \nu(\varphi) {\bf 1}$ for any $\varphi \in L$. Let $ R := P - \Pi$. By definition, we obtain $P \Pi = \Pi P = \Pi$ and $\Pi ^2 = \Pi$ which implies $\Pi R = R \Pi =0$ and $R^n = P^n - \Pi$ for any $n \ge 1$. Here we only sketch the main steps by taking into account the proof of Theorem III.1 in [@HH]. Let $\mu ^{*n}$ be the distribution of the random variable $L_n$ and set $$\begin{aligned} \label{eqn16} c(\mu^{*n}) := \sup \left\{ \int _S \frac{d(g \cdot x, g \cdot y)}{d(x,y)} d \mu^{*n}(g) : x,y \in \mathbb X, x \ne y \right\}. \end{aligned}$$ Since $c(\cdot) \le 1$, we have $c(\mu^{*n}) \le 1$. Furthermore, we can see that $c(\mu^{* (m+n)}) \le c(\mu^{* m}) c(\mu^{*n})$ for any $m,n >0$. Hence, the sequence $ ( c(\mu^{*n}))_{n \ge 1}$ is submultiplicative and satisfies $ c(\mu ^{\ast n_0}) <1$ for some $n_0 \ge 1$. Besides, we obtain $m(P^n \varphi) \le m(\varphi) c(\mu^{\ast n})$. Moreover, we also obtain $m(\varphi) \le |\varphi|_L \le 3m(\varphi)$ for any $\varphi \in Ker \Pi$. Notice that $P^n(\varphi - \Pi \varphi)$ belongs to $Ker \Pi$ for any $\varphi \in L$ and $n \ge 0$. Hence $|P^n (\varphi - \Pi \varphi)|_L \le 3 c(\mu ^{\ast n}) |\varphi|_L$ which yields $$|R^n|_{L \to L} = |P^n- \Pi|_{L \to L} = |P^n(I - \Pi)|_{L \to L} \le 3 c(\mu ^{\ast n}).$$ Therefore, the spectral radius of $R$ is less than or equal to $ \displaystyle \kappa := \lim_{n \to +\infty} \Bigl(c(\mu^{\ast n}) \Bigr)^{1\over n}$ which is strictly less than $1$ by hypothesis P3 and Proposition \[propH\] (3). The theory of the perturbation implies that for $\epsilon$ small enough and for any $t \in [-\epsilon; \epsilon]$, the operator $P_t$ may be decomposed as $P _t = \lambda _t \Pi _t + R_t$, where $\lambda _t$ is the dominant eigenvalue of $P _t$ and the spectral radius of $R_t$ is less than or equal to some $r_\epsilon \in [0,1[$. In order to control $P _t ^n$, we ask $\lambda_t ^n$ to be bounded. Notice that by Hypothesis P1, the function $t \mapsto P _t$ is analytic near $0$. To prove that the sequence $(P^n_t)_t$ is bounded in $L$, it suffices to check $|\lambda_t| \le 1$ for any $t \in [-\epsilon, \epsilon]$. When $\varphi(x) = {\bf 1} (x)$, equality (\[eqn8\]) becomes $$\begin{aligned} \label{eqn113} P _t ^n {\bf 1} (x) = \mathbb E \left[ e^{it\rho( L_n , x)}\right] = \lambda _t ^n \Pi _t {\bf 1} (x) + R_t ^n {\bf 1} (x) . \end{aligned}$$ We have the local expansion of $\lambda_t$ at $0$: $$\begin{aligned} \label{eqn115} \lambda_t = \lambda_0 + t \lambda_0 ' + \frac{t^2}{2} \lambda''_0 [1+ o(1)]. \end{aligned}$$ Taking the first derivative of (\[eqn113\]) with respect to $t$, we may write for any $n \ge 0$, $$\frac{d}{dt} P _t ^n {\bf 1} (x) = \frac{d}{dt} \Bigl( \lambda _t ^n \Pi _t {\bf 1} (x) + R_t^n {\bf 1} (x) \Bigr) = \mathbb E \left[i \rho(L_n , x) e^{it \rho(L_n,x)} \right].$$ Since $\lambda_0 = 1$, $\Pi _0 1(x) =1$ and $| R^n | _{L \to L} \le C r_\epsilon^n$, we can imply that $$\lambda_0' =\frac{i}{n} \mathbb E [\rho(L_n,x)] - \frac{\Pi_0' 1(x)}{n} - \frac{[R^n_t 1(x)]'_{t=0}}{n},$$ which yields $ \displaystyle \lambda_0' = i \lim_{n \to +\infty} {1\over n}\mathbb E [\rho(L_n,x)] = i \gamma _\mu =0 $. Similarly, taking the second derivative of (\[eqn113\]) implies $ \displaystyle \lambda_0'' = - \lim_{n \to +\infty} \frac{1}{n} \mathbb E [\rho(L_n,x) ^2]$. Denote $\sigma ^2 := \displaystyle \lim_{n \to +\infty} \frac{1}{n} \mathbb E _x [S_n^2]$ . Applying in our context of matrices with non-negative coefficients the argument developed in [@BL] Lemma 5.3, we can imply that $\sigma^2 >0$ and hence $\lambda_0'' = -\sigma^2 <0$. Therefore, in particular, for $t$ closed to $0$, expression (\[eqn115\]) becomes $$\lambda_t= 1 -{\sigma^2\over 2} t^2 [1+ o(1)]$$ which implies $\vert \lambda_t\vert \leq 1$ for $t$ small enough. In particular, inequality (\[controlnormgx\]) implies $\vert \rho(g, x)\vert \leq \log N(g)$ for any $x \in \mathbb X$. Therefore, for any $p \geq 1$, $x\in \mathbb X$ and $n\geq 1$, Hypothesis P1 yields $$\mathbb E_x \vert \rho(g_{n+1}, X_{n})\vert^p \leq {p!\over \delta_0^p } \mathbb E_x e^{\delta_0 \vert \rho(g_{n+1}, X_{n})\vert} \leq {p! \over \delta_0^p } \mathbb E N(g_{n+1})^{\delta_0} <+\infty.$$ Martingale approximation of the chain $( S_n)_{n \ge 0}$ -------------------------------------------------------- As announced in the abstract, we approximate the process $(S_n)_{n \ge 0}$ by a martingale $(M_n)_{n \ge 0}$. In order to construct the suitable martingale, we introduce the operator $\overline P$ and then find the solution of the Poisson equation as follows. First, it is neccessary to introduce some notation and basic properties. Let $g_0 = I$ and $X_{-1} :=X_0 $. The sequence $((g_n, X_{n-1}))_{n \geq 0}$ is a Markov chain on $S\times \mathbb X$, starting from $(Id, x)$ and with transition operator $\overline P$ defined by: for any $(g, x) \in S \times \mathbb X$ and any bounded measurable function $\phi : S \times \mathbb X \to \mathbb R$, $$\begin{aligned} \label{eqn27} \overline {P} \phi(g,x) := \int_{S \times \mathbb X}\phi(h,y) \overline {P} ((g,x), dh dy) = \int_{S } \phi(h,g \cdot x) \mu(dh).\end{aligned}$$ Notice that, under assumption [**P1**]{}, the quantity $\overline P\rho(g, x)$ is well defined for any $(g, x) \in S \times \mathbb X$. \[remark7.3\] The function $\displaystyle \bar \rho : x \mapsto \int_S \rho(g, x) \mu(dg) $ belongs to $L$ and for any $g \in S$, $x \in \mathbb X$ and $n \ge 1$, $$\begin{aligned} \label{import} \overline {P}^{n+1} \rho (g,x) = P ^n \overline{\rho} (g \cdot x). \end{aligned}$$ [**Proof.**]{} [**(1)** ]{} For any $x \in \mathbb X$, definition of $\rho$ and (\[lem2.21\]) yield $$\begin{aligned} |{\overline \rho} (x)| &\leq & \int_{S } |\log |gx||\mu(dg) \le \int_{S } 2 |\log N(g)|\mu(dg) \le \int_{S } 2 | N(g)|^{\delta_0}\mu(dg) < +\infty. \end{aligned}$$ Hence $|\overline \rho|_\infty < +\infty$. For any $x,y \in \mathbb X$ such that $d(x,y) > \frac{1}{2}$, we can see that $$\begin{aligned} \label{eqn821} |\rho(g,x) - \rho(g,y)| \le |\rho(g,x) - \rho(g,y)| 2 d(x,y) \le 8 \log N(g) d(x,y). \end{aligned}$$ For any $x,y \in \mathbb X$ such that $d(x,y) \le \frac{1}{2}$, applying Lemma III.1 in [@HH], we obtain $$\begin{aligned} \label{eqn822} |\rho(g,x) - \rho(g,y)| \le 2 \log \frac{1}{1-d(x,y)} \le 2C d(x,y), \end{aligned}$$ where $C$ is given in Lemma \[lemHH\]. For any $x,y \in \mathbb X$, by (\[eqn821\]) and (\[eqn822\]) we obtain $$\begin{aligned} | {\overline \rho} (x) -{\overline \rho} (y)| &\le& \int_{S} \left|\rho(g,x) - \rho(g,y) \right| \mu(dg) \\ &\le& \int_{S} [8 \log N(g) +2C ] d(x,y) \mu(dg). \end{aligned}$$ Thus $\displaystyle m(\overline \rho) = \sup_{x,y \in \mathbb X, x \ne y} \frac{| {\overline \rho} (x) -{\overline \rho} (y)|}{d(x, y)} < +\infty$. [**(2)**]{} From (\[eqn27\]) and definition of $\rho$, it is obvious that $$\begin{aligned} \overline {P} \rho (g,x) = \int_{S} \rho(h, g \cdot x) \mu(dh)= \overline{\rho} (g \cdot x), \end{aligned}$$ which yields $$\begin{aligned} \overline {P}^2 \rho (g,x)&=& \overline {P} (\overline {P} \rho ) (g,x) = \int_{S \times \mathbb X} (\overline {P} \rho) (k,y) \overline {P}((g,x), dk dy) \\ &=& \int_{S \times \mathbb X} \overline{\rho}(k \cdot y) \overline {P} ((g,x), dk dy) \\ &=&\int_{S} \overline{\rho}(k \cdot (g \cdot x)) \mu(dk) =P \overline{\rho}(g \cdot x). \end{aligned}$$ By induction, we obtain $\overline {P}^{n+1} \rho (g,x) = P ^n \overline{\rho} (g \cdot x)$ for any $n \ge 0$. Formally, the solution $\theta: S\times \mathbb X \to \mathbb R$ of the equation $\theta-\overline P \theta = \rho$ is the function $$\theta : (g, x) \mapsto \sum_{n = 0}^{+\infty} \overline P ^n \rho(g , x).$$ Notice that we do not have any spectral property for $\overline P$ and $\rho$ does not belong to $L$. However, we still obtain the convergence of this series by taking into account the important relation (\[import\]), as shown in the following lemma. The sum $\displaystyle \theta = \sum_{n=0}^{+\infty} \overline P ^n \rho$ exists and satisfies the Poisson equation $\rho = \theta - \overline {P} \theta$. Moreover, $$\begin{aligned} \label{lem4.1} |\overline P \theta |_\infty = \sup_{ g \in S, x \in \mathbb X} |\theta(g,x) - \rho(g,x)| < +\infty; \end{aligned}$$ and for any $p \ge 1$, it holds $$\begin{aligned} \label{remark7.2} \displaystyle \sup_{n \geq 0 } \sup_{x \in \mathbb X} \mathbb E _x| \theta (g_{n+1},X_n)| ^p < +\infty. \end{aligned}$$ [ **Proof.**]{} [**(1)**]{} Since $P$ acts on $(L, |\cdot|_L)$ and $\overline \rho \in L$ from Lemma \[remark7.3\], we obtain $P \overline \rho \in L$. Thanks to definition of $\rho$, (\[eqn10\]) and P4, it follows that $$\nu(\overline \rho) = \int_{\mathbb X} \overline \rho (x) \nu(dx) = \int_{S} \int_{\mathbb X} \rho (g,x) \nu(dx) \mu(dg) = \gamma _\mu =0.$$ Proposition \[spectre\] and the relation (\[import\]) yield for any $x \in \mathbb X$ and $n \ge 0$, $$\overline {P} ^{n+1} \rho (g,x) = P ^n \overline \rho (g \cdot x) = \Pi \overline \rho (g \cdot x) + R^n \overline \rho (g \cdot x) = \nu (\overline \rho) {\bf 1} (g \cdot x) +R^n \overline \rho (g \cdot x) = R^n \overline \rho (g \cdot x)$$ and there exist $C>0$ and $0 < \kappa < 1$ such that for any $x \in \mathbb X$ and $n \geq 0$, $$\left| R ^n \overline \rho (x) \right| \leq \left| R ^n \overline \rho \right|_L \leq \left| R^n \right|_{L \to L} \leq C \kappa ^n.$$ Hence for any $g \in S$ and $x \in \mathbb X$, $$\begin{aligned} \left| \sum_{n=1}^{+\infty} \overline P ^n {\rho} (g , x)\right| \le \sum_{n=0}^{+\infty} \left| P ^n \overline{\rho} (g \cdot x) \right| \leq C \sum_{n=0}^{+\infty} \kappa^n = \frac{C}{1-\kappa} < +\infty. \end{aligned}$$ Therefore, the function $\displaystyle \theta = \sum_{n=0}^{+\infty} \overline P ^n \rho$ exists and obviously satisfies the Poisson equation $\rho = \theta - \overline {P} \theta$. Finally, it is evident that $$\displaystyle \sup_{g \in S, x \in \mathbb X} |\theta(g,x) - \rho(g,x)|= \sup_{g \in S, x \in \mathbb X} \left| \sum_{n=1}^{+\infty} \overline P ^n {\rho} (g , x)\right| < +\infty.$$ [**(2)**]{} Indeed, from (\[momentsoforderp\]), (\[lem4.1\]) and Minkowski’s inequality, the assertion arrives. Now we contruct a martingale to approximate the Markov walk $(S_n)_{n \geq 0}$. Hence, from the definition of $S_n$ and the Poisson equation, by adding and removing the term $\overline P \theta (g_0, X_{-1})$, we obtain $$\begin{aligned} S_n &=& a+ \rho(g_1, X_0) + \ldots + \rho(g_n,X_{n-1}) \\ &=& a+ \overline{P} \theta(g_0, X_{-1}) - \overline{P} \theta (g_n,X_{n-1}) + \sum_{k=0}^{n-1} \left[ \theta (g_{k+1}, X_k) - \overline {P} \theta (g_k, X_{k-1}) \right]. \end{aligned}$$ Let $\mathcal F _0:= \{\emptyset, \Omega \}$ and $\mathcal F _n := \sigma \{ g_k: 0\leq k \leq n\}$ for $n \geq1$. For any $n \ge 0$, $x \in \mathbb X$, $a \ge 0$ and $p > 2$, the sequence $(M_n)_{n \geq0}$ defined by $$\begin{aligned} \label{eqn13} M_0 := S_0\,\, \mbox{and} \,\, M_n:= M_0 + \sum_{k=0}^{n-1} \left[ \theta (g_{k+1}, X_k) - \overline {P} \theta (g_k, X_{k-1}) \right] \end{aligned}$$ is a martingale in $L^p (\Omega, \mathbb P_{x,a}, (\mathcal F _n)_{n\ge 0})$ satisfying the properties: $$\begin{aligned} \label{lem4.3} \displaystyle \sup_{n \geq0} |S_n -M_n| \leq 2 |\overline {P} \theta|_\infty \quad \mathbb P _{x,a}\mbox{-a.s.} \end{aligned}$$ $$\begin{aligned} \label{lem4.4} \displaystyle \sup_{n \geq1} n ^{-\frac{p}{2}} \sup_{x \in \mathbb X} \mathbb E _{x,a} |M_n|^{p}< +\infty. \end{aligned}$$ [ **From now on, we set $A:= 2 |\overline {P} \theta|_\infty $.**]{} [ **Proof.**]{} By definition (\[eqn13\]), martingale property arrives. [**(1)**]{} From the construction of $M_n$ and (\[lem4.1\]), we can see easily that $$\sup_{n \geq0} |S_n - M_n| = \sup_{n \geq0} \left| \overline{P} \theta(g_0, X_{-1}) - \overline{P} \theta (g_n,X_{n-1}) \right| \leq 2 \left| \overline{P} \theta \right| _\infty < +\infty \quad \mathbb P_{x,a}\mbox{-a.s.}.$$ [**(2)**]{} Denote $\xi_k := \theta (g_{k+1}, X_k) - \overline {P} \theta (g_k, X_{k-1}) $. Thus $M_n = M_0 +\sum_{k=0}^{n-1} \xi _k $. Using Burkholder’s inequality, for any $p \ge 1$, there exists some positive constant $c_p$ such that for $0 \le k <n$, $$(\mathbb E _{x,a} |M_n |^p)^{\frac{1}{p}} \leq c_p \left( \mathbb E _{x,a} \left| \sum_{k=0}^{n-1} \xi _k^2 \right| ^{\frac{p}{2}} \right) ^{\frac{1}{p}}.$$ Now, with $p >2$, applying Holder’s inequality, we obtain $$\left| \sum_{k=0}^{n-1} \xi _k^2 \right| \le n^{1-\frac{2}{p}} \left(\sum_{k=0}^{n-1} |\xi _k|^p \right)^{\frac{2}{p}},$$ which implies $$\mathbb E _{x,a} \left| \sum_{k=0}^{n-1} \xi _k^2 \right| ^{\frac{p}{2}} \leq n^{\frac{p}{2} -1} \mathbb E _{x,a} \sum_{k=0}^{n-1} |\xi _k|^p \leq n^{\frac{p}{2}} \sup_{0 \leq k \leq n-1} \mathbb E _{x,a} |\xi _k|^p.$$ Since $(M_n)_n$ is a martingale, by using the convexity property, we can see that for any $k \ge 0$, $$\Big\vert \overline{P} \theta (g_k, X _{k-1}) \Big\vert ^p = \Big\vert \mathbb E _{x,a} \Bigl[ |\theta (g_{k+1}, X_k)| |\mathcal F_k \Bigr] \Big\vert ^p \le \mathbb E _{x,a} \Bigl[ |\theta (g_{k+1}, X_k)|^p |\mathcal F_k \Bigr],$$ which implies $ \mathbb E _{x,a} \left| \overline{P} \theta (g_k, X _{k-1})\right| ^p \le \mathbb E _{x,a} \left| \theta(g_{k+1},X_k)\right| ^p $. Therefore, we obtain $$\begin{aligned} \Bigl(\mathbb E _{x,a} |M_n |^p\Bigr)^{\frac{1}{p}} &\leq& c_p\left( n^{\frac{p}{2}} \sup_{0 \leq k \leq n-1} \mathbb E _{x,a} |\xi _k|^p \right) ^{\frac{1}{p}} \le c_p n^{\frac{1}{2}} \sup_{0 \leq k \leq n-1} \Bigl( \mathbb E _{x,a} |\xi_k|^p \Bigr) ^{\frac{1}{p}} \\ &\le& c_p n^{\frac{1}{2}} \sup_{0 \leq k \leq n-1} \Bigl[ \Bigl( \mathbb E _{x,a} |\theta (g_{k+1}, X_k) |^p \Bigr)^{1/p} + \Bigl( \mathbb E _{x,a} |\overline P \theta (g_k, X_{k-1})|^p \Bigr)^{1/p} \Bigr] \\ &\leq& 2 c_p n^{\frac{1}{2}} \sup_{0 \leq k \leq n-1} \Bigl( \mathbb E _{x,a} |\theta(g_{k+1}, X_k)|^p \Bigr) ^{\frac{1}{p}}. \end{aligned}$$ Consequently, we obtain $\displaystyle \mathbb E _{x,a} |M_n |^p \leq (2c_p)^p n^{\frac{p}{2}} \sup_{0 \leq k \leq n-1} \mathbb E _{x,a} |\theta(g_{k+1}, X_k)|^p$ and the assertion arrives by using (\[remark7.2\]). On the $\widetilde P _+$-harmonic function $V$ and the proof of Proposition \[theo2\] ===================================================================================== In this section we construct explicitly a $\widetilde P_+$-harmonic function $V$ and study its properties. We begin with the first time the martingale $(M_n)_{n \geq 0}$ (\[eqn13\]) visit $]-\infty, 0]$, defined by $$T = \min \{ n \geq 1: M_n \le 0\}.$$ The equality $\gamma_\mu=0 $ yields $\displaystyle \liminf_{n \to +\infty} S_n=-\infty \,\, \mathbb P_x$-a.s. for any $x \in \mathbb X$, thus $\displaystyle \liminf_{n \to +\infty} M_n=-\infty \,\, \mathbb P_x$-a.s., so that $T <+\infty \,\, \mathbb P_x$-a.s. for any $x \in \mathbb X$ and $ a \ge 0$. On the properties of $T$ and $(M_n)_n$ -------------------------------------- We need to control the first moment of the random variable $ \vert M_{T \wedge n}\vert$ under $\mathbb P_x$; we consider the restriction of this variable to the event $[T \leq n]$ in lemma \[lem5.2\] and control the remaining term in lemma \[lem5.6\]. \[lem5.2\] There exists $\varepsilon _0 >0$ and $c>0$ such that for any $\varepsilon \in (0, \varepsilon _0), n \geq1, x \in \mathbb X$ and $a \geq n^{\frac{1}{2} -\varepsilon}$, $$\mathbb E _{x,a} \Bigl[|M_{T}|; T \leq n\Bigr] \leq c \frac{a}{n^\varepsilon}.$$ [ **Proof.**]{} For any $\varepsilon >0$, consider the event $\displaystyle A_n := \{ \max_{0 \leq k \leq n-1} |\xi_k| \leq n^{\frac{1}{2} -2 \varepsilon} \}$, where $ \xi_k = \theta(g_{k+1},X_k) - \overline{P} \theta(g_k, X_{k-1})$; then $$\begin{aligned} \label{lelnvkj} \mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n\Bigr] &=& \mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n,A_n\Bigr] +\mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n,A^c_n\Bigr].\end{aligned}$$ On the event $ [T \le n]\cap A_n $, we have $| M_T | \le |\xi _{T-1}| \le n^{\frac{1}{2} - 2 \varepsilon}$. Hence for any $x \in \mathbb X$ and $a \geq n^{\frac{1}{2} -\varepsilon}$, $$\begin{aligned} \label{eqn24} \mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n,A_n\Bigr] \le \mathbb E_{x,a} \Bigl[|\xi_{T-1}|;T \leq n,A_n\Bigr] \leq n^{\frac{1}{2} -2 \varepsilon} \leq \frac{a}{n^\varepsilon}.\end{aligned}$$ Let $M^*_n := \displaystyle \max_{1 \leq k \leq n} |M_k|$; since $|M_{T}| \leq M^*_n$ on the event $[T \leq n]$, it is clear that, for any $x \in \mathbb X$ and $a \ge 0$, $$\begin{aligned} \label{Mn*ANc} \mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n,A^c_n\Bigr] &\le& \mathbb E_x [M^*_n; A^c_n] \notag \\ &\le& \mathbb E_{x,a} \Bigl[M^*_n;M^*_n > n^{\frac{1}{2} +2\varepsilon}, A^c_n\Bigr] + n^{\frac{1}{2} +2\varepsilon} \mathbb P_{x,a} ( A^c_n) \notag\\ &\le& \int^{+\infty}_{n^{\frac{1}{2}+2\varepsilon}} \mathbb P_{x,a} (M^*_n >t) dt + 2 n^{\frac{1}{2} +2\varepsilon} \mathbb P_{x,a} ( A^c_n).\end{aligned}$$ We bound the probability $\mathbb P_{x,a} (A^c_n)$ by using Markov’s inequality, martingale definition and (\[remark7.2\]) as follows: for any $p \ge 1$, $$\begin{aligned} \mathbb P_{x,a} (A^c_n) &\le& \sum_{k=0}^{n-1} \mathbb P_{x,a} \left( |\xi_k| > n^{\frac{1}{2} - 2 \varepsilon } \right)\notag \\ &\le& \frac{1}{n^{(\frac{1}{2} - 2 \varepsilon )p}} \sum_{k=0}^{n-1} \mathbb E_{x,a} |\xi_k|^p \notag \\ &\le& \frac{2^p}{n^{(\frac{1}{2} - 2 \varepsilon )p}} \sum_{k=0}^{n-1} \mathbb E _{x,a} |\theta(g_{k+1}, X_k) |^p \notag\\ &=& \frac{c_p}{n^{{p \over 2} - 1- 2 \varepsilon p}}.\end{aligned}$$ For any $a \ge n^{\frac{1}{2} - \varepsilon}$, it follows that $$\label{TYF} n^{\frac{1}{2} +2\varepsilon} \mathbb P_{x,a} ( A^c_n) \le a n^{3\varepsilon}\mathbb P_{x,a} ( A^c_n) \leq \frac{c_p a}{n^{{p \over 2} -1-2\varepsilon p -3\varepsilon}}.$$ Now we control the integral in (\[Mn\*ANc\]). Using Doob’s maximal inequality for martingales and (\[lem4.4\]), we receive for any $p \ge 1$, $$\mathbb P_x (M^*_n >t) \leq \frac{1}{t^p} \mathbb E_x \Bigl[|M_n|^p \Bigr] \leq c_p \frac{n^{{p \over 2}}}{t^p},$$ which implies for any $a \ge n^{\frac{1}{2} - \varepsilon}$, $$\begin{aligned} \label{eqn5.8} \int^{+\infty}_{n^{\frac{1}{2}+2\varepsilon}} \mathbb P_x (M^*_n >t) dt &\le& {c_p \over p-1} \frac{n^{{p \over 2}}}{n^{(\frac{1}{2}+2\varepsilon)(p-1)}} \le {c_p \over p-1} \frac{a}{n^{2\varepsilon p-3 \varepsilon}}.\end{aligned}$$ Taking (\[Mn\*ANc\]), (\[TYF\]) and (\[eqn5.8\]) altogether, we obtain for some $c_p'$, $$\label{kzrhd} \mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n,A^c_n\Bigr] \leq c_p' \left( \frac{a}{n^{2\varepsilon p-3 \varepsilon}} + \frac{ a}{n^{{p \over 2} -1-2\varepsilon p -3\varepsilon}} \right).$$ Finally, from (\[lelnvkj\]), (\[eqn24\]) and (\[kzrhd\]), we obtain for any $a \geq n^{\frac{1}{2} -\varepsilon}$, $$\mathbb E_{x,a} \Bigl[|M_{T}|;T \leq n\Bigr] \leq \frac{a}{n^\varepsilon} + c_p' \frac{a }{n^\varepsilon} \left(\frac{1}{n^{2\varepsilon p-4 \varepsilon}} + \frac{ 1}{n^{{p \over 2} -1-2\varepsilon p -4\varepsilon}} \right).$$ Fix $p >2$. Then there exist $c >0$ and $\varepsilon_0 >0$ such that for any $\varepsilon \in (0, \varepsilon_0) $ and $a \geq n^{\frac{1}{2} -\varepsilon}$, $$\mathbb E_{x,a} \Bigl[|M_{T}|, T \le n \Bigr] \leq c \frac{a}{n^\varepsilon}$$ which proves the lemma. For fixed $\varepsilon >0$ and $a \ge 0$, we consider the first time $\nu_{n, \varepsilon}$ when the process $(|M_k|)_{k \ge 1}$ exceeds $2 n ^{\frac{1}{2} - \varepsilon}$. It is connected to Lemma \[lem6.2\] where $\mathbb P (\tau_a ^{bm} >n)$ is controlled uniformly in $a$ under condition $a \le \theta_n \sqrt n$ with $\lim_{ n \to +\infty} \theta_n =0$ which we take into account here by setting $$\begin{aligned} \nu_{n, \varepsilon} := \min \{k \geq1: | M_k| \geq2 n ^{\frac{1}{2} - \varepsilon} \}.\end{aligned}$$ Notice first that for any $\varepsilon>0, x \in \mathbb X$ and $a \ge 0$ the sequence $( \nu _{n, \varepsilon})_{n \geq 1}$ tends to $+\infty$ a.s. on $(\Omega, \mathcal B(\Omega), \mathbb P_{x,a})$. The following lemma yields to a more precise control of this property. \[lem5.4\] For any $\varepsilon \in (0, \frac{1}{2})$, there exists $c_\varepsilon >0$ such that for any $x \in \mathbb X$, $a \ge 0$ and $n \geq1$, $$\mathbb P _{x,a} (\nu_{n, \varepsilon} > n^{1-\varepsilon}) \leq \exp (-c_\varepsilon n^ \varepsilon).$$ [ **Proof.**]{} Let $m = [B^2 n^{1-2 \varepsilon}]$ and $K = [n^\varepsilon / B^2]$ for some positive constant $B$. By (\[lem4.3\]), for $n$ sufficiently great such that $A \leq n^{\frac{1}{2} -\varepsilon}$, we obtain for any $x \in \mathbb X$ and $a \ge 0$, $$\begin{aligned} \label{eqn5.12} \mathbb P_{x,a} (\nu_{n, \varepsilon} > n^{1-\varepsilon}) &\le& \mathbb P_{x,a} \left( \max_{1 \leq k \leq n^{1-\varepsilon}} |M_k| \leq 2n^{\frac{1}{2} -\varepsilon}\right) \notag \\ &\le& \mathbb P_{x,a} \left( \max_{1 \leq k \leq K} |M_{km}| \leq 2n^{\frac{1}{2} -\varepsilon}\right) \notag \\ &\le& \mathbb P_{x,a} \left( \max_{1 \leq k \leq K} |S_{km}| \leq 3n^{\frac{1}{2} -\varepsilon}\right).\end{aligned}$$ Using Markov property, it follows that, for any $x \in \mathbb X$ and $a \ge 0$, from which by iterating $K$ times, we obtain $$\label{KJyazdtfhg} \mathbb P_{x,a} \left( \max_{1 \leq k \leq K} | S_{km}| \leq 3n^{\frac{1}{2} -\varepsilon}\right) \leq \left( \sup_{b \in \mathbb R, x \in \mathbb X} \mathbb P_{x,b} \left( |S_{m}| \leq 3n^{\frac{1}{2} -\varepsilon}\right) \right) ^K.$$ Denote $\mathbb B (b;r) = \{c: |b+c| \leq r \}$. Then for any $x \in \mathbb X$ and $b \in \mathbb R$ $$\mathbb P_{x,b} \left( |S_m| \leq 3n^{\frac{1}{2} - \varepsilon }\right) = \mathbb P_x \left( \frac{S_m}{ \sqrt m} \in \mathbb B (b / \sqrt m; r_n) \right),$$ where $r_n = {3 n^{\frac{1}{2} -\varepsilon} \over {\sqrt m}}$. Using the central limit theorem for $S_n$ (Theorem 5.1 property iii) [@BL]), we obtain for $n \to +\infty$, $$\begin{aligned} \sup_{b \in \mathbb R, x \in \mathbb X} \left| \mathbb P_x \left( \frac{S_m}{ \sqrt m} \in \mathbb B (b/ \sqrt m; r_n) \right) - \int_{\mathbb B (b/ \sqrt m; r_n)} \phi_{\sigma ^2} (u) du \right| \to 0,\end{aligned}$$ where $\phi_{\sigma ^2}(t) = \frac{1}{\sigma\sqrt{2 \pi}} \exp \left( - \frac{t^2}{2 \sigma ^2}\right)$ is the normal density of mean $0$ and variance $\sigma ^2$ on $\mathbb R$. Since $r_n \leq c_1 B^{-1}$ for some constant $c_1 >0$, we obtain $$\sup_{b \in \mathbb R} \int_{\mathbb B (b / \sqrt m; r_n)} \phi_{\sigma ^2} (u) du \leq \int_{-r_n}^{r_n} \phi_{\sigma ^2} (u) du \leq \frac{2r_n}{\sigma \sqrt{2 \pi}} \le \frac{2c_1}{B \sigma \sqrt{2 \pi}}.$$ Choosing $B$ and $n$ great enough, for some $q_\varepsilon <1$, we obtain $$\sup_{b \in \mathbb R, x \in \mathbb X} \mathbb P_{x,b} \left( |S_m| \leq 3n^{\frac{1}{2} - \varepsilon }\right) \leq \sup_{b \in \mathbb R} \int_{\mathbb B (b/ \sqrt m; r_n)} \phi_{\sigma ^2} (u) du +o(1)\leq q_\varepsilon .$$ Implementing this bound in (\[KJyazdtfhg\]) and using (\[eqn5.12\]), it follows that for some $c_\varepsilon >0$, $$\sup_{a >0 , x \in \mathbb X} \mathbb P_{x,a} (\nu_{n, \varepsilon} >n^{1-\varepsilon}) \leq q_\varepsilon ^K \le q_\varepsilon^{\frac{n^\varepsilon}{B^2} -1} \leq e^{- c_\varepsilon n^\varepsilon }.$$ \[lem5.5\] There exists $c >0$ such that for any $\varepsilon \in (0, \frac{1}{2})$, $ x \in \mathbb X$, $a \ge 0$ and $n \geq1$, $$\sup_{1 \leq k \leq n} \mathbb E _{x,a} [|M_k|; \nu_{n, \varepsilon} > n ^{1- \varepsilon}] \leq c (1+a) \exp (-c _\varepsilon n^ \varepsilon)$$ for some positive constant $c_\varepsilon $ which only depends on $\varepsilon$. [ **Proof.**]{} By Cauchy-Schwartz inequality, for any $x \in \mathbb X$, $a \ge 0$ and $1 \leq k \leq n$, $$\mathbb E_{x,a} \left[|M_k|; \nu_{n, \varepsilon} > n^{1-\varepsilon}\right] \leq \sqrt{ \mathbb E_{x,a} |M_k|^2 \mathbb P_{x,a} ( \nu_{n, \varepsilon} > n^{1-\varepsilon})}.$$ By Minkowsky’s inequality, (\[lem4.3\]) and the fact that $\frac{1}{n} \mathbb E_x |S_n|^2 \to \sigma^2$ as $n \to +\infty$, it yields $$\sqrt{ \mathbb E_{x,a} |M_k|^2} \leq a+ \sqrt{ \mathbb E_{x,a} [M_k^2]} \leq a+ A + \sqrt{ \mathbb E_{x,a} [S_k^2]} \leq c (a+ n^ {\frac{1}{2}})$$ for some $c >0$ which does not depend on $x$. The claim follows by Lemma \[lem5.4\]. \[lem5.6\] There exists $c >0$ and $\varepsilon_0 >0$ such that for any $\varepsilon \in (0, \varepsilon_0)$, $x \in \mathbb X$, $a \ge 0$ and $n \geq1$, $$\label{5.15} \mathbb E _{x,a} [M_n; T >n] \leq c(1+a).$$ and $$\label{lem5.10} \lim_{a \to +\infty} \frac{1}{a} \lim_{n \to +\infty} \mathbb E _{x,a} [M_n;T >n]= 1.$$ [ **Proof.**]{} [ **(1)**]{} On one hand, we claim $$\begin{aligned} \label{eqnclaim} \mathbb E_{x,a} [M_n; T >n , \nu_{n, \varepsilon} \leq n^{1-\varepsilon}] &\leq & \left( 1+ \frac{c'_\varepsilon}{n^\varepsilon} \right) \mathbb E_{x,a} \left[M_{[n^{1-\varepsilon}]}; T >[n^{1-\varepsilon}]\right]\end{aligned}$$ and delay the proof of (\[eqnclaim\]) at the end of the first part. On the other hand, by Lemma \[lem5.5\], there exists $c >0$ such that for any $\varepsilon \in (0, \frac{1}{2})$, $x \in \mathbb X$, $a \ge 0$ and $n \ge 1$, $$\begin{aligned} \label{eqn5.112} \mathbb E_{x,a} [M_n; T >n , \nu_{n, \varepsilon} > n^{1-\varepsilon}] &\le& \sup_{1 \le k \le n} \mathbb E_{x,a} \Bigl[|M_k|; \nu_{n, \varepsilon} > n^{1-\varepsilon}\Bigr] \notag \\ &\le& c(1+a) \exp (-c_\varepsilon n^\varepsilon).\end{aligned}$$ Hence combining (\[eqnclaim\]) and (\[eqn5.112\]), we obtain for any $x \in \mathbb X$ and $a \ge 0$, $$\label{5.23} \mathbb E_{x,a} [M_n; T >n] \leq \left( 1+ \frac{c'_\varepsilon}{n^\varepsilon} \right) \mathbb E_{x,a} \left[M_{[n^{1-\varepsilon}]}; T >[n^{1-\varepsilon}]\right] +c(1+a) \exp (-c_\varepsilon n^\varepsilon).$$ Let $k_j := \left[ n^{(1-\varepsilon)^j} \right]$ for $j \geq0$. Notice that $k_0 =n$ and $[k_j^{1-\varepsilon}] \le k_{j+1}$ for any $j \ge 0$. Since the sequence $((M_n){\bf 1}_{[T >n]})_{n\geq1}$ is a submartingale, by using the bound (\[5.23\]), it yields $$\begin{aligned} \mathbb E_{x,a} [M_{k_1}; T >{k_1}] &\le& \left( 1+ \frac{c'_\varepsilon}{{k_1}^\varepsilon} \right) \mathbb E_{x,a} \left[M_{[{k_1}^{1-\varepsilon}]}; T >[{k_1}^{1-\varepsilon}]\right] + c(1+a) \exp (-c_\varepsilon k_1^\varepsilon) \\ &\le& \left( 1+ \frac{c'_\varepsilon}{{k_1}^\varepsilon} \right) \mathbb E_{x,a} [ M_{k_2}; T >k_2] + c(1+a) \exp (-c_\varepsilon k_1^\varepsilon).\end{aligned}$$ Let $n_0$ be a constant and $m = m(n)$ such that $k_m = \left[ n^{(1-\varepsilon )^m}\right] \le n_0$. After $m$ iterations, we obtain $$\label{5.24} \mathbb E_{x,a} [M_n; T >n] \leq A_m \Bigl( \mathbb E_{x,a} [M_{k_m}; T >k_m] + c(1+a) B_m \Bigr),$$ where $$\begin{aligned} \label{5.25} A_m = \prod _{j=1}^m \left( 1+ \frac{c'_\varepsilon}{k_{j-1}^\varepsilon}\right) \le \exp \Bigl( {2^\varepsilon c'_\varepsilon \frac{n_0^{-\varepsilon}}{ 1- n_0^{-\varepsilon^2} }} \Bigr), \end{aligned}$$ and $$\label{5.26} B_m = \sum_{j=1}^m \frac{\exp \left( -c_\varepsilon k_{j-1}^\varepsilon \right)}{\Bigl( 1+\frac{c'_\varepsilon}{k^\varepsilon_{j-1}} \Bigr) \ldots \Bigl( 1+\frac{c'_\varepsilon}{k^\varepsilon_m} \Bigr) } \leq c_1 \frac{n_0^{-\varepsilon}} {1- n_0^{-\varepsilon^2}}$$ from Lemma 5.6 in [@GLP1]. By choosing $n_0$ sufficient great, the first assertion of the lemma follows from (\[5.24\]), (\[5.25\]) and (\[5.26\]) taking into account that $$\begin{aligned} \label{5.29} \mathbb E_{x,a} [ M_{k_m}; T >k_m] \le \mathbb E_{x,a} [ M_{n_0}; T > n_0] \leq \mathbb E_{x,a} |M_{n_0}| \leq a+c. \end{aligned}$$ Before proving (\[eqnclaim\]), we can see that there exist $c >0$ and $0 < \varepsilon_0 < {1 \over 2}$ such that for any $\varepsilon \in (0, \varepsilon_0)$, $x \in \mathbb X$ and $b \geq n^{\frac{1}{2} -\varepsilon}$, $$\label{jktbzh} \mathbb E_{x,b} [M_n;T >n] \leq \left( 1+\frac{c}{n^\varepsilon}\right)b.$$ Indeed, since $(M_n, \mathcal F_n)_{n \geq1}$ is a $\mathbb P_{x,b}$- martingale, we obtain $$\mathbb E_{x,b} [M_n;T \leq n] = \mathbb E_{x,b} [M_{T};T \leq n]$$ and thus $$\begin{aligned} \label{jkjejkva} \mathbb E_{x,b} [M_n;T>n] &=& \mathbb E_{x,b} [M_n] -\mathbb E_{x,a} [M_n;T \leq n]\notag \\ &=& b - \mathbb E_{x,b} [M_{T};T \leq n] \notag\\ &=& b + \mathbb E_{x,b} [|M_{T}|;T \leq n] .\end{aligned}$$ Hence (\[jktbzh\]) arrives by using Lemma \[lem5.2\]. For (\[eqnclaim\]), it is obvious that $$\label{5.19} \mathbb E_{x,a} \Bigl[M_n; T >n , \nu_{n, \varepsilon} \leq n^{1-\varepsilon}\Bigr] = \sum_{k=1}^{[n^{1-\varepsilon}]} \mathbb E_{x,a} \Bigl[M_n; T >n, \nu_{n, \varepsilon} =k\Bigr].$$ Denote $U_m(x, a) := \mathbb E_{x,a} [M_m; T >m]$. For any $m \geq1$, by the Markov property applied to $(X_n)_{n \geq1}$, it follows that $$\begin{aligned} \label{hhgfjkegr} \mathbb E_{x,a} \Bigl[M_n; T >n, \nu_{n, \varepsilon} =k\Bigr] &=& \int \mathbb E_{y,b} [M_{n-k}; T>n-k] \notag\\ && \qquad \qquad \mathbb P_{x,a} (X_k \in dy, M_k \in db; T >k, \nu_{n, \varepsilon }=k)\notag \\ &=& \mathbb E_{x,a} \Bigl[U_{n-k} (X_k, M_k); T >k,\nu_{n, \varepsilon } =k \Bigr].\end{aligned}$$ From the definition of $\nu_{n, \varepsilon}$, we can see that $[\nu_{n, \varepsilon} =k] \subset \left[ |M_k|\geq n^{\frac{1}{2} - \varepsilon} \right]$, and by using (\[jktbzh\]), on the event $[T >k, \nu_{n, \varepsilon} =k]$ we have $U_{n-k} (X_k, M_k) \leq \left( 1 + \frac{c}{(n-k) ^\varepsilon} \right) M_k$. Therefore (\[hhgfjkegr\]) becomes $$\begin{aligned} \label{5.22} \mathbb E_{x,a} [M_n; T >n, \nu_{n, \varepsilon}=k] \leq \left( 1+ \frac{c}{(n-k)^\varepsilon} \right) \mathbb E_{x,a} [M_k; T >k, \nu_{n, \varepsilon}=k].\end{aligned}$$ Combining (\[5.19\]) and (\[5.22\]), it follows that, for $n$ sufficiently great, $$\begin{aligned} \mathbb E_{x,a} [M_n; T >n , \nu_{n, \varepsilon} \leq n^{1-\varepsilon}] &\leq & \sum_{k=1}^{[n^{1-\varepsilon}]} \left( 1+ \frac{c}{(n-k)^\varepsilon} \right) \mathbb E_{x,a} [M_k; T >k, \nu_{n, \varepsilon} =k]\\ &\le& \left( 1+ \frac{c'_\varepsilon}{n^\varepsilon} \right) \sum_{k=1}^{[n^{1-\varepsilon}]} \mathbb E_{x,a} [M_k; T >k, \nu_{n, \varepsilon} =k],\end{aligned}$$ for some constant $c'_\varepsilon >0$. Since $(M_n{\bf 1}_{[T >n]})_{n\geq1}$ is a submartingale, for any $x \in \mathbb X$, $a \ge 0$ and $1 \leq k \leq [n^{1-\varepsilon}]$, $$\mathbb E_{x,a} [M_k; T >k, \nu_{n, \varepsilon} =k] \leq \mathbb E_{x,a} \Bigl[M_{[n^{1-\varepsilon}]}; T >[n^{1-\varepsilon}], \nu_{n, \varepsilon} =k\Bigr].$$ This implies $$\begin{aligned} \mathbb E_{x,a} [M_n; T >n , \nu_{n, \varepsilon} \leq n^{1-\varepsilon}] &\leq & \left( 1+ \frac{c'_\varepsilon}{n^\varepsilon} \right) \sum_{k=1}^{[n^{1-\varepsilon}]} \mathbb E_{x,a} \left[M_{[n^{1-\varepsilon}]}; T >[n^{1-\varepsilon}], \nu_{n, \varepsilon} =k\right]\\ &\leq & \left( 1+ \frac{c'_\varepsilon}{n^\varepsilon} \right) \mathbb E_{x,a} \left[M_{[n^{1-\varepsilon}]}; T >[n^{1-\varepsilon}]\right].\end{aligned}$$ [**(2)**]{} Let $\delta >0$. From (\[5.25\]) and (\[5.26\]), by choosing $n_0$ sufficiently great, we obtain $A_m \leq 1+\delta$ and $B_m \leq \delta$. Together with (\[5.24\]), since $(M_n {\bf 1}_{[T >n]})_{n \geq1}$ is a submartingale, we obtain for $k_m \le n_0$, $$\mathbb E_{x,a} [M_n; T >n] \leq (1+\delta) \Bigl( \mathbb E_{x,a} [ M_{n_0}; T > n_0] +c(1+a) \delta \Bigr).$$ Moreover, the sequence $\mathbb E_{x,a} [M_n; T >n]$ is increasing, thus it converges $\mathbb P_{x,a}$-a.s. and $$\lim_{n \to +\infty} \mathbb E_{x,a} [M_n; T >n] \leq (1+\delta) \Bigl( \mathbb E_{x,a} [M_{n_0}; T >n_0] +c(1+a) \delta \Bigr).$$ By using (\[jkjejkva\]), we obtain $$a \leq \lim_{n \to +\infty} \mathbb E_{x,a} [M_n; T >n] \leq (1+ \delta) \Bigl( a+ \mathbb E_x |M_{n_0}| + c(1+a) \delta \Bigr).$$ Hence the assertion follows since $\delta >0$ is arbitrary. On the stopping time $\tau$. ---------------------------- We now state some useful properties of $\tau$ and $S_\tau$. \[lem5.7\] There exists $c>0$ such that for any $x \in \mathbb X$, $a \ge 0$ and $n \geq1$, $$\mathbb E _{x,a} [ S_n , \tau >n ] \leq c(1+a).$$ [**Proof.**]{} (\[lem4.3\]) yields $\mathbb P _{x} (\tau_a \le T_{a+A}) =1$ and $ A+M_n \geq S_n >0$ on the event $[\tau >n]$. By (\[5.15\]), it follows that $$\begin{aligned} \mathbb E_{x,a} [S_n; \tau >n] &\leq& \mathbb E_{x,a} [ A+M_n ; \tau >n] \notag \\ &\leq& \mathbb E_{x,a+A} [ M_n; T >n] \\ &\leq& c_1 (1+ a+ A) \leq c_2(1+a).\end{aligned}$$ \[lem5.9\] There exists $c >0$ such that for any $x \in \mathbb X$ and $a \ge 0$, $$\begin{aligned} \mathbb E_{x,a} |S_\tau| \le c(1+a) < +\infty, \end{aligned}$$ $$\begin{aligned} \label{lem3.72} \mathbb E_{x,a} |M_\tau| \le c(1+a) < +\infty. \end{aligned}$$ [**Proof.**]{} By (\[lem4.3\]), since $(M_n)_n$ is a martingale, we can see that $$\begin{aligned} -\mathbb E_{x,a} [S_{\tau} ; \tau \leq n] &\le& -\mathbb E_{x,a} [M_{\tau} ; \tau \leq n] +A \\ &=& \mathbb E_{x,a} [M_n ; \tau > n] -\mathbb E_{x,a} [M_n] +A \\ &\le& \mathbb E_{x,a} [S_n ; \tau > n] + 2A.\end{aligned}$$ Hence by Lemma \[lem5.7\], for any $x \in \mathbb X$ and $a \ge 0$, $$\begin{aligned} \label{5.31} \mathbb E_{x,a} \left[ |S_{\tau }| ;\tau \le n \right] &\leq& \mathbb E_{x,a} \left| S_{\tau \wedge n} \right| \notag \\ &=& \mathbb E_{x,a} \left[ S_{n } ;\tau > n \right] - \mathbb E_{x,a} \left[ S_{\tau } ;\tau \leq n \right] \notag \\ &\le& 2 \mathbb E_{x,a} [S_n ; \tau > n] + 2A \notag \\ &\le& c(1+a) +2A.\end{aligned}$$ By Lebesgue’s Dominated Convergence Theorem, it yields $$\mathbb E_{x,a} |S_{\tau }| = \lim_{n \to +\infty} \mathbb E_{x,a} \left[ |S_{\tau}| ;\tau \le n \right] \leq c(1+a) +2A < +\infty.$$ By (\[lem4.3\]), the second assertion arrives. Proof of Proposition \[theo2\] ------------------------------ Denote $\tau_a := \min \{n \ge 1: S_n \le -a \}$ and $T_a := \min \{n \ge 1: M_n \le -a \}$ for any $a \ge 0$. Then $\mathbb E _{x,a} M _{\tau} = a+ \mathbb E _{x} M _{\tau_a} $ and $\mathbb P_{x,a} (\tau >n) = \mathbb P_{x} (\tau_a > n) $. [**(1)**]{} By (\[lem3.72\]) and Lebesgue’s Dominated Convergence Theorem, for any $x \in \mathbb X$ and $a \ge 0$, $$\lim_{n \to +\infty} \mathbb E_{x,a} [ M_{\tau}; \tau \leq n ] = \mathbb E_{x,a} M_{\tau} =a - V(x,a),$$ where $V(x,a)$ is the quantity defined by: for $x \in \mathbb X$ and $a \in \mathbb R$, $$\begin{aligned} V(x,a) := \left\{ \begin{array}{l} - \mathbb E _x M_{\tau _a} \quad \mbox{if} \quad a \ge 0,\\ 0 \qquad \qquad \,\, \mbox{if} \quad a < 0. \end{array} \right.\end{aligned}$$ Since $(M_n, \mathcal F_n)_{n \geq1}$ is a $\mathbb P_{x,a}$-martingale, $$\begin{aligned} \label{eqn22} \mathbb E_{x,a} [M_n; \tau >n] &=& \mathbb E_{x,a} M_n - \mathbb E_{x,a} [M_n; \tau \leq n] = a - \mathbb E_{x,a} [M_\tau; \tau \leq n],\end{aligned}$$ which implies $$\lim_{n \to +\infty} \mathbb E_{x,a} [ M_n; \tau > n ] = V(x, a).$$ Since $\left| S_n - M_n \right| \leq A$ $\mathbb P _x$-a.s. and $\displaystyle \lim_{n \to +\infty} \mathbb P_{x,a} (\tau >n) =0$, it follows that $$\lim_{n \to +\infty} \mathbb E_{x,a} [S_n; \tau >n] = \lim_{n \to +\infty} \mathbb E_{x,a} [M_n; \tau >n] = V(x, a).$$ [**(2)**]{} The assertion arrives by taking into account that $0 \le a \leq a'$ implies $\tau_a \leq \tau_{a'}$ and $$\mathbb E_x [a+S_n; \tau_a >n] \leq \mathbb E_x [a' +S_n; \tau_{a'} >n].$$ [**(3)**]{} Lemma \[lem5.7\] and assertion 1 imply that $V(x, a) \leq c(1+a)$ for any $x \in \mathbb X$ and $a \ge 0$. Besides, (\[eqn22\]) and (\[lem4.3\]) yield $$\mathbb E_{x,a} [M_n ; \tau >n] \geq a - \mathbb E_{x,a} [ S_\tau ; \tau \leq n] - A \geq a -A,$$ which implies $$\begin{aligned} \label{eqn231} V(x,a) \ge a-A.\end{aligned}$$ Now we prove $V(x,a) \ge 0$. Assertion 2 implies $V(x,0) \le V(x,a)$ for any $x \in \mathbb X$ and $a \ge 0$. From P5, let $E_\delta := \{g \in S: \forall x \in \mathbb X, \log|gx| \ge \delta \}$ and choose a positive constant $k$ such that $k\delta > 2A$. Hence, for any $g_1, \ldots, g_k \in E_\delta$ and any $x \in \mathbb X$, we obtain $\log |g_k \ldots g_1 x| \ge k \delta > 2A$. It yields $$\begin{aligned} V(x,0) &=& \lim_{n \to +\infty} \mathbb E_{x} [S_n ; \tau >n] \\ &\ge & \liminf_{n \to +\infty} \int_{E_\delta} \ldots \int_{E_\delta} \mathbb E_{g_k \ldots g_1 \cdot x, \log |g_k \ldots g_1x|} [S_{n-k} ; \tau >n-k] \mu(dg_1) \ldots \mu(dg_k) \\ &\ge & \liminf_{n \to +\infty} \int_{E_\delta} \ldots \int_{E_\delta} V(g_k \ldots g_1 \cdot x,2A) \mu(dg_1) \ldots \mu(dg_k) \\ &\ge& A \Bigl( \mu(E_\delta) \Bigr)^k >0,\end{aligned}$$ where the last inequality comes from (\[eqn231\]) by applying to $a = 2A$. [**(4)**]{} Equation (\[eqn231\]) yields $\displaystyle \lim_{a \to +\infty} \frac{V(x, a)}{a} \geq1$. By (\[lem4.3\]), it yields that $ \mathbb P_{x} (\tau_a < T_{A+a}) =1$, which implies $$\mathbb E_{x,a} [S_n; \tau >n] \leq \mathbb E_{x,a} [A+ M_n ; \tau >n] \leq \mathbb E_{x,a} [ A +M_n; T_{ A} >n] = \mathbb E_{x,a+A} [M_n; T >n].$$ From (\[lem5.10\]), we obtain $\displaystyle\lim_{n \to +\infty} \frac{V(x, a)}{a} \leq 1$. [ **(5)**]{} For any $x \in \mathbb X$, $a \ge 0$ and $n \geq 1$, we set $V_n (x, a) := \mathbb E_{x,a} [S_n; \tau >n]$. By assertion 1, we can see $\displaystyle \lim_{n \to +\infty} V_n(x,a) = V(x,a)$. By Markov property, we obtain $$\begin{aligned} V_{n+1}(x, a) &=& \mathbb E_{x,a} \left[ \mathbb E \left[ S_1 + \sum_{k=1}^{n} \rho(g_{k+1} ,X_k) ; S_1 >0, \ldots , S_{n+1} >0 |\mathcal F_1 \right] \right] \\ &=& \mathbb E_{x,a} \left[ V_n(X_1,S_1) ;\tau >1 \right]. \end{aligned}$$ By Lemma \[lem5.7\], we obtain $\displaystyle \sup_{x \in \mathbb X,a \ge 0} V_n(x, a) \leq c(1+a)$ which implies $\mathbb P$-a.s. $$V_n(X_1,S_1) {\bf 1}_{[\tau >1]} \leq c(1+S_1){\bf 1}_{[\tau >1]}.$$ Lebesgue’s Dominated Convergence Theorem and (\[eqn7\]) yield $$\begin{aligned} \label{eqn23} V(x, a) = \lim_{n \to +\infty} V_{n+1} (x,a) &=& \lim_{n \to +\infty} \mathbb E_{x,a} [ V_n (X_1, S_1); \tau >1] \notag \\ &=& \mathbb E_{x,a} [ V (X_1, S_1); \tau >1] \notag \\ &=& \widetilde P _+ V(x, a). \end{aligned}$$ Coupling argument and proof of theorems \[theo3\] and \[theo4\] =============================================================== First, we apply the weak invariance principle stated in [@GLP1] and verify that the sequence $(\rho(g_k, X_{k-1}))_{k \ge 0}$ satisfies the conditions of Theorem 2.1 [@GLP1]. The hypotheses C1, C2 and C3 of this theorem are given in terms of Fourier transform of the partial sums of $S_n$. Combining the expressions (\[eqn201\]), (\[eqn8\]), (\[eqn1.131\]) and the properties of the Fourier operators $(P_t)_t$, we verify in the next section that the conditions C1, C2 and C3 of Theorem 2.1 in [@GLP1] are satisfied in our context. This lead to the following simpler statement but sufficient. Assume P1-P4. There exist - $\varepsilon_0 >0$, and $c_0 >0$, - a probability space $(\widetilde \Omega, B(\widetilde \Omega))$, - a family $(\widetilde {\mathbb P} _x)_{x \in \mathbb X}$ of probability measures on $(\widetilde \Omega, B(\widetilde \Omega))$, - a sequence $(\tilde a _k)_k$ of real-valued random variables on $(\widetilde \Omega, B(\widetilde \Omega))$ such that $\mathcal L \Bigl((\tilde a _k)_k \slash \widetilde {\mathbb P} _x\Bigr) = \mathcal L \Bigl(( a _k)_k \slash {\mathbb P} _x\Bigr) $ for any $x \in \mathbb X$, - and a sequence $(\widetilde W _i)_{i \ge 1}$ of independent standard normal random variables on $(\widetilde \Omega, \mathcal B(\widetilde \Omega)) $ such that for any $x \in \mathbb X$, $$\label{eqn212} \widetilde{\mathbb{P} _x}\left(\sup_{1 \le k\leq n}\left\vert \sum_{i=1}^{k}(\tilde{a}_i -\sigma \widetilde W_i) \right\vert > n^{{1\over 2}-\varepsilon_0}\right) \leq c_0 n^{- \varepsilon_0}.$$ Notice that the fact (\[eqn212\]) holds true for $\varepsilon_0$ implies (\[eqn212\]) holds true for $\varepsilon $, whenever $\varepsilon \le \varepsilon _0$. In order to simplify the notations, we identify $(\widetilde \Omega, \mathcal B( \tilde \Omega))$ and $( \Omega, \mathcal B( \Omega))$ and consider that the process $(\log |L_n x|)_{n \ge 0}$ satisfies the following property: there exists $\varepsilon_0 >0$ and $c_0 >0$ such that for any $\varepsilon \in (0, \varepsilon_0]$ and $x \in \mathbb X$, $$\begin{aligned} \label{eqn6.5} \mathbb P \left( \sup_{0 \le t \le 1} | \log |L_{[nt]}x| - \sigma B_{nt}| > n^{\frac{1}{2} -\varepsilon}\right) = \mathbb P_x \left( \sup_{0 \le t \le 1} | S_{[nt]} - \sigma B_{nt}| > n^{\frac{1}{2} -\varepsilon}\right) \le c_0 n^{-\varepsilon},\end{aligned}$$ where $(B_t)_{t \ge 0}$ is a standard Brownian motion on the probability space $(\Omega, \mathcal B(\Omega), \mathbb P)$ and $\sigma >0$ is used in the proof of Proposition \[spectre\], part c). For any $a \ge 0$, let $\tau_a^{bm}$ be the first time the process $(a+ \sigma B_t)_{t\ge 0}$ becomes non-positive: $$\tau_a^{bm} = \inf \{t \ge 0: a+\sigma B_t \le 0 \}.$$ The following lemma is due to Levy [@Levy] (Theorem 42.I, pp. 194-195). \[lem6.1\] 1. For any $a\ge 0$ and $n \ge 1$, $$\begin{aligned} \mathbb P (\tau_a^{bm} >n) = \mathbb P \left( \sigma \inf_{0 \le u \le n} B_u > -a \right) = \frac{2}{\sigma \sqrt {2 \pi n}} \int_0^a \exp \left( - \frac{s^2}{2n \sigma^2}\right) ds. \end{aligned}$$ 2. For any $a, b$ such that $0 \le a < b< +\infty$ and $n \ge 1$, $$\begin{aligned} && \mathbb P(\tau_a^{bm} >n, a + \sigma B_n \in [a,b]) \notag \\ && \qquad = \frac{1}{\sigma \sqrt {2 \pi n}} \int_a^b \left[ \exp \left( - \frac{(s-a)^2}{2n \sigma^2}\right) -\exp \left( - \frac{(s+a)^2}{2n \sigma^2}\right) \right] ds. \end{aligned}$$ From lemma \[lem6.1\], we can obtain the next result. \[lem6.2\] 1. There exists a positive constant $c$ such that for any $a\ge 0$ and $n \ge 1$, $$\begin{aligned} \label{eqn6.3} \mathbb P (\tau_a^{bm} >n) \le c\frac{a}{\sigma \sqrt n}. \end{aligned}$$ 2. For any sequence of real numbers $(\alpha_n)_n$ such that $\alpha_n \to 0$ as $n\to +\infty$, there exists a positive constant $c$ such that for any $a \in [0 , \alpha_n \sqrt n]$, $$\begin{aligned} \label{eqn6.4} \Big| \mathbb P(\tau_a^{bm} >n) - \frac{2a}{\sigma \sqrt{2 \pi n}} \Big| \le c\frac{\alpha_n}{\sqrt n} a. \end{aligned}$$ We use the coupling result described in Theorem 4.1 above to transfer the properties of the exit time $\tau_a^{bm}$ to the exit time $\tau_a$ for great $a$. Proof of Theorem \[theo3\] -------------------------- Let $\varepsilon \in (0, \min\{\varepsilon_0; {1 \over 2}\}) $ and $(\theta_n)_{n \ge 1}$ be a sequence of positive numbers such that $\theta_n \to 0$ and $\theta_n n^{\varepsilon /4} \to +\infty$ as $n \to +\infty$. For any $x \in \mathbb X$ and $ a \ge 0$, we have the decomposition $$\begin{aligned} \label{eqn6.13} P_n (x,a) := \mathbb P _{x,a} (\tau >n) &=& \mathbb P _{x,a} (\tau >n, \nu_{n,\varepsilon} > n^{1 - \varepsilon}) +\mathbb P _{x,a} (\tau >n, \nu_{n,\varepsilon} \le n^{1 - \varepsilon}) .\end{aligned}$$ It is obvious that from lemma \[lem5.4\], we obtain $$\begin{aligned} \label{eqn6.14} \sup_{x \in \mathbb X, a \ge 0} \mathbb P _{x,a} (\tau >n, \nu_{n,\varepsilon} > n^{1 - \varepsilon} ) \le \sup_{x \in \mathbb X, a \ge 0} \mathbb P _{x,a} ( \nu_{n,\varepsilon} > n^{1 - \varepsilon} ) \le e^{-c_\varepsilon n^\varepsilon}.\end{aligned}$$ For the second term, by Markov’s property, $$\begin{aligned} \label{eqn6.15} \mathbb P _{x,a} (\tau >n, \nu_{n,\varepsilon} \le n^{1 - \varepsilon}) &= & \mathbb E_{x,a} \left[ P _{n - \nu _n} (X_{\nu_{n,\varepsilon}}, S_{\nu_{n,\varepsilon}}); \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] \label{4.67} \\ & =& I_n(x,a) + J_n(x,a), \notag \end{aligned}$$ where $$\begin{aligned} I_n(x,a) := \mathbb E_{x,a} \left[ P _{n - \nu _n} (X_{\nu_{n,\varepsilon}}, S_{\nu_{n,\varepsilon}}); S_{\nu_{n,\varepsilon}} \le \theta_n n^{\frac{1}{2}} , \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right], \\ \mbox{and}\, \,J_n(x,a) := \mathbb E_{x,a} \left[ P _{n - \nu _n} (X_{\nu_{n,\varepsilon}}, S_{\nu_{n,\varepsilon}}); S_{\nu_{n,\varepsilon}} > \theta_n n^{\frac{1}{2}} , \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right]. \end{aligned}$$ Now we control the quantity $P _{n - \nu _n} (X_{\nu_{n,\varepsilon}}, S_{\nu_{n,\varepsilon}})$ by using the following lemma. The proofs of the lemmas stated in this subsection are postponed to the next subsection. \[lem6.4\] 1. There exists $c >0$ such that for any $n$ sufficiently great, $x \in \mathbb X$ and $a \in [n^{\frac{1}{2} -\varepsilon}, \theta_n n^{\frac{1}{2}}] $, $$\begin{aligned} \label{eqn6.41} \left| \mathbb P_{x,a} (\tau >n)- \frac{2a}{ \sigma \sqrt{2 \pi n}} \right| \le c \frac{a\theta_n }{\sqrt{n}}. \end{aligned}$$ 2. There exists $c >0$ such that for any $x \in \mathbb X$, $a \ge n^{\frac{1}{2} -\varepsilon}$ and $n \ge 1$, $$\begin{aligned} \label{eqn6.42} \mathbb P _{x,a} (\tau >n) \le c \frac{a}{\sqrt n}. \end{aligned}$$ Notice that for any $x \in \mathbb X, a \ge 0$ and $0 \le k \le n^{1-\varepsilon}$, $$\begin{aligned} \label{eqn6.17} P_n(x,a) \le P_{n-k}(x,a) \le P_{n -[n^{1-\varepsilon}]} (x,a).\end{aligned}$$ By definition of $\nu_{n,\varepsilon}$ and (\[lem4.3\]), as long as $A \le n^{\frac{1}{2}-\varepsilon}$, we have $\mathbb P _{x,a}$-a.s. $$\begin{aligned} \label{eqn6.16} S_{\nu_{n,\varepsilon}} \ge M_{\nu_{n,\varepsilon}}- A \ge 2 n^{\frac{1}{2}- \varepsilon} -A \ge n^{\frac{1}{2}- \varepsilon} .\end{aligned}$$ Using (\[eqn6.41\]) and (\[eqn6.17\]), (\[eqn6.16\]) with $\theta_n$ replaced by $\theta_n \left(\frac{n}{n - n^{1-\varepsilon}} \right) ^{\frac{1}{2}}$, for $n$ sufficiently great, on the event $ \left[ S_{\nu_{n,\varepsilon}} \le \theta_n n^{\frac{1}{2}}, \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1 - \varepsilon}\right]$, we obtain $\mathbb P_{x,a}$-a.s. $$P_{n - \nu_{n,\varepsilon}} (X_{\nu_{n,\varepsilon}}, S_{\nu_{n,\varepsilon}}) = \frac{2(1+ o(1)) S_{\nu_{n,\varepsilon}}}{\sigma \sqrt {2 \pi n}} .$$ Let $$\begin{aligned} I_n' (x,a) &:=& \mathbb E_{x,a} \left[ S_{\nu_{n,\varepsilon}}; \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] \label{I'}, \\ J_n' (x,a) &:=& \mathbb E_{x,a} \left[ S_{\nu_{n,\varepsilon}}; S_{\nu_{n,\varepsilon}} > \theta_n n^{\frac{1}{2}}, \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right]. \label{J'} \end{aligned}$$ Hence $$\begin{aligned} I_n(x,a) &=& \frac{2(1+o(1))}{\sigma \sqrt{2\pi n}} \mathbb E_{x,a} \left[ S_{\nu_{n,\varepsilon}}; S_{\nu_{n,\varepsilon}} \le \theta_n n^{\frac{1}{2}}, \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] \\ &=& \frac{2(1+o(1))}{\sigma \sqrt{2\pi n}} \left[ I_n' (x,a) -J_n'(x,a) \right],\\ J_n(x,a) &=& \frac{c(1+ o(1))}{\sqrt n} J_n'(x,a).\end{aligned}$$ Therefore (\[eqn6.13\]) becomes $$\Big| \mathbb P_{x,a} (\tau >n) - \frac{2(1+o(1))}{\sigma \sqrt{2\pi n}} I_n' (x,a) \Big| \le C \left( n^{-\frac{1}{2}} J_n'(x,a) \right) + C' \left( e^{-c_\varepsilon n^\varepsilon} \right).$$ The first assertion of Theorem \[theo3\] immediately follows by noticing that the term $J_n'$ is negligible and $\mathbb P_{x,a} (\tau >n)$ is dominated by the term $I_n'$ as shown in the lemma below. \[lem6.5\] $$\lim_{n \to +\infty} I_n'(x,a) = V(x,a) \quad \mbox{and} \quad \lim_{n \to +\infty} n^{2 \varepsilon} J_n' =0,$$ where $I_n'$ and $J_n'$ are defined in (\[I’\]) and (\[J’\]). [**(2)**]{} By using Proposition \[theo2\] (3), it suffices to prove $ \sqrt n \mathbb P_{x,a} (\tau >n) \le c(1+a)$ for $n$ great enough. For $n$ sufficiently great, using (\[eqn6.42\]) and (\[eqn6.16\]), we obtain $\mathbb P_{x,a}$- a.s. $$P_{n -[n^{1-\varepsilon}]} (X_{\nu_{n,\varepsilon}}, S_{\nu_{n,\varepsilon}}) \le c \frac{ S_{\nu_{n,\varepsilon}}}{\sqrt n}.$$ Combined with (\[4.67\]), it yields $$\begin{aligned} \label{eqn6.43} \mathbb P_{x,a} (\tau >n, \nu_{n,\varepsilon} \le n^{1-\varepsilon}) \le \frac{c}{\sqrt n} I_n'.\end{aligned}$$ Since $\tau_a < T_{a+A}$ $\mathbb P$-a.s. and (\[5.15\]), it follows that $$I'_n(x,a) \le \mathbb E_{x,a+A} [M_{\nu_{n,\varepsilon}}; T > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} ] \le c(1+a+A).$$ Hence (\[eqn6.43\]) becomes $$\begin{aligned} \label{eqn6.44} \mathbb P_{x,a} (\tau >n, \nu_{n,\varepsilon} \le n^{1-\varepsilon}) \le \frac{c}{\sqrt n} (1+a+A).\end{aligned}$$ Combining (\[eqn6.13\]), (\[eqn6.14\]) and (\[eqn6.44\]), we obtain for $n$ great enough, $$\mathbb P_{x,a} (\tau >n) \le e^{-c_\varepsilon n^\varepsilon} + \frac{c}{\sqrt n} (1+a+A) \le c'(1+a).$$ Proof of Theorem \[theo4\] -------------------------- Let us decompose $\mathbb P_{x,a} (S_n \le t \sqrt n|\tau >n)$ as follows: $$\begin{aligned} \label{eqn7.11} \frac{\mathbb P_{x,a} (S_n \le t \sqrt n,\tau >n)} {\mathbb P_{x,a} (\tau >n)} = D_{n,1} + D_{n,2} + D_{n,3},\end{aligned}$$ where $$\begin{aligned} D_{n,1} &:=& \frac{ \mathbb P_{x,a} (S_n \le t \sqrt n,\tau >n, \nu_{n, \varepsilon} > n^{1-\varepsilon})}{\mathbb P_{x,a} (\tau >n)}, \\ D_{n,2} &:=& \frac{ \mathbb P_{x,a} (S_n \le t \sqrt n,\tau >n, S_n > \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon})}{\mathbb P_{x,a} (\tau >n)}, \\ D_{n,3} &:=& \frac{ \mathbb P_{x,a} (S_n \le t \sqrt n,\tau >n, S_n \le \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon})}{\mathbb P_{x,a} (\tau >n)}.\end{aligned}$$ Lemma \[lem5.4\] and Theorem \[theo3\] imply $$\begin{aligned} \label{eqn7.13} \lim_{n \to +\infty} D_{n,1} =0.\end{aligned}$$ Theorem \[theo3\] and Proposition \[theo2\] (3) imply $$\begin{aligned} D_{n,2} &\le & \frac{ \mathbb P_{x,a} ( \tau >n, S_n > \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon})}{\mathbb P_{x,a} (\tau >n)} \notag \\ &= & \frac{1}{\mathbb P_{x,a} (\tau >n)} \mathbb E_{x,a} \Bigl[ P_{n- \nu_{n,\varepsilon}}(X_{\nu_{n, \varepsilon}}, S_{\nu_{n,\varepsilon}}) ; \tau > \nu_{n, \varepsilon}, S_{\nu_{n, \varepsilon}} > \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon} \Bigr] \notag \\ & \le& c \frac{ \mathbb E_{x,a} \Bigl[ 1 +S_{\nu_{n,\varepsilon}} ; \tau > \nu_{n, \varepsilon}, S_{\nu_{n, \varepsilon}} > \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon} \Bigr] }{ \mathbb P_{x,a} (\tau >n)\sigma \sqrt{n - n^{1 - \varepsilon}} } \notag \\ &\le & c' \frac{\mathbb E_{x,a} \Bigl[ S_{\nu_{n, \varepsilon} } ; \tau > \nu_{n, \varepsilon}, S_{\nu_{n, \varepsilon}} > \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon} \Bigr] + \mathbb P_{x,a}(\tau > \nu_{n, \varepsilon})}{V(x,a)\sqrt{1 - n^{ - \varepsilon}} }.\end{aligned}$$ Since $\mathbb P_{x,a} (\tau <+ \infty) =1$ and $\mathbb P_{x,a} (\nu_{n, \varepsilon} <+ \infty) =0$, Lemma \[lem6.5\] yields $$\begin{aligned} \label{eqn7.12} \lim_{n \to +\infty} D_{n,2} =0.\end{aligned}$$ Now we control $D_{n,3}$. Let $H_m (x,a) := \mathbb P_{x,a} ( S_m \le t \sqrt n, \tau >m)$. We claim the following lemma and postpone its proof at the end of this section. \[lem7.1\] Let $\varepsilon \in (0,\varepsilon_0), t >0$ and $(\theta_n)_{n \ge 1}$ be a sequence such that $\theta_n \to 0$ and $\theta_n n^{\varepsilon/4} \to +\infty$ as $n \to +\infty$. Then for any $x \in \mathbb X$, $n^{1/2 - \varepsilon} \le a \le \theta_n \sqrt n$ and $1 \le k \le n^{1- \varepsilon}$, $$\begin{aligned} \mathbb P_{x,a} \Bigl( S_{n-k} \le t \sqrt n, \tau >n-k \Bigr) = \frac{2a}{\sigma^3 \sqrt{2 \pi n}} \int_0^t u \exp \Bigl(-\frac{u^2}{2 \sigma^2} \Bigr) du (1+o(1)). \end{aligned}$$ It is noticeable that on the event $ [\tau > k, S_k \le \theta_n \sqrt n, \nu_{n, \varepsilon} =k ] $, the random variable $H_{n-k} (X_k,S_k)$ satisfies the hypotheses of Lemma \[lem7.1\]. Hence $$\begin{aligned} && \mathbb P_{x,a} (S_n \le t \sqrt n,\tau >n, S_n \le \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon}) \\ &=& \mathbb E_{x,a} \Bigl[ H_{n- \nu_{n, \varepsilon}} (X_{\nu_{n, \varepsilon}}, S_{\nu_{n, \varepsilon}} ); \tau > \nu_{n, \varepsilon}, S_{\nu_{n, \varepsilon}} \le \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1-\varepsilon} \Bigr] \\ &=& \sum_{k=1}^{[n^{1- \varepsilon}]} \mathbb E_{x,a} \Bigl[ H_{n- k} (X_k, S_k ); \tau > k, S_k \le \theta_n \sqrt n, \nu_{n, \varepsilon} =k \Bigr] \\ &=& \frac{2 (1+o(1))}{\sigma^3 \sqrt{2 \pi n }} \int_0^t u \exp \left( - u^2 \over 2 \sigma^2 \right) du \, \mathbb E_{x,a} \Bigl[ S_{\nu_{n, \varepsilon}} ; \tau > \nu_{n, \varepsilon}, S_{\nu_{n, \varepsilon}} \le \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1- \varepsilon} \Bigr] . \end{aligned}$$ Lemma \[lem6.5\] yield as $n \to +\infty$, $$\mathbb E_{x,a} \Bigl[ S_{\nu_{n, \varepsilon}} ; \tau > \nu_{n, \varepsilon}, S_{\nu_{n, \varepsilon}} \le \theta_n \sqrt n, \nu_{n, \varepsilon} \le n^{1- \varepsilon} \Bigr] = V(x,a) (1+o(1)).$$ Therefore, Theorem \[theo3\] yields $$\begin{aligned} \label{eqn7.14} D_{n,3} &=& \frac{2 V(x,a) (1+o(1))}{ \mathbb P_{x,a} (\tau >n) \sigma^3 \sqrt{2 \pi n}} \int_0^t u \exp \left( - u^2 \over 2 \sigma^2 \right) du \notag \\ &=& \frac{1+o(1)}{\sigma^2} \int_0^t u \exp \left( - u^2 \over 2 \sigma^2 \right) du. \end{aligned}$$ The assertion of the theorem arrives by combining (\[eqn7.11\]), (\[eqn7.13\]), (\[eqn7.12\]) and (\[eqn7.14\]). Proof of Lemma [\[lem6.4\]]{} ----------------------------- [**(1)**]{} Fix $\varepsilon >0$ and let $$A_{n, \varepsilon}:= \left[ \sup_{0 \le t \le 1} | S_{[nt]} - \sigma B_{nt}| \le n^{\frac{1}{2} -2\varepsilon} \right].$$ For any $x \in \mathbb X$, (\[eqn6.5\]) implies $\mathbb P_x (A_{n, \varepsilon}^c) \le c_0 n^{-2\varepsilon}$. Denote $a^ {\pm} := a \pm n^{\frac{1}{2} -2\varepsilon} $ and notice that for $a \in [n^{\frac{1}{2} -\varepsilon}, \theta_n \sqrt n ]$, $$\begin{aligned} \label{eqn6.9} 0 \le a^{\pm} \le 2 \theta_n \sqrt n.\end{aligned}$$ Using (\[eqn6.4\]) and (\[eqn6.9\]), for any $x \in \mathbb X$ and $a \in [n^{\frac{1}{2} -\varepsilon}, \theta_n \sqrt n]$, we obtain $$\begin{aligned} \label{eqn6.10} - \frac{c a^{\pm} \theta_n}{\sqrt n} \pm \frac{2n^{-2 \varepsilon}}{\sigma \sqrt {2 \pi}} \le \mathbb P_x (\tau_{a^\pm}^{bm} >n) - \frac{2a}{\sigma \sqrt{2 \pi n}} \le \frac{c a^{\pm} \theta_n}{\sqrt n} \pm \frac{2n^{-2 \varepsilon}}{\sigma \sqrt {2 \pi}}.\end{aligned}$$ For any $a \ge n^{\frac{1}{2} -\varepsilon}$, we have $ \left[\tau_{a^-}^{bm} >n \right] \cap A_{n, \varepsilon}^c \subset \left[\tau_a >n \right] \cap A_{n, \varepsilon}^c \subset \left[ \tau_{a^+}^{bm} >n \right] \cap A_{n, \varepsilon}^c, $ which yields $$\mathbb P_x (\tau_{a^-}^{bm} >n) - \mathbb P_x (A^c_{n, \varepsilon}) \le \mathbb P_x (\tau_a >n) \le \mathbb P_x (\tau_{a^+}^{bm} >n) + \mathbb P_x (A^c_{n, \varepsilon})$$ for any $x \in \mathbb X$. It follows that $$\begin{aligned} \label{eqn6.8} \left\{ \begin{array}{l} \mathbb P_x (\tau_a >n) - \mathbb P_x (\tau_{a^+}^{bm} >n) \le c_0 n^{-2\varepsilon}, \\ \mathbb P_x (\tau_{a^-}^{bm} >n) - \mathbb P_x (\tau_a >n) \le c_0 n^{-2\varepsilon}. \end{array} \right.\end{aligned}$$ The fact that $\theta_n n^{\varepsilon/4} \to +\infty$ yields for $n$ great enough $$\begin{aligned} \label{eqn6.11} \theta_n \frac{a}{\sqrt n} \ge \frac{n^{\frac{1}{2} -\varepsilon}}{n^\varepsilon \sqrt n} = n^{-2\varepsilon}.\end{aligned}$$ From (\[eqn6.10\]), (\[eqn6.8\]) and (\[eqn6.11\]), it follows that for any $a \in [n^{\frac{1}{2} -\varepsilon}, \theta_n \sqrt n]$, $$\left| \mathbb P_x (\tau_a > n) - \frac{2a}{\sigma \sqrt{2 \pi n}} \right| \le c(1+ \theta_n) n^{-2\varepsilon} + c_1\frac{\theta_n a}{\sqrt n} \le c_2 \frac{\theta_n a}{\sqrt n}.$$ [**(2)**]{} For $n$ great enough, condition $a \ge n^{\frac{1}{2} -\varepsilon}$ implies $a^+ \le 2a$. From (\[eqn6.3\]) and (\[eqn6.8\]) , since $n^{-2\varepsilon} \le \frac{a}{\sqrt n}$, for any $x \in \mathbb X$, $$\mathbb P_x (\tau_a >n) \le c \frac{a}{\sigma \sqrt n} + c_0 n^{-2\varepsilon} \le c_1 \frac{a}{\sqrt n}.$$ Proof of Lemma \[lem6.5\] ------------------------- [**(1)**]{} We prove that $\displaystyle \lim_{n \to +\infty} \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon}}; \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] = V(x,a)$. Then, the assertion arrives by using (\[lem4.3\]) and taking into account that $\mathbb P_x (\tau_a < +\infty) =1$ and $\displaystyle \mathbb P_x (\lim_{n \to +\infty} \nu_{n,\varepsilon} = +\infty) =1$. For $x \in \mathbb X$ and $a \ge 0$, we obtain $$\begin{aligned} \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon}}; \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] &=& \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}]}; \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] \notag \\ &=& \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}]}; \tau > \nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}] \right] \notag \\ & & \,\, - \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}]}; \tau > \nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}], \nu_{n,\varepsilon} > n^{1-\varepsilon} \right] \notag.\end{aligned}$$ By using Lemma \[lem5.5\], $$\begin{aligned} \label{eqm6.25} \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}]}; \tau > \nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}], \nu_{n,\varepsilon} > n^{1-\varepsilon} \right] \le c(1+a) e^{-c_\varepsilon n^\varepsilon}. \end{aligned}$$ Using the facts that $(M_n)_{n \ge 0}$ is a martingale and $\displaystyle \mathbb P_x \left(\lim_{n \to +\infty}\nu_{n,\varepsilon} = + \infty\right) =1$, we obtain $$\begin{aligned} \lim_{n \to +\infty} \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon}}; \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] &=& \lim_{n \to +\infty} \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}]}; \tau > \nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}] \right] \notag \\ &=& a- \lim_{n \to +\infty} \mathbb E_{x,a} \left[ M_{\nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}]}; \tau \le \nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}] \right] \notag \\ &=& a- \lim_{n \to +\infty} \mathbb E_{x,a} \left[ M_\tau; \tau \le \nu_{n,\varepsilon} \wedge [n^{1 -\varepsilon}] \right] \notag \\ &=& a - \mathbb E _{x,a} [M_\tau]= V(x,a).\end{aligned}$$ [**(2)**]{} Let $b = a +A$. Remind that $M^*_n = \displaystyle \max_{1 \leq k \leq n} |M_k|$. We obtain $$\begin{aligned} \mathbb E_{x,a} \left[ S_{\nu_{n,\varepsilon}}; S_{\nu_{n,\varepsilon}} > \theta_n n^{\frac{1}{2}}, \tau > \nu_{n,\varepsilon}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] &\le & \mathbb E_{x,b} \left[ M_{\nu_{n,\varepsilon}}; M_{\nu_{n,\varepsilon}} > \theta_n n^{\frac{1}{2}}, \nu_{n,\varepsilon} \le n^{1-\varepsilon} \right] \\ &\le & \mathbb E_{x,b} \left[ M^*_{[n^{1-\varepsilon}]}; M^*_{[n^{1-\varepsilon}]} > \theta_n n^{\frac{1}{2}} \right] .\end{aligned}$$ Since $\theta_n n^{\varepsilon/4} \to +\infty$ as $n \to +\infty$, it suffices to prove that for any $\delta >0$, $ x \in \mathbb X$ and $b \in \mathbb R$, $$\begin{aligned} \label{eqn6.32} \lim_{n \to +\infty} n^{2 \varepsilon} \mathbb E _x \left[b+M^*_n; M^*_n > n^{\frac{1}{2} +\delta} \right] =0.\end{aligned}$$ Obviously, by (\[eqn5.8\]), $$\begin{aligned} \mathbb E _x \left[b+M^*_n; M^*_n > n^{\frac{1}{2} +\delta} \right] &\le& b \mathbb P _x \left( M^*_n > n^{\frac{1}{2} +\delta} \right) +\mathbb E_x \left[M^*_n; M^*_n > n^{\frac{1}{2} +\delta} \right] \\ &=& \left( b+ n^{\frac{1}{2} +\delta} \right) \mathbb P _x \left( M^*_n > n^{\frac{1}{2} +\delta} \right) + \int^{+\infty}_{n^{\frac{1}{2} +\delta}} \mathbb P_x (M^*_n >t) dt\\ &\le& c\left( b+n^{\frac{1}{2}+\delta}\right) n^{-p\delta} + c n ^{-p\delta + \frac{1}{2} +\delta}.\end{aligned}$$ Since $p$ can be taken arbitrarily great, it follows that $\displaystyle \lim_{n \to +\infty} n^{2 \varepsilon} J_n' =0$. Proof of lemma \[lem7.1\] ------------------------- Recall that $a^\pm = a \pm n^{1/2 - 2 \varepsilon}$ and denote $t^\pm = t \pm 2 n^{-2 \varepsilon}$. For any $1 \le k \le n^{1- \varepsilon}$, $$\{\tau_{a^-}^{bm} \} \cap A_{n, \varepsilon} \subset \{ \tau_a > n-k\} \cap A_{n, \varepsilon} \subset \{\tau_{a^+}^{bm} \} \cap A_{n, \varepsilon}$$ and $$\{a^-+\sigma B_{n-k} \le t^-\sqrt n \} \cap A_{n, \varepsilon} \subset \{a+S_{n-k} \le t\sqrt n \} \cap A_{n, \varepsilon} \subset \{a^+ +\sigma B_{n-k} \le t^+\sqrt n \} \cap A_{n, \varepsilon},$$ which imply $$\begin{aligned} \label{eqn7.2} && \mathbb P_x (\tau_{a^-}^{bm} >n-k, a^- +\sigma B_{n-k} \le t^-\sqrt n) - \mathbb P_x (A^c_{n, \varepsilon}) \notag \\ &&\qquad \qquad \qquad \le \mathbb P_x (\tau_a >n-k, a+ S_{n-k} \le t \sqrt n) \le \\ && \qquad \qquad \qquad \qquad \qquad \mathbb P_x (\tau_{a^+}^{bm} >n-k, a^+ +\sigma B_{n-k} \le t^+\sqrt n) +\mathbb P_x (A^c_{n, \varepsilon}).\notag \end{aligned}$$ Moreover, by Lemma \[lem6.1\], we obtain $$\begin{aligned} \label{eqn7.8} \mathbb P_x \Bigl(\tau_{a^+}^{bm} >n-k, a^+ +\sigma B_{n-k} \le t^+\sqrt n \Bigr) = \frac{2a}{\sigma^3 \sqrt{2 \pi n}} \int_0^t u \exp \Bigl(-\frac{u^2}{2 \sigma^2} \Bigr) du (1+o(1))\end{aligned}$$ and similarly, $$\begin{aligned} \label{eqn7.9} \mathbb P_x \Bigl(\tau_{a^-}^{bm} >n-k, a^- +\sigma B_{n-k} \le t^-\sqrt n \Bigr) = \frac{2a}{\sigma^3 \sqrt{2 \pi n}} \int_0^t u \exp \Bigl(-\frac{u^2}{2 \sigma^2} \Bigr) du (1+o(1)).\end{aligned}$$ Therefore, from (\[eqn7.2\]), (\[eqn7.8\]), (\[eqn7.9\]) and $\mathbb P_x (A^c_{n, \varepsilon})\le c n^{-2\varepsilon}$, it follows that $$\begin{aligned} \mathbb P_x \Bigl(\tau_a >n-k, a + S_{n-k} \le t \sqrt n \Bigr) = \frac{2a}{\sigma^3 \sqrt{2 \pi n}} \int_0^t u \exp \Bigl(-\frac{u^2}{2 \sigma^2} \Bigr) du (1+o(1)).\end{aligned}$$ On conditions C1-C3 of Theorem 2.1 in [@GLP1] ============================================= Let $k_{gap}, M_1, M_2 \in \mathbb N$ and $j_0 < \ldots < j_{M_1 +M_2}$ be natural numbers. Denote $a_{k + J_m} = \sum_{l \in J_m} a_{k+l}$, where $J_m = [j_{m-1}, j_m), m= 1, \ldots, M_1 +M_2$ and $k \ge 0$. Consider the vectors $\bar {a}_1 = (a_{J_1}, \ldots, a_{J_{M_1}})$ and $\bar {a}_2 = (a_{k_{gap}+J_{M_1 +1}}, \ldots, a_{k_{gap}+J_{M_1 + M_2}})$. Denote by $\phi_x (s,t) = \mathbb E e^{is\bar {a}_1 + it \bar {a}_2 }$, $\phi_{x,1} (s) = \mathbb E e^{is\bar {a}_1}$ and $\phi_{x,2} (s) = \mathbb E e^{it\bar {a}_2}$ the characteristic functions of $(\bar {a}_1, \bar {a}_2 )$, $\bar {a}_1$ and $\bar {a}_2 $, respectively. For the sake of brevity, we denote $\phi_1(s) = \phi_{x,1}(s), \phi_2(t) = \phi_{x,2}(t)$ and $\phi(s,t) = \phi_{x}(s,t)$. We first check that conditions C1-C3 hold and then prove the needed lemmas. Statement and proofs of conditions C1-C3 ---------------------------------------- [**C1**]{}: There exist positive constants $\varepsilon_0 \le 1, \lambda_0, \lambda_1, \lambda_2$ such that for any $k_{gap} \in \R, M_1, M_2 \in \N$, any sequence $j_0 < \ldots < j_{M_1 + M_2}$ and any $s \in \R ^{M_1}, t \in \R ^{M_2}$ satisfying $|(s,t)|_\infty \le \varepsilon_0$, $$|\phi(s,t) - \phi_1(s) \phi_2(t)| \le \lambda_0 \exp (-\lambda_1 k_{gap}) \left( 1+ \max _{m= 1, \ldots, M_1+M_2} card(J_m)\right)^{\lambda_2(M_1+M_2)}.$$ [**C2**]{}: There exists a positive constant $\delta $ such that $\sup_{n \ge 0} | a_n|_{L^{2 + 2 \delta}} < +\infty$.\ [**C3**]{}: There exist a positive constant $C$ and a positive number $\sigma$ such that for any $\gamma >0$, any $x \in \X$ and any $n \ge 1$, $$\sup_{m \ge 0} \left| n^{-1} Var_{\P_x} \left( \sum_{i=m}^{m+n-1} a_i\right) - \sigma^2 \right| \le C n^{-1 + \gamma}.$$ \[C1\] Condition 1 is satisfied under hypotheses P1-P5. [**Proof.**]{} First, we prove the following lemma. \[lem3\] There exist two positive constants $C$ and $\kappa$ such that $0< \kappa <1$ and $$|\phi(s,t) - \phi_1(s) \phi_2(t)| \le C C_P^{M_1 +M_2} \kappa^{k_{gap}},$$ where $C_P$ is defined in Proposition \[spectre\]. [**Proof.**]{} In fact, the characteristic functions of the ramdom variables $\bar{a}_1, \bar{a}_2$ and $(\bar{a}_1,\bar{a}_2)$ can be written in terms of operator respectively as follows: $$\begin{aligned} \phi_1(s) &=& \E _x [e^{is \bar a_1}] = P ^{j_0 -1} P_{s_1}^{|J_1|} \ldots P_{s_{M_1}}^{|J_{M_1}|} {\bf 1} (x) ,\notag \\ \phi_2(t) &=& \E _x [e^{it \bar a_2}] = P ^{k_{gap} +j_{M_1} -1} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} (x), \\ \phi(s,t) &=& \E _x [e^{is \bar a_1+ it \bar a_2}] = P ^{j_0 -1} P_{s_1}^{|J_1|} \ldots P_{s_{M_1}}^{|J_{M_1}|} P ^{k_{gap}} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} (x). \notag\end{aligned}$$ Now we decompose $\phi(s,t)$ into the sum of $\phi_\Pi (s,t)$ and $\phi_R (s,t)$ by using the spectral decomposition $P = \Pi + R$ in Proposition \[spectre\], where $$\begin{aligned} \phi_\Pi (s,t) &=& P ^{j_0 -1} P_{s_1}^{|J_1|} \ldots P_{s_{M_1}}^{|J_{M_1}|} \Pi P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} (x), \notag \\ \phi_R (s,t) &=& P ^{j_0 -1} P_{s_1}^{|J_1|} \ldots P_{s_{M_1}}^{|J_{M_1}|} R ^{k_{gap}} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} (x). \notag \end{aligned}$$ Since $\Pi(\varphi) = \nu (\varphi) {\bf 1}$ for any $\varphi \in L$ and $P_t$ acts on $L$, we obtain $$\phi_\Pi (s,t) = P ^{j_0 -1} P_{s_1}^{|J_1|} \ldots P_{s_{M_1}}^{|J_{M_1}|} {\bf 1} (x) \nu\left(P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1}\right).$$ Then setting $\psi_2(t) = \nu(P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1}) $ yields $$\begin{aligned} \phi(s,t) &= \phi_1(s) \psi_2(t) + \phi_R(s,t) \notag \\ &= \phi_1(s) \phi_2(t) + \phi_1(s) [ \psi_2(t) - \phi_2(t)] + \phi_R(s,t), \notag \end{aligned}$$ which implies $$\begin{aligned} \label{46} |\phi (s,t)- \phi_1(s) \phi_2(t) | \le |\phi_1(s) | |\psi_2(t) - \phi_2(t) | + |\phi_R(s,t) |.\end{aligned}$$ On the one hand, we can see that $|\phi_1(s)| = \left| \left( P ^{j_0 -1} P_{s_1}^{|J_1|} \ldots P_{s_{M_1}}^{|J_{M_1}|} {\bf 1} \right) (x) \right| \le C_P^{1+M_1}$ and $|\phi_R(s,t)| \le C_P^{1+M_1 + M_2} C_R \kappa^{k_{gap}}$. On the other hand, since $\nu $ is $P$-invariant measure and $(\nu - \delta_x) ({\bf 1} ) =0$, by using again the expression $P = \Pi +R$, we obtain $$\begin{aligned} |\psi_2(t) - \phi_2(t) | &=& \left| (\nu - \delta_x) \left( P ^{k_{gap} +j_{M_1} -1} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} \right) \right| \notag \\ &\le& \left| (\nu - \delta_x) \left( \Pi P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} \right) \right| \notag \\ && \qquad \qquad \qquad + \left| (\nu - \delta_x) \left( R ^{k_{gap} +j_{M_1} -1} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} \right)\right| \notag \\ &=& \left| (\nu - \delta_x) ({\bf 1} ) \nu \left( P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} \right) \right| \notag \\ && \qquad \qquad \qquad + \left| (\nu - \delta_x) \left( R ^{k_{gap} +j_{M_1} -1} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} \right)\right| \notag \\ &=& \left| (\nu - \delta_x) \left( R ^{k_{gap} +j_{M_1} -1} P_{t_1}^{|J_{M_1+1}|} \ldots P_{t_{M_2}}^{|J_{M_1 +M_2}|} {\bf 1} \right) \right| \notag \\ &\le& C C_P^{M_2} \kappa ^{k_{gap} +j_{M_1} -1} .\end{aligned}$$ Therefore, (\[46\]) follows. Second, let $\lambda_2 = \max \{1 , \log_2 C_P \}$. Since $\displaystyle \max_{m=1, \ldots, M_1 +M_2} card (J_m) \ge 1$, we obtain $$C_P^{M_1 +M_2} \le 2^{\lambda_2 ( M_1 +M_2)} \le \left( 1+ \max_{m=1, \ldots, M_1 +M_2} card (J_m)\right) ^{\lambda_2 ( M_1 +M_2)},$$ which implies that $$|\phi(s,t) - \phi_1(s) \phi_2(t)| \le C \kappa^{k_{gap}} \left( 1+ \max_{m=1, \ldots, M_1 +M_2} card (J_m)\right) ^{\lambda_2 ( M_1 +M_2)}.$$ Finally, let $\lambda_0 =C$ and $\lambda_1 = - \log \kappa$. Then the assertion arrives. \[C2\] Condition 2 is satisfied under hypotheses P1-P5. [**Proof.**]{} Condition P1 implies that there exists $\delta_0>0$ such that $\mathbb E [N(g)^{\delta_0}] < + \infty $ and since $\displaystyle \mathbb E [N(g)^{\delta_0}] = \mathbb E [\exp (\delta_0 \log N(g))] = \sum_{k=0}^{+\infty} \frac{\delta_0^k}{k!} \mathbb E [(\log N(g))^k] $, we obtain $\mathbb E |a_n|^k \le \mathbb E [(\log N(g))^k] < +\infty $ for any $n \ge 0$ and any $k \ge 0$. \[C3\] Condition 3 is satisfied under hypotheses P1-P5. More precisely, there exists a positive constant $\sigma $ such that for any $x \in \mathbb X$ and any $n \ge 1$, $$\begin{aligned} \label{C3} \sup_{m \ge 0} \left| Var_{\mathbb P_x} \left( \sum_{k=m}^{m+n-1} a_k \right) - n \sigma^2 \right| < +\infty. \end{aligned}$$ [**Proof.**]{} For any integer $m,n \ge 0$, we denote $S_{m,n} = \sum_{k=m}^{m+n-1}a_k$, $V_x (X) = Var_{\mathbb P_x} (X) = \mathbb E_x (X^2)- (\mathbb E_x X)^2$ and $Cov_x(X,Y) = Cov_{\mathbb P_x} (X,Y)$. Then $$\begin{aligned} \label{e1} V_x (S_{m,n}) &=& \sum_{k=m}^{m+n-1} V_x (a_k ) + 2 \sum_{k=m}^{m+n-1} \sum_{l=1}^{m+n-k-1} Cov_x (a_k, a_{k+l})\end{aligned}$$ and (\[C3\]) becomes $\sup_{m \ge 0} |V_x (S_{m,n}) - n \sigma^2 | < +\infty$. We claim two lemmas and postpone their proofs until the end of this section. \[lem1\] There exist $C>0$ and $0 < \kappa <1$ such that for any $x \in \mathbb X$, any $k \ge 0$ and any $l \ge 0$, $$\begin{aligned} \label{e3} \left| Cov_x (a_k, a_{k+l})\right| \le C \kappa^l. \end{aligned}$$ \[lem2\] There exist $C>0$, $0 < \kappa <1$ and a sequence $(s_n)_{n \ge 0}$ of real numbers such that for any $x \in \mathbb X$, any $k \ge 0$ and any $l \ge 0$, $$\begin{aligned} \left| Cov_x (a_k, a_{k+l}) - s_l \right| & \le C \kappa^k, \label{e4} \\ |s_l| \le C \kappa^l. \label{e5} \end{aligned}$$ For the first term of the right side of (\[e1\]), by combining Lemma \[lem1\] and Lemma \[lem2\], we obtain $$\begin{aligned} \label{e6} \left| Cov_x (a_k, a_{k+l}) - s_l \right| \le C \kappa^{\max\{k,l \}} .\end{aligned}$$ Inequality (\[e4\]) implies $ |V_x (a_k) -s_0 | \le C \kappa^k$, which yields for any integer $m,n \ge 0$, $$\begin{aligned} \label{e7} \left|\sum_{k=m}^{m+n-1} V_x(a_k) - n s_0 \right| \le \sum_{k=m}^{m+n-1} |V_x(a_k)- s_0 | \le C \sum_{k=m}^{m+n-1} \kappa^k \le {C \over 1-\kappa} < +\infty.\end{aligned}$$ For the second term of the right side of (\[e1\]), we can see that $$\begin{aligned} \label{e8} & &\left| \sum_{k=m}^{m+n-1} \sum_{l=1}^{m+n-k-1} Cov_x(a_k, a_{k+l}) - \sum_{k=m}^{m+n-1} \sum_{l=1}^{+\infty} s_l \right| \notag \\ &\le& \sum_{k=m}^{m+n-1} \sum_{l=1}^{m+n-k-1} \left| Cov_x(a_k, a_{k+l}) - s_l \right| + \sum_{k=m}^{m+n-1} \sum_{l=m+n-k}^{+\infty} |s_l| \notag\\ &=& \Sigma_1(x, m ,n) + \Sigma_2(x, m ,n).\end{aligned}$$ On the one hand, by (\[e4\]) and (\[e6\]), we can see that for any $x \in \X$, any $m\ge 0$ and any $n \ge 1$, $$\begin{aligned} \label{e9} \Sigma_1(x, m ,n) &\le& \sum_{k=0}^{+\infty} \sum_{l=1}^{k}C \kappa^k + \sum_{k=0}^{+\infty} \sum_{l=k+1}^{+\infty} C \kappa ^l \notag \\ &\le& \sum_{k=0}^{+\infty} C k \kappa^k + \sum_{k=0}^{+\infty} C {\kappa^{k+1} \over 1 - \kappa} < +\infty.\end{aligned}$$ Similarly, on the other hand, by (\[e5\]) we obtain for any $x \in \X$, any $m\ge 0$ and any $n \ge 1$, $$\begin{aligned} \label{e10} \Sigma_2(x, m ,n) \le \sum_{k=0}^{n-1} \sum_{l=n-k}^{+\infty} C \kappa^{l} \le {C \over (1 - \kappa)^2} < +\infty.\end{aligned}$$ Combining (\[e1\]),(\[e7\]),(\[e8\]),(\[e9\]) and (\[e10\]), we obtain $$\begin{aligned} \sup_{m \ge 0} \left|V_x(S_{m,n}) - n \sum_{l=0}^{+\infty} s_l \right| < +\infty.\end{aligned}$$ In fact, by using Lemma 2.1 in [@LPP], Theorem 5 in [@H2] implies that the sequence $(\frac{S_n}{\sqrt n} )_{n \ge 1}$ converges weakly to a normal law with variance $\sigma^2$. Meanwhile, under hypothesis P2, Corollary 3 in [@H2] implies that the sequence $(|R_n|)_{n \ge 1}$ is not tight and thus $\sigma^2 >0$, see [@H2] for the definition and basic properties. Therefore, we can see that $Var_x S_n \sim n \sigma^2$ with $\sigma^2 >0$, which yields $\sum_{l=0}^{+\infty} s_l = \sigma^2$. Proof of Lemma \[lem1\] ----------------------- Let $g(x) = \left\{ \begin{array}{l} x \ \ \mbox{if} \ |x| \le 1,\\ 0\ \ \mbox{if} \ |x| > 2. \end{array} \right.$ such that $g$ is $C^\infty$ on $\R$ and $|g(x)| \le |x|$ for any $x \in \R$. Then $g \in L^1(\R)\cap C^1_{c}(\R)$. Therefore, the Fourier transform of $g$ is $\hat g$ defined as follows: $$\hat g (t) := \int_{\R}e^{-itx} g(x) dx,$$ and the Inverse Fourier Theorem yields $$g(x) = \frac{1}{2 \pi}\int_{\R} e^{itx} \hat g(t) dt.$$ Let $g_T(x) := Tg({x \over T})$ for any $T >0$. Then $|\hat g_T|_1 = T |\hat g|_1 < +\infty$. Let $h_T (x,y) = g_T (x) g_T(y)$. Then $\hat h_T(x,y) = \hat g_T(x) \hat g_T(y)$. Let $V$ and $V'$ be two i.i.d. random variables with mean $0$, independent of $a_l$ for any $l \ge 0$ whose characteristic functions have the support included in the interval $[-\varepsilon_0, \varepsilon_0]$ for $\varepsilon_0 $ defined in C1. Assume that $\E |V|^n < +\infty$ for any $n >0$. Let $Z_k = a_k +V$ and $Z'_{k+l} = a_{k+l} +V'$ and denote by $\widetilde \phi _1(s), \widetilde \phi_2 (t)$ and $\widetilde \phi (s,t)$ the characteristic functions of $Z_k, Z'_{k+l}$ and $(Z_k, Z'_{k+l})$, respectively. We use the same notations introduced at the beginning of this section by setting $\phi_1(s) = \E _x [e^{isa_k}], \phi_2(t) = \E _x [e^{ita_{k+l}}]$ and $\phi(s,t) = \E _x [e^{is a_k + it a_{k+l}}]$. We also denote $\varphi$ the characteristic function of $V$, that yields $$\begin{aligned} \label{1.23} \widetilde \phi_1(s) &=& \E [e^{isZ_k}] = \E [e^{isa_k}] \E [e^{isV}] =\phi_1(s) \varphi(s) ,\notag \\ \widetilde \phi_2(t) &=& \E [e^{itZ'_{k+l}}] = \E [e^{ita_{k+l}}] \E [e^{itV'}] =\phi_2(t) \varphi(t), \\ \widetilde \phi(s,t) &=& \E [e^{isZ_k + itZ'_{k+l} }] = \E [e^{isa_k + ita_{k+l} }] \E [e^{isV}] \E [e^{itV'}] =\phi(s,t) \varphi(s) \varphi(t) .\notag\end{aligned}$$ Then we can see that $\widetilde \phi_1$ and $\widetilde \phi_2$ have the support in $[-\varepsilon_0, \varepsilon_0]$. We perturb $a_k$ and $a_{k+l}$ by adding the random variables $V$ and $V'$ with mean $0$ and the support of their characteristic functions are on $[-\varepsilon_0, \varepsilon_0]$. We explicit the quantity $Cov_x (a_k, a_{k+l})$: $$\begin{aligned} \label{11} Cov_x (a_k, a_{k+l}) = \E _x [a_k, a_{k+l}] - \E _x a_k \E _x a_{k+l}.\end{aligned}$$ On the one hand, we can see that $$\begin{aligned} \label{12} \E _x [a_k a_{k+l}] = \E _x [Z_k Z'_{k+l}] &=& \E _x [h_T(Z_k; Z'_{k+l})] +\E _x [Z_k Z'_{k+l}] - \E _x [h_T(Z_k; Z'_{k+l})] \notag \\ &=&\frac{1}{(2 \pi)^2} \E _x \int \int \hat h _T (s,t) e^{isZ_k + it Z'_{k+l}} ds dt + R_0 \notag \\ &=&\frac{1}{(2 \pi)^2} \int \int \hat h _T (s,t) \E _x \left[ e^{isZ_k + it Z'_{k+l}} \right] ds dt + R_0 \notag \\ &=&\frac{1}{(2 \pi)^2} \int \int \hat h _T (s,t) \widetilde \phi (s,t) ds dt + R_0, \notag \\\end{aligned}$$ where $R_0 = \E _x [Z_k Z'_{k+l}] - \E _x [h_T(Z_k; Z'_{k+l})]$. On the other hand, we obtain $$\begin{aligned} \label{13} \E _x a_k = \E _x Z_k &=& \E _x g_T(Z_k) +\E _x Z_k - \E _x g_T(Z_k) \notag \\ &=& \frac{1}{2 \pi} \int \hat g _T (s) \widetilde \phi_1 (s)ds + R_1, \end{aligned}$$ where $R_1 =\E _x Z_k - \E _x g_T(Z_k) $ and $$\begin{aligned} \label{14} \E _x a_{k+l} = \E _x Z'_{k+l} &=& \E _x g_T(Z'_{k+l}) +\E _x Z'_{k+l} - \E _x g_T(Z'_{k+l}) \notag \\ &=& \frac{1}{2 \pi} \int \hat g _T (t) \widetilde \phi_2 (t)dt + R_2, \end{aligned}$$ where $R_2 = \E _x Z'_{k+l} - \E _x g_T(Z'_{k+l}) $. From (\[11\]), (\[12\]), (\[13\]) and (\[14\]), since $\hat h_T(s,t) = \hat g_T(s) \hat g_T(t)$, we obtain $$\begin{aligned} \label{1.28} Cov_x (a_k, a_{k+l}) &=& \frac{1}{(2 \pi)^2} \int \int \hat h _T (s,t) \widetilde \phi (s,t) ds dt + R_0 \notag \\ & & \qquad -\left(\frac{1}{2 \pi} \int \hat g _T (s) \widetilde \phi_1 (s)ds + R_1 \right) \left( \frac{1}{2 \pi} \int \hat g _T (t) \widetilde \phi_2 (t)dt + R_2 \right) \notag \\ &=& \frac{1}{(2 \pi)^2} \int \int \hat h _T (s,t) \left[ \widetilde \phi (s,t) - \widetilde \phi_1(s) \widetilde \phi_2(t) \right] ds dt +R\end{aligned}$$ where $\displaystyle R=R_0 - R_1 R_2 - R_1 \frac{1}{2 \pi}\int \hat g _T (t) \widetilde \phi_2 (t)dt - R_2 \frac{1}{2 \pi}\int \hat g _T (s) \widetilde \phi_1 (s)ds $. Since $\hat g _T \in L_1(\R)$ and applying Lemma \[lem3\] for $ j_0 =k, j_1 = k+1, j_2 =k+2, k_{gap} =l, M_1=M_2 =1$, we obtain $$\begin{aligned} \label{15} |Cov_x (a_k, a_{k+l}) | &\le &\frac{1}{(2 \pi)^2} \int \int \left| \hat h _T (s,t) \right| \left| \widetilde \phi (s,t) - \widetilde \phi_1(s) \widetilde \phi_2(t) \right| ds dt + \left| R \right| \notag \\ &\le &\frac{1}{(2 \pi)^2} \int \int \left| \hat h _T (s,t) \right| \left| \phi (s,t) \varphi(s) \varphi(t) - \phi_1(s) \phi_2(t) \varphi(s) \varphi(t) \right| ds dt + \left| R \right| \notag \\ &\le & \sup_{ |s| , |t| \le \varepsilon_0} \left| \phi (s,t) - \phi_1(s) \phi_2(t) \right| \left( \int \left| \hat g _T (s) \right| ds \right) ^2 +|R|\notag \\ &\le & C T^2 \kappa ^l +|R|.\end{aligned}$$ It remains to bound of $|R|$. On the one hand, we can see that $$\begin{aligned} &\bullet& \qquad |R_1| = \left| \E _x [Z_k - g_T(Z_k) ] \right| = \E _x \left| [Z_k - g_T(Z_k) ]{\bf 1}_{[|Z_k| >T]} \right| \le 2T^{-1}\E _x |Z_k|^2, \notag \\ &\bullet& \qquad \left| \frac{1}{2 \pi}\int \hat g _T (s) \widetilde \phi_1 (s)ds \right| =\left| \E_x g_T(Z_k) \right| \le \E_x \left| Z_k \right| \le \E_x |a_k| + \E_x |V| \le C, \notag \\ &\bullet& \qquad |R_2| = \left| \E _x [Z'_{k+l} - g_T(Z'_{k+l}) ] \right| \le 2T^{-1}\E _x |Z'_{k+l}|^2, \notag \\ &\bullet& \qquad \left| \frac{1}{2 \pi}\int \hat g _T (t) \widetilde \phi_2 (t)dt \right| = \left| \E_x g_T(Z'_{k+l}) \right| \le \E_x |a_{l+k}| + \E_x |V'| \le C. \notag \end{aligned}$$ On the other hand, similarly for $|R_0|$, we obtain $$\begin{aligned} |R_0| &=& \E_x \left[ \left|Z_k Z'_{k+l} - h_T(Z_k, Z'_{k+l} ) \right| \left( {\bf 1}_{[|Z_k| >T]} + {\bf 1}_{[|Z_k| \le T]} \right) \left( {\bf 1}_{[|Z'_{k+l}| >T]} + {\bf 1}_{[|Z'_{k+l}| \le T]} \right) \right] \notag \\ &\le& \E_x \left[ \left|Z_k Z'_{k+l} - h_T(Z_k, Z'_{k+l} ) \right| \left( {\bf 1}_{[|Z_k| >T]} + {\bf 1}_{[|Z'_{k+l}| >T]} \right) \right] \notag \\ &\le& 2 \E_x \left| Z_k Z'_{k+l} {\bf 1}_{[|Z_k| >T]} \right| + 2 \E_x \left| Z_k Z'_{k+l} {\bf 1}_{[|Z'_{k+l}| >T]} \right| .\end{aligned}$$ For any positive $\delta$, let $q_\delta = {\delta+1 \over \delta}$, by Holder’s inequality, we obtain $$\begin{aligned} \E_x \left| Z_k Z'_{k+l} {\bf 1}_{[|Z_k| >T]} \right| \le \left( \E _x |Z_k|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} \left( \E _x |Z'_{k+l}|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} \P _x(|Z_k| >T)^{1 \over q_\delta}.\end{aligned}$$ By Minkowski’s inequality, $$\begin{aligned} \left( \E _x |Z_k|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} & \le \left( \E _x |a_k|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} + \left( \E _x |V|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} < C, \notag \\ \left( \E _x |Z'_{k+l}|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} & \le \left( \E _x |a_{l+k}|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} + \left( \E _x |V'|^{2 +2 \delta}\right) ^{1 \over 2+2 \delta} < C\notag .\end{aligned}$$ By Markov’s inequality, $$\begin{aligned} \P _x (|Z_k| >T) & \le {1 \over T^{ q_\delta}} \E _x |Z_k|^{ q_\delta} \le {C \over T^{ q_\delta} }, \notag \\ \P _x (|Z'_{k+l}| >T) & \le {1 \over T^{ q_\delta}} \E _x |Z'_{k+l}|^{ q_\delta} \le {C \over T^{ q_\delta} } \notag .\end{aligned}$$ Hence $|R_0| \le C T^{-1}$ for $T>1$ and thus $|R| \le CT^{-1}$. Thus, (\[15\]) becomes $|Cov_x (a_k, a_{k+l}) | \le C T^2 \kappa ^l + C T^{-1}$. By choosing $T= \kappa ^{- \alpha}$ with $\alpha >0$, we obtain $$|Cov_x (a_k, a_{k+l}) | \le C \kappa ^{l - 2 \alpha} + C \kappa ^\alpha \le C' \max \{\kappa ^{l - 2 \alpha}, \kappa ^\alpha \}.$$ Now we choose $\alpha >0$ such that $l - 2 \alpha >0$, for example, let $\alpha = {l \over 4}$, we obtain $$|Cov_x (a_k, a_{k+l}) | \le C \kappa ^{l \over 4}.$$ Proof of Lemma [\[lem2\]]{} --------------------------- Inequality (\[e5\]) follows by setting $k=l$ in (\[e3\]) and (\[e4\]). It suffices to prove (\[e4\]). Recall the definition in (\[1.23\]) and let $$\begin{aligned} \label{defifi} \psi(s) &=& \nu(P_s {\bf 1})\varphi(s) , \notag \\ \psi(s,t; l) &=& \nu(P_s P^{l-1} P_t {\bf 1}) \varphi(s) \varphi(t) , \notag \\ \widetilde \psi (s,t; l) &=& \psi(s,t; l) - \psi(s) \psi(t), \\ \widetilde \phi_0 (s,t) &=& \widetilde \phi (s,t) - \widetilde \phi_1 (s) \widetilde \phi_2 (t) ,\notag \\ s_{l,T}&=& \frac{1}{(2 \pi)^2} \int \int \hat h_T(s,t) \widetilde \psi (s,t; l) ds dt .\notag \end{aligned}$$ Then (\[1.28\]) implies $$\left| Cov_x (a_k, a_{k+l}) - s_{l,T} \right| \le \left| \frac{1}{(2 \pi)^2} \int \int \hat h_T(s,t) [\widetilde \phi_0 (s,t) - \widetilde \psi (s,t; l)] ds dt \right| + |R|.$$ We claim that $$\begin{aligned} \label{cla} \left| \frac{1}{(2 \pi)^2} \int \int \hat h_T(s,t) [\widetilde \phi_0 (s,t) - \widetilde \psi (s,t; l)] ds dt \right| \le C \kappa^{k-1} T^2,\end{aligned}$$ which implies $$\begin{aligned} \label{12.1} \left| Cov_x (a_k, a_{k+l}) - s_{l,T} \right| &\le& C \kappa^{k-1} T^2 + C T^{-1},\end{aligned}$$ which yields for any $k,m \ge 1$, $$\begin{aligned} \left| Cov_x (a_k, a_{k+l}) - Cov_x (a_m, a_{m+l}) \right| \le C \kappa^{\min \{k-1,m-1 \} } T^2 + C T^{-1}.\end{aligned}$$ By choosing $T= \kappa^{-{1 \over 4}\min \{k-1,m-1 \} }$, we obtain $$\begin{aligned} \label{12.2} \left| Cov_x (a_k, a_{k+l}) - Cov_x (a_m, a_{m+l}) \right| \le C \kappa^{\min \{{k-1 \over 4},{m-1 \over 4} \} }.\end{aligned}$$ Hence we can say that $( Cov_x (a_k, a_{k+l}))_l$ is a Cauchy sequence, thus it converges to some limit, denoted by $s_l(x)$. When $k \to +\infty$, (\[12.1\]) becomes $$|s_l(x)- s_{l,T}| \le C T^{-1}.$$ Now let $T=T(\ell) = \kappa^{-\ell} $, we obtain $|s_l(x)- s_{l,T(\ell)}| \le C \kappa^\ell$. Let $\ell \to +\infty$, we can see that $s_{l,T(\ell)} \to s_l(x)$. Since $s_{l,T(\ell)}$ does not depend on $x$, so is $s_l(x)$, i.e. $s_l(x) = s_l$. Now let $m \to +\infty$ in (\[12.2\]), we obtain $$|Cov_x(a_k,a_{k+l}) -s_l| \le C \kappa^{k-1 \over 4}.$$ Now we prove the claim (\[cla\]). By definitions in (\[1.23\]) and (\[defifi\]), we obtain $$\begin{aligned} \label{11.1} \left|\widetilde \phi_0 (s,t) - \widetilde \psi (s,t; l) \right| \le \left| \widetilde \phi (s,t) - \psi (s,t; l)\right| + \left| \widetilde \phi_1(s) \widetilde \phi_2 (t) - \psi(s) \psi(t) \right|.\end{aligned}$$ On the one hand, we can see that $$\begin{aligned} \label{1.35} & &\left| \widetilde \phi (s,t) - \psi (s,t; k) \right| \notag \\ &=& \left| P^{k-1} P_s P^{l-1} P_t {\bf 1} (x) \varphi(s) \varphi(t) - \nu (P_s P^{l-1} P_t {\bf 1} )\varphi(s) \varphi(t)\right| \notag \\ &=& \left| \Pi P_s P^{l-1} P_t {\bf 1} (x) + R^{k-1} P_s \Pi P_t {\bf 1} (x) + R^{k-1} P_s R^{l-1} P_t {\bf 1} (x) - \nu (P_s P^{l-1} P_t {\bf 1} ) \right| \notag \\ &=& \left| R^{k-1} P_s {\bf 1} (x) \nu(P_t {\bf 1}) + R^{k-1} P_s R^{l-1} P_t {\bf 1} (x) \right| \le C \kappa^{k-1}.\end{aligned}$$ On the other hand, $$\begin{aligned} \left| \widetilde \phi_1(s) \widetilde \phi_2 (t) - \psi(s) \psi(t) \right| &=& \left| [\widetilde \phi_1(s) - \psi(s)] \widetilde \phi_2(t) + \psi(s) [\widetilde \phi_2(t) -\psi(t)] \right| \notag \\ &\le& \left|\widetilde \phi_1(s) - \psi(s) \right| + \left|\widetilde \phi_2(t) - \psi(t) \right| \notag \\ &\le& \left| \phi_1(s) \varphi(s) - \psi(s) \right| + \left| \phi_2(t) \varphi(t) - \psi(t) \right| \notag ,\end{aligned}$$ where as long as $k \ge 2$, $$\begin{aligned} \left| \phi_1(s) \varphi(s) - \psi(s) \right| &=& \left| \left[ \Pi P_s{\bf 1} (x) + R^{k-1}P_s{\bf 1} (x) \right]\E _x [e^{isV}] - \nu(P_s {\bf 1}) \E _x [e^{isV}]\right| \notag \\ &\le& \left| [\Pi P_s{\bf 1} (x) - \nu(P_s {\bf 1}) ] + R^{k-1}P_s{\bf 1} (x)\right| \notag \\ &=& \left| R^{k-1}P_s{\bf 1} (x) \right| \le C \kappa ^{k-1}.\end{aligned}$$ Similarly, we obtain $$\begin{aligned} \label{1.37} \left| \widetilde \phi_1(s) \widetilde \phi_2 (t) - \psi(s) \psi(t) \right| \le C \kappa^{k-1}.\end{aligned}$$ Therefore, (\[11.1\]), (\[1.35\]) and (\[1.37\]) imply $\left| \widetilde \phi_0(s,t) - \widetilde \psi(s,t; l) \right| \le C \kappa^{k-1} $ which yields the assertion of the claim. 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[**25**]{}, no. [**4**]{}, 1545–1587. <span style="font-variant:small-caps;">H. Hennion</span> (1984) Loi des grands nombres et pertubations pour des produits reductibles de matrices aléatoires indépendantes, Z. Wahrsch. Verw. Gebiete, [**67**]{}, 265-278. <span style="font-variant:small-caps;">H. Hennion and L. Hervé</span> (1984) Stable laws and products of positive random matrices,Journal of Theoretical Probability [**21**]{}, no. [**4**]{}, 966–981. \(1982) Théorèmes limites pour les produits de matrices aléatoires, Springer Lecture Notes [**928**]{}, 258–303. \(1937) Théory de l’addition des variables aléatoires. Gauthier-Villars [^1]: Université Fr. Rabelais Tours, LMPT UMR CNRS 7350, Tours, France.\ email : Thi-Da-Cam.Pham@lmpt.univ-tours.fr [^2]: The author thanks the [**Vietnam Institute for Advanced Studies in Mathematics**]{} (VIASM) in Ha Noi for generous hospitality and financial support during the first semester 2017.

--- abstract: 'Shear viscosity $\eta$ is calculated for the nuclear matter described as a system of interacting nucleons with the van der Waals (VDW) equation of state. The Boltzmann-Vlasov kinetic equation is solved in terms of the plane waves of the collective overdamped motion. In the frequent-collision regime, the shear viscosity depends on the particle-number density $n$ through the mean-field parameter $a$, which describes attractive forces in the VDW equation. In the temperature region $T=15 - 40$ MeV, a ratio of the shear viscosity to the entropy density $s$ is smaller than 1 at the nucleon number density $n =(0.5 - 1.5)\,n^{}_0$, where $n^{}_0=0.16\,$fm$^{-3}$ is the particle density of equilibrium nuclear matter at zero temperature. A minimum of the $\eta/s$ ratio takes place somewhere in a vicinity of the critical point of the VDW system. Large values of $\eta/s\gg 1$ are, however, found in both the low-density, $n\ll n^{}_0$, and high-density, $n>2n^{}_0$, regions. This makes the ideal hydrodynamic approach inapplicable for these densities.' author: - 'A. G. Magner' - 'M. I. Gorenstein' - 'U. V. Grygoriev' - 'V. A. Plujko' date: 'October, 9th 2016' title: Shear viscosity of nuclear matter --- INTRODUCTION ============ The shear viscosity $\eta$ and its ratio to the entropy density $s$ became recently attractive (see, e.g., Refs.[@csernai; @shaefer2; @wiranata-prc-12; @kapusta] and references therein) in connection with a development of the hydrodynamic approach to the relativistic nucleus-nucleus collisions. Chapman and Enskog (CE) obtained [@chapman; @uhlenbeck; @huang; @silin; @fertziger; @LLv10] the shear viscosity $\eta$ in a gas of non-relativistic particles by using the Boltzmann kinetic equation (BKE) for the phase-space distribution function $f(\r,\p,t) $, where $\r$ and $\p$ are the particle coordinate and momentum, respectively, and $t$ denotes the time variable. The BKE was solved within the frequent-collision (FC) regime for which one can use a perturbation expansion in a small parameter, e.g., $\omega/\nu$, where $\nu$ is the collision frequency and $\omega$ measures the characteristic dynamical variations of the distribution function $\delta f(\r,\p,t)$. In this case the Boltzmann integral collision term is dominant as compared to other collisionless terms. For their calculations the local-equilibrium distribution function $f_{\rm l.e.}$ was used in a standard form in terms of the evolution of particle-number density $n(\r,t)$, temperature $T(\r,t)$, and collective velocity $\u(\r,t)$. The hydrodynamic variables $n(\r,t)$ and $\u(\r,t)$ are defined as the zero and first moments of the distribution function in the momentum space. Thus, the evolution derivative of the distribution function, $\d f/\d t$, as one of the local equilibrium distribution, $\d f_{\rm l.e.}/\d t$, in solving the BKE at the first order in $\omega/\nu $ can be decomposed into terms proportional to that of $n(\r,t) $, $T(\r,t)$, and $\u(\r,t)$. Using then the standard closed system of hydrodynamical equations and condition $\delta T=0$, one obtains [@chapman; @silin; @fertziger; @LLv10] the expression for the shear viscosity $\eta$. For a gas of elastic scattering balls with the diameter $d$, at the first approximation in $\omega/\nu$, one finds [@chapman] ł[etaCE]{} \^\_[CE]{}=, where $m$ is the particle mass. The shear viscosity $\eta^{}_{\rm CE}$ appears to be independent of the particle-number density $n$ because, at the first order in small parameter $\omega/\nu$, the attractive interaction on large distances between particles was neglected to simplify the CE viscosity calculations. Extensions of the CE method to the relativistic high energy-density problems are given in Refs. [@prakash-pr93; @wiranata-prc-12]. In particular, Eq. (\[etaCE\]) was reproduced in the nonrelativistic limit within the CE approach in Ref.[@wiranata-prc-12]. The mixture of different hadron species was considered in Ref. [@gorenstein]. Several investigations were devoted to go beyond the hydrodynamical approach [@chapman]; see, e.g., Refs. [@abrkhal; @brooksyk; @balescu; @baympeth; @kolmagpl; @magkohofsh; @pethsmith2002; @kolomietz1996; @kolomietz1998; @plujko1999; @plujko2001a; @plujko2001b; @kolomietz2004; @spiegel_BVKE-visc_2003; @smith2005; @review]. In contrast to the CE approach, the main problem solved in these works was to take into account a self-consistent mean field in calculations of the viscosity of Fermi liquids within the Landau quasiparticle theory. In the present paper, we use the Boltzmann-Vlasov kinetic equation (BVKE) for a system of interacting nucleons with the van der Waals (VDW) equation of state. Therefore, both scattering of particles owing to the hard-core repulsions and Vlasov self-consistent mean field, owing to the VDW attractive interaction, are taken into account in solving the BVKE. In our consideration, the small dynamical variations $\delta f$ are found from a linearized BVKE in the simplified form, ł[dfge]{} f(,, t) = f(,, t)- f\^\_[0]{}(p), where $f^{}_{0}(p)$ is the static global-equilibrium distribution function ł[maxwell]{} f\^\_[0]{}(p)= (- ). The function $f^{}_{0}(p)$ (\[maxwell\]) is taken in the Maxwell form with constant values of $T$ and $n$ and zero value of the collective velocity, $\u=0$. The damping plane wave (DPW) solutions are assumed to be a good approximation to the dynamical variations $\delta f$ at finite frequencies $\omega$ within the FC regime (large-enough collision frequency $\nu$). These dynamical deviations allow us to take into account analytically the attractive VDW interactions through the self-consistent Vlasov mean field in the BVKE. As a result, we obtain the shear viscosity dependence on the particle number density at the leading order in a small parameter $\omega/\tau$. The overdamped (see, e.g., Refs. [@magkohofsh; @review]) attenuation of the DPW will be considered. Our approach is based on the methods applied earlier for calculations of the viscosity of the Fermi liquids [@abrkhal; @brooksyk; @baympeth; @kolmagpl; @review]. In the present paper, the shear viscosity in the first order over small parameter $\omega/\nu$ is calculated analytically for the nuclear matter considered as a gas of interacting nucleons with the VDW equation of state. The paper is organized as follows. In Sec. \[sec-vdw\] we remind the basic properties of thermodynamically equilibrated systems with the VDW equation of state. In Sec. \[sec-kinetic\] we outlook the kinetic approach based on the BVKE and give general definitions of the shear viscosity coefficient. In Sec. \[sec-disp\], the solution to the BVKE and its perturbation expansion is presented in terms of the plane waves accounting for a strong attenuation owing to the particle collisions. Finally, this section is devoted to the main results for the VDW viscosity. The obtained results are discussed in Sec. \[sec-disc\] and summarized in Sec. \[sec-concl\]. Some details of our calculations can be found in Appendixes A–C. VDW EQUATION OF STATE {#sec-vdw} ===================== The VDW equation of state presents the system pressure $P$ in terms of the particle number density $n$ and temperature $T$ as [@LL], ł[vdw-p-n]{} P(T,n) =  - an\^2 , where $a>0$ and $b>0$ are the VDW parameters that describe attractive and repulsive interactions, respectively. The first term on the right-hand side of Eq. (\[vdw-p-n\]) contains the excluded volume correction ($b=2 \pi d^3/3$, with $d$ being the particle hard-core diameter), while the second term comes from the mean-field description of attractive interactions. The entropy density $s$ and energy density $\varepsilon$ for the VDW system are calculated as [@LL] \[s\] s(T,n) &=& n + n,\ (T,n)&=& n .\[E\] In Eq. (\[s\]) $m$ is the particle mass and $g$ is the degeneracy factor ($g=4$ for nucleons; two spin and two isospin states). Note that the VDW entropy density (\[s\]) is independent of the attractive mean-field interaction parameter $a$, whereas the energy density (\[E\]) does not depend on the particle repulsion constant $b$. The VDW equation of state contains the first-order liquid-gas phase transition with a critical point [@LL]: \[Tc\] T\_c =  ,     n\_c =  ,     P\_c =  . To study the phase coexistence region which exists below the critical temperature, $T<T_c$, the VDW isotherms should be corrected by the well-known Maxwell construction of equal areas. The VDW equation of state was recently applied to a description of nuclear matter in Ref. [@marik2]. In the present study we fix the VDW parameters for the system of interacting nucleons as $d=1$ fm, i.e., $b\cong 2.1$ fm$^{3}$, and $a=100$ MeVfm$^{3}$. This gives $n_c\cong n_0=0.16$ fm$^{-3}$ and $T_c\cong 14$ MeV ($n_0=0.16$ fm$^{-3}$ corresponds to the nucleon number density of the normal nuclear matter at zero temperature). In what follows we restrict our analysis of the kinetic properties of the VDW system of nucleons to $T>T_c$. In this region of the phase diagram the VDW equation of state describes a homogeneous one-phase system, and all criteria of the thermodynamical stability are satisfied. We do not consider too large temperatures by taking $T \siml 40$ MeV. This allows us to neglect a production of new particles (pions and baryonic resonances) in the system of interacting nucleons. In addition, this restriction guarantees a good accuracy of the nonrelativistic approximation adopted in the present study. Note also that at $T\rightarrow 0$ the quantum statistics effects neglected in the present study should be taken into account (see Ref. [@marik2]). KINETIC APPROACH {#sec-kinetic} ================ For calculations of the shear viscosity, we start with the BVKE linearized near the static distribution function (\[maxwell\]) for the dynamical variations of the distribution function $\delta f(\r,\p,t)$ (\[dfge\]) : ł[Boltzlin]{} + - = St. The dynamical part of the attractive potential $\delta U$ from the VDW forces is defined self-consistently as ł[effpot]{} U(,t)=-a f(,,t). In Eq. (\[Boltzlin\]), the collision term $\delta St$ is taken in the standard Boltzmann form [@chapman; @silin], ł[dstdef]{} St  =   \_1 |\_1-| Q, where ł[dQ]{} Q& &f\^\_[0]{}(p’)f(,\_1’,t)+ f\^\_[0]{}(p\_1’)f(,’,t)\ &-&f\^\_[0]{}(p)f(,\_1,t) -f\^\_[0]{}(p\_1)f(,,t) is the variation of $f(\r,\p',t) f(\r,\p_1',t)-f(\r,\p,t) f(\r,\p_1,t)$ over $\delta f$ and $\b$ the impact parameter for two-body collisions. ![[The geometry of the collision of two hard spheres in the center mass system; $\q=\p_1-\p$, $\q'=\p_1'-\p'$, $OO_1=d$, $d$ is the sphere particle diameter, $\beta$ is the impact parameter; $\theta_{p'}$ is the scattering angle. ]{}[]{data-label="fig1"}](fig1.eps){width="49.00000%"} Figure \[fig1\] shows the collision geometry for two hard-core sphere scattering in the center-of-mass coordinates. The relationship between the impact parameter, $\beta$, and the cross section, $\sigma=\pi d^2$, where $d$ is the diameter of the particle, is given in Appendix A for calculations of the integral term (\[dstdef\]). The shear viscosity $\eta$ can be defined through the dynamical components of the momentum flux tensor $\Pi_{\mu \nu}(\r,t)$, ł[dpidef]{} \_ = -\_ + \_ + P\_, where $\delta \sigma_{\mu \nu}$ is a traceless stress tensor. Other terms are diagonal kinetic and interaction pressures. The stress tensor $\delta \sigma_{\mu \nu}$ can be determined through the second $\p$ moment of the distribution function linearized over $\delta f$ as ł[dsigmadef]{} \_=- p\_p\_ f(,,t) + \_. In Eq. (\[dpidef\]), the quantities $\delta \mathcal{P}$ and $\delta P $ are calculated as ł[presskin]{}  & = &   p\^2 f(,,t),\ P   &=&   -2 a n f(,,t). The shear viscosity $\eta$ is defined as a coefficient in the relationship between the dynamical component of the stress tensor $\delta \sigma_{\mu \nu}$ \[Eq. (\[dpidef\])\] and the traceless tensor $\mathcal{U}_{\mu \nu}$ of the coordinate derivatives of the velocity field $\u(\r,t)$ [@LLv10; @balescu; @brooksyk; @kolmagpl; @LLv6], ł[etadef]{} \_(,t)= \_(,t), where ł[veltens]{} \_= ( + - u\_). The velocity field $\u$ is defined through the first $\p$ moment of $\delta f(\r,\p,t)$, ł[udf]{} = f(,,t). Note that our method can also be presented within the linear response-function theory [@balescu; @hofmann; @review] (cf. the Kubo formulas for the diffusion, thermal conductivity, and viscosity; see also the recent article [@wiranata-jp-14]). DISPERSION RELATION AND VISCOSITY {#sec-disp} ================================= We suggest to calculate the shear viscosity $\eta$ by directly solving the BVKE (\[Boltzlin\]) in terms of the plane-wave representation for the dynamical distribution-function variations $\delta f(\r,\p,t)$ \[Eq. (\[dfge\])\] in the following rather general form [@brooksyk; @kolmagpl; @review], ł[planewavesol]{} f(,,t)= f\^\_[0]{}(p) () (-i t + i), where $\omega$ and $\k$ are a frequency and a wave vector of the DPW, respectively. As unknown yet, amplitudes $\varphi(\hat{p})$ are functions of the momentum angle variable $\hat{p}=\p/p$. It is naturally to find solutions of the BVKE as proportional to the static distribution function, $f^{}_{0}(p)$, specifying the dependence of $\delta f$ on the modulus of momentum $p$ because the derivative of $f^{}_{0}(p)$ (\[maxwell\]) over momentum in Eq. (\[Boltzlin\]) and the variations of the collision integral (\[dstdef\]) are proportional to $f^{}_{0}(p)$. Then, one can reduce the problem for solving the BVKE (\[Boltzlin\]) to a function of angles $\varphi(\hat{p})$ which, however, depends on the uknown frequency $\omega~$. \[We shall leave out the argument $\omega$ in $\varphi(\hat{p})$ for simplicity of the notations.\] Note that any physical quantity, in particular the viscosity coefficient, is independent of the direction of the unit wave vector $\hat{k}=\k/k$ of the DPW spreading in infinite nuclear matter. Therefore, it is convenient to use the spherical phase-space coordinate system with the polar axis directed to this vector $\hat{k}$. The solution for the plane-wave distribution function $\delta f(\r,\p,t)$ \[Eq. (\[planewavesol\])\] or more precisely $\varphi(\hat{p})$, and the frequency $\omega $ depends only on the wave vector length $k$. For convenience, one may write the frequency $\omega$ through the wave number $k$ and the dimensionless sound velocity $c$, ł[freqom]{} = k v = kv\^\_[T]{} c, where $v=v^{}_{T} c$ is the DPW speed, and $c$ its dimensionless value given in units of the most probable thermal velocity $v^{}_{T}$ of particles at a given temperature $T$, $v^{}_{T}=\sqrt{2T/m}$. The viscosity $\eta$ is related to an attenuation of the DPW (\[planewavesol\]) measured by the collision term $\delta St$ (\[dstdef\]). Following Refs. [@chapman; @silin; @fertziger; @brooksyk], one applies the perturbation expansion of the dynamical distribution-function variations $\delta f$ through their amplitudes $\varphi(\hat{p})$, ł[varphipertexp]{} ()=\^[(0)]{}() + \^[(1)]{}() + \^2 \^[(2)]{}() + ..., and similarly, in addition to Ref. [@chapman], for the frequency $\omega$, ł[ompertexp]{} =\^[(0)]{} + \^[(1)]{} + \^2 \^[(2)]{} + ..., in a small parameter, ł[eps]{} =/=. Here, $\tau$ is the relaxation time[^1] defined by the collision term through the time-dependent rate $\nu$ (collision frequency) of the damping of distribution function $\delta f$, ł[taudef]{} =1/. Expansions (\[varphipertexp\]) and (\[ompertexp\]) are defined within the standard perturbation method [@silin; @brooksyk; @review; @madelung] for the eigenfunction, $\varphi(\hat{p})$, and eigenvalue, $\omega$, problem. In these perturbation expansions, the coefficients $\varphi^{(n)}(\hat{p})$ and $\omega^{(n)}$ are assumed to be independent of $\epsilon $. By using the BVKE with this perturbation method, they can be found at each order of $\epsilon$. Note that $\omega$ in the definition of the small parameter $\epsilon$ (\[eps\]) is determined consistently at any given order of the perturbation expansions (\[varphipertexp\]) and (\[ompertexp\]); see Appendix B for details. A smallness of $\epsilon$ can be achieved by increasing the collision frequency $\nu$ for a given $\omega$ (Appendix A). Substituting the plane-wave representation (\[planewavesol\]) for the distribution function $\delta f$ into the BVKE (\[Boltzlin\]), for convenience, one can also expand $\varphi(\hat{p})$ in series over the spherical harmonics $Y_{\ell 0}(\hat{p})$, ł[varphiexp]{} ()=\_[=0]{}\^ \^\_Y\_[0]{}(),=. This reduces the integro-differential BVKE to much more simple linear algebraic equations (\[Boltzeqfin\]) for the partial multipole amplitudes $\varphi^{}_{\ell}$ at each order in $\epsilon$ \[Eq. (\[eps\]) and Appendix B\]. As shown in Appendixes A and B, in the FC regime, $|\epsilon| \ll 1$ \[Eq. (\[eps\])\], one can truncate the multipole expansion (\[varphiexp\]) over $\ell$ at $\ell=2$ because of a good convergence in the small parameter $\epsilon~$. At this leading approximation to viscosity calculations, for the collision term $\delta St$ \[Eq.  (\[dstdef\])\], one obtains (Appendix A) the simple expression ł[tauapprox]{} St = - f\^\_2(,,t), where ł[nu]{} ,=d\^2, $\sigma$ is the cross section for a two elastic hard-core sphere scattering, as introduced above, ł[df2]{} f\_2(,,t)=f\^\_[0]{}(p)\^\_2Y\_[20]{}() (-i t + i). As shown in Appendix A, within the accuracy about   6%, this value agrees with its mean effective quantity $\nu_{av}$ (\[nuav\]), evaluated through the momentum average of the collision term, $\langle \delta St \rangle_{\rm av}$, over particle momenta $p$ with the help of the Maxwell distribution $f^{}_{0}(p)$ (\[maxwell\]). Multiplying then the BVKE (\[Boltzlin\]), with the quadrupole collisional term (\[tauapprox\]), by the spherical function $Y_{L 0}(\hat{p})$ ($L=0,1,2,...$), one can integrate the BVKE term by term over angles ($\hat{p}$) of the momentum $\p$. Thus, one obtains the linear homogeneous equations (\[Boltzeqfin\]) with respect to coefficients $\varphi^{}_\ell$ of the expansion (\[varphiexp\]) in the plane-wave amplitudes $\varphi(\hat{p})$ at any order in $\epsilon$ in Eqs. (\[varphipertexp\]) and (\[ompertexp\]). This system has nontrivial solutions in the quadrupole approximation $\ell \leq 2$, valid at the leading (linear in $\epsilon$) approximation in expansions (\[varphipertexp\]) and (\[ompertexp\]). They obey the cubic dispersion equation for $c=\omega/(k v^{}_{T})$ (expansion of $c$ is similar to Eq.(\[ompertexp\]); see also Appendix B), ł[dispeq]{} & [det]{}\_2 c\^3 + i c\^2 - c\ & - (1 - )=0, where $\mathcal{F}$ is the dimensionless VDW interaction parameter, ł[Fant]{} =an/T. The truncated (at $\ell=2 $) $3 \times 3$ matrix $\mathcal{A}^{(2)}_{L\ell}(c)$ is given by Eq. (\[matrix2\]). For convenience, we introduced also the dimensionless collisional rate (\[nu\]): ł[gamma]{} == = . The FC perturbation parameter $\epsilon$ \[Eq. (\[eps\])\] can be expressed in terms of the $\gamma$ and $c$ as ł[FCcond]{} =c/,|c/| 1. The cubic dispersion equation (\[dispeq\]) has still two limit solutions with respect to the complex velocity, $c=c_r+ic_i$ for real $k$ (or equivalently, a complex wave number $k=k_r+ik_i$ for a real velocity $c$, both related by the same $\omega=k c v^{}_{T}=\omega_r + i \omega_i$, where low subscripts denote the real and imaginary parts). One of them can be called as the underdamped (weakly damped) first sound mode for which the imaginary part of $c$, $c_i$, is much smaller than the real one $c_r$, $|c_i/c_r| \ll 1$, while in the opposite case $|c_i/c_r| \gg 1$, one has the overdamped motion. In the first underdamped sound case ($|c_i/c_r| \ll 1 $), the collision term can be considered as small with respect to the left-hand side (LHS) of the BVKE, $|\gamma/c| \sim 1/|\omega \tau| \ll 1$, that is, the rare collision (RC) regime. For the overdamped motion (FC case) the collision term is dominant. In our DPW derivations below one can use also the frequency expansion (\[ompertexp\]) over the same small parameter $|c/\gamma| \sim |\omega \tau| \ll 1$ \[ Eqs. (\[gamma\]) and (\[FCcond\])\]. In the present study we consider the overdamped motion, while the underdamped case will be studied in separate publications. Expanding the LHS of the truncated (quadrupole) dispersion equation (\[dispeq\]) for $c$ in powers of $\epsilon$ \[see Eqs.  (\[eps\]) and (\[FCcond\])\] in the FC perturbation expansions (\[varphipertexp\]) and (\[ompertexp\]), one can divide all of its terms by $\gamma^3$. Then, one can neglect the relatively small cubic \[$(c/\gamma)^3\sim \epsilon^3$\] and quadratic ($\sim \epsilon^2$) terms as compared to the last two linear (in $\epsilon$) ones depending explicitly on the interaction parameter $\mathcal{F}$ \[Eq. (\[Fant\])\]. At this leading order, one results in the explicit quadrupole solution for the velocity $c$ \[Eq. (\[sol0soundr\])\], ł[linsols0]{} c = i c\_i =- . To get small corrections of the real sound velocity $c_r$, one has to take into account the quadratic and cubic in $c$ terms of the dispersion equation (\[dispeq\]). Note that formally, one can consider the real DPW velocity $c$ but the complex wave number $k$ within the same complex frequency $\omega$, which are both almost pure imaginary ones. The latter describes the sound attenuation as the exponential decrease of the DPW amplitude, $\delta f \propto \exp(- t/\mathcal{T})$ \[Eq. (\[planewavesol\])\] with the damping time $\mathcal{T}$, ł[damptime]{} . This time was obtained as the imaginary part of the complex frequency, $\omega = -i/\mathcal{T}$, through Eq. (\[linsols0\]), formally introduced above (finally, all physical quantities will be determined by taking their real parts). Note also that the relaxation time $\tau$ \[Eqs. (\[taudef\]) and (\[nu\])\], ł[reltime]{} =, differs from the damping time, $\mathcal{T}$ \[Eq. (\[damptime\])\]. In particular, this time $\mathcal{T}$, being of the order of $\tau$, depends on the interaction constant $\mathcal{F}$. Note that the FC condition (\[FCcond\]) can be satisfied for the interaction parameter $\mathcal{F}$ of the order of one. However, as shown below, one finds a reasonable result even in the limit $\mathcal{F}\to 0$. ![[Shear viscosity $\eta$ \[Eq. (\[viscSDfc\])\] in the frequent-collision regime in units of the CE value $\eta^{}_{\rm CE}$ \[Eq. (\[etaCE\])\] versus particle-number density $n$ in units of the normal density $n_0=0.16$ fm$^{-3}$ of nuclear matter; $m\cong 938$ MeV; $d=1~$fm, $a=100$ MeV fm$^3$. ]{}[]{data-label="fig2"}](fig2.eps){width="49.00000%"} Using the DPW solutions (\[planewavesol\]) for $\delta f$ of the BVKE (\[Boltzlin\]), and Eqs. (\[USzz\]) for $\mathcal{U}_{zz}$ and (\[Szztilde\]) for $\sigma_{zz}$, for the definition of the shear viscosity $\eta$ \[Eq. (\[etadef\])\], one finds the FC expansions \[(\[varphipertexp\]) and (\[ompertexp\])\] of $\eta$ in powers of small $\epsilon$ \[see Eq. (\[eps\]), and Appendixes C and B\]. As shown in Appendix C, the leading term of this FC shear viscosity $\eta$ at first order in $\epsilon$ is approximately a constant, independent of $\omega$ (or $k$), and proportional to $1/\nu$, i.e., to the relaxation time $\tau$ \[Eq. (\[reltime\])\]. Finally, up to relatively high (second) order terms in the small parameter $\epsilon$ we arrive at ł[viscSDfc]{} &=& (1-59 ) = (1-59 )\^\_[CE]{}\ &=& 1.018(1-59 )\^\_[CE]{}. In these derivations we used, at the leading first order in $\epsilon$, the quadrupole multipolarity truncation of rapidly converged series (\[varphiexp\]); see Eqs. (\[phieq\]) for the amplitudes $\varphi^{}_\ell$ and (\[linsols0\]) for the sound velocity $c$ ($\ell \leq 2$) within the dispersion equation (\[dispeq\]). As seen from Eq. (\[viscSDfc\]), within the present accuracy, the shear viscosity $\eta$ differs in 2 % from the CE result $\eta^{}_{\rm CE}$ \[Eq. (\[etaCE\])\] at zero attractive mean field, $a \to 0$. Note that a more exact CE result is $\eta=1.016 \eta_{\rm CE}$ (see Ref. [@chapman], Chap. 12.1). Formula (\[viscSDfc\]) for the shear viscosity can be presented in a more traditional way through the relaxation time $\tau$ \[Eq. (\[reltime\])\], ł[etatau2]{} = (1 - ) m n v\^2\_[T]{}. This relationship, $\eta \propto \tau$, is typical for the FC regime, in contrast to the rare collision one, $\eta \propto 1/\tau$, which should be expected for the perturbation expansion at leading order in the opposite small parameter $1/\epsilon$; see Refs. [@brooksyk; @baympeth; @kolmagpl; @review]. {width="80.00000%"} {width="80.00000%"} Note that the perturbation method for the eigenfunctions $\varphi(\hat{p})$ (or $\varphi^{}_\ell$, Eq. (\[varphipertexp\]) as in Ref. [@chapman])), and in addition, eigenvalues $\omega$ allows us to obtain in a regular way high-order corrections in $\epsilon$. In this way, one has to go beyond the quadrupole multipolarity ($\ell \leq 2$) approximation taking into account, consistently at a given $\epsilon$, higher order terms, $\ell > 2$, in expansion (\[varphiexp\]) for $\varphi(\hat{p})$. DISCUSSION OF THE RESULTS {#sec-disc} ========================= Equation (\[viscSDfc\]) for $\eta$ has the same classical hydrodynamical dependence on the temperature $T$ and diameter $d$, $\eta \propto \sqrt{mT}/d^2$ \[cf.  with Eq. (\[etaCE\])\], because of using the FC approximation as in both the molecular kinetic theory [@LLv10] and the CE approach [@chapman]. In this approximation for the overdamped case (Appendix B) the dominating contribution into the viscosity yields from the collision term which mainly determines both the classical hydrodynamical solutions (Ref. [@chapman]) and our DPW ones \[Eq. (\[planewavesol\])\] for the distribution function to the BVKE. Therefore, as expected, in the limit $a \rightarrow 0$ ($\mathcal{F} \ll 1$), one finds the number constant \[in front of $\sqrt{mT}/d^2$; see Fig. \[fig2\] and Eq.  (\[viscSDfc\])\] that approximately coincides within the accuracy of 2% with the CE result (\[etaCE\]). The difference between the hydrodynamical (\[etaCE\]) and overdamped DPW (\[viscSDfc\]) viscosities in the zero interaction constant limit should be, indeed, small as compared to the leading collisional term. Figure \[fig2\] shows the shear viscosity $\eta$ \[Eq. (\[viscSDfc\])\] for a few temperatures above the critical value $T_c$ \[Eq. (\[Tc\])\]. From Fig. \[fig2\], one can clearly see that the shear viscosity $\eta$ differs significantly from the classical hydrodynamical formula (\[etaCE\]) by the particle density dependence. It appears through the VDW parameter $\mathcal{F}$ \[Eq. (\[Fant\])\], owing to accounting for dynamical variations of the mean-field interaction (\[effpot\]) in our derivations. As displayed in this figure, the significant effects originate by the Vlasov self-consistent attractive-interaction terms of the BVKE. In our approach this is achieved by solving the BVKE (\[Boltzlin\]) in terms of the DPW nonlocal-equilibrium distribution function $\delta f$ \[see Eq. (\[planewavesol\])\] and using, therefore, the perturbation expansion (\[ompertexp\]) for the frequency $\omega$ as a solution of the dispersion equation, in addition to Eq. (\[varphipertexp\]). This is in contrast to the CE approach based on the dynamical local-equilibrium distribution-function variations and hydrodynamical equations, used on the left-hand side of the BVKE. The interaction term of the Boltzmann kinetic equation containing $\delta U$ \[Eqs. (\[effpot\])\] is neglected in the CE method [@chapman] as compared to the integral collision term of the BVKE at the leading first order in $\epsilon$. Therefore, there is no particle density corrections to the shear viscosity in the CE approach at this order. To obtain these corrections, we found another alternative DPW solution (\[planewavesol\]) through the self-consistent interaction term of the BVKE. Note also that with increasing attractive interaction parameter $a$ ($a>0$), one finds a linearly decreasing viscosity $\eta$ through the dimensionless parameter $\mathcal{F}$. Figures \[fig3\] and \[fig4\] show the ratio, $\eta/s$, of the viscosity $\eta$ to the entropy density $s$ \[Eq. (\[s\])\] given by Eqs. (\[viscSDfc\]) and (\[etaCE\]), respectively, in the $n-T$ plane for temperatures $T$ above the critical value $T_c$ \[Eq. (\[Tc\])\]. As seen from comparison of the two overdamped viscosities in units of the entropy density in these figures, the ratio $\eta/s$ takes form of a minimum with values $\eta/s \siml 1$ at densities $(0.5 - 2)n^{}_0$, somewhere in a vicinity of the critical point ($T_c$,$n_c$). This minimum is significantly smaller and moves to smaller temperatures in our DPW calculations (Fig. \[fig3\]) as compared to the CE ones (Fig. \[fig4\]) though they are both smaller than 1. Note also a weak sensitivity of these properties depending of the size of the hard spheres $d$ around $d=1$ fm for its deflections in about 20%. However, $\eta/s\gg 1$ both at small ($n \ll n_0^{}$) and large ($n \simg 2n^{}_0$) particle density, which makes the ideal hydrodynamic approach inapplicable for these densities. We should emphasize that the BVKE can be applied for enough dilute system of particles where the mean free path is large as compared to particles’ interaction region (in our example, of the order of the size of particles $d$). This gas condition should be satisfied for all desired densities. CONCLUSIONS {#sec-concl} =========== The shear viscosity of a nucleon gas is derived by solving the BVKE for the FC regime with taking into account the van der Waals interaction parameters for both the hard-elastic sphere scattering and attractive mean-field interaction. The viscosity $\eta$ depends on the particle density $n$ through the dynamical mean-field forces measured by the VDW parameter, $~an/T~$, which is positive for the attractive long-distance mean-field interaction. Therefore, the viscosity $\eta$ decreases with the interaction constant $a >0$ through the VDW parameter $\mathcal{F}$. The ratio of the FC viscosity to the entropy density, $\eta/s$, as function of the particle density $n$ and temperature $T$ is found to have a minimum which is essentially smaller than one. The viscosity is significantly smaller at this minimum which moves to smaller temperatures toward the critical temperature owing to the long-distance interaction, as compared to the classical hydrodynamical CE result. Our DPW viscosity calculations have the same overdamped behavior (strong attenuation) such that the collisional term is dominating above all of other parts of the BVKE. Note that the viscosity coefficient can be consider as a response (Ref. [@balescu; @review; @hofmann; @review2]) of the stress tensor $\sigma_{\mu\nu}$ for the shear pressure on the velocity derivative tensor $\mathcal{U}_{\mu\nu}$; see, e.g., Eq. (\[etadef\]). See also the Green’s–Kubo formula for the shear viscosity, as for the conductivity coefficient [@balescu; @wiranata-jp-14]. Our results might be interesting for the kinetic and hydrodynamic studies of nucleus-nucleus collisions at laboratory energies of a few hundreds MeV per nucleon. The ideal hydrodynamics can be a fairly good approximation for a system of the interacting nucleons in the region of $n$ and $T$ that corresponds at least to $\eta/s\ll 1$. However, the classical hydrodynamical approach for both the dilute nucleon gas with $n\ll n_0$ and the nuclear-dense matter with $n \simg 2n_0$ seems to be rather questionable to use. As a different perturbation theory has to be used in expansions over small $\omega \tau$ for the FC and small $1/(\omega \tau)$ in the RC regime, we should expect very different dependencies of the viscosity (and other transport coefficients) on the particle density $n$ in these two opposite limits. For instance, the RC regime is important to study a weak absorption of the DPW in the gas system with small far-acting interactions, especially for ultrasonic absorption [@bhatia; @spiegel_Ultrasonic_2001]. Therefore, in the case when the contributions of collisions into the BVKE dynamics are changed from the dominant (small $\epsilon $) to almost collisionless process (small $1/\epsilon$) with increasing DPW frequency for a given collision frequency $\nu$, a transition from the FC to RC regimes should be accounted beyond the classical hydrodynamical approach. This can be realized for small $n/n^{}_0$ and large $\eta/s$, in the corresponding $n-T$ regions of the phase diagram for analysis of the nucleus-nucleus collisions. Our approach can be applied to calculations of the thermal conductivity and diffusion coefficients in nuclear physics, as well as those and viscosity in nuclear astrophysics, and to study different phenomena in the electron-ion plasma. [**ACKNOWLEDGMENTS**]{} We thank D.V. Anchishkin, S.N. Reznik and A.I. Sanzhur for fruitful discussions. The work of M.I.G. was supported by the Program of Fundamental Research of the Department of Physics and Astronomy of National Academy of Sciences of Ukraine. One of us (A.G.M.) is very grateful for the financial support of the Program of Fundamental Research to develop further cooperation with CERN and JINR “Nuclear matter in extreme conditions” by the Department of Nuclear Physics and Energy of National Academy of Sciences of Ukraine, Grant No. CO-2-14/2016, for nice hospitality during his working visit to the Nagoya Institute of Technology, and also for financial support from the Japanese Society of Promotion of Sciences, Grant No. S-14130. COLLISION TERM CALCULATIONS =========================== ł[appA]{} We shall neglect approximately an influence of the effective potential $\delta U[n(\r,t)]$ (\[effpot\]) of the long-range particle interaction during a two-particle collision of hard-core sphere particles of the gas in the FC regime. Using the multipole expansion (\[varphiexp\]) of the amplitude factor $\varphi(\hat{p})$, one can simplify the linearized collisional term, $\delta St~$ \[Eqs. (\[dstdef\]) and (\[planewavesol\])\], in the BVKE (\[Boltzlin\]), ł[dst]{} &&St= \_ \^\_ \_1 |\_1-| \_[p’]{}\ &&{f\^\_[0]{}(p\_1’)f\^\_[0]{}(p’).\ &-& . f\_[0]{}(p\_1)f\_[0]{}(p)}, where $f^{}_{0}(p)$ is the static distribution function (\[maxwell\]), ł[ftildel]{} \^\_=\^\_ (-i t +i), and $\varphi^{}_\ell$ is the $\ell$ coefficient in the expansion (\[varphiexp\]) for amplitudes $\varphi(\hat{p})$. One finds the relationship between the impact parameter $\b$ in the center-of-mass coordinate system \[see Fig. \[fig1\]\] and the scattering angle $\theta_{p'}$ (and $ \beta \d \beta$ to $\d\Omega_{p'}$), ł[impactpar]{} &=&(\_[p’]{}/2) d,=\ &=& \_[p’]{}= \_[p’]{} \_[p’]{} \_[p’]{}. The Boltzmann collision term (\[dst\]) is defined in such a way that its zero and first $\p$ moments have to be zero because of the particle-number conservation (related to the continuity equation), and momentum conservation, ł[momcons]{} + \_1 = ’ + ’\_1, (associated with the momentum continuity equation) during a two-body collision. We take also into account that the distribution function (\[maxwell\]) is located within a small momentum interval $(2mT)^{1/2}$. Within this range the momentum vectors are approximately changed only by their direction angles, ł[momconsang]{} + \_1 ’ + ’\_1, and one can use also the kinetic-energy conservation equation, ł[conservkinen]{} p\_\^[2]{} + p\_1\^[2]{} = p\_\^[ 2]{} + p\_1\^[ 2]{}. Substituting Eq. (\[maxwell\]) for the static distribution function, $f^{}_{0}(p)$, and using the conservation equations (\[momcons\])–(\[conservkinen\]) in Eq. (\[dst\]), one finds $f^{}_{0}(p_1')f^{}_{0}(p')=f^{}_{0}(p_1)f^{}_{0}(p)$. Therefore, from Eq. (\[dst\]), one obtains ł[dst1]{} &&St=f\_[0]{}(p) \_\^\_ \_1 |\_1-| f\_[0]{}(p\_1)\ && \_[p’]{} . Thus, the collisional term (\[dst\]) ensures all necessary (particle number, momentum, and energy) conservation laws. In particular, one can check that there is no $\ell=0$ and $1$ terms in the sum over $\ell$ of Eq. (\[dst1\]). By that reason, because of zero two first moments of the collision term $\delta St$ (\[dst1\]), there are no contributions from (\[dst1\]) into the continuity equation \[zero $\p$ moment of the Boltzmann equation (\[Boltzlin\])\] and, explicitly, into the momentum equation \[the first $\p$ moment of (\[Boltzlin\])\]. This term $\delta St$ will affect only on the momentum flux tensor $\delta \Pi_{\mu\nu}$ (\[dpidef\]) through the solutions (\[planewavesol\]) for the distribution function $\delta f$ \[see, e.g., Eq. (\[dsigmadef\])\] in terms of the viscosity coefficients \[Eq. (\[etadef\])\]. For the integration over $\p_1$ in Eq. (\[dst1\]), it is convenient to use the system of the center of mass for a given two-body collision, with the symmetry $z$ axis directed along the relative motion of projectile particle having the reduced mass (Fig. \[fig1\]). We transform the integral over $\p_1$ to the relative momentum $\q=\p_1-\p$. Then, using the spherical-coordinate system with the symmetry $z$ axis directed along the vector $\q$ of the relative motion of projectile particle (Fig. \[fig1\]), $\d \q=q^2 \d q\; \sin \theta_q\; \d \theta_q\; \d \varphi_q$ and ($x_q=\cos\theta_q$), one finds from (\[dst1\]) ł[dst2]{} &&St\^\_L \_p Y\_[L 0]{}()St()\ &=&f\_[0]{}(p)f\_[0]{}(p) \_\^\_\ &&\_[-1]{}\^[1]{}x P\_L(x) \_[-1]{}\^[1]{} x\_q\_0\^q\^3q\ &&\_[-1]{}\^[1]{} x’ . Here, $P_\ell(\cos\theta_p)=(4 \pi/(2 \ell+1))^{1/2}\;Y_{\ell 0}(\hat{p})$ is the Legendre polynomial of $\ell$ order, and several transformations of the angle coordinates are performed: ł[xdefs]{} x&=&\_[p]{}x\_1=\_[p\_1]{},\ x’&=&\_[p’]{}x\_1’=\_[p\_1’]{}. Note that the integration over the azimuthal angle of the relative momentum was taken from zero to $\pi$ [@abrkhal]. For the integration over the modulus of the relative momentum $q$, for the fixed $x$ and $x'$, one can change the angle variables to functions of the relative $x_q$: ł[transangles]{} x\_1’&=&x\_1 + x - x’,\ x\_1&=&z x\_q/p +x. For the fixed $x$ and $x'$ we integrate first analytically over the variable $z=q/p^{}_{T}$, where $p^{}_{T}=\sqrt{2 m T}$, and then over $x_q$. ![Integral $y^\lambda\mathcal{J}_{St}(y)$ \[Eq. (\[Istfun\]), solid\] vs $y$ for powers $\lambda=2$ (red), $3$ (blue), and $4$ (black); dashed lines show those with the corresponding asymptotics \[Eq. (\[Istfunexplargey\])\] up to fourth order. []{data-label="fig5"}](fig5.eps){width="49.00000%"} Integrating then, e.g., the $\ell=2$ term, $\delta St_2$ of Eq. (\[dst2\]), explicitly over remaining angles $x$ and $x'$, one obtains ł[integration]{} St\^\_2 = p\^[4]{}\_[T]{}\^\_2 f\_[0]{}(p) I\_[St]{}(p), where ł[intSt]{} I\_[St]{}(p)&=& f\^\_[0]{}(p)\_[-1]{}\^[1]{} x P\_2(x) \_[-1]{}\^[1]{} x’ \_[-1]{}\^[1]{}x\_q\ && \_[0]{}\^ z\^3 z\ &&\ &=& () +\ && (-) = \_[St]{}(p/p\^\_[T]{}), with the error function $\mbox{erf}(y)=2 \int_0^y \d z \exp(-z^2)/\sqrt{\pi}$, ł[Istfun]{} &\_[St]{}(y)=(y)\ &+ (-y\^2),y=p/p\^\_[T]{}. To reduce the BVKE to the perturbation eigenvalue problem \[Eqs. (\[varphipertexp\]) and (\[ompertexp\])\] for the eigenfunctions $\varphi(\hat{p})$ and eigenvalues $c=\om/(k v^{}_{T})$ as solutions of the linear homogeneous equations for $\varphi(\hat{p})$, and dispersion equation for $c$ (Appendix B), we may derive now the accurate constant (independent of $y$) approximations to the function $\mathcal{J}_{\rm St}(y)$ \[Eqs. (\[Istfun\])\]. Using these approximations, one obtains Eqs. (\[tauapprox\]) with (\[nu\]) for the collision term $\delta St$ (\[dst2\]). Indeed, we may note that for the derivation of such approximations the collision term $St$ \[Eq. (\[dst1\])\] can be considered through all of its $\p$ moments. They are integrals over the modulus $p$, which are taken up to the constant from the product of $\mathcal{J}_{\rm St}(p)$ \[Eq. \[Istfun\]\] and the power $p^\lambda$ at $\lambda\geq 2$, in addition to the Maxwell distribution function $f_{0}(p)$, ł[momiintcol]{} \_0\^ p p\^St\^\_2 \^\_2 \_0\^ y y\^\_[St]{}(y)f\_[0]{}(y p\^\_[T]{}). Figure \[fig5\] shows a fast convergence of the product $y^\lambda\;\mathcal{J}_{\rm St}(y)$ \[Eq. (\[Istfun\])\] of the integrand in Eq. (\[momiintcol\]) to its asymptotics at large $y$ in powers of $1/y$ taking enough many terms, ł[Istfunexplargey]{} \_[St]{}(y)= y + + () , y 1, up to fourth-order terms for all $y$ values owing to the power factor $p^\lambda$ ($\lambda \geq 2$). Evaluating a smooth asymptotical function $\mathcal{J}_{\rm St}(y)$ (\[Istfunexplargey\]) with respect to the Maxwell distribution function $f^{}_{0}(y p^{}_{T})$ at the maximum contribution into the integrals (\[momiintcol\]) at $y \approx 1$ ($p \approx p^{}_{T}$), one obtains approximately the damping rate $\nu$ of the collisional term (\[nu\]): ł[nu1]{} = n v\^\_[T]{}\_[St]{}(1) . Note that the second exponent term in Eq. (\[Istfun\]) for $\mathcal{J}_{\rm St}$ was exactly canceled by the second term of the error function expansion, that leads to a good relative accuracy (about 6%) after neglecting terms of the order of $1/y^4$ in asymptotics (\[Istfunexplargey\]). This accuracy can be checked by comparison of (\[nu1\]) with calculations of the exact function $\mathcal{J}_{\rm St}(y)$, and its average $\langle \mathcal{J}_{\rm St}(y)\rangle_{\rm av}$ over $y$ with the static distribution $f^{}_{0}$ (\[maxwell\]), ł[Istfunav]{} \_[St]{}(y)\_[av]{}= = . Calculating $\nu$ traditionally [@abrkhal; @pethsmith2002; @smith2005] through the averaged value (\[Istfunav\]) of the collision term \[ or $\mathcal{J}_{\rm St}(y)$ (\[Istfun\])\] over all momenta $\p$ (or $y$), one obtains ł[nuav]{} && \_[av]{}= n v\^\_[T]{}\_[St]{}(y)\_[av]{}\ && n v\^\_[T]{} d\^2. Thus, both approximations for $\nu$, Eqs. (\[nu\]) and (\[nuav\]), are almost the same within a good relative precision mentioned above. DERIVATIONS OF DISPERSION EQUATION ================================== ł[appB]{} To derive the dispersion equation (\[dispeq\]) for the ratio $c=\omega/(k v^{}_{T})$ with respect to $c$ in the FC regime, one may specify a small perturbation parameter $\epsilon$ \[Eq. (\[eps\])\] in perturbation expansion for $\varphi(\hat{p})$ \[Eqs. (\[varphipertexp\]) and (\[ompertexp\])\]. Then, in the FC regime (small $\epsilon $), one can truncate the expansion of $\varphi(\hat{p})$ (\[varphiexp\]) over spherical functions $Y_{\ell 0}(\hat{p})$ in the plane-wave distribution function $\delta f$ (\[planewavesol\]) at the quadrupole value of $\ell$, $\ell \leq 2~$, because of a fast convergence of the sum (\[varphiexp\]) over $\ell~$ [@brooksyk]. Substituting the plane-wave solution (\[planewavesol\]) with the multipole expansion (\[varphiexp\]) for $\varphi(\hat{p})$ in $\delta f$ into the BVKE (\[Boltzlin\]), after simple algebraic transformations, one finally arrives (within the same approximations used in Appendix A) at the following linear equations ($L=0,1,2,...$) for $\varphi^{}_\ell$: ł[Boltzeqfin]{} \_ \_[L]{}(c)\^\_=0, c=/(k v\^\_[T]{}), where ł[Boltzeqcoef]{} &\^\_[L]{}(c) c \^\_[L]{} - C\_[1;L]{} + \_[L1]{}\_[0]{}\ &+ i \_[L]{} (1-\_[0]{})(1-\_[1]{}), ł[angintclebsh]{} C\_[1;L]{}&=& \_p Y\_[L0]{}() Y\_[10]{}()Y\_[0]{}()\ &=& (C\_[0,10]{}\^[L0]{})\^2, $C_{\ell 0,10}^{L0}$ is the Clebsh-Gordan coefficients [@varshalovich], and $\gamma$ is given by Eq. (\[gamma\]) (Appendix A). We multiplied the BVKE (\[Boltzlin\]) by $Y_{L0}(\hat{p})$, and integrated term by term over angles $\d\Omega_{p}$ of the unit momentum vector $\hat{p}$ in the spherical coordinate system with the polar $z$ axis along the unit wave vector $\hat{k}$. The integrals can be calculated explicitly by using the orthogonal properties of spherical functions and Clebsh-Gordan techniques for calculations of a few spherical function products in the integrand. The matrix $\mathcal{A}_{L\ell}$ has a simple structure. At the diagonal, one finds non-zero values $\mathcal{A}_{\ell\ell}$ depending on the sound velocity $c$. There are also two $L=\ell \pm 1$ lines, parallel to the diagonal, above and below it, with the nonzero number coefficients, depending on the Clebsh-Gordan coefficients through Eq. (\[angintclebsh\]). They are independent of the velocity $c$ and the dimensionless collisional rate $\gamma$. Other matrix elements are zero. The isotropic mean field $\delta U$ (\[effpot\]) influences, through the interaction constant $\mathcal{F}$ (\[Fant\]), on only one matrix element, $\mathcal{A}_{01}=\mathcal{F}/\sqrt{3}-C_{01,0}$. The damping rate constant $\gamma$ related to the collision integral \[Eq. (\[tauapprox\])\] are placed only in the main diagonal $\mathcal{A}_{\ell \ell}$ at $\ell\geq 2$, $\mathcal{A}_{\ell \ell}=c+i \gamma$ because of the conservation conditions, as explained above (Appendix A). For the FC regime, because of large $\gamma$, one notes the convergence of the coefficients $\varphi^{}_\ell$ of the expansion in multipolarities (\[varphiexp\]): Any $\varphi^{}_\ell$ at $\ell \geq 2$ is smaller than $\varphi^{}_{\ell-1}$ by factor $1/(c+i \gamma)$ [@brooksyk; @kolmagpl]. See more explicit expressions for ratios of the amplitudes $\varphi^{}_\ell$ in Appendix C \[Eq. (\[phieq\])\] in the case of the quadrupole truncation of the characteristic matrix $\mathcal{A}~$. Truncating this matrix at the quadrupole value $\ell \leq 2$ and $L \leq 2$, one obtains the following simple $3\times 3$ matrix ł[matrix2]{} \^[(2)]{}=( [ccc]{} c & -C\_[11;0]{} & 0\ /-C\_[01;1]{} & c & -C\_[21;1]{}\ 0 & -C\_[11;2]{} & c + i ), with $~C_{0 1;1}=C_{1 1;0}=1/\sqrt{3}~$ and $~C_{2 1;1}= C_{11;2}=2/\sqrt{15}~$. Accounting for Eq. (\[gamma\]) for $\gamma$, and explicit expressions for these constants $C_{\ell 1;L}$ (\[angintclebsh\]), in the quadrupole FC case, one obtains the condition of existence of nonzero solutions \[det$\mathcal{A}^{(2)}(c)=0$\] of linear equations (\[Boltzeqfin\]), that is the cubic equation (\[dispeq\]) with respect to $c~$. Substituting $c=c_r+ic_i$ into the dispersion equation (\[dispeq\]), one can use the overdamped conditions within the FC regime, ł[fccond0s]{} |c/|=|| 1, |c\_r/c\_i|1. Then, at leading order one obtains (for $\gamma \neq 0$) ł[eqover]{} - i \_1 - \_1 -i \_2=0, where $\mathcal{F}_1=3/5-\mathcal{F}/3$ and $\mathcal{F}_2=(1-\mathcal{F})/3$, and $\gamma$ is given by Eq. (\[gamma\]). Separating real and imaginary parts, at leading order within the conditions (\[fccond0s\]), one finds the overdamped solution, ł[sol0soundr]{} c\_r=0,c\_i=-, that is identical to Eq. (\[linsols0\]). MOMENTS OF THE DISTRIBUTION FUNCTION AND VISCOSITY ================================================== ł[appC]{} For the shear viscosity $\eta$ \[Eq. (\[etadef\])\], one has to calculate the matrices $\mathcal{U}_{\mu\nu}$ \[Eq. (\[veltens\])\] and $\delta \sigma_{\mu\nu}$ \[Eq. (\[dsigmadef\])\]. Taking the polar axis of the spherical coordinate system in the momentum space along the unit wave vector $\hat{k}=\k/k$, we note that these matrices are symmetric with zero nondiagonal terms, and ł[Uxyz]{} \_[xx]{}=\_[yy]{}=-12\_[zz]{}, ł[sigmaxyz]{} \_[xx]{}=\_[yy]{}=-12\_[zz]{}. We find easy these relations using the symmetry arguments and properties of the integrals of the plane-wave solution (\[planewavesol\]) for $\delta f$ over the angles $\d \Omega_p$ of vector $\p~$. Therefore, from Eqs. (\[veltens\]), (\[dsigmadef\]), (\[planewavesol\]) and (\[varphiexp\]) one has to obtain only the simplest $zz$ components, ł[USzz]{} \_[zz]{}&=&2-23 =43 i k \_[z]{}(-i t +i),\ \_[zz]{}&=&\_[zz]{}(-i t +i), where ł[Uzztilde]{} \_[z]{}&=& p\_z f\^\_[0]{}(p) ()=\^\_1,\ \_&=&\_p Y\_[0]{}()(), and ł[Szztilde]{} &&\_[zz]{}=- (3 p\_z\^2 -p\^2) f\^\_[0]{}(p)()\ &=& - p\^4 p f\^\_[0]{}(p) \_p Y\_[20]{}() ()\ &=&-\^\_2. We calculated explicitly the Gaussian-like integrals over $p$ using the static distribution function $f^{}_{0}$ \[Eq. (\[maxwell\])\], ł[intpnu]{} I\_=\_0\^p p\^ f\^\_[0]{}(p) =(), where $\Gamma(x)$ is the $\Gamma$ function. 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--- abstract: 'We discuss the effects of a gauge freedom in representing quantum information processing devices, and its implications for characterizing these devices. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gate-set, leading to different error rates. Consequently, it can be misleading to attach a concrete operational meaning to figures of merit for individual gate-set elements. We propose an alternative operational figure of merit for a gate-set, the mean variation error, and a protocol for measuring this figure.' author: - Junan Lin - Brandon Buonacorsi - Raymond Laflamme - 'Joel J. Wallman' bibliography: - 'references.bib' title: On the freedom in representing quantum operations --- Knowing how to characterize one’s control over a quantum system is of utmost importance in quantum information processing. An experimentalist requires protocols and metrics that appropriately describe the error rate of their quantum processor. Quantum mechanics allows us to assign representations to describe the state of quantum objects and processes, and many figures of merit have been developed to evaluate them based on their representation [@gilchrist2005distance]. For example, the fidelity and trace distance are two commonly quoted measures. Conventional quantum tomography unrealistically assumes that the states and/or measurements being used to probe the unknown operation are ideal. Recently, gate-set tomography (GST) has been developed to avoid making such assumptions by self-consistently inferring all gate-set elements from experimentally estimated probabilities [@Merkel2013; @blume2013robust]. Relaxing these assumptions results in a non-unique representation of the gate-set due to a gauge freedom [@blume2013robust; @blume2017demonstration]. Many conventional measures depend on the particular representation for quantum operations. Therefore, assessing the quality of a quantum device in terms of these metrics applied to non-unique representations may be inaccurate. Despite the broad conceptual importance of representing quantum operations, the impact of the gauge freedom has only occasionally been analyzed, and primarily in the context of gate-dependent noise in randomized benchmarking [@proctor2017randomized; @rudnicki2017gauge; @wallman2018randomized]. In this work we clarify how this gauge freedom affects experimental descriptions and demonstrate some of its implications for interpreting experimentally reconstructed representations of quantum objects. In \[exampleSection\] we demonstrate with an experimentally motivated example that the gauge freedom makes assigning errors to individual operations ambiguous and demonstrates that the gauge freedom is a separate issue from gate-dependent noise. In \[gaugeSection\] we give definitions for gauges and gauge transformations, as well as their role in representing quantum operations as mathematical objects. In \[implicationSection\] we discuss the implications brought by this gauge freedom, in particular addressing why many figures of merit such as the diamond norm distance between a measured gate and a target do not have a concrete operational meaning. We also mention some common practices in tomography that are related to this problem. Lastly, in \[newMeasureSection\], we define and motivate the mean variation error (MVE), a gauge-invariant figure of merit for gate-sets. We provide a protocol to experimentally measure the MVE and demonstrate its behaviour relative to randomized benchmarking through numerical simulations. Assigning errors to operations {#exampleSection} ============================== We now illustrate the gauge freedom with a simple, experimentally relevant example, namely, amplitude damping. A gate-set is a mathematical description of the possible actions executable in an experiment, typically consisting of models for initial states ($\mathds{S}$), gate operations ($\mathds{G}$), and measurements ($\mathds{M}$). If an experimentalist with an ideal quantum system could initialize a qubit in the state $\ket{0}$, apply an arbitrary unitary gate, and measure the expectation value of $Z$, then their control can be represented by the gate-set $$\Phi = \left\{ \mathds{S}_\Phi = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix},\ \mathds{G}_\Phi = \rm{SU}(2) ,\ \mathds{M}_\Phi = Z\right\}.$$ Now suppose that the experimentalist prepares a mixed initial $Z$ state with polarization $\epsilon_1$ and performs a measurement with signal-to-noise ratio $\epsilon_2$. Suppose further that before each gate is applied, the system undergoes amplitude damping with strength $\gamma$ but that the target Hamiltonian is still implemented perfectly. The Pauli-Liouville representation (see Appendix or e.g., [@greenbaum2015introduction]) of the noisy gate-set is then $$\label{gateSetPauli} \Theta = \left\{\mathds{S}_\Theta = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ \epsilon_1 \end{pmatrix},\ \mathds{G}_\Theta = \{\mc{UA}_\gamma: U\in\rm{SU}(2)\},\ \mathds{M}_\Theta =\begin{pmatrix} 0 & 0 & \frac{\epsilon_2}{\sqrt{2}} & 0 \end{pmatrix} \right\}$$ where $\mathcal{U} = U\rho U^\dagger$ denotes the unitary channel acting via conjugation and $$\begin{aligned} \mc{A}_\gamma = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \sqrt{1 - \gamma} & 0 & 0\\ 0 & 0 & \sqrt{1 - \gamma} & 0\\ \gamma & 0 & 0 & 1 - \gamma \end{pmatrix}.\end{aligned}$$ The expectation value of an operator $M$ given an input state $\rho$ is the vector inner product between the Pauli-Liouville representations of the state and measurement operators, $$\text{prob} = {\langle\!\langle M \vert \rho \rangle\!\rangle}.$$ If $m$ gates $\mathcal{G}_1,\ldots,\mathcal{G}_m\in\mathds{G}$ are applied to the state in chronological order before the measurement takes place, the expectation value becomes $$\text{prob} = {\langle\!\langle M \rvert} \mathcal{G}_{m:1} {\lvert \rho \rangle\!\rangle}$$ where we use the shorthand notation $$\mathcal{G}_{b:a} \coloneqq \begin{cases} \mathcal{G}_b \mathcal{G}_{b-1}...\mathcal{G}_a & \text{if}\ b \geq a \\ \mathcal{I} & \text{otherwise.} \end{cases}$$ The above probabilities are preserved under the family of gate-set transformations $$\label{GaugeEqn} {\lvert \rho \rangle\!\rangle} \rightarrow B {\lvert \rho \rangle\!\rangle},\ {\langle\!\langle M \rvert} \rightarrow {\langle\!\langle M \rvert}B^{-1},\ \mathds{G}_\Phi \rightarrow B\mathds{G}_\Phi B^{-1}$$ for some invertible matrix $B$. Because these probabilities are the only experimentally accessible quantities, the same experimental results can be predicted equally well by these two gate-sets. This is the gauge freedom inherent in mathematically representing quantum experiments, in analogy with concepts in thermodynamics and electromagnetism [@jackson2017nonholonomic], with $B$ being called a gauge transformation matrix. The analogy arises from the fact that changing the gauge does not result in observable effects in an experiment, just as changing the electromagnetic gauge would not result in any difference in the measurable electric or magnetic fields. Generally, a gate-set is considered valid if all quantum states can be represented as density matrices, measurements as expectation values of Hermitian operators, and quantum gates as completely-positive, trace-preserving (CPTP) maps as these conditions ensure that probabilities for arbitrary experiments are positive. Gauge transformations do not generally preserve these *canonical constraints*, although the resulting gate-set is nevertheless an equally valid mathematical description of the same experiment. We now present a simple, physically motivated, gauge transformation that yields a gate-set that suggests a different physical interpretation of the experimental system. Applying the gauge transformation matrix $$\label{gauge} B=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & q & 0 & 0\\ 0 & 0 & q & 0\\ 0 & 0 & 0 & q \end{pmatrix}$$ for any $q\in[-1,1]$ to the noisy gate-set in \[gateSetPauli\] yields the equivalent gate-set $$\label{gateSetPauli2} \Theta_q = \left\{\mathds{S}_{\Theta_q} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ q \epsilon_1 \end{pmatrix},\ \mathds{G}_{\Theta_q} = \{\mc{U} \mc{A}_{\gamma,q}: U\in\rm{SU}(2)\},\ \mathds{M}_{\Theta_q} =\begin{pmatrix} 0 & 0 & 0 & \frac{\epsilon_2}{q\sqrt{2}} \end{pmatrix} \right\},$$ where $$\begin{aligned} \mc{A}_{\gamma,q} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \sqrt{1 - \gamma} & 0 & 0\\ 0 & 0 & \sqrt{1 - \gamma} & 0\\ q\gamma & 0 & 0 & 1 - \gamma \end{pmatrix}\end{aligned}$$ and we have used the fact that $\mc{U}$ commutes with $B$ for any $U\in\rm{SU}(2)$. The gauge transformation results in equivalent statistics but suggests a different noise model, namely, relaxation to a mixed state rather than a pure state (corresponding to a different effective temperature). As long as $|q|\in[|\epsilon_2|,1]$, the states, measurements, and transformations all satisfy the canonical constraints for gate-set elements. Note that this gauge freedom does not change the average gate fidelity as $\tr \mc{A}_{\gamma,q}$ is independent of $q$ [@Kimmel2014 Eq. 2.5]. However, the diamond distance from the identity depends on $q$, with $\|\mc{A}_{\gamma,1} - \mc{I}\|_\diamond \approx 2 \|\mc{A}_{\gamma,0} - \mc{I}\|_\diamond $ for $\gamma\in[0,1]$ [@Kueng2016]. Moreover, this example illustrates that noise can be artificially reassigned to different objects, as the state in \[gateSetPauli\] is closer to pure than the one in \[gateSetPauli2\]. Note that the range of gauge transformations is constrained by $\epsilon_2$ and so cannot significantly change the effective temperature for systems with high quality readout. We could have added larger errors by considering a non-unital (e.g., $\mc{A}_{\gamma,q}B$) or unitary gauge transformation at the cost of making the errors gate-dependent and consequently giving a more complicated example. We did not do this as our intent is to clarify that the gauge freedom is more than a basis mis-match [@carignan2018randomized] and is distinct from the issue of gate-dependent noise [@proctor2017randomized; @wallman2018randomized; @carignan2018randomized]. In particular, we note that the full effect of the gauge freedom for states and measurements is unknown. Gauge and Representation of Quantum States {#gaugeSection} ========================================== We have seen that under realistic circumstances, the same experiment can be described by distinct gate-sets that suggest different physical noise models due to a gauge freedom. In this section we illustrate how representations of quantum states are related to the concepts of gauges and gauge transformations. For clarity we focus on the representation for quantum states, but similar arguments can be made about gate and measurement operations. From the point of view of scientific realism, the apparatus (e.g., a qubit) has a physical existence and properties (which may be relative to the environment) independent of our representation. We describe the abstract state of this physical object as a *noumenal state* following the terminology in [@brassard2017equivalence], denoted as $\mathds{N}$ in Figure \[VennDiag\]. Here we slightly change their definition to include in $\mathds{N}$ both physically allowed (denoted as $P$) and forbidden states, such that the set $\mathds{N}$ contains both states that the system can be in, and ones that it cannot be in based on the physics. Quantum mechanics allows us to assign to each noumenal state a mathematical *representation* which is an element of a Hilbert space $\mathcal{H}^d$: for example, one can associate the system with a matrix that summarizes its properties, and the set of all $d \times d$ matrices is called $\mathds{R}$ in the same figure. Such an association is what we call a *gauge* $\Gamma$, which is a bijective map from $\mathds{N}$ to $\mathds{R}$: the bijectivity of the map should be clear from our inclusion of physically-forbidden states in $\mathds{N}$, which allows assigning “some state” to every $d \times d$ matrix. Different choices of $\Gamma$ thus correspond to different mathematical descriptions of the noumenal states. The common formulation of quantum mechanics says that every state of a quantum object can be described by a density operator [@hardy2001quantum], which belongs to a subset of the *canonical constraints* defined at the beginning of \[exampleSection\]. This means that there exists a *canonical gauge* $$\Gamma_1:\ N \rightarrow R,\ \Gamma_1(\mathds{P}) = \mathds{D}_d$$ where $\mathds{D}_d$ is the set of $d \times d$ density operators. In fact, there exists a family of canonical gauges that are all related to $\Gamma_1$ through unitary gauge transformations which preserves the shape of $\mathds{D}_d$. Satisfying the canonical constraint implies that we should work in one of these canonical gauges. Now, consider another gauge $\Gamma_2$ which can be converted from $\Gamma_1$ with a *gauge transformation* $\mathfrak{B}_{12}$, defined by $$\mathfrak{B}_{12} \coloneqq \Gamma_2 (\Gamma_1^{-1}),\ \mathfrak{B}_{12}(r_{*}^{1}) = r_*^2$$ where the $r$’s are members in $\mathds{R}$ and the superscript denotes the gauge in which they are represented. In the light of \[GaugeEqn\], this transformation can be represented in Pauli-Liouville representation as $$\label{gauge B12} {\lvert \mathfrak{B}_{12} (\rho) \rangle\!\rangle} \coloneqq B_{12} {\lvert \rho \rangle\!\rangle},\ {\langle\!\langle \mathfrak{B}_{12}(M) \rvert} \coloneqq {\langle\!\langle M \rvert}B_{12}^{-1},\ \mathds{G}_{\mathfrak{B}_{12}(\Phi)} \coloneqq B_{12}\mathds{G}_{\Phi} B_{12}^{-1}$$ As a subset of $\mathds{R}$, $\Gamma_1(\mathds{P})$ is generally not invariant under an arbitrary gauge transformation: consider a general trace-preserving transformation given by the following transformation matrix $$B_{12} = \begin{pmatrix} 1 & 0\\ \vec{x} & y \end{pmatrix}$$ where $\vec{x}$ is a $(d^2-1) $ by $ 1$ real vector and $y$ is a $(d^2-1) $ by $ (d^2-1)$ real matrix: the image of this affine transformation of $\Gamma_1(\mathds{P})$ is a different subset of $\mathds{R}$. Such a gauge is perfectly valid in principle, provided that *all* the gates and measurement operators are transformed according to \[gauge B12\] as well, even though $\Gamma_2(\mathds{P})$ is no longer the set of density operators. The existence of a non-canonical gauge implies, for example, that a physical state may or may not be represented by a density operator: as illustrated in \[VennDiag\], $r_1^2 \in \Gamma_1(\mathds{P})$ whereas $r_2^2 \notin \Gamma_1(\mathds{P})$. Similarly, a density operator in a non-canonical gauge does not necessarily correspond to a physical state, as $r_3^2 \in \Gamma_1(\mathds{P})$ but $n_3 \notin \mathds{P}$. One example for the state $n_3$ is a qubit state represented as $\frac{1}{2} (I+\frac{10}{9} \sigma_z)$ in a canonical gauge. It is not positive semidefinite and thus lies outside $I_1$, representing an abstract state the qubit cannot be in. Now, using $B_{12} = B$ from \[gauge\] with $q = \frac{9}{10}$, the image of $n_3$ under $\Gamma_2$ becomes $\frac{1}{2} (I+ \sigma_z)$, which *is* a density operator, but only as a consequence of this non-canonical gauge. We conclude that if the gauge is unknown, the mathematical representation does not imply the noumenal state is physically possible. Representations satisfying the canonical constraints are easier to work with, so it is often implicitly assumed that all gate-set elements (obtained from a tomography experiment, for example) are expressed in a canonical gauge. However, this assumption can only be verified by performing perfect experiments, which are axiomatically the operations specified by the canonical constraints (up to a unitary change of basis). Operational interpretations of figures of merit {#implicationSection} =============================================== The existence of this gauge freedom has direct implications for figures of merit used to evaluate quantum operations. The main problem is that there is no way to know whether an experimentally-determined gate-set element is expressed in a canonical gauge. We have already seen in \[exampleSection\] that by changing the gauge, the states can appear as having different expressions; the same holds true for gates and measurement operators. From quantum information theory, we have successfully attached some operational meanings to various distance metrics: an important example is the interpretation for the diamond norm distance between two channels $\mc{A}$ and $\mc{B}$ as the maximum distinguishability between output states under a fixed input [@watrous2018theory]. Mathematically, $$\label{optimalInputOutput} \frac{1}{2} \norm{\mc{A} - \mc{B}}_\diamond = \max_{M\in\Gamma(\mathds{M}), \rho\in\Gamma(\mathds{P})} {\langle\!\langle M_1 \rvert} (\mc{A} - \mc{B}) \otimes I) {\lvert \rho \rangle\!\rangle}$$ where $\mathds{P}$ and $\mathds{M}$ are the set of physically possible states and measurements respectively. This operational meaning is gauge invariant, provided one consistently transforms $\mc{A}$, $\mc{B}$, $\Gamma(\mathds{P})$, and $\Gamma(\mathds{M})$. However, when $\mc{A}$ is an experimentally reconstructed gate and $\mc{B}$ is its ideal target, $\Gamma(\mathds{P})$ and $\Gamma(\mathds{M})$ are unknown and so the above maximization that leads to its operational meaning cannot be performed. To obtain concrete numbers, people calculate $$\frac{1}{2} \norm{\mc{A} - \mc{B}}_\diamond = \max_{M\in \mu(\mathds{D}_{d}), \rho\in\mathds{D}_{d}} {\langle\!\langle M_1 \rvert} (\mc{A} - \mc{B}) \otimes I) {\lvert \rho \rangle\!\rangle}$$ where $\mu(\mathds{D}_{d})$ is the set of all POVMs. However, this assumes that the reconstructed $\mc{A}$ and the ideal target $\mc{B}$ are expressed in a canonical gauge. While $\mc{B}$ is an ideal gate, its representation may not be unitary in a non-canonical (and unknown) gauge. Other works have reported that the quantity $\frac{1}{2} \norm{\mc{A} - \mc{B}}_\diamond$ can be changed by changing the gauge and used this to minimize reported error rates [@blume2017demonstration], however, such changes are obtained by implicitly changing the set of physically allowed states and measurements. Note that even in one special case of interest where $\mc{B} = \mc{I}$, which is gauge invariant, $\mc{A}$ is still reconstructed in an unknown gauge. We briefly discuss several common practices related to this gauge freedom in quantum tomography. The process known as “gauge optimization” is commonly adopted in GST experiments whereby the gauge transformation matrix $B$ is varied to minimize the distance from the target gate-set according to a (non-gauge-invariant) weighted distance measure [@blume2017demonstration]. However, this optimized gauge is just as arbitrary as any other gauge, and one still cannot know whether the resultant gate-set is a faithful representation of the apparatus, in particular, whether the states and measurements that satisfy the canonical constraints are actually the images of the set of physically possible states and measurements respectively. Moreover, such optimization undermines a common use of tomography, namely assessing the performance of a system against some external threshold (e.g., a fault-tolerance threshold). Altering the gauge to make the tested channel similar to its target will artificially reduce the distance between the two. Additionally, assigning different weights on SPAM and gates during gauge optimization will result in a difference in the output, and such weights are only based on rough initial guess about the relative quality of these components. Another common approach is Maximum Likelihood Estimation (MLE), which takes the estimated gate-set to be the one that maximizes the likelihood function of obtaining the experimental data, while restricting the gate-set elements to satisfy the canonical physicality constraints [@medford2013self; @brida2012quantum]. MLE does not resolve the gauge ambiguity either, because all gauge-equivalent gate-sets are equally likely to produce the data by definition. In the process of optimization, one will find that the likelihood function profile has the same value wherever two points are related by a gauge-transformation, and the actual output is largely a matter of the optimization algorithm and the initial parameters [@blume2015turbocharging]. A Gauge-Invariant Measure for Gate-Sets {#newMeasureSection} ======================================= The gauge freedom prevents one from using conventional distance measures to faithfully evaluate the quality of individual quantum operations. Note that our discussion is carried out in the absence of any additional errors such as finite-counting, and in a real experiment the situation becomes even more complicated. Fundamentally, this problem is due to the limited information that can be gained from experimental probabilities. A gauge-transformation re-assigns state, gate, and measurement “errors” by adjusting their relative appearance in different representations, while keeping the experimental measurables unchanged, although some degrees of freedom can be fixed by convention (e.g., that the state preparation is diagonal in the $Z$ eigenbasis). We now propose a gauge-invariant figure of merit for a *gate-set*. As far as we know, this is the first fully gauge-invariant measure, addressing a problem raised in Ref. [@blume2015turbocharging]. Let $\Phi$ denote the gate-set $\{\mathds{S}, \mathds{G}, \mathds{M}\}$ and $C$ denote a particular experiment with input state $\rho \in \mathds{S}$, measurement $M \in \mathds{M}$, and a set of $m$ gates $\mc{G}_1...\mc{G}_m$ each selected from $\mathds{G}$. The only observable properties of the experiment $C$ is the probability distribution over outcomes. We can quantify the error of the experiment by the total variation distance between the observed and ideal distributions over outcomes, $$\delta d (C,\tilde{C}) \coloneqq \frac{1}{2} \sum_i \abs{\Tr[\tilde{M}_i^\dagger \tilde{G}_{m:1} (\tilde{\rho})] - \Tr[M_i^\dagger G_{m:1} (\rho)]}$$ where the tilde represents real versions of the operations. The total variation distance is a metric over probability distributions. Denoting the set of all experiments with $m$ gates by $\mathds{A}_{m}$, we further define the *Mean Variation Error* (MVE) over $\mathds{A}_{m}$ with the underlying gate-set $\Phi$ as $$\label{MVE definition} x(\Phi, m) \coloneqq \frac{1}{|\mathds{A}_{m}|}\sum_{C\in\mathds{A}_m} [\delta d (C, \tilde{C})]$$ Note that although in each $C$ only one state and one measurement is allowed (in order for the final outcome to be a valid probability distribution), there is no constraint on how many are included in the gate-set. The size of $\mathds{A}_{m}$ is thus given by $\abs{\mathds{A}_{m}} = \abs{\mathds{S}} \abs{\mathds{G}}^m \abs{\mathds{M}}$. +\[error bars/.cd,y dir=both,y explicit\] table \[x=m, y=mean, y error=std, only marks, col sep=comma\] [depolarizing\_identity.csv]{}; +\[error bars/.cd,y dir=both,y explicit\] table \[x=m, y=mean, y error=std, only marks, col sep=comma\] [depolarizing\_general.csv]{}; +\[error bars/.cd,y dir=both,y explicit\] table \[x=m, y=mean, y error=std, only marks, col sep=comma\] [unitary\_identity.csv]{}; +\[error bars/.cd,y dir=both,y explicit\] table \[x=m, y=mean, y error=std, only marks, col sep=comma\] [unitary\_general.csv]{}; +\[error bars/.cd,y dir=both,y explicit\] table \[x=m, y=mean, only marks, col sep=comma\] [shortunitary\_identity.csv]{}; +\[error bars/.cd,y dir=both,y explicit\] table \[x=m, y=mean, only marks, col sep=comma\] [shortunitary\_general.csv]{}; The MVE quantifies how well the apparatus performs an average experiment from the gate-set. In the case where the measurement is a projective measurement in the basis of the initial state (i.e., $\rho$ = $M_i$ for some $i$) and the gate sequence is self-inverting, $\delta d(C,\tilde{C})$ can be simplified as $$\begin{aligned} \delta d (C, \tilde{C}) &= \frac{1}{2} \left(\abs{\Tr[\tilde{M}_i^\dagger \tilde{G}_{m:1} (\tilde{\rho})] - 1} + \sum_{j \neq i} \abs{\Tr[(I - \tilde{M}_{j})^\dagger \tilde{G}_{m:1} (\tilde{\rho})] - 0} \right)\\ &= 1 - \Tr[\tilde{M}_i^\dagger \tilde{G}_{m:1} (\tilde{\rho})] \end{aligned}$$ whose average over $\mathds{A}_m$ is just 1 minus the “survival probability” plotted in a conventional randomized benchmarking experiment. When $\mathds{G}$ is a unitary 2-design, the MVE restricted to self-inverting gate sequences is well-approximated by a linear relation to first order in the average error rate [@wallman2018randomized; @proctor2017randomized]. However, for generic gate sequences, the MVE behaves differently depending on the underlying error model. This behavior provides additional information about the underlying error mechanism compared to a conventional randomized benchmarking experiment [@wallman2015bounding]. To illustrate this, we simulated random circuits of varying length $m$ sampled from the gate-set $\Phi = \{\mathds{S} = \ketbra{0}, \mathds{G} = \textit{Cl}_1, \mathds{M} = \ketbra{0}\}$ (with $\textit{Cl}_1$ denoting the set of 1-qubit Clifford gates), where erroneous gates are represented as $\tilde{\mathcal{G}} =\mathcal{E} \mathcal{G}$ for a fixed error channel $\mathcal{E}$. We simulated two types of random circuits: circuits from the entire set of possible experiments allowed by the gate-set, and circuits restricted to self-inverting gate sequences. In both simulations, the state and measurements are assumed to be error-free. The results are shown in \[New-metric-plot\]. When the error is a depolarizing channel, the MVE scales linearly with the gate sequence length $m$ for both random and self-inverting circuits, with the slope for random circuits being $\sim 1/3$ the slope for self-inverting circuits. This is because when the state is transformed onto the xy-plane of the Bloch sphere right before measurement (which happens about 2/3 of the time), the depolarizing channel does not affect the outcome probability of a z-axis measurement, resulting in an MVE of 0 for those circuit sequences. Additionally, there is no statistical error present for the self-inverting circuit under this error model because all circuits of the same sequence length have exactly the same overall error, as the error channel commutes with all the gates in $Cl_1$. In contrast, for a gate-independent unitary error, the scaling remains linear for the self-inverting circuits but exhibits a $\sqrt{m}$ scaling for generic circuits. This occurs because when the state system is in the xy-plane before measurement, each error contributes a random sign to the probability of each outcome, whereas when the system is on the $z$ axis each error has to contribute a systematic sign [@wallman2015bounding]. As shown in \[New-metric-plot\], this implies that restricting to self-inverting circuits can underestimate the MVE by over an order of magnitude in the small-$m$ regime, which is relevant for near-term quantum computer applications. Unlike other distance measures where an improvement in quality can be caused by a bias in choosing a gauge, a decrease in MVE is unequivocally an improvement due to its gauge-invariance and because, by definition, the output probability distribution gets closer to the ideal distribution. Furthermore, the MVE captures the relevant behavior for generic circuits, rather than just self-inverting circuits which, by design, perform no useful computation. A protocol for estimating the MVE of a gate-set $\Phi = \{\mathds{S}, \mathds{G}, \mathds{M}\}$ is as follows: 1. Select $N_m$ random experiments $C\in\mathds{A}_m$, for some $N_m$ large enough to accurately approximate the average. 2. Repeat each experiment $C$ $K_m$ times to estimate $\langle \tilde{M}_i, \tilde{G}_{m:1}(\tilde{\rho}) \rangle$ for each $C$. 3. Compute the ideal probabilities $\langle M_i, G_{m:1}(\rho) \rangle$ for each observed outcome of $\tilde{C}$. 4. Calculate $\delta d(C, \tilde{C})$ for each experiment $C$, average over them to estimate $x(\Phi, m)$. 5. Repeat step 1–4 for different values of $m$ to measure the scaling behaviour of MVE. Note that if $\mathds{G}$ is a unitary 2-design and the states and measurements are chosen appropriately, the applied operations are identical to those used to estimate the unitarity [@Wallman2015]. The primary difference in the protocol is that it is more scalable, more general, and has different post-processing. The scalability of the above protocol is affected by the number of experiments $N_m$, the number of repetitions for each experiment $K_m$, and the complexity of calculating the ideal probabilities. The number of experiments determine the accuracy of the MVE, and can be estimated using Hoeffding’s inequality independently of the number of qubits [@hoeffding1963probability]. The complexity in the protocol is determined by the complexity of calculating probabilities and by the number of repetitions required to estimate $\delta d(C,\tilde{C})$ to a fixed precision. The number of repetitions required to estimate $\delta d(C,\tilde{C})$ to a fixed precision is polynomial in the number of outcomes [@chan2014]. To efficiently characterize multi-qubit gate-sets (where the number of raw outcomes grows exponentially with the number of qubits), we can coarse-grain the measurements over sets of outcomes. The computational complexity of calculating each probability will depend on the gate-set in question. The ideal probabilities can be efficiently computed if $\mathds{G}$ is the $N$-qubit Clifford group [@gottesman1998heisenberg; @nest2008classical]. For gate-sets containing only one- and two-qubit gates and product states and measurements, the MVE can be computed for small values of $m$. However, for a generic gate-set, each probability will be hard to compute. Of course, for small systems with a few qubits, this procedure can nonetheless be performed quickly on a classical computer. An experimentalist can perform a feedback loop whereby they update the control parameters, rerun the MVE evaluation experiment (potentially for some fixed value of $m$) and compare to the previous result to see if the error has decreased. Protocols that use feedback from experimental outcomes to improve control over quantum devices have been proposed before, such as in [@kelly2014optimal] where control parameters were optimized by maximizing the randomized benchmarking survival probability for a fixed sequence length. Optimizing the MVE instead of the randomized benchmarking survival probability corresponds to minimizing the effect of errors on generic quantum circuits, rather than minimizing the effect of errors on self-inverting circuits. As demonstrated in \[New-metric-plot\], errors in self-inverting circuits may be substantially smaller than those in generic circuits because such circuits suppress coherent errors and implement a form of randomized dynamical decoupling [@viola2005]. JJW acknowledges helpful discussions with Robin Blume-Kohout. This research was supported by the U.S. Army Research Office through grant W911NF-14-1-0103, the Government of Ontario, and the Government of Canada through CFREF, NSERC and Industry Canada. Pauli-Liouville representation ============================== Here we present the Pauli-Liouville representation for gate-set elements. For simplicity we focus on system of $n$ qubits where quantum states can be represented as $2^{n} \times 2^{n}$ Hermitian operators. It is commonly known that the set of (tensor products of) normalized Pauli matrices, which we denote as $\mathbf{P}_n$, form an orthonormal basis for all $2^{n} \times 2^{n}$ Hermitian operators. Elements in $\mathbf{P}_n$ are of the form $$P = \bigotimes_{k} \left(\frac{\sigma_k}{\sqrt{2}}\right)$$ and each $\sigma_k$ is a member from the single-qubit Pauli group, $\mathbf{P}_1 \coloneqq \{\sigma_0 = I,\ \sigma_1 = X,\ \sigma_2 = Y,\ \sigma_3 = Z\}$. The orthonormality is defined with respect to the Hilbert-Schmidt inner product $$\langle P_i,\ P_j \rangle_{HS} \coloneqq \Tr[P_i^\dagger P_j] = \delta_{ij},\ P_i, P_j \in \mathbf{P}_n$$ Any $2^n$ by $2^n$ Hermitian matrix can be represented as a real linear combination of Pauli basis matrices. Writing these inner products as components of a vector will define a representation in the space of $2^{2n} \times 1$ real vectors, which is isomorphic to the set of $2^n \times 2^n$ real matrices. For every $2^n$ by $2^n$ Hermitian matrix $\rho$, we define its Pauli-Liouville representation as follows: $${\lvert \rho \rangle\!\rangle} \coloneqq \sum_{i} \Tr[\rho P_i] {\lvert i \rangle\!\rangle}$$ and define an element $\sigma$ in the dual space (e.g., representing a measurement operator) as $${\langle\!\langle \sigma \rvert} \coloneqq \sum_{i} \Tr[P_i \sigma]^* {\langle\!\langle i \rvert}$$ where ${\lvert i \rangle\!\rangle}$ and ${\langle\!\langle i \rvert}$ are standard computational (column and row) basis vectors with $1$ in the $i$-th entry and $0$ elsewhere. We see that the Hilbert-Schmidt inner product is now transformed into an Euclidean inner product: $$\begin{split} {\langle\!\langle \sigma \vert \rho \rangle\!\rangle} &= \sum_{i,j} \Tr[\rho P_i] \Tr[ P_j \sigma]^* {\langle\!\langle j \vert i \rangle\!\rangle}\\ &= \sum_{i,j} \Tr[\rho P_i] \Tr[P_j \sigma]^* \Tr[P_i P_j]\\ &= \Tr[\sum_{i} \Tr[\rho P_i] P_i \sum_j \Tr[P_j \sigma]^* P_j]\\ &= \Tr[\sigma^\dagger \rho] \end{split}$$ where the following equation is used: $$\begin{split} \sum_i \Tr[P_i A]^* P_i & = \sum_i \Tr[P_i^\dagger A]^* P_i\\ &= \sum_i \Tr[\overline{(A^\dagger P_i)^\dagger}] P_i\\ &= \sum_i \Tr[(A^\dagger P_i)^T] P_i\\ &= \sum_i \Tr[A^\dagger P_i] P_i = A^\dagger \end{split}$$ Now, define the Pauli-Liouville representation of a (linear) map $\mc{G}$ as $\mc{A}_{\mc{G}}$, which has components $$(\mc{A}_{\mc{G}})_{i j} \coloneqq \Tr[P_{i} \mc{G}(P_{j})]$$ then the post-state of $\mc{G}$ acting on a state $\rho$, written all in the Pauli basis, can be shown to be equal to a matrix multiplication: $$\label{PauliProduct} \begin{split} {\lvert \mc{G}(\rho) \rangle\!\rangle} & = \sum_{i} \Tr[\mc{G}(\rho) P_{i}] {\lvert i \rangle\!\rangle}\\ &= \sum_{i} \Tr[\mc{G}(\sum_{j} \Tr[\rho P_{j}] P_{j}) P_{i}] {\lvert i \rangle\!\rangle}\\ &= \sum_{i j} \Tr[\rho P_{j}] \Tr[\mc{G}( P_{j}) P_{i}] {\lvert i \rangle\!\rangle}\\ &= \sum_{i j} (\mc{A}_{\mc{G}})_{i j} ({\lvert \rho \rangle\!\rangle})_{j} {\lvert i \rangle\!\rangle}\\ &= \mc{A}_{\mc{G}} {\lvert \rho \rangle\!\rangle} \end{split}$$ Thus, series of gates are conveniently expressed as matrix multiplications (from the left) in the Pauli-Liouville representation.

--- abstract: | **Background:** A common survival strategy of microorganisms subjected to stress involves the generation of phenotypic heterogeneity in the isogenic microbial population enabling a subset of the population to survive under stress. In a recent study, a mycobacterial population of *M. smegmatis* was shown to develop phenotypic heterogeneity under nutrient depletion. The observed heterogeneity is in the form of a bimodal distribution of the expression levels of the Green Fluorescent Protein (GFP) as reporter with the *gfp* fused to the promoter of the *rel* gene. The stringent response pathway is initiated in the subpopulation with high *rel* activity. **Results:** In the present study, we characterise quantitatively the single cell promoter activity of the three key genes, namely, *mprA, sigE* and *rel*, in the stringent response pathway with *gfp* as the reporter. The origin of bimodality in the GFP distribution lies in two stable expression states, i.e., bistability. We develop a theoretical model to study the dynamics of the stringent response pathway. The model incorporates a recently proposed mechanism of bistability based on positive feedback and cell growth retardation due to protein synthesis. Based on flow cytometry data, we establish that the distribution of GFP levels in the mycobacterial population at any point of time is a linear superposition of two invariant distributions, one Gaussian and the other lognormal, with only the coefficients in the linear combination depending on time. This allows us to use a binning algorithm and determine the time variation of the mean protein level, the fraction of cells in a subpopulation and also the coefficient of variation, a measure of gene expression noise. **Conclusions:** The results of the theoretical model along with a comprehensive analysis of the flow cytometry data provide definitive evidence for the coexistence of two subpopulations with overlapping protein distributions. --- [**Phenotypic Heterogeneity in Mycobacterial Stringent Response** ]{}\ Sayantari Ghosh$^{1}$, Kamakshi Sureka$^{2}$, Bhaswar Ghosh$^{3}$, Indrani Bose$^{1\ast}$, Joyoti Basu$^{2}$, Manikuntala Kundu$^{2}$ \ Background {#background .unnumbered} ========== Microorganisms are subjected to a number of stresses during their lifetime. Examples of such stresses are: depletion of nutrients, environmental fluctuations, lack of oxygen, application of antibiotic drugs etc. Microorganisms take recourse to a number of strategies for survival under stress and adapting to changed circumstances [@1; @2; @3; @4]. A prominent feature of such strategies is the generation of phenotypic heterogeneity in an isogenic microbial population. The heterogeneity is advantageous as it gives rise to variant subpopulations which are better suited to persist under stress. Bistability refers to the appearance of two subpopulations with distinct phenotypic characteristics [@5; @6]. In one of the subpopulations, the expression of appropriate stress response genes is initiated resulting in adaptation . There are broadly two mechanisms for the generation of phenotypic heterogeneity [@7; @8]. In “responsive switching” cells switch phenotypes in response to perturbations associated with stress. In the case of “spontaneous stochastic switching”, transitions occur randomly between the phenotypes even in the absence of stress. Responsive switching may also have a stochastic component as fluctuations in the level of a key regulatory molecule can activate the switch once a threshold level is crossed [@1; @3; @5; @6]. The pre-existing phenotypic heterogeneity, an example of the well-known “bet-hedging-strategy, keeps the population in readiness to deal with future calamities. Using a microfluidic device, Balaban et al. [@9] have demonstrated the existence of two distinct subpopulations, normal and persister, in a growing colony of *E. coli* cells. The persister subpopulation constitutes a small fraction of the total cell population and is distinguished from the normal subpopulation by a reduced growth rate. Since killing by antibiotic drugs like ampicillin depends on the active growth of cell walls, the persister cells manage to survive when the total population is subjected to antibiotic treatment. The normal cells, with an enhanced growth rate are, however, unable to escape death. Once the antibiotic treatment is over, some surviving cells switch from the persister to the normal state so that normal population growth is resumed [@9; @11]. A simple theoretical model involving transitions between the normal and persister phenotypes explains the major experimental observations well [@9; @10]. In the case of environmental perturbations, Thattai and Oudenaarden [@12] have shown through mathematical modelling that a dynamically heterogeneous bacterial population can under certain circumstances achieve a higher net growth rate conferring a fitness advantage than a homogeneous one. Mathematical modelling further shows that responsive switching is favoured over spontaneous switching in the case of rapid environmental fluctuations whereas the reverse is true when environmental perturbations are infrequent [@7]. Another theoretical prediction that cells may tune the switching rates between phenotypes to the frequency of environmental changes has been verified in an experiment by Acar et al. [@13] involving an engineered strain of S. cerevisiae which can switch randomly between two phenotypes. The major feature of all such studies is the coexistence of two distinct subpopulations in an isogenic population and their interconversions in the presence/absence of stress. Bistability, i.e., the partitioning of a cell population into two distinct subpopulations has been experimentally observed in a number of cases [@1; @3; @5]. Some prominent examples include: lysis/lysogeny in bacteriophage $\lambda$ [@14], the activation of the lactose utilization pathway in *E. coli* [@15] and the galactose utilization genetic circuit in S. cerevisiae [@16], competence development in *B. subtilis* [@6; @17; @18] and the stringent response in mycobacteria [@19]. The mycobacterial pathogen *M. tuberculosis*, the causative agent of tuberculosis, has remarkable resilience against various physiological and environmental stresses including that induced by drugs [@20; @21; @22]. On tubercular infection, granulomas form in the host tissues enclosing the infected cells. Mycobacteria encounter a changed physical environment in the confined space of granulomas with a paucity of life-sustaining constituents like nutrients, oxygen and iron [@23; @24]. The pathogens adapt to the stressed conditions and can survive over years in the so-called latent state. In vitro too, *M. tuberculosis* has been found to persist for years in the latent state characterised by the absence of active replication and metabolism [@25]. Researchers have developed models simulating the possible environmental conditions in the granulomas. One such model is the adaptation to nutrient-depleted stationary phase [@26]. The processes leading to the slowdown of replicative and metabolic activity constitute the stringent response. In mycobacteria, the expression of *rel* initiates the stringent response which leads to persistence. The importance of Rel arises from the fact that it synthesizes the stringent response regulator ppGpp (guanosine tetraphosphate) [@27] and is essential for the long-term survival of *M. tuberculosis* under starvation [@28] and for prolonged life of the bacilli in mice [@29]. Key elements of the stringent response and the ability to survive over long periods of time under stress are shared between the mycobacterial species *M. tuberculosis* and *M. smegmatis* [@30]. Recent experiments provide knowledge of the stress signaling pathway in mycobacteria linking polyphosphate (poly P), the two-component system MprAB, the alternate sigma factor SigE and Rel [@31]. In an earlier study [@19], we investigated the dynamics of *rel-gfp* expression (*gfp* fused with *rel* promoter) in *M. smegmatis* grown upto the stationary phase with nutrient depletion serving as the source of stress. In a flow cytometry experiment, we obtained evidence of a bimodal distribution in GFP levels and suggested that positive feedback in the stringent response pathway and gene expression noise are responsible for the creation of phenotypic heterogeneity in the mycobacterial population in terms of the expression of *rel-gfp*. Positive feedback gives rise to bistability [@5; @6], i.e., two stable expression states corresponding to low and high GFP levels. We further demonstrated hysteresis, a feature of bistability, in *rel-gfp* expression. The mathematical model developed by us to study the dynamics of the stringent response pathway predicted bistability in a narrow parameter regime which, however, lacks experimental support. In general, to obtain bistability a gene circuit must have positive feedback and cooperativity in the regulation of gene expression. Recently, Tan et al. [@32] have proposed a new mechanism by which bistability arises from a noncooperative positive feedback circuit and circuit-induced growth retardation. The novel type of bistability was demonstrated in a synthetic gene circuit. The circuit, embedded in a host cell, consists of a single positive feedback loop in which the protein product X of a gene promotes its own synthesis in a noncooperative fashion. The protein decay rate has two components, the degradation rate and the dilution rate due to cell growth. In the circuit considered, production of X slows down cell growth so that at higher concentrations of X, the rate of dilution of X is reduced. This generates a second positive feedback loop since increased synthesis of X proteins results in faster accumulation of the proteins so that the protein concentration is higher. The combination of two positive feedback loops gives rise to bistability in the absence of cooperativity. A related study by Klumpp et al [@35] has also suggested that cell growth inhibition by a protein results in positive feedback. Results {#results .unnumbered} ======= In this paper, we develop a theoretical model incorporating the effect of growth retardation due to protein synthesis [@32; @33]. We provide some preliminary experimental evidence in support of the possibility. In our earlier study [@19], bimodality in the *rel-gfp* expression levels was observed. As a control, GFP expression driven by the constitutive *hsp60* promoter was monitored as a function of time. A single bright population was observed at different times of growth (Figure S4 of [@19]). The unimodal rather than bimodal distribution ruled out the possibility that clumping of mycobacterial cells and cell-to-cell variation of plasmid copy number were responsible for the observed bimodal fluorescence intensity distribution of *rel* promoter driven GFP expression. In the present study, we perform flow cytometry experiments to monitor *mprA-gfp* and *sigE-gfp* expression levels. The distribution of GFP levels in each case is found to be bimodal. We determine the probability distributions of the two subpopulations associated with low and high expression levels at different time points in the three cases of *mprA-gfp*, *sigE-gfp* and *rel-gfp* expression. In each case, the total distribution is a linear combination of two invariant distributions with the coefficients in the linear combination depending on time. The results of hysteresis experiments are also reported. Mathematical modeling of the stress response pathway {#mathematical-modeling-of-the-stress-response-pathway .unnumbered} ----------------------------------------------------  Figure 1 shows a sketch of the important components of the stress response pathway in *M. smegmatis* subjected to nutrient depletion [@19; @31]. The operon *mprAB* consists of two genes *mprA* and *mprB* which encode the histidine kinase sensor MprB and its partner the cytoplasmic response regulator MprA respectively. The protein pair responds to environmental stimuli by initiating adaptive transcriptional programs. Polyphosphate kinase 1 (PPK1) catalyses the synthesis of polyphosphate (poly P) which is a linear polymer composed of several orthophosphate residues. Mycobacteria possibly encounter a phosphate-limited environment in macrophages. Sureka et al. [@31] proposed that poly P could play a critical role under ATP depletion by providing phosphate for utilisation by MprAB. A recent experiment [@33] on a population of *M. tuberculosis* has established that the MTB gene *ppk1* is significantly upregulated due to phosphate starvation resulting in the synthesis of inorganic poliphosphate (poly P). The two-component regulatory system SenX3-RegX3 is known to be activated on phosphate starvation in both *M. smegmatis* [@34] and *M. tuberculosis* [@33]. In the latter case, RegX3 has been shown to regulate the expression of *ppk1*, a feature expected to be shared by *M. smegmatis*. In both the mycobacterial populations, poly P regulates the stringent response via the *mprA-sigE-rel* pathway [@31]. In our experiments, nutrient depletion possibly gives rise to phosphate starvation. On activation of the *mprAB* operon, MprB autophosphorylates itself with poly P serving as the phosphate donor [@31; @33]. The phosphorylated MprB-P phosphorylates MprA via phosphotransfer reactions. There is also evidence that MprB functions as a MprA-P (phosphorylated MprA) phosphatase. MprA-P binds the promoter of the *mprAB* operon to initiate transcription. A positive feedback loop is functional in the signaling network as the production of MprA brings about further MprA synthesis. The *mprAB* operon has a basal level of gene expression independent of the operation of the positive feedback loop. Once the *mprAB* operon is activated, MprA-P regulates the transcription of the alternate sigma factor gene *sigE*, which in turn controls the transcription of *rel*. We construct a mathematical model to study the dynamics of the above signaling pathway. The new feature included in the model takes into account the possibility that the production of stress-induced proteins like MprA and MprB slows down cell growth. This effectively generates a positive feedback loop as explained in Refs. [@32; @33].  **** Figure 2(a) shows the mean amount of GFP fluorescence in the total mycobacterial population as measured in a flow cytometry experiment (*mprA* promoter fused with *gfp*) versus time. Figure 2(a) shows the specific growth rate of the cell population versus time. The inset shows the experimental growth curve for the mycobacterial population. The growth was monitored by recording the absorbance values at 600 nm spectrophotometrically (see Methods). The specific growth rate at time $t$ is given by $\frac{1}{N(t)}\frac{dN(t)}{dt}$ where $N(t)$ is the number of mycobacterial cells at time $t$. Nutrient depletion limits growth and proliferation and culminates in the activation of stress response genes. It appears that in many cases rapid growth and stress response are mutually exclusive so that the production of a stress response protein gives rise to a slower growth rate [@36]. The balance between the expressions of growth-related and stress-induced genes determines the cellular phenotype with respect to growth rate and stress response. Persister cells in both *E. coli* [@9; @11] and mycobacteria [@21; @22] have slow growth rates. In the case of *M. smegmatis*, we have already established that the slower growing persister subpopulation has a higher level of Rel, the initiator of stringent response, as compared to the normal subpopulation [@19]. The new addition to our mathematical model [@19] involves nonlinear protein decay rates arising from cell growth retardation due to protein synthesis. We briefly discuss the possible origins of the nonlinearity and its mathematical form [@32; @33]. The temporal rate of change of protein concentration is a balance between two terms: rate of synthesis and rate of decay. The decay rate constant $(\gamma_{eff})$ has two components: the dilution rate due to cell growth ($\mu$) and the natural decay rate constant ($\gamma$), i.e., $\gamma_{eff}=\mu+\gamma$ where $\mu$ is the specific growth rate. In many cases, the expression of a protein results in cell growth retardation [@32; @33]. The general form of the specific growth rate in such cases is given by $$\mu=\frac{\phi}{1+\theta x}$$ where $x$ denotes the protein concentration and $\phi,\theta$ are appropriate parameters. In Ref. [@32], the expression for $\mu$ (Eq. (1)) is arrived at in the following manner. The Monod model [@37] takes into account the effect of resource or nutrient limitation on the growth of bacterial cell population. The rate of change in the number of bacterial cells is $$\frac{dN}{dt}=\mu N$$ where the specific growth rate $\mu$ is given by $$\mu=\mu_{max\:}\frac{s}{k+s}$$ In (3), $s$ is the nutrient concentration and $k$ the half saturation constant for the specific nutrient. When $s=k$, the specific growth rate attains its half maximal value ($\mu_{max}$ is the maximum value of specific growth rate). The metabolic burden of protein synthesis affecting the growth rate is modeled by reducing the nutrient amount $s$ by $\epsilon$, i.e., $$\mu=\frac{\mu_{max}}{1+\frac{k}{s(1-\epsilon)}}$$ The magnitude of $\epsilon$ is assumed to be small and proportional to the protein concentration $x$. Following the procedure outlined in the Supplementary Information of [@32], namely, applying Taylor’s expansion to (4) and putting $\epsilon=\lambda x$ ($\lambda$ is a constant), one obtains the expression in Eq. (1) with $\phi\mbox{ = }\frac{\mu_{max}s}{s+k}$ and $\theta=\frac{k\lambda}{s+k}$ . Thus, the decay rate of proteins has the form $-\gamma_{eff}x=-(\gamma+\mu)\, x$ where $\mu$ is given by Eq. (1). There are alternative explanations for the origin of the nonlinear decay term, e.g., the synthesis of a protein may retard cell growth if it is toxic to the cell [@35]. In the case of mycobacteria, there is some experimental evidence of cell growth retardation brought about by protein synthesis. The response regulator MprA has an essential role in the stringent response pathway leading to persistence of mycobacteria under nutrient deprivation. Inactivation of the regulator in an *mprA* insertion mutant resulted in reduced persistence in a murine model but the growth of the mutant was proved to be significantly higher than that observed in the cases of the wild-type species [@38; @39].  Our experimental data (Figure 3) provide further support to the hypothesis that MprA synthesis leads to reduced specific growth rate. The data points represent GFP fluorescence intensity with *gfp* fused to the *mprA* promoter. The GFP acts as a reporter of the *mprA* promoter activity culminating in MprA (also MprB) synthesis. The data points shown in Figure 3 are those that correspond to the growth period of 16-23 hours in Figure 2. The data points are fitted by an expression similar to that in Eq. (1) with $\mu_{max}^{GFP}=0.94$ and $\theta^{GFP}=0.317$. The differential equations describing the temporal rates of change of key protein concentrations in our model are described in the Additional File 1. Solving the equations, one finds the existence of bistability, i.e., two stable expression states in an extended parameter regime. Figures S1 A-C (Additional File 1) show the plots for bistability and hysteresis for the proteins MprA, SigE and Rel versus the autophosphorylation rate. In the deterministic scenario and in the bistable regime, all the cells in a population are in the same steady state if exposed to the same environment and with the same initial state. The experimentally observed heterogeneity in a genetically identical cell population is a consequence of stochastic gene expression. The biochemical events involved in gene expression are inherently probabilistic [@40; @41] in nature. The uncertainty introduces fluctuations (noise) around mean expression levels so that the single protein level of the deterministic case broadens into a distribution of levels. In the case of bistable gene expression, the distribution of protein levels in a population of cells is bimodal with two distinct peaks. Bimodal Expression of *mprA*, *sigE* and *rel* in *M. smegmatis* {#bimodal-expression-of-mpra-sige-and-rel-in-m.-smegmatis .unnumbered} ---------------------------------------------------------------- In the earlier study [@19], we investigated the dynamics of *rel* transcription in individual cells of *M. smegmatis* grown in nutrient medium up to the stationary phase, with nutrient depletion serving as the source of stress. We employed flow cytometry to monitor the dynamics of green fluorescent protein (GFP) expression in *M. smegmatis* harboring the *rel* promoter fused to *gfp* as a function of time. The experimental signature of bistaility lies in the coexistence of two subpopulations. We now extend the study to investigate the dynamics of *mprA* and *sigE* transcription in individual *M. smegmatis* cells in separate flow cytometry experiments. Figures 4(a) and (b) show the time course of *mprA*-GFP and *sigE*-GFP expressions respectively as monitored by flow cytometry. In both the cases, the distribution of GFP-expressing cells is bimodal indicating the existence of two distinct subpopulations. In each case, the cells initially belong to the subpopulation with low GFP expression. The fraction of cells with high GFP expression increases as a function of time. The two subpopulations with low and high GFP expression are designated as L and H subpopulations respectively. In the stationary phase, the majority of the cells belong to the H subpopulation. The presence of two distinct subpopulations confirms the theoretical prediction of bistability.  We analysed the experimental data shown in Figure 4 and found that at any time point the distribution $P(x,t)$ of GFP levels in a population of cells is a sum of two overlapping and time-independent distributions, one Gaussian ($P_{1}(x)$) and the other lognormal ($P_{2}(x)$), i.e., $$P(x,t)=C_{1}(t)P_{1}(x)+C_{2}(t)P_{2}(x)$$ The coefficients $C_{i}$’s (i=1, 2) depend on time whereas $P_{1}(x)$ and $P_{2}(x)$ are time-independent. The general forms of $P_{1}(x)$ and $P_{2}(x)$ are, $$P_{1}(x)=\frac{exp(-(\frac{x-x_{01}}{w_{01}})^{2})}{w_{01}\,\sqrt{\frac{\pi}{2}}}$$ $$P_{2}(x)=\frac{exp(-\frac{1}{2}(\frac{lnx-x_{02}}{w_{02}})^{2})}{x\, w_{02}\,\sqrt{2\pi}}$$ Figures S2(a) and (b) in Additional File 1 illustrate the typical forms of the Gaussian and lognormal distributions. The Gaussian distribution has a symmetric form whereas the lognormal distribution is asymmetric and long-tailed.  Figure 5 shows the experimental data for cell count versus GFP fluorescence intensity at selected time points in the cases when *gfp* is fused with *mprA* and *sigE* promoters in separate experiments. The dotted curves represent the individual terms in the r.h.s. of Eq. (5) and the solid curve denotes the linear combination $P(x,t)$. The different parameters of $P_{1}(x)$ and $P_{2}(x)$ have the values $x_{01}=97.3366\:(145.86181),w_{01}=103.0731\:(154.67381),x_{02}=5.95526\:(6.1171),w_{02}=0.17618\:(0.2509)$ when *gfp* is fused with *mprA* (*sigE*). The ratio of the coefficients, $C_{1}(t)/C_{2}(t)$, has the value listed by the side of each figure. Figure S3 displays a similar analysis of the experimental data when *gfp* is fused to the *rel* promoter. In the earlier study [@19], the total cell population was divided into L and H subpopulations depending on whether the measured GFP fluorescence intensity was less or greater than a threshold intensity. In the present study, we have obtained approximate analytic expressions for the distributions of GFP fluorescence intensity in the L and H subpopulations. The two distributions, Gaussian and lognormal, have overlaps in a range of fluorescence intensity values (Figure 5 and Figure S3 in Additional File 1). We next used the binning algorithm developed by Chang et al. [@42] to partition the cells of the total population into two overlapping distributions, one Gaussian (Eq. (6)) and the other lognormal (Eq. (7)). At time t, let N(t) be the total number of cells. For each cell, the data $x_{j}$ for the fluorescence intensity is used to calculate the ratios, $$g_{1}(x_{j})=\frac{P_{1}(x_{j})}{P_{1}(x_{j})+P_{2}(x_{j})}\:,\: g_{2}(x_{j})=\frac{P_{2}(x_{j})}{P_{1}(x_{j})+P_{2}(x_{j})}$$ where $P_{1}(x)$ and $P_{2}(x)$ are the distributions in Eqs. (6) and (7). A random number r is generated and the cell $j$ is assigned to the L subpopulation if $0\leq r<g_{1}(x_{j})$ , the cell belongs to the H subpopulation otherwise. Once the total population is partitioned into the L and H subpopulations, one can calculate the following quantities: $$\begin{aligned} \omega_{i}(t) & = & \frac{N_{i}(t)}{N(t)}\;\;\;(i=1,2) \nonumber \\ \mu_{i}(t) & = & \sum x_{ji}(t)/N_{i}(t) \\ \sigma_{i}^{2}(t) & = &\sum(x_{ji}(t)-\mu_{i}(t))^{2}/(N_{i}(t)-1) \nonumber \end{aligned}$$ The indices $i=1,2$ correspond to the L and H subpopulations, $\omega_{i}(t)$ is the fraction of cells in the $i$th subpopulation at time t, $\mu_{i}(t)$ is the mean fluorescence intensity for the $i$th subpopulation and $\sigma_{i}^{2}(t)$ the associated variance.  Figure 6 shows the results of the data analysis. Figure 6(a) shows the plots of mean GFP fluorescence level for the L subpopulation (basal level) versus time in the three cases of *gfp* fused with the promoters of *mprA*, *sigE* and *rel* respectively. Figure 6(b) displays the data for the fractions of cells, $\omega_{2}(t)$ , versus time in the three cases and Figure 6(c) shows the transition rate versus time along with the coefficients of variation CV (CV= standard deviation/mean) of the protein levels in the L subpopulation versus time. Figure S4 (Additional File 1) shows the plots of mean GFP fluorescence level for the total population versus time in the three cases of *gfp* fused with the promoters of *mprA*, *sigE* and *rel* respectively. As in the case of the basal level versus time data (Figure 6(a)), the plots are sigmoidal in nature. We solved the differential equations of the theoretical model described in Additional File 1 and obtained the concentrations of MprA, MprB, SigE, MprA-P, MprB-P and GFP versus time. Some of these plots are shown in Figure S5 (Additional File 1) and reproduce the sigmoidal nature of the experimental plots. We note that the sigmoidal nature of the curves is obtained only when the non-linear nature of the degradation rate is taken into account. As we have already discussed, the distribution of GFP levels in the mycobacterial cell population is a linear combination of two invariant distributions, one Gaussian and the other lognormal, with only the coefficients in the linear combination dependent on time. Friedman et al. [@43] have developed an analytical framework of stochastic gene expression and shown that the steady state distribution of protein levels is given by the gamma distribution. The theory was later extended to include the cases of transcriptional autoregulation as well as noise propagation in a simple genetic network. While experimental support for gamma distribution has been obtained earlier [@44], a recent exhaustive study [@45] of the *E. coli* proteome and transcriptome with single-molecule sensitivity in single cells has established that the distributions of almost all the protein levels out of the 1018 proteins investigated, are well fitted by the gamma distribution in the steady state. The gamma distribution was found to give a better fit than the lognormal distribution for proteins with low expression levels and almost similar fits for proteins with high expression levels. We analysed our GFP expression data to compare the fits using lognormal and gamma distributions. For all the three sets of data (*gfp* fused with the promoters of *mprA*, *sigE* and *rel*), the lognormal and gamma distribution give similar fits at the different time points. Figure S6 (Additional File 1) shows a comparison of the fits for the case of *gfp-mprA*. The lognormal appears to give a somewhat better fit than the gamma distribution, specially at the tail ends. Hysteresis in *gfp* expression {#hysteresis-in-gfp-expression .unnumbered} ------------------------------ Some bistable systems exhibit hysteresis, i.e., the response of the system is history-dependent. In the earlier study, experimental evidence of hysteresis was obtained with *gfp* fused to the promoter of *rel*. The experimental procedure followed for the observation of hysteresis is as follows. In PPK-KO, the *ppk1* knockout mutant, the *ppk1* gene was introduced under the control of the *tet* promoter. We grew PPK-KO carrying the tetracycline-inducible *ppk1* and *rel-gfp* plasmid in medium with increasing concentration of tetracycline (inducer). For each inducer concentration, the distribution of cells expressing *gfp* was analysed by flow cytometry in the stationary phase (steady state) and the mean GFP level was measured. A similar set of experiments was carried out for decreasing concentrations of tetracycline. In the present study, hysteresis experiments in the manner described above were carried out in the two cases of *gfp* fused to *mprA* and *sigE* promoters respectively.  Figure 7 shows the hysteresis data (mean GFP fluorescence versus inducer concentration) in the two cases for increasing (branch going up) and decreasing (branch going down) inducer concentrations. The existence of two distinct branches is a confirmation of hysteresis in agreement with theoretical predictions (Figures S1 A-C).  Figure 8 shows the GFP distributions in the stationary phase for two sets of experiments with different histories, one in which the inducer concentration is increased from low to a specific value (indicated as “Low” in black) and the other in which the same inducer concentration is reached by decreasing the inducer concentration from a high value (indicated as “High” in red). The distributions show that two regions of monostability are separated by a region of bistability. In the cases of monostability, the distributions with different histories more or less coincide. In the region of bistability, the distributions are distinct indicating a persistent memory of initial conditions. Discussion {#discussion .unnumbered} ========== The development of persistence in microbial populations subjected to stress has been investigated extensively in microorganisms like *E. coli* and mycobacteria [@9; @11; @21; @22; @28; @29]. In an earlier study [@19], we demonstrated the roles of positive feedback and gene expression noise in generating phenotypic heterogeneity in a population of *M. smegmatis* subjected to nutrient depletion. The heterogeneity was in terms of two distinct subpopulations designated as L and H subpopulations. The subpopulations corresponded to persister and non-persister cell populations with the stringent response being initiated in the former. In the present study, we have undertaken a comprehensive single cell analysis of the expression activity of the three key molecular players in the stringent response pathway, namely, MprA, SigE and Rel. This has been done by fusing *gfp* to the respective genes in separate experiments and monitoring the GFP levels in a population of cells via flow cytometry. The distribution has been found to be bimodal in each case. In our earlier study [@19], with only the positive autoregulation of the *mprAB* operon taken into account, bistability was obtained in a parameter regime with restricted experimental relevance. The inclusion of the effective positive feedback loop due to growth retardation by protein synthesis gives rise to a considerably more extended region of bistability in parameter space. The persister cells with high stringent response regulator levels are known to have slow growth rates [@21; @22; @28; @29]. This is consistent with the view that stress response diverts resources from growth to stress-related functions resulting in the slow growth of stress-resistant cells [@36]. Figures 2 and 3 provide experimental evidence that the mean intensity of GFP fluorescence monitoring *mprA-gfp* expression increases with time while the specific growth rate $\mu$ of the *M. smegmatis* population decreases in the same time interval. The reciprocal relationship between the two quantities is represented by an expression similar to that in Eq. (1). Since our knowledge of the detailed genetic circuitry involved in the stringent response is limited, we have not attempted to develop a model to explain the origin of cell growth retardation due to protein synthesis. Further experiments (e.g., sorting of the mycobacterial cell population into two subpopulations) are needed to provide conclusive evidence that increased protein synthesis retards cell growth. The stringent response pathway involving MprA and MprB is initiated when the mycobacterial population is subjected to stresses like nutrient depletion. There is now experimental evidence of complex transcriptional, translational, and posttranslational regulation of SigE in mycobacteria [@46; @47; @48; @49]. A double positive feedback loop arises due to the activation of transcription initiation of *sigE* by MprA-P and the activation of the transcription of the *mprAB* operon by the SigE-RNAP complex. Posttranslational regulation of SigE is mediated by RseA, an anti-sigma factor. Barik et al. [@49] have identified a novel positive feedback involving SigE and RseA which becomes functional under surface stress. More experiments need to be carried out to obtain insight on the intricate control mechanisms at work when mycobacteria are subjected to stresses like nutrient deprivation. This will lead to a better understanding of the major contributory factors towards the generation of phenotypic heterogeneity in mycobacterial populations subjected to stress. Conclusions {#conclusions .unnumbered} =========== In the present study, we have characterised quantatively the single cell promoter activity of three key genes in the stringent response pathway of the mycobacterial population *M. smegmatis*. Under nutrient depletion, a “responsive switching” occurs from the L to the H subpopulation with low and high expression levels respectively. A comprehensive analysis of the flow cytometry data demonstrates the coexistence of two subpopulations with overlapping protein distributions. We have further established that the GFP distribution at any time point is a linear superposition of a Gaussian and a lognormal distribution. The coefficients in the linear combination depend on time whereas the component distributions are time-invariant. The Gaussian and lognormal distributions describe the distribution of protein levels in the L and H subpopulations respectively. The two distributions overlap in a range of GFP fluorescence intensity values. We also find that the experimental data for the H subpopulation can be fitted very well by the gamma distribution though the lognormal distribution gives a slightly better fit. In the case of skewed positive data sets, the two distributions are often interchangeable [@50]. An analytical framework similar to that in Ref. [@43] is, however, yet to be developed for the mycobacterial stringent response pathway studied in the paper. The major components in the pathway are the two-component system $mprAB$ and multiple positive feedback loops. The two-component system is known to promote robust input-output relations [@51] and persistence of gene expression states [@52] which may partly explain the good fitting of the experimental data by well-known distributions. Further quantitative measurements combined with appropriate stochastic modeling are needed to characterise the experimentally observed subpopulations more uniquely. We used the binning algorithm developed in [@42] to partition the experimental cell population into the L and H subpopulations. This enabled us to compute quantities like the mean protein level in the L subpopulation, the fraction of cells in the H subpopulation and the CV of GFP levels in the L subpopulation as a function of time. The picture that emerges from the analysis of experimental data is that of bistability, i.e., the coexistence of two distinct subpopulations and stochastic transitions between the subpopulations resulting in the time evolution of the fraction of cells in the H subpopulation. As pointed out in the earlier study [@19], the rate of transition to the H subpopulation and the CV of the L subpopulation levels attain their maximum values around the same time point (Figure 5(c)) indicating the role of gene expression noise in bringing about the transition from the L to the H subpopulation. We have not attempted to develop theoretical models describing the time evolution of the relative weights, $\omega_{i}$’s ($i=1,2$), of the two subpopulations (Eq. (9)). A simple model of two interacting and evolving subpopulations with linear first order kinetics [@12], cannot explain the sigmoidal nature of the time evolution. A model with nonlinear growth kinetics has been proposed in [@42] but lacking definitive knowledge on the origin of nonlinearity in the growth of mycobacterial subpopulations we defer the task of model building to a future publication. Methods {#methods .unnumbered} ======= Strains {#strains .unnumbered} -------- *M. smegmatis* mc$^{2}$155 was grown routinely in Middle Brook (MB) 7H9 broth (BD Biosciences) medium supplemented with 2% glucose and 0.05% Tween 80. Construction of plasmids for fluorescence measurements {#construction-of-plasmids-for-fluorescence-measurements .unnumbered} ------------------------------------------------------- The *mprAB* promoter was amplified from the genomic DNA of *M. smegmatis* using the sense and antisense primers, 5-AA**GGTACC**GCGCAACACCACAAAAAGCG-3 and 5-TA**GGATCC**AGTTTTGACTCACTATCTGAG-3 respectively and cloned into the promoter-less replicative *gfp* vector pFPV27 between the KpnI and BamHI sites (in bold). The *sigE* and *rel* promoters fused to *gfp* have been described earlier [@19; @31]. The resulting plasmids were electroporated into *M. smegmatis* mc$^{2}$155 for further study. For the study of hysteresis, expression of *ppk1* under a tetracycline-inducible promoter in an *M. smegmatis* strain inactivated in the *ppk1* gene (PPK-KO), has been described earlier [@19]. FACS analysis {#facs-analysis .unnumbered} ------------- *M. smegmatis* cells expressing different promoters fused to GFP were grown in medium supplemented with kanamycin (25 $\mu g/ml$) and analysed at different points of time on a FACS Caliber (BD Biosciences) flow cytometer as described earlier [@19]. Briefly, cells were washed, resuspended in PBS and fluorescence intensity of 20,000 events was measured. The data was analyzed using Cell Quest Pro (BD Biosciences) and WINMIDI software. The flow cytometry data is represented in histogram plots where the x-axis is a measure of fluorescence intensity and the y-axis represents the number of events. Measurement of growth rate {#measurement-of-growth-rate .unnumbered} -------------------------- *M. smegmatis* expressing promoter-*gfp* fusion constructs were grown in Middle Brook (MB) 7H9 broth supplemented with glucose and Tween 80, and kanamycin (25 *$\mu g/ml$*). Growth at different time points was measured by recording absorbance values at 600 nm (a value of 1 OD$_{600}$ is equal to $10^{8}$ cells or 200 $\mu g$ dry weight of cells). A growth curve was generated by plotting absorbance values against time (inset of Figure 2). The specific growth rate $\mu$ (Eq. (2)) at different time points is determined by taking derivatives of the growth curve at the different time points (Figure 2). Authors contributions {#authors-contributions .unnumbered} ===================== IB, JB and MK conceptualised, supervised and coordinated the study. KS, JB and MK carried out the experiments. SG, BG and IB developed the theoretical model, performed the data analysis and interpreted the data. 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Mathematial model {#mathematial-model .unnumbered} ================= The reaction scheme describing the processes shown in Figure 1 is given by $$B_{2}+B_{2}\:\begin{array}{c} k_{1}\\ \rightleftharpoons\:\\ k_{2}\end{array}B_{2}-B_{2}\:\overset{k_{3}}{\longrightarrow\:}B_{1}+B_{1}$$ $$A_{2}+B_{1}\begin{array}{c} k_{4}\\ \:\rightleftharpoons\:\\ k_{5}\end{array}A_{2}-B_{1}\overset{k_{6}}{\:\longrightarrow}\: A_{1}+B_{2}$$ $$A_{1}+B_{2}\begin{array}{c} k_{7}\\ \:\rightleftharpoons\\ k_{8}\end{array}\: A_{1}-B_{2}\overset{k_{9}}{\:\longrightarrow\:}A_{2}+B_{2}$$ $$G_{AB}+A_{1}\begin{array}{c} k_{a}\\ \:\rightleftharpoons\:\\ k_{d}\end{array}G_{AB}^{*}\overset{\beta}{\:\longrightarrow\:}A_{2}+B_{2}$$ $$G_{AB}+A_{1}\overset{s}{\:\longrightarrow\:}A_{2}+B_{2}$$ In the equations, $A_{1}(A_{2})$ represents the phosphorylated (unphosphorylated) form of MprA and $B_{1}(B_{2})$ denotes the phosphorylated (unphosphorylated) form of MprB. The inactive and active states of the *mprAB* operon are represented by $G_{AB}$ and $G_{AB}$[\*]{} respectively. In the inactive state, MprA and MprB proteins are synthesized at a basal rate s and in the active state protein production occurs at an enhanced rate $\beta$. Eq. (1) describes the autophosphorylation reaction of MprB with the poly P chain serving as a source of phosphate groups. Eq. (2) describes the transfer of the phosphate group from the phosphorylated MprB to MprA. Eq. (3) corresponds to dephosphorylation of phosphorylated MprA by unphosphorylated MprB which thus acts as a phosphatase (in the earlier study [@key-1], phosphorylated MprB was assumed to act as a phosphatase which is not consistent with experimental evidence). Eqs. (4) and (5) describe activation of the *mprAB* operon by phosphorylated MprA and basal expression of the operon respectively. Refs. [@key-2; @key-3; @key-4] provide experimental justification for the reaction scheme shown in Eqs. (1)-(5). Using standard mass action kinetics, we write down the rate equations for the concentration of each of the key molecular species participating in the biochemical events. The equations are: $$\frac{d[A_{1}]}{dt}\:= \: k_{6}[A_{2}-B_{1}]-k_{7}[A_{1}][B_{2}]+k_{8}[A_{1}-B_{2}]-\gamma[A_{1}]- \frac{\phi[A_{1}]}{1+\theta_{1}[A_{1}]}$$ $$\frac{d[A_{2}]}{dt}\:= \: s+\beta\frac{[A_{1}]/k}{1+[A_{1}]/k}+k_{9}[A_{1}-B_{2}]-k_{4}[A_{2}][B_{1}]+k_{5}[A_{2}-B_{1}] -\gamma_{1}[A_{2}]-\frac{\phi[A_{2}]}{1+\theta_{2}[A_{2}]}$$ $$\frac{d[B_{1}]}{dt}\:=\: k_{3}[B_{2}-B_{2}]-k_{4}[A_{2}][B_{1}]+k_{5}[A_{2}-B_{1}]-\gamma[B_{1}]-\frac{\phi[B_{1}]}{1+\theta_{1}[B_{1}]}$$ $$\frac{d[B_{2}]}{dt}\:=\: s+\beta\frac{[A_{1}]/k}{1+[A_{1}]/k}+k_{2}[B_{2}-B_{2}]-k_{1}[B_{2}]^{2}+k_{6}[A_{2}-B_{1}]-k_{7}[A_{1}][B_{2}]+(k_{8}+k_{9})[A_{1}-B_{2}]-\gamma[B_{2}]-\frac{\phi[B_{2}]}{1+\theta_{2}[B_{2}]}$$ $$\frac{d[B_{2}-B_{2}]}{dt}\:=\:-k_{2}[B_{2}-B_{2}]+k_{1}[B_{2}]^{2}-k_{3}[B_{2}-B_{2}]$$ $$\frac{d[A_{2}-B_{1}]}{dt}\:=\: k_{4}[A_{2}][B_{1}]-k_{5}[A_{2}-B_{1}]-k_{6}[A_{2}-B_{1}]$$ $$\frac{d[A_{1}-B_{2}]}{dt}\:=\: k_{7}[A_{1}][B_{2}]-(k_{8}+k_{9})[A_{1}-B_{2}]$$ $$\frac{d[SigE]}{dt}\:=\: s_{1}+\beta_{1}\frac{[A_{1}]/k'}{1+[A_{1}]/k'}-\delta_{1}[SigE]$$ $$\frac{d[GFP]}{dt}\:=\: s_{2}+\beta_{2}\frac{[SigE]/k''}{1+[SigE]/k''}-\delta_{2}[GFP]$$ Eq. (13) represents SigE synthesis due to transcriptional activation of the *sigE* gene by phosphorylated MprA-P. Eq. (14) describes GFP production due to the activation of the *rel* promoter by SigE. The rate constants $\gamma$, $\gamma_{1}$, $\delta_{1}$ and $\delta_{2}$ are the degradation rate constants. The last terms in Eqs. (6)-(9) represent the nonlinear decay rates the genesis of which is explained in the main text (see Eq. 4) [@key-5]. Eqs. (6)-(14) correspond to the case where *gfp* is fused to the *rel* promoter. In the other cases when *gfp* is fused to the *mprA* or *sigE* promoter, appropriate modifications in the set of equations are required. The steady state solution of Eqs. (6)-(14) is obtained by setting all the rates of change to be zero. In the case of bistability, there are three steady state solutions, two stable and one unstable [@key-6; @key-7; @key-8]. In the steady state, one has to solve the following set of coupled nonlinear algebraic equations: $$\alpha_{1}[A_{2}][B_{1}]-\alpha_{2}[A_{1}][B_{2}]-\gamma[A_{1}]-\frac{\phi[A_{1}]}{1+\theta_{1}[A_{1}]}=0$$ $$s+\beta\frac{[A_{1}]/k}{1+[A_{1}]/k}-\alpha_{1}[A_{2}][B_{1}]+\alpha_{2}[A_{1}][B_{2}]-\gamma_{1}[A_{2}]-\frac{\phi[A_{2}]}{1+\theta_{2}[A_{2}]}=0$$ $$\alpha[B_{2}]^{2}-\alpha_{1}[A_{2}][B_{1}]-\gamma[B_{1}]-\frac{\phi[B_{1}]}{1+\theta_{1}[B_{1}]}=0$$ $$s+\beta\frac{[A_{1}]/k}{1+[A_{1}]/k}-\alpha[B_{2}]^{2}+\alpha_{1}[A_{2}][B_{1}]-\gamma[B_{2}]-\frac{\phi[B_{2}]}{1+\theta_{2}[B_{2}]}=0$$ $$s_{1}+\beta_{1}\frac{[A_{1}]/k'}{1+[A_{1}]/k'}-\delta_{1}[SigE]=0$$ $$s_{2}+\beta_{2}\frac{[SigE]/k''}{1+[SigE]/k''}-\delta_{2}[GFP]=0$$ where,$$\alpha=\frac{k_{1}k_{3}}{k_{2}+k_{3}},\;\alpha_{1}=\frac{k_{4}k_{6}}{k_{5}+k_{6}},\;\alpha_{2}=\frac{k_{7}k_{9}}{k_{8}+k_{9}},\; k=\frac{k_{d}}{k_{a}}$$ The solutions of Eqs. (15)-(20) are obtained with the help of Mathematica. Figures S1 A-C show the steady state solutions generated by varying the parameter $\alpha$ (associated with the autophosphorylation of MprB). The parameters have values:$\alpha_{1}=2.4,\alpha_{2}=2.8,\gamma=0.1,s=0.14,\beta=4,k=1,\gamma_{1}=1,\phi=0.5,\theta_{1}=1,\theta_{2}=10,s_{1}=0.02,\beta_{1}=4,k'=10,\delta_{1}=1,$ $s_{2}=0.12,\beta_{2}=4,k''=2,\delta_{2}=0.1$ in appropriate units. In each of the Figures S1 A-C, the solid and dotted branches represent stable and unstable steady states respectively. Bistability is obtained over a wide range of parameter values due to the inclusion of the non-linear decay term in Eqs. (6)-(9). In the hysteresis experiments, the inducer tetracycline was used to control the level of PPK1 and therefore the synthesis of the poly P chain. Since the latter acts as the source of phosphate groups for the autophosphorylation of MprB (Eq. (1)), the rate constant k, in Eq. (1) is effectively proportional to the inducer (or the PPK1) concentration. As the parameter $\alpha$ (Eq. (21)) includes the rate constant $k_{1}$ , a varying inducer concentration is equivalent to varying the parameter $\alpha$. There is some experimental evidence that MprA-P regulates the expression of the *mprAB* operon in the form of dimers [@key-9]. Inclusion of this feature in our model makes the bistable behaviour more prominent.  **Figure S1:** Bistability and hysteresis in the deterministic model. Steady state concentrations of MprA, SigE and Rel versus the parameter $\alpha$ (Eq. (21)).  **Figure S2:** (a) Gaussian and (b) lognormal distributions which describe the distribution of GFP leads in the L and H subpopulations respectively.  **Figure S3:** Experimental data for cell count versus GFP fluorescence intensity at selected time points when *gfp* is fused with *rel* promoter. The solid curve represents $P(x,t)$ in Eq. (5) and the dotted curves are the individual terms on the r.h.s. The different parameters of $P_{1}(x)$ and $P_{2}(x)$ have the values $x_{01}=157.14748,\; w_{01}=150.43575,\; x_{02}=6.10036,\; w_{02}=0.1847$ when *gfp* is fused with *rel*.  **Figure S4:** Mean GFP fluorescence level for the total population versus time in the three cases of *gfp* fused with the promoters of (a) *mprA*, (b) *sigE* and (c) *rel*.  **Figure S5:** Concentrations of MprA, SigE and GFP (in arbitrary units) versus time. The values of the concentrations are obtained by solving Eqs. (6)-(14) in Text S1.  **Figure S6:** Comparison of fits of experimental data for cell count versus GFP fluorescence intensity at selected time points when *gfp* is fused with *mprA* promoter, with lognormal (Eq. (7)) and gamma distributions. The gamma distribution has the form $P(x)=\frac{x^{a-1}\, exp(-\frac{x}{b})}{b^{a}\,\Gamma(a)}$ , where the parameters *a* and *b* have the values $a=33.246,$ $b=11.59$ and $\Gamma(a)$ is the gamma function. [9]{} Sureka K, Ghosh B, Dasgupta A, Basu J, Kundu M and Bose I (2008) Positive feedback and noise activate the stringent response regulator Rel in mycobacteria. PLoS ONE 3: e 1771 Sureka K, Dey S, Datta P, Singh AK, Dasgupta A, et al. (2007) Polyphosphate kinase is involved in stress-induced mprAB-sigE-rel signaling in mycobacteria. Mol Microbiol 65: 261-276. Zahrt TC, Wozniak C, Jones D, Trevett A (2003) Functional analysis of the Mycobacterium tuberculosis MprAB two-component signal transduction system. Infect Immun 71: 6962-6870. Ferrell JE Jr. (2002) Self-perpetuating states in signal transduction: positive feedback, double-negative feedback and bistability. Curr. Opin. Cell Biol. 14: 140-148. Tan C, Marguet P and You L (2009) Emergent bistability by a growth-modulating positive feedback circuit. Nat. Chem. Biol. 5 : 842-848 Ferrell JE Jr. (2002) Self-perpetuating states in signal transduction: positive feedback, double-negative feedback and bistability. Curr. Opin. Cell Biol. 14: 140-148. Veening J-W, Smits WK and Kuipers OP (2008) Bistability, epigenetics and bet-hedging in bacteria. Annu. Rev. Microbiol. 62: 193-210 Pomerening JR (2008) Uncovering mechanisms of bistability in biological systems. Curr. Opin. Biotechnol. 19:381-8. He H, Zahrt TC (2005) Identification and characterization of a regulatory sequence recognized by Mycobacterium tuberculosis persistence regulator MprA. J Bacteriol 187: 202-212.

--- abstract: 'We present a elementary approach to asymptotic behavior of generalized functions in the Cesàro sense. Our approach is based on Yosida’s subspace of Mikusiński operators. Applications to Laplace and Stieltjes transforms are given.' address: - 'Department of Mathematics, California State University, Stanislaus, One University Circle, Turlock, CA 95382, USA' - 'Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, 9000 Gent, Belgium' author: - Dennis Nemzer - Jasson Vindas title: An elementary approach to asymptotic behavior in the Cesàro sense and applications to the Laplace and Stieltjes transforms --- [^1] [^2] Introduction ============ Asymptotic analysis is a very much studied topic within generalized function theory and has shown to be quite useful for the understanding of structural properties of a generalized function in connection with its local behavior as well as its growth properties at infinity. Several applications have been developed in diverse areas such as Tauberian and Abelian theory for integral transforms, differential equations, number theory, and mathematical physics. There is a vast liteturature on the subject, see the monographs [@EstradaKanwal; @ML; @PilStanTak; @PilStanVindas; @vdz] and references therein. The purpose of this paper is to present an elementary approach to asymptotic behavior in the Cesàro sense. The Cesàro behavior for Schwartz distributions was introduced by Estrada in [@Estrada] (see also [@EstradaKanwal]). The approach we develop in this article uses Yosida’s algebra $\mathcal{M}$ of operators [@yosida], which provides a simplified but useful version of Mikusiński’s operational calculus [@Mikusinski]. While Schwartz distribution theory is based on the duality theory of topological vector spaces, the construction of Yosida’s space $\mathcal{M}$ is merely algebraic, making only use of elementary notions from calculus. That is why we call our approach to Cesàro asymptotics *elementary*. The plan of the article is as follows. In Section \[preliminaries\], we recall the construction of Yosida’s space $\mathcal{M}$. We study some useful localization properties of elements of $\mathcal{M}$ in Section \[localization\]. The asymptotics in the Cesàro sense is defined in Section \[cesaro\] and its properties are investigated. As an application, we conclude the article with some Abelian and Tauberian theorems for Stieltjes and Laplace transforms in Section \[applications\]. It should be mentioned that Abelian and Tauberian theorems for Stieltjes and Laplace transforms of generalized functions have been extensively investigated by several authors, see, e.g., [@Estrada-Vindas; @ML; @PilStanTak; @PilStanVindas; @vdz]. Preliminaries ============= We recall in this section the construction of Yosida’s space of operators $\mathcal{M}$ and explain some of its properties. See [@yosida] for more details about $\mathcal{M}$. Let $C^{n}_+({{\mathbb R}})$ denote the space of all $n$-times continuously differentiable functions on ${{\mathbb R}}$ which vanish on the interval $(-\infty , 0)$. We write $C_+({{\mathbb R}})=C^{0}_+({{\mathbb R}})$. For $f, g \in C_+({{\mathbb R}})$, the convolution is given by $$(f \ast g)(x) = \int_0^x f(x-t)g(t) dt.$$ Let $H$ denote the Heaviside function. That is, $H(x) = 1$ for $x \ge 0$ and zero otherwise. For each $n \in {{\mathbb N}}$, we denote by $H^n$ the function $H \ast \dots \ast H$ where $H$ is repeated $n$ times. One has $H^{n}\in C^{n-2}_{+}(\mathbb{R})$, $n\geq 2$. Note that if $f \in C_+({{\mathbb R}})$, then $$(H^m \ast f)(x) = \frac{1}{(m-1)!} \int_0^x (x-t)^{m-1} f(t) dt.$$ The space ${{\mathcal M}}$ is defined as follows, $${{\mathcal M}}= \left\{ \frac{f}{H^k} : f \in C_+({{\mathbb R}}),\: k \in {{\mathbb N}}\right\}.$$ Two elements of ${{\mathcal M}}$ are equal, denoted $\displaystyle \frac{f}{H^n} = \frac{g}{H^m}$ , if and only if $H^m \ast f = H^n \ast g$. Addition, multiplication (using convolution), and scalar multiplication are defined in the natural way, and ${{\mathcal M}}$ with these operations is a commutative algebra with identity $ \delta = \displaystyle\frac{H^2}{H^2}$, the Dirac delta. We can embed $C_{+}({{\mathbb R}})$ into $\mathcal{M}$. Indeed, for $f \in C_+({{\mathbb R}})$, we set $W_f= \displaystyle\frac{H \ast f}{H}$. Obviously, $f\mapsto W_{f}$ is injective. Let $W = \displaystyle\frac{f}{H^k} \in {{\mathcal M}}$. The generalized derivative of $W$ is defined as $ DW =\displaystyle\frac{f}{H^{k+1}}.$ The product of $x$ and $W$ is given by $$xW = \frac{xf-kH \ast f}{H^k} \quad (k \ge 2).$$ In the last formula $k\geq2$ is no restriction because $\displaystyle\frac{f}{H^{k}}=\frac{H\ast f}{H^{k+1}}$. Clearly, these definitions do not depend on the representative of $W$. The generalized derivative and product by $x$ satisfy: 1. $xW_f = W_{xf}$, $f\in C_{+}({{\mathbb R}})$. 2. $DW_f = W_{f'},$ $f \in C^1_+({{\mathbb R}})$. 3. $D(xW) = W + xDW, \,\, W \in {{\mathcal M}}$. For (a), $$\begin{aligned} xW_f &= \frac{x(H^2 \ast f)}{H^2} - \frac{2H^2 \ast f}{H} = \frac{xH^2 \ast f}{H^2} + \frac{H^2 \ast xf}{H^2} - \frac{2H^2 \ast f}{H}\\ &= \frac{2H^3 \ast f}{H^2} + \frac{H^2 \ast xf}{H^2} - \frac{2H^2 \ast f}{H}\\ & = \frac{H \ast xf}{H} \quad = \quad W_{xf}\:.\end{aligned}$$ If $f'\in C_{+}({{\mathbb R}})$, we have $\displaystyle DW_f = \frac{H \ast f}{H^2} = \frac{f}{H} = \frac{H \ast f'}{H} = W_{f'},$ which shows (b). Next, let $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$. Then, $$W + xDW = \frac{f}{H^k} + \left( \frac{xf}{H^{k+1}} - \frac{(k+1)f}{H^k} \right) = \frac{xf}{H^{k+1}} - \frac{kf}{H^k} = D(xW).$$ Notice by identifying $f \in L^1_{loc}({{\mathbb R}}^+)$ with $\displaystyle\frac{H \ast f}{H} \in {{\mathcal M}}$, the space $L^1_{loc}({{\mathbb R}}^+)$ can be considered a subspace of ${{\mathcal M}}$. Also, for the construction of ${{\mathcal M}}$, the space of locally integrable functions which vanish on $(-\infty , 0)$ could have been used instead of $C_+({{\mathbb R}})$. Localization ============ We discuss in this section localization properties of elements of $\mathcal{M}$. Let $W = \displaystyle \frac{f}{H^k} \in {{\mathcal M}}$. $W$ is said to vanish on an open interval $(a, b)$, denoted $W(x) = 0$ on $(a, b)$, provided there exists a polynomial $p$ with degree at most $k-1$ such that $p(x) = f(x)$ for $a < x < b$. Two elements $W, V \in {{\mathcal M}}$ are said to be equal on $(a, b)$, denoted $W(x) = V(x)$ on $(a, b)$, provided $W - V$ vanishes on $(a, b)$. The support of $W \in {{\mathcal M}}$, denoted supp$\,W$, is the complement of the largest open set on which $W$ vanishes. The degree of a polynomial $p$ will be denoted by $\operatorname*{deg} p$ in the sequel. Recall $ \delta = \displaystyle\frac{H^2}{H^2}$. Notice that $H^2(x) = x$ on the open interval $(0, \infty)$. Thus, $\delta(x) = 0$ on $(0, \infty)$. Also, $H^2(x) = 0$ on $(-\infty, 0)$. So, $\delta(x) = 0$ on $(-\infty, 0)$. Therefore, $\operatorname*{supp}\delta = \{0\}$. Let $W \in {{\mathcal M}}$. 1. If $W(x) = 0$ on $(a,b)$, then $DW(x) = 0$ on $(a,b)$. 2. If $DW(x)=0$ on $(a,b)$, then $W$ is constant on $(a,b)$. Part (a) follows immediately from definitions. See [@Nem Thm. 4.1] for (b). \[WOnInterval\] Let $W \in {{\mathcal M}}$. 1. If $W(x) = 0$ on $(a,b)$, then $xW(x) = 0$ on $(a,b)$. 2. Suppose $xW(x) = 0$ on $(a,b)$. Then, 1. $W(x) = 0$ on $(a,b)$, provided $0 \notin (a,b)$. 2. $W(x) = 0$ on $(a,0) \cup (0,b)$, provided $0 \in (a,b)$. Let $W =\displaystyle \frac{f}{H^k} \in {{\mathcal M}}$ ($k \ge 2$). *Part (a).* Since $W(x) = 0$ on $(a,b)$, there exist $a_0, a_1, \dots, a_{k-1} \in {{\mathbb C}}$ such that $f(x) = a_0 + a_1x + \dots + a_{k-1}x^{k-1}$ for $a<x<b$. Now, there exists $A \in {{\mathbb C}}$ such that $$xf(x) - k(H \ast f)(x) = A + a_0(1-k)x + a_1\left(1-\frac{k}{2}\right)x^2 + \dots + a_{k-2}\left(1-\frac{k}{k-1}\right)x^{k-1},$$ for $a<x<b$. Since $\displaystyle xW = \frac{xf-kH\ast f}{H^k}$, the above yields $xW(x) = 0$ on $(a,b)$. *Part (b)*. Suppose $xW(x) = 0$ on $(a,b)$. 1. If $a<0$, then the conclusion is clearly true. So assume $a \ge 0$. Now, there exists a polynomial $p$ with $\deg p \le k-1$ such that $$xf(x) - k\int_0^xf(t)dt = p(x) \quad \mbox{ for } a<x<b.$$ Thus, $f \in C^1(a,b)$ and $xf'(x) + (1-k)f(x) = p'(x)$ for $a<x<b.$ Solving this differential equation, it follows that $f(x) = q(x)$ for $a<x<b,$ where $q$ is a polynomial with $\deg q \le k-1.$ Therefore, $W(x) = 0$ on $(a,b)$. 2. Since for all $W \in {{\mathcal M}}$, we have $W(x) = 0$ on $(-\infty, 0)$, we only need to show that $W(x) = 0$ on $(0,b)$. Since $0 \in (a,b)$, one has $$\label{eq} xf(x) - k\int_0^x f(t)dt = 0$$ for $a<x<b$, in particular, for $0<x<b$. Similarly as in the proof of part (i), we obtain $f(x) = Ax^{k-1}$ for $0<x<b,$ where $A \in {{\mathbb C}}.$ Thus, $W(x) = 0$ on $(0,b)$. Therefore, $W(x) = 0$ on $(a,0) \cup (0,b)$. Let $W \in {{\mathcal M}}$. 1. If $xW(x) = 0$ on $(-\infty, \infty)$, then $W = \alpha \delta$ for some $\alpha \in {{\mathbb C}}$. 2. If $W = \displaystyle \frac{f}{H^k} \,\, (k \ge 2)$ and $W(x) = 0$ on $(0, \infty)$, then $W = \sum_{n=0}^{k-2} \alpha_n \delta^{(n)}$, for some $\alpha_n \in {{\mathbb C}},$ $n = 0, 1, 2, \dots, k-2$. Let $W =\displaystyle \frac{f}{H^k} \, \, (k \ge 2)$. *Part (a)*. Suppose $xW(x) =0$ on $(-\infty, \infty)$. Similarly as in the proof of Proposition \[WOnInterval\](b), we obtain (\[eq\]) on $(-\infty, \infty)$ It follows that $f(x)= \beta x^{k-1}$ on $(0, \infty)$, for some $\beta \in {{\mathbb C}}.$ Since $f$ is continuous on ${{\mathbb R}}$ and vanishes on $(-\infty, 0)$, we have $f(x) = \beta x^{k-1} \,\, \mbox{ on } [0, \infty).$ Therefore, $$W = \frac{f}{H^k} = (k-1)! \, \beta \, \frac{H^k}{H^k} = \alpha \delta \quad \mbox{where } \alpha = (k-1)! \beta.$$ *Part (b)*. Suppose $W(x) = 0$ on $(0,\infty)$. Then, $f(x) = a_0 + a_1 x + \dots + a_{k-1} x^{k-1}$ on $(0, \infty)$, where $a_0, a_1, \dots, a_{k-1} \in {{\mathbb C}}$. Since $f$ is continuous and vanishes on $(-\infty,0),$ we obtain $a_0 = 0$. Thus, $$\begin{aligned} W & = a_1 \frac{H^2}{H^k} + 2! a_2\, \frac{H^3}{H^k} + \dots + (k-1)!\, a_{k-1}\,\frac{H^k}{H^k} \\ & = a_1\, \delta^{(k-2)} + 2!\, a_2\, \delta^{(k-1)} + \dots + (k-1)!\, a_{k-1}\, \delta. \end{aligned}$$ Asymptotics in the Cesàro sense {#cesaro} =============================== We now introduce and study asymptotics in the Cesàro sense for elements of $\mathcal{M}$. As usual, $\Gamma$ stands for the Euler Gamma function and $o$ stands for the little $o$ growth order symbol of Landau [@EstradaKanwal Chap. 1]. \[def3.1\] Let $W \in {{\mathcal M}}$. For $\alpha \in {{\mathbb R}}\backslash\{-1, -2, \dots\}$, define $$W(x) \sim \frac{\gamma \, x^\alpha}{\Gamma(\alpha + 1)} \quad (C),$$ if and only if there is $k\in\mathbb{N}$ such that $W = \displaystyle \frac{f}{H^k}$ with $$\label{eqC1} \frac{\Gamma(\alpha + k +1) f(x)-p(x)}{x^{\alpha + k}} \to \gamma, \quad x \to \infty,$$ where $p$ is some polynomial with $\deg p \le k-1$. That is, $$\Gamma(\alpha + k +1) f(x) = p(x) + \gamma x^{\alpha + k} + o(x^{\alpha + k}), \quad \mbox{ as } x \to \infty.$$ If $\alpha >-1$, then the polynomial $p$ is not needed. The next theorem tells us that Definition \[def3.1\] is consistent with the choice of representatives. \[aclaim\] If $W = \displaystyle \frac{f}{H^k} \in {{\mathcal M}}$ is such that (\[eqC1\]) holds, then the continuous function $H^{m}\ast f$ in the representation $W=\displaystyle\frac{H^m \ast f}{H^{k+m}} \quad (m \in {{\mathbb N}})$ satisfies $$\label{eqC2} \frac{\Gamma(\alpha + k+m +1) (H^{m}\ast f)(x)-q(x)}{x^{\alpha + k+m}} \to \gamma, \quad x \to \infty,$$ for some polynomial $q$ of degree at most $k+m-1.$ Suppose that (\[eqC1\]) holds for some polynomial $p$ with $\deg p \le k-1$. Consider $$\Gamma(\alpha +k+2) \int_0^x \left( f(t) - \frac{p(t)}{\Gamma(\alpha + k + 1)} \right)dt + K,$$ with $K$ a constant to be determined. Suppose $\alpha + k < -1 \Rightarrow x^{\alpha+k} \in L^1(1, \infty)$. So, (4.1) implies $ f- p/\Gamma(\alpha+k+1)\in L^1(0, \infty)$. Therefore there exists a constant $K$ such that $$\Gamma(\alpha + k + 2) \int_0^x \left(f(t) - \frac{p(t)}{\Gamma(\alpha + k + 1)} \right)dt + K \to 0, \quad \mbox{ as } x \to \infty.$$ Thus, using L’Hospital’s rule and (\[eqC1\]) $$\displaystyle\lim_{x \to \infty} \frac{\Gamma(\alpha + k + 2) \int_0^x \left(f(t) - \frac{p(t)}{\Gamma(\alpha + k + 1)} \right)dt + K}{x^{\alpha + k + 1}} = \lim_{x \to \infty} \frac{\Gamma(\alpha + k + 1)f(x) - p(x)}{x^{\alpha + k}} = \gamma.$$ Let $q(x) = (\alpha + k+1)\int_0^x p(t)dt - K.$ Then $q$ is a polynomial with $\deg q \le k$. Moreover, $$\frac{\Gamma(\alpha + k + 2)(H \ast f)(x) - q(x)}{x^{\alpha + k + 1}} \to \gamma, \quad \mbox{ as } x \to \infty.$$ Suppose $\alpha + k > -1.$ Assume $\gamma \ne 0$. Then (\[eqC1\]) implies $$\int_0^x \left(\Gamma(\alpha + k + 1) f(t) - p(t) \right)dt \to \pm \infty, \quad \mbox{ as } x \to \infty.$$ Using L’Hospital’s Rule and (\[eqC1\]), $$\frac{\Gamma(\alpha + k + 2) \int_0^x\left( f(t) - \frac{p(t)}{\Gamma(\alpha+k + 1)} \right)dt}{x^{\alpha + k + 1}} = \frac{(\alpha + k + 1) \int_0^x (\Gamma(\alpha+k+1)f(t)-p(t))}{x^{\alpha+k+1}} \to \gamma,$$ as $x \to \infty.$ For the case $\gamma = 0$ (and $\alpha + k > -1)$, we set $g(x) = \Gamma(\alpha + k + 1)f(x) - p(x)$. It is well known that if $x^{-\alpha-k}g(x)\to 0$, then $x^{-\alpha - k - 1} \int_0^x g(t) dt \to 0$ (see e.g. [@EstradaKanwal Chap. 1]). The above shows that the claim is true for $m=1$. By using induction, the result follows. Unless otherwise stated, we assume from now on $\alpha \in {{\mathbb R}}\backslash \{-1. -2, \dots\}$.\ The following provides an alternative to Definition \[def3.1\]. Let $W \in {{\mathcal M}}$ and $\gamma \in {{\mathbb C}}.$ Then $$\label{eqC} W(x) \sim \frac{\gamma x^\alpha}{\Gamma(\alpha + 1)} \quad (C), \quad x \to \infty$$ if and only if there exist $n \in {{\mathbb N}}$ with $\alpha +n >0$, $g \in C_+({{\mathbb R}})$, and $b>0$ such that $W(x) = D^ng(x)$ on $(b, \infty)$ and $g(x)/x^{\alpha+n}\, \to \, \gamma/\Gamma(\alpha+n+1)$ as $x \to \infty$. We leave the verification of this fact to the reader. \[Thm1\] We have: 1. If $f \in C_+(\mathbb{R})$ such that $f(x) \sim \gamma x^\alpha$ as $x \to \infty$, then $W_f(x) \sim \gamma x^\alpha \quad (C),$ $x \to \infty$. 2. Let $W \in {{\mathcal M}}$ satisfy $(\ref{eqC})$. Then, $$\label{eqDW} DW(x) \sim \frac{\alpha \gamma x^{\alpha -1}}{\Gamma(\alpha+1)} \quad (C), \quad x \to \infty .$$ and $$\label{eqxW} xW(x) \sim \frac{\gamma \, x^{\alpha+1}}{\Gamma(\alpha + 1)} \quad (C), \quad x \to \infty.$$ We only prove (\[eqxW\]) since the other parts of the theorem follow from the definitions. We first assume a stronger condition on the polynomial $p$. Suppose $\displaystyle V = \frac{g}{H^n} \in {{\mathcal M}}\, \, (n \ge 2)$ with $$\label{gammaWithp} \frac{\Gamma(\alpha + n + 1) g(x) - p(x)}{x^{\alpha+n}} \to \gamma \quad \mbox{ as } x \to \infty,$$ for some polynomial $p$ with $\deg p \le n-2$. Then there exists a polynomial $q$ with $\deg q \le n-1$ such that $$\frac{\Gamma(\alpha+n+2)x g(x) - q(x)}{x^{\alpha+n+1}} \to (\alpha +n+1)\gamma \quad \mbox{ as }\quad x \to \infty.$$ Therefore, $$\frac{xg}{H^n} \sim \frac{(\alpha + n + 1)\gamma x^{\alpha+1}}{\Gamma(\alpha + 2)} \quad (C),\quad \, \, x \to \infty.$$ Also, from (\[gammaWithp\]) it follows that $$\frac{-ng}{H^{n-1}} \sim \frac{-n \gamma x^{\alpha + 1}}{\Gamma(\alpha +2)} \quad (C), \quad \, \, x \to \infty.$$ Thus, $$xV(x) \sim \frac{\gamma x^{\alpha + 1}}{\Gamma(\alpha+1)} \quad (C), \quad \, \, x \to \infty.$$ We now remove the stronger condition that $\deg p \le n-2$ and complete the proof of the theorem. Let $W =\displaystyle \frac{f}{H^k} \in {{\mathcal M}}$ such that (\[eqC\]) holds. That is, there exists $a_0, a_1, \dots, a_{k-1} \in {{\mathbb C}}$ such that $$\label{Existas} \frac{\Gamma(\alpha + k + 1) f(x) - (a_0+a_1 x + \dots + a_{k-1}x^{k-1})}{x^{\alpha+k}} \to \gamma \mbox{ \quad as } x \to \infty$$ Let $\displaystyle V = \frac{f(x) - \beta x^{k-1}}{H^k} \, \in {{\mathcal M}}$, where $\beta = a_{k-1}/\Gamma(\alpha + k + 1)$. Then, from (\[Existas\]) it follows that, $$V(x) \sim \frac{\gamma x^\alpha}{\Gamma(\alpha+1)} \quad (C), \quad \, \, x \to \infty.$$ And, by the first part of the proof, $$xV(x) \sim \frac{\gamma x^{\alpha +1}}{\Gamma(\alpha + 1)} \quad (C), \quad \, \, x \to \infty.$$ Since, $xW = xV + \beta (k-1)! x \delta = xV$, it follows that $$xW(x) \sim \frac{\gamma x^{\alpha+1}}{\Gamma(\alpha+1)} \quad (C), \quad \, \, x \to \infty.$$ By Theorem \[Thm1\], we obtain the following theorem. If $\displaystyle(H \ast W)(x) \sim \frac{\gamma \, x^\alpha}{\Gamma(\alpha + 1)} \quad (C),$ then $\displaystyle(xW)(x) \sim \frac{\alpha \gamma \, x^\alpha}{\Gamma(\alpha + 1)} \quad (C), \\ x \to \infty$. The proofs of the next proposition and corollary follow from the definitions.\ If $V$ has compact support, then $\displaystyle V(x) \sim \frac{0}{\Gamma(\alpha+1)} x^\alpha \ \ (C),$ $x \to \infty$. Asymptotics in the Cesàro sense is a local property. Let $W, V \in {{\mathcal M}}$. Suppose that $W$ has Cesàro asymptotics $(\ref{eqC})$ and $W(x) = V(x)$ on $(a, \infty)$. Then, $\displaystyle V(x) \sim \frac{\gamma \, x^\alpha}{\Gamma(\alpha + 1)} \quad (C), \quad x \to \infty$. Applications ============ In this last section we give some Abelian and Tauberian theorems for Stieltjes and Laplace transforms of elements of $\mathcal{M}$. We start by defining the Stieltjes transform [@Nem]. Let $r>-1$ and suppose $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$, where $x^{-r-k+\sigma}f(x)$ is [*bounded as*]{} $x \to \infty$ [*for some*]{} $\sigma >0$. The Stieltjes transform of $W$ of index $r$ is given by $$\Lambda_r W(z) = (r+1)_k \int_0^\infty \, \frac{f(x)}{(x+z)^{r+k+1}} \, dx \, , \quad z \in {{\mathbb C}}\backslash(-\infty, 0],$$ where $(r+1)_k = \frac{\Gamma(r+k+1)}{\Gamma(r+1)} = (r+1)(r+2) \dots (r+k).$ Notice that $\Lambda_rW(z)$ is holomorphic in the variable $z$, as one readily verifies.\ The following is a classical Abelian theorem for the Stieltjes transform. \[Carm\] If $f \in C_+({{\mathbb R}})$ such that $x^{-\nu}f(x)\to A$ as $x \to \infty$, with $\nu > -1$, then for $\rho > \nu$, $$\lim_{\stackrel{z \to \infty}{|\arg z| \le \theta < \pi/2}} \frac{z^{\rho - \nu}\, \Gamma(\rho+1) S_\rho f(z) }{\Gamma(\rho-\nu)\Gamma(\nu + 1)} \, = \, A,$$ where $S_\rho f(z) = \int_0^\infty \, \frac{f(x)}{(x+z)^{\rho+1}} \, dx.$\ Let $W \in {{\mathcal M}}$ and $r > -1$. Suppose that $\displaystyle W(x) \sim \frac{\gamma x^\alpha}{\Gamma(\alpha+1)} \quad (C), \quad x \to \infty$. Then: 1. If $r > \alpha > -1$, then $\Lambda_r W(z)$ is well-defined and has asymptotic behavior $$\lim_{\stackrel{z \to \infty}{|\arg z| \le \theta < \pi/2}} \frac{z^{r-\alpha}\, \Gamma(r+1) \Lambda_r W(z)}{\Gamma(r-\alpha)} \, = \, \gamma,$$ 2. If $\alpha <-1, \, \alpha \notin \{-2, -3, -4, \dots\}$, then $\Lambda_r W(z)$ is well-defined and there are constants $A_1, \dots, A_k$ such that $$\label{Limitgamma} \lim_{\stackrel{z \to \infty}{|\arg z| \le \theta < \pi/2}} \frac{z^{r-\alpha}\, \Gamma(r+1)}{\Gamma(r-\alpha)} \left[ \Lambda_r W(z) - \sum_{j=1}^k \frac{A_j}{z^{r+j}} \right] \, = \, \gamma.$$ \(i) Let $\alpha > -1$ and $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$ such that $$\frac{\Gamma(\alpha+k+1) f(x)}{x^{\alpha+k}} \to \gamma \quad \mbox{ as } x \to \infty.$$ It follows that for $r > \alpha, f(x)x^{-r-k+\sigma}$ is bounded as $x \to \infty$ for some $ \sigma > 0$. Now, by substituting $r+k$ for $\rho$, $\alpha + k$ for $\nu$, and $\frac{\gamma}{\Gamma(\alpha + k + 1)}$ for $A$ in the above classical Abelian theorem, we obtain $$\lim_{\stackrel{z \to \infty}{|\arg z| \le \theta < \pi/2}} \frac{z^{r - \alpha} {\Gamma(r+k+1) S_{r+k} f(z)} }{\Gamma(r-\alpha)} \, = \, \gamma.$$ Now, using the fact that $\Lambda_r W(z) = (r+1)_k S_{r+k} f(z)$, the result follows. \(ii) Suppose that $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$ with $k+\alpha >-1$ and that $f$ can be written as $f(x) = p(x) + g(x),$ where $g \in C_+({{\mathbb R}})$ satisfies $\lim_{x \to \infty}x^{-\alpha - k} g(x) =\gamma/\Gamma(\alpha + k + 1)$ and $p(x) = \sum_{j=0}^{k-1} a_j x^j$. It follows that $|f(x)| \le Cx^{k-1}$ for some constant $C$ and thus $f(x)x^{-r-k+\sigma}$ is bounded for any $0 < \sigma \le 1+r$. Observe that $$\begin{aligned} S_{r+k}\,f(z) &= S_{r+k}\,g(z) + \sum_{j=0}^{k-1}a_j \int_0^\infty \frac{x^j}{(x+z)^{r+k+1}} \, dx\\ & = S_{r+k}\,g(z) + \sum_{j=0}^{k-1} \frac{j! \Gamma(r+k-j)}{\Gamma(r+k+1)z^{j+k+r}} \, \, a_j.\end{aligned}$$ By Theorem \[Carm\], we have that $z^{r-\alpha}S_{r+k} \, g(z) \to \frac{\gamma \Gamma(r-\alpha)}{\Gamma(r+k+1)}$ as $z \to \infty$ on sectors $|\arg z| \le \theta < \frac{\pi}{2}$. Since $\Lambda_rW(z) = (r+1)_k S_{r+k}\,f(z)$, we obtain (\[Limitgamma\]) with $A_j = (k-j)!(r+1)_{j-1} a_{k-j}$. We illustrate our ideas with the ensuing example, a deduction of Stirling’s formula for the Gamma function. [**(Stirling’s formula)**]{} Recall that the digamma function $\psi$ is defined as the logarithmic derivative of $\Gamma$. By using the product formula for $\Gamma$, namely, $$\Gamma(z)= \frac{e^{- \gamma z}}{z} \prod_{n=1}^{\infty} \left(1+\frac{z}{n}\right)^{-1}e^{\frac{z}{n}},$$ one has $$\psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} = -\gamma + \sum_{n=0}^\infty \left( \frac{1}{n+1} - \frac{1}{n+z} \right), \quad z \in {{\mathbb C}}\backslash \{0,-1,-2, \dots\},$$ where $\gamma$ is the Euler-Mascheroni constant. We define $\displaystyle W = \frac{f}{H^2}$, where $f(x) = \int_0^x \left( \lfloor t \rfloor - t+ \frac{1}{2} \right) dt$ (here $\lfloor x \rfloor$ stands for the integer part of $x$). Set $g(x) = \lfloor x \rfloor - x + \frac{1}{2}$ and note that $g$ is periodic with period 1, $|g(x)| \le \frac{1}{2}$ for all $x \in {{\mathbb R}}$, and $\int_n^{n+1} g(x) dx = 0$ for all $n \in {{\mathbb N}}$. This implies that $\left| \int_0^x g(t) dt \right| \le \frac{1}{2}$ for all $x \ge 0$. Consequently, $W(x) \sim 0 \cdot x^{-\sigma} \quad (C), \quad x \to \infty$ for any $0 < \sigma < 2$. Theorem 4 now yields (from the proof it is clear that the constants $A_1 = A_2 = 0$ in this case because $f$ is bounded) $$\label{LimitZero} \lim_{\stackrel{z \to \infty}{|\arg z| \le \theta < \pi/2}}z^\sigma \Lambda_0 W(z) = 0,$$ for any $0<\sigma<2$. From now on we will work with $1 < \sigma <2$. We compute an explicit expression for $ \Lambda_0 W(z)$, $$\begin{aligned} \Lambda_0 W(z) & = 2 \int_0^\infty \frac{f(x)}{(x+z)^3}\, dx = \int_0^\infty \frac{(\lfloor x \rfloor - x + \frac{1}{2} )}{(x+z)^2} \, dx = \frac{1}{2z} + \lim_{N \to \infty} \int_0^N \frac{\lfloor x \rfloor - x}{(x+z)^2} \, dx \\ & = \frac{1}{2z} + \lim_{N \to \infty} \left( \int_0^N \frac{d \lfloor x \rfloor}{x+z} - \frac{N}{N+z} + \frac{N}{N+z} - \int_0^N \frac{dx}{x+z} \right)\\ &=\ln z + \frac{1}{2z} + \lim_{N \to \infty} \sum_{n=1}^N \frac{1}{n+z} - \ln N\\ &= \ln z + \frac{1}{2z} - \psi(z).\end{aligned}$$ The limit (\[LimitZero\]) then yields $$\label{psi} \psi(z) = \ln z + \frac{1}{2z} + o \left(\frac{1}{z^\sigma} \right), \quad \, \, z \to \infty,$$ for $z$ in the sectors $|\arg z| \le \theta < \frac{\pi}{2}$. Note that integration of (\[psi\]) implies for any $0 < \tau < 1$, $$\ln \Gamma(z) = z(\ln z - 1) + \frac{1}{2} \ln z + C + o \left( \frac{1}{z^\tau} \right), \quad z \to \infty$$ on $|\arg z| \le \theta < \frac{\pi}{2}$, which is Stirling’s asymptotic formula for the Gamma function except for the evaluation of the constant $C$. The constant is of course well known to be $C = \sqrt{2\pi}$. We refer to [@EstradaKanwal p. 43] for an elementary proof of the latter fact. We now consider the Laplace transform [@AtanNem]. If $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$, where $f(x) e^{-\sigma x}$ [*is bounded as* ]{} $x \to \infty$ [*for some* ]{} $ \sigma \in {{\mathbb R}}$, then the Laplace transform of $W$ is given by $${{\mathcal L}}W(z) = z^k \int_0^\infty \, e^{-zx} \,f(x) \, dx, \, \, \Re e\: z > \sigma.$$ Let $W \in {{\mathcal M}}$. Assume that $\displaystyle W(x) \sim \frac{\gamma x^\alpha}{\Gamma(\alpha + 1)} \quad (C), \, \, x \to \infty$. Then $W$ is Laplace transformable and 1. If $\alpha > -1$, then $$\lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}}z^{\alpha + 1} {{\mathcal L}}W(z) = \gamma.$$ 2. If $\alpha <-1, \, \alpha \notin \{-2, -3, \dots\}$, then there are constants $A_0, \dots, A_{k-1}$ such that $$\label{Laplacegamma} \lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}}\, z^{\alpha + 1}\left( {{\mathcal L}}W(z) - \sum_{j=0}^{k-1} A_j z^j \right) = \gamma$$ In the terminology of finite part limits ([@EstradaKanwal Sect. 2.4]), the limit (\[Laplacegamma\]) might be rewritten as $$\mbox{F.p.} \lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}} z^{\alpha + 1} {{\mathcal L}}W(z) = \gamma.$$ \(i) Let $\alpha > -1$ and $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$ such that $$\frac{\Gamma(\alpha+k+1) f(x)}{x^{\alpha+k}} \to \gamma \quad \mbox{ as } x \to \infty.$$ It follows that there exists $\sigma > 0$ such that $f(x)e^{-\sigma x}$ is bounded as $x \to \infty$. Now, by a well known classical Abelian theorem for the Laplace transform [@Doetsch], we obtain $$\lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}}\frac{z^{\alpha + k + 1} {{\mathcal L}}f(z)}{\Gamma(\alpha + k + 1)} = \frac{\gamma}{\Gamma(\alpha + k + 1)} \, .$$ So, $$\hspace{.75in} \lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}}z^{\alpha + 1} {{\mathcal L}}W(z) = \lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}} z^{\alpha + k + 1} {{\mathcal L}}f(z) = \gamma .$$ \(ii) Suppose $\displaystyle W = \frac{f}{H^k} \in {{\mathcal M}}$ with $k + \alpha > -1$ and that $f$ can be written as $f(x) = p(x) + g(x), $ where $g \in C_+({{\mathbb R}})$ has asymptotic behavior $\lim_{x \to \infty} x^{-\alpha-k}g(x) = \gamma/\Gamma(\alpha+k+1)$ and $p(x) = \sum_{j=0}^{k-1} \alpha_j x^j$. Note that $W = W_1+W_2$, where $\displaystyle W_1 = \frac{g}{H^k}$ and $\displaystyle W_2 = \frac{p}{H^k}$. Exactly as above, one verifies that $$\lim_{\stackrel{z \to 0}{|\arg z| \le \theta < \pi/2}}z^{\alpha + 1} {{\mathcal L}}W_1(z) = \gamma.$$ It remains to observe that ${{\mathcal L}}W_2(z) = \sum_{j=0}^{k-1} A_j z^j,$ with $A_j = \alpha_{k-j-1}/(k-j-1)!$. \[Tauberian\] [**(Tauberian Theorem)**]{} Let $\displaystyle W= \frac{f}{H^k} \in {{\mathcal M}}$, where in addition to $f \in C_+({{\mathbb R}})$, $f$ is real-valued and nonnegative. If ${{\mathcal L}}W(s) \sim \gamma s^{-\alpha - 1}, \, s \to 0^+$ (for some $\alpha > -1$), then $\displaystyle W(x) \sim \frac{\gamma x^\alpha}{\Gamma(\alpha+1)} \quad (C), \, x \to \infty$. It follows that $$\label{Tauberian1} {{\mathcal L}}f(s) \sim \gamma s^{-(\alpha + k + 1)}, \quad s \to 0^+.$$ Also, $$\label{Tauberian2} {{\mathcal L}}f(s) = \int_0^\infty e^{-st} d(H \ast f)(t), \quad \, s>0.$$ Since $H \ast f$ is nondecreasing , by (\[Tauberian1\]) and (\[Tauberian2\]) and Hardy-Littlewood-Karamata Tauberian Theorem [@BingGoldTeug; @vdz], it follows that $$(H \ast f)(x) \sim \frac{\gamma}{\Gamma(\alpha + k + 2)} \, x^{\alpha + k + 1}, \quad \,\, x \to \infty.$$ That is, $$\frac{\Gamma(\alpha + (k+1) + 1)(H \ast f)(x)}{x^{\alpha+(k+1)}} \to \gamma, \quad \, \, x \to \infty.$$ Since $\displaystyle W = \frac{H \ast f}{H^{k+1}}$, the above yields $\displaystyle W(x) \sim \frac{\gamma x^\alpha}{\Gamma(\alpha+1)} \quad (C), \, \, x \to \infty. $ [99]{} D. Atanasiu, D. Nemzer, *Extending the Laplace transform*, Math. Student **77** (2008), 203–212. N. Bingham, C. Goldie, J. Teugels, *Regular variation*, Cambridge University Press, Cambridge, 1987. R. D. 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--- abstract: | We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form $A\:u_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$, where $A,B,C\in\Rset^{n,n}$ and the matrix $A$ is singular. We prove that under certain conditions the temporal convergence order of the fully discrete scheme depends on the time index of the partial differential-algebraic equation. In particular, fractional orders of convergence in time are encountered. Furthermore we show that the fully discrete scheme suffers an order reduction caused by the boundary conditions. Numerical examples confirm the theoretical results. address: - 'Darmstadt University of Technology, Department of Mathematics, Schlo[ß]{}gartenstra[ß]{}e 7, D-64283 Darmstadt, Germany' - 'Martin-Luther-Universität Halle-Wittenberg, FB Mathematik and Informatik, Institut für Numerische Mathematik, Postfach, D-06099 Halle (Saale), Germany' author: - 'K. Debrabant' - 'K. Strehmel' title: 'Convergence of Runge-Kutta Methods Applied to Linear Partial Differential-Algebraic Equations ' --- Partial differential-algebraic equations ,Coupled systems ,Implicit Runge-Kutta methods ,Convergence estimates Introduction ============ In this paper we consider linear partial differential-algebraic equations (PDAEs) of the form $$\begin{aligned} \label{pdaglgsys1} A\:u_t(t,x) +B\:\left( u_{xx}(t,x)+ru_{x}(t,x)\right) + C\: u(t,x) = f(t,x),\end{aligned}$$ where $t\in (t_0 , t_e)$, $x \in \Omega=(-l,l) \subset\Rset,$ $A,B,C \in \Rset^{n,n}$ are constant matrices, $r\in\Rset,$ $u,f: [t_0 , t_e]\times \overline{\Omega}\to \Rset^{n}$. We are interested in cases where the matrix $A$ is singular. The singularity of $A$ leads to the differential-algebraic aspect.\ It will always be tacitly assumed that the exact solution is as often differentiable as the numerical analysis requires. In contrast to parabolic initial boundary value problems with regular matrices $A$ and $B$, here we cannot prescribe initial and boundary values for all components of the solution vector, they have to fulfill certain consistency conditions. We consider one example: Superconducting coil (see Marszalek/Trzaska, Campbell/Marszalek [@MarszalekTrzaska; @CampbellMarszalek.b]): $$\begin{pmatrix}{~\;0}&0\\{-\frac{LC}{l^2}}&{\frac{L}D} \end{pmatrix} u_{tt}-u_{xx} +\begin{pmatrix}0&1\\0&0 \end{pmatrix} u=0$$ with $~x\in(0,l),~t>0$. $u_1(t,x)$ denotes the voltage, $u_2(t,x)$ denotes the divergence of the electric field strength within the coil. $l$ is the length of the whole winding. $L$, $C$ and $D$ are further coil parameters. Transformation to a partial differential-algebraic system of first order in $t$ yields $$\label{GlgSystemSupraMagnetSpule} \begin{pmatrix} 0 & 0 &0 &0\\ 0 & 0 &-\frac{LC}{l^2}~~ & \frac LD\\ {1} & {0} & {0} & {0}\\ {0} & {1} & {0} & {0} \end{pmatrix} u_t- \begin{pmatrix} 1 & 0 &0 &0\\ 0 & 1 & 0 &0\\ {0} & {0} & {0} & {0}\\ {0} & {0} & {0} & {0} \end{pmatrix} u_{xx} + \begin{pmatrix} 0 & 1 &~\;0 &~\;0\\ 0 & 0 & ~\;0 &~\;0\\ {0} & {0} & {-1} & {~\;0}\\ {0} & {0} & {~\;0} & {-1} \end{pmatrix} u=0.$$ As initial conditions we choose $$u_1(0,x)=\left(\frac{E}l-\frac{CDEl}6\right) x+\frac{CDE}{6l}x^3,~u_3(0,x)=0$$ and as boundary conditions $$u_1(t,0)=u_2(t,0)=0,~u_1(t,l)=E,~u_2(t,l)=CDE,$$ where $E$ is the energizing source voltage at the input of the coil.\ As the boundary values of $u_1$ and $u_2$ are constant, we get from the third and fourth equation of [(\[GlgSystemSupraMagnetSpule\])]{} that $u_3$ and $u_4$ fulfill homogeneous boundary conditions. From the initial condition of $u_1$ and the first equation we derive With $u_3(0,x)=0$ and the third equation it follows and therefore . With the first equation this implies and with the fourth equation we get finally $u_{4}(0,x)=0$.\ Here we have chosen the prescribed initial and boundary values such that all initial and boundary values are compatible. For further examples considering the determination of the initial and boundary values which cannot be prescribed see Lucht/S./Eichler-Liebenow [@LuStreEiL]. In the following we assume that for the numerical computation all initial values $$u(t_0,x)= \varphi(x),\;x\in\bar{\Omega},$$ and all boundary values entering into the space discretization are known, $$\begin{aligned} 4 B u(t,x)&=\;&&\psi(t,x),~&&x\in\partial\Omega, ~&&t\in [t_0 , t_e],\end{aligned}$$ where we restrict ourselves to Dirichlet boundary conditions to simplify the presentation.\ Investigations of the convergence of Runge-Kutta methods applied to abstract parabolic differential equations can be found for example in Brenner/Crouzeix/Thomée [@BrennerCrouzeixThomee], Lubich/Ostermann [@LubichOstermann1993] and Ostermann/Thalhammer [@OstermannThalhammer]. The approach used there cannot be carried forward directly to the class of problems considered here because the matrix $A$ is singular. This paper is organized as follows: In Section \[Abschnitt2\] we derive a semi-discrete system based on finite differences. The result is a method-of-lines-DAE (MOL-DAE).\ Section \[Abschnitt3\] is devoted to the Runge-Kutta approximation of the MOL-DAE. Under a regular transformation, the MOL-DAE of dimension $nN$ is decoupled into $N$ systems of dimension $n$, where $N$ denotes the number of grid points on the $x$-axis. Furthermore, a Weierstrass-Kronecker transformation is used to decouple each of these systems into an ODE-system and an algebraic system. We introduce the differential time index of the linear PDAE and give the Runge-Kutta approximation to these subsystems.\ In Section \[Abschnitt4\] we prove the convergence of $L$-stable Runge-Kutta discretizations with constant step sizes. The attained order of convergence in time depends on the differential time index of the PDAE and on the boundary conditions (homogeneous or inhomogeneous) which enter into the space discretization.\ Numerical experiments are finally presented in Section \[Abschnitt5\]. We illustrate our convergence results for the backward Euler method and the 3-stage Radau IIA method. Space discretization {#Abschnitt2} ==================== The discretization in space of problem (\[pdaglgsys1\]) by means of finite-differences results in a differential-algebraic equation (MOL-DAE) $$\label{ZMOL} M\dot{U}=DU(t)+\tilde{F}(t), \quad t_0\leq t\leq t_e,$$ where $U(t)$ is an $Nn$-dimensional real vector consisting of approximations to $u$ at the grid points. Here $N$ denotes the number of grid points on the $x$-axis. The matrix $M$ is given by $M=I_N\otimes A$ and the matrix $D$ originates from the discretization of the differential operator $B\frac{\partial^2}{\partial x^2}$ by second order difference-approximations, from the discretization of the differential operator $rB\frac{\partial }{\partial x}$ by second ($\delta=\frac12$) or first order difference-approximations ($\delta\in[0,1]\setminus\{\frac12\}$) and from the matrix $C$, i.e., $D$ is given by $$D=-\frac1{h^2}P\otimes{B}-I_N\otimes{C},$$ where $I_{N}$ is the $N$-dimensional identity matrix, $$P=\begin{pmatrix} -(2-hr(1-2\delta)) & 1+hr\delta & \\ 1+hr(\delta -1)& -(2-hr(1-2\delta))& 1+hr\delta \\ \dots & \dots & \dots \\ \dots & \dots & 1+hr(\delta -1)& -(2-hr(1-\delta)) \end{pmatrix}$$ and $h=\frac{2l}{N+1}$ denotes the constant grid size. The $Nn$-dimensional real vector $\tilde{F}(t)$ arises from the right hand side $f$ of (\[pdaglgsys1\]) and the boundary values which enter into the discretization.\ We denote by $U_h(t)$ the restriction of $u(t,x)$ to the spatial grid and by $\alpha_h(t)$ the space truncation error defined by $$\label{alpha} \alpha_{h}(t):=M\dot{U}_{h}(t) -D U_{h}(t)-\tilde{F}(t).$$ By Taylor expansion of the exact solution we get $$\begin{aligned} 2\label{alpha2} \alpha_h(t)&= h^{p_x} (I_{N}\otimes B) \gamma_{h}(t) &\quad \text{with}\quad \|\gamma_{h}(t)\|_\infty&\leq K,\end{aligned}$$ where $p_x\in\{1,2\}$ is the order of approximation of the space discretization and $K$ is a positive constant, i.e., $$\alpha_h(t)=\mathcal{O}(h^{p_x})\quad \text{as}\quad h\to 0.$$ Furthermore, we can show that there exists a regular matrix $Q$ with $$\begin{aligned} 2\label{ShiGlg} Q^{-1}\frac{1}{h^2}PQ=\text{diag}\{\lambda_{1},\dots,\lambda_{N}\},&\quad\end{aligned}$$ where $$\lambda_{j}=-\frac{ 2-hr(1-2\delta)}{h^2}+2\frac{1+hr\delta }{h^2} \sqrt{\frac{1+hr(\delta -1)}{1+hr\delta}}\cos\frac{j\pi}{N+1}.$$ In the discrete $L_2$-norm we have $$\label{CKonst} \max\{\|Q\|,\|Q^{-1}\|\}\leq C_{1}$$ with a positive constant $C_1$ independent of $h$. Therefore, in the following this norm is used.\ Runge-Kutta approximations {#Abschnitt3} ========================== In order to numerically advance in time the solution of the MOL-DAE (\[ZMOL\]), we employ an $s$-stage Runge-Kutta method $$\begin{aligned} {2} U_{m+1}^{(i)}&=U_m+\tau\sum_{i=1}^sa_{ij}K_{m+1}^{(i)},&~ MK_{m+1}^{(i)}&=DU_{m+1}^{(i)}+\tilde{F}(t_m+c_i\tau),~ 1\leq i\leq s,\\ U_{m+1}&=U_m+\tau\sum_{i=1}^sb_iK_{m+1}^{(i)},\end{aligned}$$ where $a_{ij},b_i,c_i\in\mathbb{R}$ are the coefficients of the method and $\tau=\frac{t_e-t_0}{M_e}$ the time step size.\ For the investigation of the convergence of the method, it is useful to introduce the Runge-Kutta matrix $\mbox{$\mathfrak{A}$}=(a_{ij})_{ij=1}^s$ and the vector notation ${1\kern-0.25em{\rm l}}_s=(1,\dots,1)^{\top}\in \mathbb{R}^s,$\ Then, with the Kronecker product, we obtain the compact scheme \[NLRKGlg\] $$\begin{aligned} 1 U_{m+1}&=U_m+\tau\left( b^{\top}\otimes I_{Nn}\right) K_{m+1}\label{NLRKGlg1},\\ S_{m+1}&={1\kern-0.25em{\rm l}}_s\otimes U_m+\tau\left(\mbox{$\mathfrak{A}$}\otimes I_{Nn}\right) K_{m+1}\label{NLRKGlg2},\\ \left( I_s\otimes M\right) K_{m+1}&=\left( I_s\otimes D\right) S_{m+1}+\bar{F}(t_{m+1}),\label{NLRKGlg3}~m=0,\dots,M_e-1,\end{aligned}$$ where $S_{m+1}=\left({U_{m+1}^{(1)}}^{\top},\dots,{U_{m+1}^{(s)}}^{\top}\right)^{\top},$ $K_{m+1}=\left({K_{m+1}^{(1)}}^{\top},\dots,{K_{m+1}^{(s)}}^{\top}\right)^{\top}$ and $\bar{F}(t_{m+1})=\left({\tilde{F}\left( t_m+c_1\tau\right)}^{\top},\dots,{\tilde{F}\left( t_m+c_s\tau\right)}^{\top}\right)^{\top}.$ By the regular transformation [(\[ShiGlg\])]{}, the MOL-DAE [(\[ZMOL\])]{} can be decoupled into $N$ DAEs $$\label{DASys_entk} A\dot{U}_{Qk}(t)=D_kU_{Qk}(t)+\tilde{F}_{Qk}(t),\quad k=1,\dots,N,$$ with $D_k=-{\lambda_k} B-C$ and $$\begin{aligned} {1} \left( U_{Q1}(t)^\top,\dots,U_{QN}(t)^\top\right)^\top&=\left( Q^{-1}\otimes I_n\right) U(t),\quad\\ \left( \tilde{F}_{Q1}(t)^\top,\dots,\tilde{F}_{QN}(t)^\top\right)^\top&=\left( Q^{-1}\otimes I_n\right)\tilde{F}(t).\end{aligned}$$ In the following we assume that the matrix pencil $\{D+\lambda M\}$, ${\lambda\in\Cset}$, is regular, which is equivalent to the regularity of all the matrix pencils $\{ D_k+\lambda A\}$. Suppose that all matrix pencils $\{D_k+\lambda A\}$, $k=1,\dots,N$, are regular and have the same index $\nu_{dt}$. Then the differential time index of the linear PDAE [(\[pdaglgsys1\])]{} is defined to be $\nu_{dt}$. According to Weierstrass and Kronecker there exist regular matrices $P_{k}$ and $Q_{k}$ with \[WKTransf\] $$\begin{aligned} P_{k}AQ_{k}&=&\mathrm{diag}\{I_{n_{k 1}},\dots,I_{n_{k s_k}},N_{m_{k 1}},\dots,N_{m_{k l_k}}\},\\ P_{k}D_{k}Q_{k}&=&\mathrm{diag}\{R_{k 1},\dots,R_{k s_k},I_{m_{k 1}},\dots,I_{m_{k l_k}}\}\label{Dkvekdiag},\end{aligned}$$ where $$\label{RkiNkiDef} R_{k i}= \left(\begin{array}{cccc} \varkappa_{k i} & 1 & & 0 \\ & \ddots & \ddots & \\ & & \varkappa_{k i} & 1 \\ 0 & & & \varkappa_{k i} \end{array} \right)\in\Cset^{n_{k i},n_{k i}},~ N_{m_{k i}}= \left(\begin{array}{cccc} 0 & 1 & & 0 \\ & \ddots & \ddots & \\ & & 0 & 1 \\ 0 & & & 0 \end{array} \right)\in\Cset^{m_{k i},m_{k i}}$$ (see Hairer/Wanner [@HairerWanner]), and for the differential time index of the PDAE it follows $$\nu_{dt}=\max\limits_{k}\{m_{k i}:i=1,\dots,l_k\}.$$ Therefore, DAE [(\[DASys\_entk\])]{} is decoupled into systems of the form $$\begin{aligned} 3\label{WK1} \dot{U}_{1kl}(t)&=R_{kl}U_{1kl}(t)+\Tilde{F}_{1kl}(t),\quad &l&=1,\dots,s_k,\\ N_{m_{kl}}\dot{U}_{2kl}(t)&=U_{2kl}(t)+\Tilde{F}_{2kl}(t),\quad &l&=1,\dots,l_k\label{WK2}\end{aligned}$$ with $$\left( U_{1k1}(t)^\top,\dots,U_{1ks_k}(t)^\top,U_{2k1}(t)^\top,\dots,U_{2kl_k}(t)^\top\right)^\top =Q_k^{-1}U_{Qk}$$ and $$\left(\Tilde{F}_{1k1}(t)^\top,\dots,\Tilde{F}_{1ks_k}(t)^\top,\Tilde{F}_{2k1}(t)^\top,\dots,\Tilde{F}_{2kl_k}(t)^\top\right)^\top =P_k\Tilde{F}_{Qk}.$$ Similarly, DAE [(\[alpha\])]{} can be transformed to $$\begin{aligned} 3 \label{alphaWK1} \dot{U}_{h1kl}(t)&=R_{kl}U_{h1kl}(t)+\Tilde{F}_{1kl}(t)+\alpha_{h1kl}(t),\quad &l&=1,\dots,s_k,\\ N_{m_{kl}}\dot{U}_{h2kl}(t)&=U_{h2kl}(t)+\Tilde{F}_{2kl}(t)+\alpha_{h2kl}(t),\quad &l&=1,\dots,l_k.\end{aligned}$$ Runge-Kutta methods are invariant under the transformations [(\[ShiGlg\])]{} and [(\[WKTransf\])]{}. Therefore, to analyze convergence it is sufficient to apply them to systems of the form [(\[WK1\])]{} and [(\[WK2\])]{}. Application to [(\[WK1\])]{} yields \[NLRKGlgWK1\] $$\begin{aligned} 1 U_{1kl,m+1}&=U_{1kl,m}+\tau\left( b^{\top}\otimes I_{n_{kl}}\right) K_{1kl,m+1}\label{NLRKGlg1WK1},\\ S_{1kl,m+1}&={1\kern-0.25em{\rm l}}_s\otimes U_{1kl,m}+\tau\left(\mbox{$\mathfrak{A}$}\otimes I_{n_{kl}}\right) K_{1kl,m+1}\label{NLRKGlg2WK1},\\ K_{1kl,m+1}&=\left( I_s\otimes R_{kl}\right) S_{1kl,m+1}+\bar{F}_{1kl}(t_{m+1}),\label{NLRKGlg3WK1}~m=0,\dots,M_e-1,\end{aligned}$$ and to [(\[WK2\])]{} \[NLRKGlgWK2\] $$\begin{aligned} 1 U_{2kl,m+1}&=U_{2kl,m}+\tau\left( b^{\top}\otimes I_{n_{kl}}\right) K_{2kl,m+1}\label{NLRKGlg1WK2},\\ S_{2kl,m+1}&={1\kern-0.25em{\rm l}}_s\otimes U_{2kl,m}+\tau\left(\mbox{$\mathfrak{A}$}\otimes I_{n_{kl}}\right) K_{2kl,m+1}\label{NLRKGlg2WK2},\\ \left( I_s\otimes N_{m_{kl}}\right) K_{2kl,m+1}&=S_{2kl,m+1}+\bar{F}_{2kl}(t_{m+1}),\label{NLRKGlg3WK2}~m=0,\dots,M_e-1.\end{aligned}$$ Now we start our convergence investigations. Convergence estimates {#Abschnitt4} ===================== At first we introduce the global (space-time discretization) error $e_{m+1}$ and the residual (space-time discretization) errors $\delta_{m+1}$ and $\Delta_{m+1}$ at the time level $t=t_{m+1}$. The global error $e_{m+1}$ at $t_{m+1}$ is defined by $$e_{m+1}:=U_h(t_{m+1})-U_{m+1}$$ and the residual errors $\delta_{m+1},\Delta_{m+1}$ are given by $${\delta_{m+1}}:=U_{h}(t_m+\tau)-U_{h}(t_m)- \tau\left( b^{\top}\otimes I_{Nn}\right)\hat{K}_{m+1}, \label{ResGlg2}$$ $$\label{ResGlg1} \Delta_{m+1} := \hat{S}_{m+1}-{1\kern-0.25em{\rm l}}_s\otimes U_h(t_m)-\tau\left(\mbox{$\mathfrak{A}$}\otimes I_{Nn}\right)\hat{K}_{m+1},$$ where $\hat{S}_{m+1}$ and $\hat{K}_{m+1}$ are defined by the exact solution $U_h(t)$ of the PDAE, i.e., $$\begin{aligned} 1 \hat{S}_{m+1}&:=\left( U_h(t_m+c_1\tau)^{\top},\dots U_h(t_m+c_s\tau)^{\top}\right)^{\top},\\ \hat{K}_{m+1}&:=\left(\dot{U}_h(t_m+c_1\tau)^{\top},\dots\dot{U}_h(t_m+c_s\tau)^{\top}\right)^{\top}.\end{aligned}$$ The discretization scheme (\[NLRKGlg\]) is convergent of order $(p_x,p^\star)$, if the global error satisfies $$\| e_{m+1}\| =\mathcal{O}(h^{p_x})+\mathcal{O}(\tau^{p^\star}) \quad \mathrm{for} \quad (m+1)\tau=\mathrm{const.},~\tau,\, h\to 0,$$ whenever $u(t,x)$ is sufficiently often differentiable. With the components $e_{Qk,m+1}$ defined by $$\left( e_{Q1,m+1}^\top,\dots,e_{QN,m+1}^\top\right)^\top:=(Q^{-1}\otimes I_n)e_{m+1}$$ and [(\[CKonst\])]{} we obtain the estimate $$\label{eqzsglg} \frac1{C_1}\sqrt{h\sum_{k=1}^N\|e_{Qk,m+1}\|^2}\leq \|e_{m+1}\|\leq C_1\sqrt{h\sum_{k=1}^N\|e_{Qk,m+1}\|^2}.$$ Letting $\alpha_{h 1kl, m+1}=\left(\alpha_{h 1kl} (t_m+c_1\tau)^{\top},\dots,\alpha_{h 1kl}(t_m+c_s\tau)^{\top}\right)^{\top} $ we get with [(\[alphaWK1\])]{} and [(\[NLRKGlg3WK1\])]{} $$\hat{K}_{1kl,m+1}-K_{1kl,m+1}= \left( I_s\otimes R_{kl}\right)\left(\hat{S}_{1kl,m+1}-S_{1kl,m+1}\right)+\alpha_{h1kl, m+1}.$$ Using (\[NLRKGlg2WK1\]), the transformed components $e_{1kl,m}=U_{h1kl}(t_{m})-U_{1kl,m}$ of the global discretization error and the transformed components $\Delta_{1kl,m+1}$ of (\[ResGlg1\]) we obtain $$\hat{S}_{1kl,m+1}-S_{1kl,m+1}={1\kern-0.25em{\rm l}}_s\otimes e_{1kl,m} +\tau\left(\mbox{$\mathfrak{A}$}\otimes I_{n_{kl}}\right)\left(\hat{K}_{1kl,m+1}-K_{1kl,m+1}\right) +\Delta_{1kl,m+1}\label{StGlgDiff2}.$$ Combining the last two equations leads to $$\label{Stern1} G(\tau R_{kl})\left(\hat{K}_{1kl,m+1}-K_{1kl,m+1}\right)= \left({1\kern-0.25em{\rm l}}_s\otimes R_{kl}\right) e_{1kl,m} +\left( I_s\otimes R_{kl}\right)\Delta_{1kl,m+1}+\alpha_{h1kl, m+1},$$ where $G(z)=I_s-z\mbox{$\mathfrak{A}$}$. For a (matrix-valued) function $f(z):\Cset\to\Cset^{m,n}$ which is analytic in a neighbourhood of $\kappa_{ki}$, the matrix function $f(R_{ki})$ with $R_{ki}$ given in [(\[RkiNkiDef\])]{} is defined by (see Golub/van Loan [@GolubvanLoan]) $$f(R_{ki}):= \left(\begin{array}{cccc} f_{i_1i_2}(\varkappa_{k i}) & \frac{f_{i_1i_2}^{(1)}(\varkappa_{k i})}{1!} & \dots & \frac{f_{i_1i_2}^{(n_{ki}-1)}(\varkappa_{k i})}{(n_{ki}-1)!} \\ & \ddots & \ddots & \vdots \\ & & f_{i_1i_2}(\varkappa_{k i}) & \frac{f_{i_1i_2}^{(1)}(\varkappa_{k i})}{1!} \\ 0 & & & f_{i_1i_2}(\varkappa_{k i}) \end{array} \right)_{i_1=1,\dots,m,~i_2=1,\dots,n}.$$ In the following we assume that the Runge-Kutta method is A-stable and $\Re(\varkappa_{kl})\leq0$ or $|\kappa_{kl}|\leq C_2$ for all $h\in(0,h_0]$ with a positive constant $C_2$. Then for sufficiently small $\tau$ the matrix $G(\tau R_{kl})$ is regular, and the Runge-Kutta system [(\[NLRKGlgWK1\])]{} has a unique solution. Using (\[NLRKGlg1WK1\]), (\[Stern1\]) and the transformed components $\delta_{1kl,m+1}$ of (\[ResGlg2\]) we obtain the recursion $$\begin{aligned} 1 e_{1kl,m+1} &=R(\tau R_{kl})e_{1kl,m} +L(\tau R_{kl})\Delta_{1kl,m+1} +\tau J(\tau R_{kl})\alpha_{h1kl, m+1} +\delta_{1kl,m+1}\label{Fehlerrekursion}\end{aligned}$$ for the discretization error $e_{1kl,m+1}$, where we have used the abbreviations $$J(z)=b^{\top}G(z)^{-1}, \quad R(z)=1+J(z){1\kern-0.25em{\rm l}}_sz, \quad L(z)=J(z)z$$ ($R(z)$ equals the classical stability function of the Runge-Kutta method). Solving the recursion (\[Fehlerrekursion\]) with $e_0=0$ leads to $$\begin{aligned} 1\nonumber e_{1kl,m+1} =& \sum_{i=0}^mR(\tau R_{kl})^iL(\tau R_{kl})\Delta_{1kl,m+1-i} \\&\label{emp1glgn1} +\tau\sum_{i=0}^mR(\tau R_{kl})^iJ(\tau R_{kl})\alpha_{h1kl, m+1-i}+\sum_{i=0}^mR(\tau R_{kl})^i\delta_{1kl,m+1-i}.\end{aligned}$$ Now we assume that the Runge-Kutta method under consideration has (classical) order $p$ and stage order $q$ ($p\geq q$). Then the simplifying conditions (see Hairer/Wanner [@HairerWanner]) $$\begin{aligned} 1 B(p):~&\sum_{i=1}^sb_ic_i^{k-1}=\frac{1}{k},~k=1,\dots,p,\\ C(q):~&\sum_{j=1}^sa_{ij}c_j^{k-1}=\frac{1}{k}c_i^{k},~i=1,\dots,s,~k=1,\dots,q,\end{aligned}$$ are fulfilled.\ With a Taylor expansion of $U_{h}(t_m+c_j\tau)$ and $\dot{U}_{h}(t_m+c_j\tau),~j=1,\dots,s$, around $t_m$ up to the order $p$ we obtain for the $j$-th component of the residual error $\Delta_{m+1}$ the equation $$\Delta_{j,m+1} = \sum_{r=q+1}^p \frac{\tau^r}{r!} \left( \tilde{c}^r -r\mathfrak{A}\tilde{c}^{r-1} \right)_j U_{h}^{(r)}(t_m) +r_{\Delta_j,m+1},\quad\|r_{\Delta_j,m+1}\|=\mathcal{O}(\tau^{p+1})$$ with $\tilde{c}^i=(c_1^i,\dots,c_s^i)^\top$. Therefore, with $r_{\Delta1kl,m+1}=\left( r_{\Delta_11kl,m+1}^\top,\dots,r_{\Delta_s1kl,m+1}^\top\right)^\top$ and $$W_r(z)=\frac{L(z)\left[\tilde{c}^r-r\mathfrak{A}\tilde{c}^{r-1}\right]}{1-R(z)},$$ the error equation (\[emp1glgn1\]) can be written as $$\begin{aligned} 1\nonumber e_{1kl,m+1}~=~& \tau\sum_{i=0}^mR(\tau R_{kl})^iJ(\tau R_{kl})\alpha_{h1kl, m+1-i}+\sum_{i=0}^mR(\tau R_{kl})^i\delta_{1kl,m+1-i}\\ & +\nonumber \sum_{i=0}^mR(\tau R_{kl})^iL(\tau R_{kl})r_{\Delta1kl,m+1}\\ & +\underbrace{\sum_{i=0}^mR(\tau R_{kl})^i \left( I_{n_{kl}}-R(\tau R_{kl}) \right) \sum_{r=q+1}^p \frac{\tau^{r}}{r!} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{m-i})}_{=\kappa}.\label{emglg2}\end{aligned}$$ The function $W_{r}(z)$ was introduced by Ostermann/Roche [@OstermannRoche] to investigate the convergence of Runge-Kutta methods for abstract scalar parabolic differential equations. For the subsequent error estimate, the term $\kappa$ is transformed in the following manner: By exchanging the order of summation we get $$\begin{aligned} 1 \kappa \nonumber =&~\nonumber \sum_{r=q+1}^p \frac{\tau^{r}}{r!} \Bigg( \sum_{i=0}^{m-1} R(\tau R_{kl})^{m-i} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{i}) + W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{m}) \\&~\nonumber - \sum_{i=1}^{m} R(\tau R_{kl})^{m-i+1} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{i}) - R(\tau R_{kl})^{m+1} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{0}) \Bigg).\end{aligned}$$ From this we obtain $$\begin{aligned} 1 \kappa =&~\nonumber \sum_{r=q+1}^p \frac{\tau^{r}}{r!} \Bigg( \sum_{i=0}^{m-1} R(\tau R_{kl})^{m-i} W_{r}(\tau R_{kl}) \left( U_{h1kl}^{(r)}(t_{i})-U_{h1kl}^{(r)}(t_{i+1})\right) \\&~\nonumber + W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{m}) - R(\tau R_{kl})^{m+1} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{0}) \Bigg).\end{aligned}$$ Therefore it holds $$\begin{aligned} 1 \kappa =&~\nonumber \sum_{r=q+1}^p \frac{\tau^{r}}{r!} \Bigg( -\sum_{i=0}^{m-1} R(\tau R_{kl})^{m-i} W_{r}(\tau R_{kl}) \int\limits_{t_i}^{t_{i+1}} U_{h1kl}^{(r+1)}(s)~ds \\ &~ + W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{m}) - R(\tau R_{kl})^{m+1} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{0}) \Bigg).\end{aligned}$$ A similar transformation can be found in Brenner/Crouzeix/Thomée [@BrennerCrouzeixThomee]. Inserting this into [(\[emglg2\])]{} results in $$\begin{aligned} 1\nonumber e_{1kl,m+1}~=~& \tau\sum_{i=0}^m R(\tau R_{kl})^iJ(\tau R_{kl})\alpha_{h1kl, m+1-i} +\sum_{i=0}^mR(\tau R_{kl})^i\delta_{1kl,m+1-i}\\ & +\nonumber \sum_{i=0}^mR(\tau R_{kl})^iL(\tau R_{kl})r_{\Delta1kl,m+1}\\ &+ \nonumber \sum_{r=q+1}^p \frac{\tau^{r}}{r!} \Bigg( -\sum_{i=0}^{m-1} R(\tau R_{kl})^{m-i} W_{r}(\tau R_{kl}) \int\limits_{t_i}^{t_{i+1}} U_{h1kl}^{(r+1)}(s)~ds \\ & + W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{m}) - R(\tau R_{kl})^{m+1} W_{r}(\tau R_{kl})U_{h1kl}^{(r)}(t_{0}) \Bigg).\end{aligned}$$ Assuming that the Runge-Kutta matrix $\mbox{$\mathfrak{A}$}$ is regular we can derive an analogous equation for the components $e_{2kl,m+1}$ of the transformed global discretization error $$\begin{aligned} 1\nonumber e_{2kl,m+1}~=~& \tau\sum_{i=0}^m \tilde{R}(N_{kl})^i\tilde{J}(N_{kl})\alpha_{h2kl, m+1-i} +\sum_{i=0}^m\tilde{R}(N_{kl})^i\delta_{2kl,m+1-i}\\ & +\nonumber \sum_{i=0}^m\tilde{R}(N_{kl})^i\tilde{L}(N_{kl})r_{\Delta2kl,m+1}\\ &+ \nonumber \sum_{r=q+1}^p \frac{\tau^{r}}{r!} \Bigg( -\sum_{i=0}^{m-1} \tilde{R}(N_{kl})^{m-i} \tilde{W}_{r}(N_{kl}) \int\limits_{t_i}^{t_{i+1}} U_{h2kl}^{(r+1)}(s)~ds \\ & + \tilde{W}_{r}(N_{kl})U_{h2kl}^{(r)}(t_{m}) - \tilde{R}(N_{kl})^{m+1} \tilde{W}_{r}(N_{kl})U_{h2kl}^{(r)}(t_{0}) \Bigg)\end{aligned}$$ with the abbreviations $$\begin{aligned} 3 \tilde{J}(z)&=b^{\top}(I_sz-\tau\mbox{$\mathfrak{A}$})^{-1},\qquad&\tilde{R}(z)&=1+\tau\tilde{J}(z){1\kern-0.25em{\rm l}}_s,\\ \tilde{L}(z)&=\tau\tilde{J}(z),\qquad& \tilde{W}_r(z)&= \frac{\tilde{L}(z)\left[\tilde{c}^r-r\mathfrak{A}\tilde{c}^{r-1}\right]}{1-\tilde{R}(z)}.\end{aligned}$$ Finally, using (\[alpha2\]), we get for $e_{Qk,m+1}=Q_k(e_{1k1,m+1}^\top,\dots,e_{1ks_k,m+1}^\top,e_{2k1,m+1}^\top,\dots,e_{2kl_k,m+1}^\top)^\top$ the equation $$\begin{aligned} 1 e_{Qk,m+1}=&\nonumber h^{p_x}\sum_{j=1}^s \tau \sum_{i=0}^m Q_{k} \mathrm{diag}\{ \dots,R(\tau R_{k j_1})^iJ_j(\tau R_{k j_1}),\dots, \\&\nonumber \hspace{3cm}\tilde{R}(N_{m_{k j_2}})^i\tilde{J}_j(N_{m_{k j_2}}),\dots \} P_{k}B\gamma_{hQk}(t_{m-i}+c_j\tau)\\\nonumber &+\sum_{i=0}^m Q_{k} \mathrm{diag}\{ \dots,R(\tau R_{k j_1})^i,\dots,\tilde{R}(N_{m_{k j_2}})^i,\dots \} Q_{k}^{-1} \delta_{Qk,m+1-i} \\&\nonumber +\sum_{j=1}^s \sum_{i=0}^m Q_{k} \mathrm{diag}\{ \dots,R(\tau R_{k j_1})^iL_j(\tau R_{k j_1}),\dots, \\&\nonumber \hspace{3cm}\tilde{R}(N_{m_{k j_2}})^i\tilde{L}_j(N_{m_{k j_2}}),\dots \} Q_{k}^{-1} r_{\Delta_jQ,m+1}\\\nonumber & +\sum_{r=q+1}^p \frac{\tau^{r}}{r!} Q_k\Bigg( \mathrm{diag}_k\{\dots, W_{r}(\tau R_{k j_1}),\dots, \tilde{W}_{r}(N_{m_{k j_2}}),\dots \} Q_{k}^{-1} U_{Qk}^{(r)}(t_m) \\\nonumber& -\sum_{i=0}^{m-1} \int\limits_{t_i}^{t_{i+1}} \mathrm{diag}_k\{\dots, R(\tau R_{k j_1})^{m-i} W_{r}(\tau R_{k j_1}) ,\dots, \tilde{R}(N_{m_{k j_2}})^{m-i}\tilde{W}_{r}(N_{m_{k j_2}}),\dots \}\\\nonumber &\hspace{3cm} Q_{k}^{-1} U_{Qk}^{(r+1)}(s)~ds \\\nonumber & -\mathrm{diag}_k\{\dots, R(\tau R_{k j_1})^{m+1}W_{r}(\tau R_{k j_1}),\dots, \tilde{R}(N_{m_{k j_2}})^{m+1}\tilde{W}_{r}(N_{m_{k j_2}}),\dots \}\\ &\hspace{3cm} \label{emglg5} Q_{k}^{-1} U_{Qk}^{(r)}(t_0) \Bigg).\end{aligned}$$ Now we can estimate the different terms in [(\[emglg5\])]{}. For that purpose we assume in the following that the matrix norms \[QkQkhm1QkPkBvBed\] $$\label{QkQkhm1Bed} \|Q_k\mathrm{diag}\{N^i_{n_{k1}},\mathfrak0,\dots,\mathfrak0\}Q_k^{-1}\|,\dots,\|Q_k\mathrm{diag}\{\mathfrak0,\dots,N^i_{m_{k l_k}}\}Q^{-1}_k\|$$ and $$\label{QkPkBvBed} \|Q_k\mathrm{diag}\{N^i_{n_{k1}},\mathfrak0,\dots,\mathfrak0\}P_k B\|,\dots,\|Q_k\mathrm{diag}\{\mathfrak0,\dots,\mathfrak0,N^i_{m_{k l_k}}\}P_k B\|$$ are bounded for $i=0,\dots,\max\{\nu_{dt},n_{kj_1}:~j_1=1,\dots,s_k\}-1$ and all $h\in(0,h_0]$, where $\mathfrak0$ denotes a zero matrix.\ Because of the $A$-stability of the Runge-Kutta method and $\Re(\varkappa_{k j_1})\leq0$ or $|\varkappa_{kj_1}|\leq C_2$ for all $h\in(0,h_0]$ we have that $\|R(\tau R_{kj_1})^i\|$, $\|J_j(\tau R_{k j_1})\|$ and $\|L_j(\tau R_{k j_1})\|$ are bounded for sufficiently small $\tau$. We assume further that $R(it)\neq1$ for $t\in\Rset\setminus\{0\}$ and $\lim\limits_{z\to-\infty}R(z)\neq1$. Then $W_r(\tau R_{kj_1})$ exists and is bounded. Moreover, as it is shown in Ostermann/Roche [@OstermannRoche], one has $$\|\tau^rW_r(\tau R_{kj_1})\|=\mathcal{O}(\tau^{\min\{p,q+2+\alpha\}})\|R_{kj_1}^{\max\{0,\min\{p-r,q+2+\alpha-r\}\}}\|$$ with $\alpha\in\Rset$, $\alpha\geq-1$. Assuming that $|\varkappa_{k j_1}|\leq C_3(1+|\lambda_k|)$ one can show (cf. D.[@Diss], the proof relies on the Mean Value Theorem and Abel’s partial summation formula) that $$\|R_{kj_1}^{1+\alpha}\|~\|U_{Qk}^{(r)}(t_m)\|=k^{1+2\alpha}\mathcal{O}(h^{-\frac12}).$$ Altogether the terms in [(\[emglg5\])]{} that originate from $e_{1kl,m+1}$ are of order $$\label{GlgOrdNr1} \mathcal{O}(h^{p_x})+\mathcal{O}(\tau^{\min\{p,q+2+\alpha\}})h^{-\frac12}k^{1+2\alpha}.$$ With the Taylor expansion $$\gamma_{hQk}(t_{m-i}+c_j\tau) =\sum_{l=0}^{\nu_{dt}-2}\sum_{k=0}^{l}\frac{\tau^{l}(-i)^{l-k}}{k!(l-k)!}c_j^k\frac{\partial^l}{\partial t^{l}}\gamma_{hQk}(t_m)+\mathcal{O}(\tau^{\nu_{dt}-1}),$$ the term $$\begin{aligned} 1 a=&h^{p_x}\sum_{j=1}^s\tau \sum_{i=0}^m Q_{k} \mathrm{diag}\{0, \dots,0, \tilde{R}(N_{m_{k j_2}})^i\tilde{J}_j(N_{m_{k j_2}}),0,\dots, \} P_{k}B\gamma_{hQk}(t_{m-i}+c_j\tau)\end{aligned}$$ of $e_{Qk,m+1}$ can be written as $$\begin{aligned} 1 a=&\sum_{j=0}^{\nu_{dt}-1}\frac1{j!}\sum_{l=0}^{\nu_{dt}-2} Q_{k} \mathrm{diag}\{0, \dots,0, \frac{N_{m_{k j_2}}^j}{\tau^{j-l}}\sum_{k=0}^l\frac1{k!(l-k)!} \\&\nonumber \qquad\qquad\qquad \sum_{i=0}^m(-i)^{l-k}\left(\tau^{j+1}\tilde{R}(z)^i\tilde{J}(z)\tilde{c}^k\right)^{(j)}(0), 0,\dots,0 \} P_{k}B\frac{\partial}{\partial t^{l}}\gamma_{hQk}(t_m) \\&\nonumber +\sum_{j=0}^{\nu_{dt}-1}\frac1{j!} \sum_{i=0}^m Q_{k} \mathrm{diag}\{0, \dots,0, N_{m_{k j_2}}^j \left(\tau^{j+1}\tilde{R}(z)^i\tilde{J}(z)\right)^{(j)}(0),0,\dots,0 \} P_{k}B \mathcal{O}(\tau^{\nu_{dt}-1}) ,\end{aligned}$$ where $(\dots)^{(j)}$ denotes the $j$-th derivative w.r.t. $z$.\ For $L$-stable Runge-Kutta methods with regular coefficient matrix $\mbox{$\mathfrak{A}$}$ we have $R(\infty)=1-b^\top\mbox{$\mathfrak{A}$}^{-1}{1\kern-0.25em{\rm l}}=0$ and therefore $\tilde{R}(0)=0$. If the matrix norms in [(\[QkQkhm1Bed\])]{} are bounded, then $\|a\|$ is bounded if $$\label{BedRhochiJalphabeschr} \sum_{k=0}^l\frac1{k!(l-k)!} \sum_{i=0}^j(-i)^{l-k}\left(\tau^{j+1}\tilde{R}(z)^i\tilde{J}(z)\tilde{c}^k\right)^{(j)}(0)=0$$ for $l=0,\dots,j-1,~j=1,\dots,\nu_{dt}-1$.\ The remaining terms in the equation [(\[emglg5\])]{} yield the classical order $p_{\nu_{dt}}$ of the Runge-Kutta method applied to a linear DAE of index $\nu_{dt}$ with constant coefficients. Thus, altogether we have $$e_{Qk,m+1}=\mathcal{O}(h^{p_x})+\mathcal{O}(\tau^{p_{\nu_{dt}}})+\mathcal{O}(\tau^{\min\{p,q+2+\alpha\}})h^{-\frac12}k^{1+2\alpha}.$$ From [(\[eqzsglg\])]{} it follows that we have to choose $\alpha$ such that $\sum\limits_{k=1}^Nk^{2(1+2\alpha)}$ is bounded for $N\to\infty$. This implies $\alpha=-\frac34-\varepsilon$, $\varepsilon>0$, and we have $$\|e_{m+1}\|=\mathcal{O}(h^{p_x})+\mathcal{O}(\tau^{\min\{p_{\nu_{dt}},q+1.25-\varepsilon\}}).$$ If the derivatives of order $(q+1)$ w.r.t. the time of the boundary conditions that enter into the space discretization are homogeneous, i.e. $$\label{qp1Ablhomogen} B\frac{\partial^{q+1}u}{\partial t^{q+1}}=0 ,$$ (\[alphaWK1\]) yields $$R_{kl}U_{h1kl}^{(r)}(t)={U}_{h1kl}^{(r+1)}(t)-\Tilde{F}_{1kl}(t)+\alpha_{h1kl}(t),$$ and instead of (\[GlgOrdNr1\]) we obtain the order $$\mathcal{O}(h^{p_x})+\mathcal{O}(\tau^{\min\{p,q+2+\alpha\}})h^{-\frac12}k^{2\alpha-1},$$ which implies $\alpha=\frac14-\varepsilon$, and therefore $$\|e_{m+1}\|=\mathcal{O}(h^{p_x})+\mathcal{O}(\tau^{\min\{p_{\nu_{dt}},q+2.25-\varepsilon\}}).$$ Summarized, we have the following convergence result for smooth enough solutions $u(t,x)$ of the PDAE ($\max\{p+1,p+\nu_{dt}-1\}$ times differentiable with respect to $t$ in $[t_0,t_e]$ and $p_x+2$ times differentiable with respect to $x$ in $[-l,l]$): \[RKVWK\] Let the following assumptions be fulfilled for $h\to0$ and $k=1,\dots,N$: 1. for the matrix pencils $D_k+\lambda A$ there exist Weierstrass-Kronecker decompositions according to [(\[WKTransf\])]{}, and the matrix norms in [(\[QkQkhm1QkPkBvBed\])]{} are bounded, 2. $\Re(\varkappa_{k j_1})\leq0$ or $|\varkappa_{k j_1}|\leq C_2$ for all $h\in(0,h_0]$, 3. $|\varkappa_{k j_1}|\leq C_3(1+|\lambda_k|)$, 4. if $\nu_{dt}>2$ then [(\[BedRhochiJalphabeschr\])]{} is fulfilled for $l=0,\dots,j-1,~j=1,\dots,\nu_{dt}-1$. Furthermore let the Runge-Kutta method be of consistency order $p$, stage order $q$ and $L$-stable (if $\nu_{dt}=0$ or $\nu_{dt}=1$, it suffices $A$-stability with $\lim\limits_{z\to-\infty}R(z)<1$) with a regular matrix $\mbox{$\mathfrak{A}$}$ and $R(it)\neq1$ for $t\in\Rset\setminus\{0\}$. Let $p_{\nu_{dt}}$ be the classical order of the Runge-Kutta method applied to a linear DAE of index $\nu_{dt}$ with constant coefficients.\ Then the discretization method (\[NLRKGlg\]) converges for linear PDAEs after $\nu_{dt}$ time steps with the order $(p_x,p^\star)$ in the discrete $L_2$-norm in space and in the maximum norm in time with $$p^\star=\left\{ \begin{array}{l@{:~}l} \min\{p_{\nu_{dt}},q+1.25-\varepsilon\}&\text{inhomog. boundary conditions according to (\ref{qp1Ablhomogen})}\\ \min\{p_{\nu_{dt}},q+2.25-\varepsilon\}&\text{homog. boundary conditions according to (\ref{qp1Ablhomogen})} \end{array} \right.$$ and $\varepsilon>0$ arbitrary small. \[BemKonvSatz\] 1. Stage order $q\geq\nu_{dt}-2$ implies condition d) in Theorem 7 for $\nu_{dt}=3$ or $\nu_{dt}=4$. 2. The assumptions on the Runge-Kutta method are fulfilled, e.g., for the Radau IIA and the Lobatto IIIC methods and in the case of $\nu_{dt}\leq1$ also for the implicit midpoint rule. 3. If $p\geq p^\star+1$, then for $L$-stable Runge-Kutta methods with $zL(1,z)$ bounded for $\Re(z)\leq0$, the condition that [(\[QkQkhm1Bed\])]{} is bounded can be replaced by the boundedness of the matrix norms \[QkQkhm1Bed2\] $$\begin{aligned} 1 &\|\frac1{\kappa_{k1}}Q_k\mathrm{diag}\{N^i_{n_{k1}},\mathfrak0,\dots,\mathfrak0\}Q_k^{-1}\|,\dots, \|\frac1{\kappa_{k s_k}}Q_k\mathrm{diag}\{\mathfrak0,\dots,\mathfrak0,N^i_{n_{k s_k}},\mathfrak0,\dots,\mathfrak0\}Q_k^{-1}\|,\\ &\|Q_k\mathrm{diag}\{\mathfrak0,\dots,\mathfrak0,N^i_{m_{k 1}},\mathfrak0,\dots,\mathfrak0\}Q^{-1}_k\|, \|Q_k\mathrm{diag}\{\mathfrak0,\dots,N^i_{m_{k l_k}}\}Q^{-1}_k\|.\end{aligned}$$ 4. If we choose $\varepsilon=0$, then we get for $p^\star<p_{\nu_{dt}}$ $$\|e_{m+1}\|=\mathcal{O}(h^{p_x})+ \mathcal{O}\left(\sqrt{|\ln h|}\tau^{p^\star}\right).$$ \[BemEuler\] For a given Runge-Kutta method, Theorem \[RKVWK\] can be specialized. E.g., if we take the implicit Euler method, the resulting BTCS method is convergent of time order 1 for arbitrary time index, if only the conditions a) and b) of Theorem \[RKVWK\] are fulfilled. For the Radau IIA methods with $s\geq2$ stages we get $$p^\star=\left\{ \begin{array}{l@{:~}l} \min\{s+1.25-\varepsilon,2s-1\}&\nu_{dt}=0,1, \text{inhomog. b.c.s according to (\ref{qp1Ablhomogen})}\\ \min\{s+2.25-\varepsilon,2s-1\}&\nu_{dt}=0,1, \text{homog. b.c.s according to (\ref{qp1Ablhomogen})}\\ s+2-\nu_{dt}&\nu_{dt}\geq 2 \end{array} \right.$$ as temporal order of convergence, provided that the assumptions a)-d) of Theorem \[RKVWK\] are fulfilled. Numerical examples {#Abschnitt5} ================== The numerical examples given below illustrate our convergence results. For the time integration we use the backward Euler method and the code RADAU5, which is a variable step size implementation of the 3-stage Radau IIA method, see Hairer/Wanner [@HairerWanner]. The Euler and Radau IIA methods are of great importance in applications. The backward Euler method is given by the parameters $$s=1,\mbox{$\mathfrak{A}$}=(1),b=c=(1),p=q=1.$$ We consider the linear PDAE $$\underbrace{\begin{pmatrix} 0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\\ \end{pmatrix}}_{=A} u_t+ \underbrace{\left( \begin{array}{crr} 0& 0& -1\\ 0& -1& -1\\ 0& 0& 0\\ \end{array} \right)}_{=B} u_{xx}+ \underbrace{ \left( \begin{array}{rrr} -1& -1& -1\\ 0& -1& 0\\ 0& 0& -1\\ \end{array} \right)}_{=C} u=f,$$ a coupled system of two parabolic equations and one algebraic equation, with $x\in [- 0.5,0.5]$, . The right-hand side, initial and Dirichlet boundary values are chosen such that $$u(t,x)= \left( \begin{array}{c} x(x-1)\sin(t)\\ x(x-1)\cos(t)\\ x(x-1)(e^t+t^5)\\ \end{array} \right)$$ is the exact solution. It holds $$D_k=-{\lambda_k} B-C=\begin{pmatrix} 1& 1~& 1+{\lambda_k}\\ 0& 1+{\lambda_k}& {\lambda_k}\\ 0& 0~& 1 \end{pmatrix}.$$ With $$P_k=\left( \begin{array}{crc} {\lambda_k}+1& -1& -{\lambda_k}-{\lambda_k}^2-1\\ 0& 1& -{\lambda_k}-1\\ 0& 0& 1\\ \end{array} \right)\text{ and } Q_k=\begin{pmatrix} \frac1{{\lambda_k}+1}& ~0& ~0\\ 0& \frac{~1}{{\lambda_k}+1}& \frac{~1}{{\lambda_k}+1}\\ 0& ~0& ~1\\ \end{pmatrix},$$ we obtain the Weierstrass-Kronecker decomposition $$P_kAQ_k=\begin{pmatrix} 0&1&0\\0&0&1\\0&0&0 \end{pmatrix},~P_kD_kQ_k=\begin{pmatrix} 1&0&0\\0&1&0\\0&0&1 \end{pmatrix}.$$ Therefore, the PDAE has differential time index 3, and the assumptions [(\[QkQkhm1QkPkBvBed\])]{} are fulfilled. Remark \[BemEuler\] yields that the BTCS method is convergent after three steps of time order $1$. This is confirmed by the numerical experiment, Table \[Tabelle1\] shows the observed order of convergence in time at $(x=1,t_e=1)$. $ 0.1\tau^{-1}$ $2^{ 2}$ $2^{ 3}$ $2^{ 4}$ $2^{ 5}$ $2^{ 6}$ $2^{ 7}$ ------------------ ---------- ---------- ---------- ---------- ---------- ---------- $0.1h^{-1}$ $2^{ 2}$ 0.81 0.91 0.96 0.98 0.99 0.99 $2^{ 3}$ 0.81 0.91 0.96 0.98 0.99 0.99 $2^{ 4}$ 0.81 0.91 0.96 0.98 0.99 0.99 : Numerically observed order of convergence in the discrete $L_2$-norm.\[Tabelle1\] The notation of the first element 0.81 denotes the observed order when refining the grid from $ (h=0.1/2^2,\tau=0.1/2)$ to $(h=0.1/2^2,\tau=0.1/2^2)$, i.e., where $\xi$ denotes the ratio of the error with $(h=0.1/2^2,\tau=0.1/2)$ to the error with $(h=0.1/2^2,\tau=0.1/2^2)$. Furthermore, we see that a simultaneous refinement of $h$ and $\tau$ yields no order reduction. \[Bsp\_KonvOrd4.25\] We consider the 3-stage Radau IIA method with consistency order $p=5$ and stage order $q=3$, and the linear PDAE $$\underbrace{\left( \begin{array}{rrr} 0&2&0\\1&-1&0\\1&-1&0 \end{array} \right)}_{=A}u_t+ \underbrace{\left( \begin{array}{rrr} -1&0&0\\0&0&0\\0&0&-1 \end{array} \right)}_{=B}u_{xx}+ \underbrace{\left( \begin{array}{rrr} 0&0&0\\0&-1&0\\0&0&1 \end{array} \right)}_{=C}u = f(t,x)$$ with $x\in[-1,1]$ and $t\in[0,1]$. This example shows the dependence of the time order on the boundary values. 1. We choose the right-hand side such that $$u(t,x)=\left( x^2e^{-t},x^2e^{-\frac12t},x^2\sin t \right) ^\top$$ is the exact solution. Then we have inhomogeneous boundary values $$u(t,\mp1)=\left( e^{-t},e^{-\frac12t},\sin t \right) ^\top .$$ Furthermore it holds $$D_k=\left(\begin{array}{crc}{\lambda_k}&0&0\\0&1&0\\0&0&{\lambda_k}-1\end{array}\right),~ P_kAQ_k=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}, ~P_kD_kQ_k=\begin{pmatrix}\frac{2{\lambda_k}}{-{\lambda_k}-\eta_k}&0&0\\0&\frac{2{\lambda_k}}{-{\lambda_k}+\eta_k}&0\\0&0&1\end{pmatrix}$$ with $$P_k= \begin{pmatrix} \frac{{\lambda_k}-\eta_k}{4{\lambda_k}}&1&0\\ \frac{{\lambda_k}+\eta_k}{4{\lambda_k}}&1&0\\ 0&\frac1{1-{\lambda_k}}&\frac1{{\lambda_k}-1} \end{pmatrix},~ Q_k= \begin{pmatrix} \frac{4{\lambda_k}}{({\lambda_k}+\eta_k)\eta_k}&-\frac{4{\lambda_k}}{({\lambda_k}-\eta_k)\eta_k}&0\\ -\frac{{\lambda_k}}{\eta_k}&\frac{{\lambda_k}}{\eta_k}&0\\ -\frac{{\lambda_k}}{\eta_k({\lambda_k}-1)}&\frac{{\lambda_k}}{\eta_k({\lambda_k}-1)}&1 \end{pmatrix},~ \eta_k=\sqrt{{\lambda_k}^2+8{\lambda_k}}$$ ($|\lambda_k+8|>\frac12$ for $N>3$, i.e. $h<\frac12$).\ The PDAE has therefore differential time index 1, and the conditions a)-c) of Theorem \[RKVWK\] are fulfilled which yields convergence of time order $4.25-\varepsilon$. This is confirmed by the numerical experiment, see Table \[Tabelle2\]. $ 0.1\tau^{-1}$ $2^{ 1}$ $2^{ 2}$ $2^{ 3}$ ------------------ ---------- ---------- ---------- -- $0.2h^{-1}$ $2^{ 3}$ 4.27 4.28 4.30 $2^{ 4}$ 4.26 4.26 4.26 $2^{ 5}$ 4.26 4.26 4.25 $2^{ 6}$ 4.26 4.26 4.25 : Numerically observed order of convergence in the discrete $L_2$-norm for inhomogeneous boundary values.\[Tabelle2\] 2. If instead the right-hand side is chosen such that $$u(t,x)= \left( (x^2-1)e^{-t}+t^3\cos^2x,x^2e^{-\frac12t},(x^2-1)\sin t+t^3\sin^2x \right)^\top$$ is the exact solution, then we have inhomogeneous boundary values $$u(t,\mp1)= \left( t^3\cos^21,e^{-\frac12t},t^3\sin^21 \right)^\top ,$$ but the derivatives of $Bu(t,x)$ of order 4 w.r.t. the time vanish. Therefore, we obtain the convergence order 5 in time, see Table \[Tabelle4\]. $ 0.1\tau^{-1}$ $2^{ 2}$ $2^{ 3}$ ------------------ ---------- ---------- -- $0.2h^{-1}$ $2^{ 3}$ 5.00 5.00 $2^{ 4}$ 5.00 5.00 $2^{ 5}$ 5.00 5.00 $2^{ 6}$ 5.00 5.00 : Numerically observed order of convergence in the discrete $L_2$-norm for inhomogeneous boundary values where the derivatives of $Bu$ of order 4 w.r.t. the time vanish.\[Tabelle4\] We consider the 3-stage Radau IIA method and the linear PDAE [(\[GlgSystemSupraMagnetSpule\])]{} describing the superconducting coil. It holds $$D_k=\begin{pmatrix} {\lambda_k}&-1&0&0&\\0&~\;{\lambda_k}&0&0&\\0&~\;0&1&0\\0&~\;0&0&1 \end{pmatrix},~P_kAQ_k= \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&0&0 \end{pmatrix},~P_kD_kQ_k= \begin{pmatrix} -\frac{i{\lambda_k}}{\sqrt{1-{\lambda_k}}}&0&0&0\\~\;0&\frac{i{\lambda_k}}{\sqrt{1-{\lambda_k}}}&0&0\\~\;0&0&1&0\\~\;0&0&0&1 \end{pmatrix}$$ with $$P_k=\begin{pmatrix} -\frac{i}{\sqrt{1-{\lambda_k}}}&-\frac{i\sqrt{1-{\lambda_k}}}{{\lambda_k}}&1&-1\\ ~\;\frac{i}{\sqrt{1-{\lambda_k}}}&~\;\frac{i\sqrt{1-{\lambda_k}}}{{\lambda_k}}&1&-1\\ ~\;0&~\;0&1&-\frac1{\lambda_k}\\ ~\;\frac1{\lambda_k}&~\;0&0&~\;0& \end{pmatrix},~ Q_k=\begin{pmatrix} \frac1{2(1-{\lambda_k})}&\frac1{2(1-{\lambda_k})}&\;~0&-\frac{\lambda_k}{1-{\lambda_k}}\\ \frac{\lambda_k}{2(1-{\lambda_k})}&\frac{\lambda_k}{2(1-{\lambda_k})}&\;~0&-\frac{\lambda_k}{1-{\lambda_k}}\\ -\frac{i{\lambda_k}}{2(1-{\lambda_k})^{\frac32}}&\;~\frac{i{\lambda_k}}{2(1-{\lambda_k})^{\frac32}}&-\frac{\lambda_k}{1-{\lambda_k}}&\;~0\\ -\frac{i{\lambda_k}^2}{2(1-{\lambda_k})^{\frac32}}&\;~\frac{i{\lambda_k}^2}{2(1-{\lambda_k})^{\frac32}}&-\frac{\lambda_k}{1-{\lambda_k}}&\;~0 \end{pmatrix}.$$ The coil PDAE has therefore differential time index 2, and the conditions of Theorem \[RKVWK\] (with the matrix norms [(\[QkQkhm1Bed\])]{} replaced by [(\[QkQkhm1Bed2\])]{}) are fulfilled which yields an order of convergence in time of $3$. This is confirmed by the numerical experiment, see Table \[Tabelle6\]. $ 0.1\tau^{-1}$ $2^{ 4}$ $2^{ 5}$ $2^{ 6}$ ------------------ ---------- ---------- ---------- $0.2h^{-1}$ $2^{ 2}$ 3.00 3.00 3.00 $2^{ 3}$ 3.00 3.00 3.00 $2^{ 4}$ 3.00 3.00 3.00 : Numerically observed order of convergence in the discrete $L_2$-norm for the coil PDAE.\[Tabelle6\] Conclusion {#Abschnitt6} ========== The attention has here been restricted to a class of linear partial differential-algebraic equations. We have given convergence results in dependence on the type of boundary values and the time index. When the error is measured in the discrete $L_2$-norm over the whole domain, the convergence order in time of the Runge-Kutta method for a smooth solution is in general non-integer and smaller than the order expected for differential-algebraic equations of the same index. Some numerical examples were presented and confirm the theoretical convergence results.\ The extension of the analysis to the case of space $d$ dimensional linear partial differential-algebraic equations of the form $$A\:u_t(t,\vec{x}) +\sum_{i=1}^d B_i\:\left( u_{x_ix_i}(t,\vec{x})+r_iu_{x_i}(t,\vec{x})\right) + C\: u(t,\vec{x}) = f(t,\vec{x})$$ with $\vec x=(x_1,\cdots,x_d)^\top$ and a cuboid as domain is possible, see D. [@Diss], but becomes rather technical and offers no new insight. Furthermore, the consideration of periodic boundary values is also possible. Here we could show as temporal convergence order the order of an ordinary differential-algebraic equation. In the case of Neumann boundary conditions, the temporal convergence order lies in between the order obtained for Dirichlet- and the order obtained for periodic boundary conditions.\ Future work in this area will be concerned with convergence investigations for semi-linear partial differential-algebraic equations. [**Acknowledgements**]{} The authors are very grateful to the referee for his comments and fruitful suggestions. [00]{} P. Brenner, M. Crouzeix, V. Thomée: Single step methods for inhomogeneous linear differential equations in Banach space. R.A.I.R.O. Anal. Numér. 16 (1982) 5-26. S. L. Campbell, W. Marszalek: The Index of an Infinite Dimensional Implicit System. Mathematical and Computer Modelling of Dynamical Systems 5 (1999) 18-42. K. Debrabant: Numerische Behandlung linearer and semilinearer partieller differentiell-algebraischer Systeme mit Runge-Kutta-Verfahren. Dissertation, Martin-Luther-Universität Halle-Wittenberg, 2004. G. H. Golub, C. F. van Loan: Matrix Computations. Third Edition. The John Hopkins University Press. Baltimore, London, 1996. E. Hairer, G. Wanner: Solving Ordinary Differential Equations II. Stiff and Differential - Algebraic Problems. Springer-Verlag Berlin. Heidelberg, 1996. Ch. Lubich, A.Ostermann: Runge-Kutta Methods for parabolic equations and convolution quadrature. Math. Comp. 60 (1993) 105-131. W. Lucht, K. Strehmel and C. Eichler-Liebenow: Indexes and special discretization methods for linear partial differential algebraic equations. BIT 39 (1999) 484-512. W. Marszalek, Z. W. Trzaska: Analysis of implicit hyperbolic multivariable systems. Appl. Math. Modelling 19 (1995) 400-410. A. Ostermann, M. Roche: Runge-Kutta Methods for Partial Differential Equations and Fractional Order of Convergence. Math. Comp. 59 (1992) 403-420. A. Ostermann, M. Thalhammer: Convergence of Runge-Kutta methods for nonlinear parabolic equations, Appl. Numer. Math. 42 (2002) 367-380.

--- abstract: 'A recent proof-of-principle study proposes an energy- and charge-conserving, nonlinearly implicit electrostatic particle-in-cell (PIC) algorithm in one dimension [\[]{}Chen et al, *J. Comput. Phys.*, **230** (2011) 7018[\]]{}. The algorithm in the reference employs an unpreconditioned Jacobian-free Newton-Krylov method, which ensures nonlinear convergence at every timestep (resolving the dynamical timescale of interest). Kinetic enslavement, which is one key component of the algorithm, not only enables fully implicit PIC a practical approach, but also allows preconditioning the kinetic solver with a fluid approximation. This study proposes such a preconditioner, in which the linearized moment equations are closed with moments computed from particles. Effective acceleration of the linear GMRES solve is demonstrated, on both uniform and non-uniform meshes. The algorithm performance is largely insensitive to the electron-ion mass ratio. Numerical experiments are performed on a 1D multi-scale ion acoustic wave test problem.' address: - 'Los Alamos National Laboratory, Los Alamos, NM 87545' - 'University of Colorado Boulder, Boulder, CO 80309' - 'University of New Mexico, Albuquerque, NM 87131' author: - 'G. Chen' - 'L. Chacón' - 'C. Leibs' - 'D. Knoll' - 'W. Taitano' bibliography: - 'kinetic.bib' title: 'Fluid preconditioning for Newton-Krylov-based, fully implicit, electrostatic particle-in-cell simulations ' --- electrostatic particle-in-cell ,implicit methods ,direct implicit ,implicit moment ,energy conservation ,charge conservation ,physics based preconditioner ,JFNK solver Introduction ============ The Particle-in-cell (PIC) method solves Vlasov-Maxwell’s equations for kinetic plasma simulations [@birdsall-langdon; @hockneyeastwood]. In the standard approach, Maxwell’s equations (or in the electrostatic limit, Poisson equation) are solved on a grid, and the Vlasov equation is solved by method of characteristics using a large number of particles, from which the evolution of the probability distribution function (PDF) is obtained. The field-PDF description is tightly coupled. Maxwells equations (or a subset thereof) are driven by moments of the PDF such as charge density and/or current density. The PDF, on the other hand, follows a hyperbolic equation in phase space, whose characteristics are self-consistently determined by the fields. To date, most PIC methods employ explicit time-stepping (e.g. leapfrog scheme), which can be very inefficient for long-time, large spatial scale simulations. The algorithmic inefficiency of standard explicit PIC is rooted in the presence of numerical stability constraints, which force both a minimumtion of the smallest Debye length). Moreover, a fundamental issue with explicit schemes is numerical heating due to the lack of exact energy conservation in a discrete setting This problem is particularly evident for realistic ion-to-electron mass ratios. Implicit methods hold the promise of nonlinear convergence, which leads to inconsistencies between fields and particle moments. As a result, significant numerical heating is often observed in long term simulations [@cohen-jcp-89-di_pic]. There has been significant recent work exploring fully implicit, fully nonlinear PIC algorithms, either Picard-based [@taitano-sisc-13-ipic] (following the implicit moment method school) or using Jacobian-Free Newton-Krylov (JFNK) methods [@chen-jcp-11-ipic; @markidis2011energy] (more aligned with the direct implicit school). In contrast to earlier studies, these nonlinear approaches enforce nonlinear convergence to a specified tolerance at every timestep. Their fully implicit character enables one to build in exact discrete conservation properties, such as energy and charge conservation [@chen-jcp-11-ipic; @taitano-sisc-13-ipic]. In these studies, particle orbit integration is sub-stepped for accuracy, and to ensure automatic charge conservation. The purpose of this study is to demonstrate the effectiveness of fluid (moment) equations to accelerate a JFNK-based kinetic solver (moment acceleration in a Picard sense has already been demonstrated in Ref. [@taitano-sisc-13-ipic]). An enabling algorithmic component of the JFNK-based algorithm is the enslavement of particles to the fields, which removes particle quantities from the dependent variable list of the JFNK solver. With particle enslavement, memory requirements of the nonlinear solver are dramatically reduced. Particle equations of motion are orbit-averaged and evolve self-consistently with the field. The kinetic-enslaved JFNK not only makes the fully implicit PIC algorithm practical, but also makes the fluid preconditioning of the algorithm possible. It is worth pointing out that the preconditioned JFNK approach proposed here can be conceptually viewed as an optimal combination of the direct implicit and moment implicit approaches. The fluid preconditioner is derived by taking the first two moments of the Vlasov equation, and then linearizing them into a so-called “delta-form” [@knoll2004jacobian]. Textbook linear analysis shows that such a system includes stiff electron modes in an electrostatic plasma. Although taking large timesteps for low-frequency field evolutions is desirable, previous work [@chen-jcp-11-ipic] indicates that the implicit CPU speedup over explicit PIC is largely insensitive to the timestep size for large enough timesteps owing to particle sub-cycling for orbit resolution. It is thus sufficient in this context to target the stiffest time scales supported, i.e., electron time scales. Therefore, we base our fluid preconditioner on electron moment equations only. The implicit timestep is chosen to resolve the ion plasma wave frequency. This is to resolve ion waves of all scales (including the Debye length scale which is physically relevant for some non-linear ion waves). For consistency with the orbit averaging of the kinetic solver, we take the time-average of the linearized moment equations in the preconditioner. We show that the fluid preconditioner is asymptotic preserving in the sense that it is well behaved in the quasineutral limit (as in Ref. [@degond-jcp-10-ap_pic]). However, beyond the study in Ref. [@degond-jcp-10-ap_pic], the algorithm proposed here is also well behaved for arbitrary electron-ion mass ratios. The rest of the paper is organized as follows. Section \[sec:Kinetic-enslavement\] motivates and introduces the concept to kinetic enslavement in the implicit PIC formulation. Section \[sec:JFNK-method\] introduces the mechanics of the JFNK method and preconditioning. Section \[sec:formulation-pc\] formulates the fluid preconditioner of an electrostatic plasma system in detail, with an extension to 1D non-uniform meshes. Linear analysis of electron and ion waves, together with an asymptotic analysis of the preconditioner are also provided. Section \[sec:Numerical-experiments\] presents numerical parametric experiments to test the performance of the preconditioner. Finally, we conclude in Sec. \[sec:Conclusion\]. Kinetically Enslaved Implicit PIC \[sec:Kinetic-enslavement\] ============================================================= We consider a collisionless electrostatic plasma system (without magnetic field) described by the Vlasov-Ampere equations in one dimension (1D) in both position ($x$) and velocity ($v$) [@chen-jcp-11-ipic]: $$\begin{aligned} \frac{\partial f_{\alpha}}{\partial t}+v\frac{\partial f_{\alpha}}{\partial x}+\frac{q_{\alpha}}{m_{\alpha}}E\frac{\partial f_{\alpha}}{\partial v} & = & 0,\label{eq:vlasov}\\ \epsilon_{0}\frac{\partial E}{\partial t}+j & = & \left\langle j\right\rangle ,\label{eq:ampere}\end{aligned}$$ where $f_{\alpha}(x,v)$ is the particle distribution function of species $\alpha$ in phase space, $q_{\alpha}$ and $m_{\alpha}$ are the species charge and mass respectively, $E$ is the self-consistent electric field, $j$ is the current density, $\left\langle j\right\rangle =\int jdx/\int dx$, and $\epsilon_{0}$ is the vacuum permittivity. The evolution of Vlasov equation is solved by the method of characteristics, represented by particles evolving according to Newton’s equations of motion, $$\begin{aligned} \frac{dx_{p}}{dt} & = & v_{p},\label{eq:dxpdt}\\ \frac{dv_{p}}{dt} & = & a_{p}.\label{eq:dvpdt}\end{aligned}$$ Here $x_{p}$, $v_{p}$, $a_{p}$ are the particle position, velocity, and acceleration, respectively, and $t$ denotes time. As a starting point, we may discretize Eqs. \[eq:ampere\], \[eq:dxpdt\], and \[eq:dvpdt\] by a time-centered finite-difference scheme, to find: $$\begin{aligned} \epsilon_{0}\frac{E_{i}^{n+1}-E_{i}^{n}}{\Delta t}+j_{i}^{n+1/2} & = & \left\langle j\right\rangle ,\label{eq:discete-Ampere}\\ \frac{x_{p}^{n+1}-x_{p}^{n}}{\Delta t}-v_{p}^{n+1/2} & = & 0,\label{eq:discrete-xp}\\ \frac{v_{p}^{n+1}-v_{p}^{n}}{\Delta t}-\frac{q_{p}}{m_{p}}E(x_{p}^{n+1/2}) & = & 0,\label{eq:discrete-vp}\end{aligned}$$ where a variable at time level $n+\nicefrac{1}{2}$ is obtained by the arithmetic mean of the variable at $n$ and $n+1$, the subscript $i$ denotes grid-index and subscript $p$ denotes particle-index, $j_{i}=\sum_{p}q_{p}v_{p}S(x_{p}-x_{i})$, $E(x_{p})=\sum_{i}E_{i}S(x_{i}-x_{p})$, and $S$ is a B-spline shape function [@christensen2010functions]. It is critical to realize that solving the complete system of field-particle equations (i.e., with the field and particle position and velocity as unknowns) in a Newton-Krylov-based solver is impractical, due to the excessive memory requirements of building the required Krylov subspace. To overcome the memory challenges of JFNK for implicit PIC, the concept of kinetic enslavement has been introduced [@chen-jcp-11-ipic; @markidis2011energy]. With kinetic enslavement, the JFNK residual is formulated in terms of the field equation only, nonlinearly eliminating Eqs. \[eq:discrete-xp\] and \[eq:discrete-vp\] as auxiliary computations. The resulting JFNK implementation has memory requirements comparable to that of a fluid calculation. A single copy of particle quantities is still needed for the required particle computations. One important implication of kinetic enslavement is that the enslaved particle pusher has the freedom of being adaptive in its implementation. This can be effectively exploited to overcome the accuracy shortcomings of using a fixed timestep $\Delta t$ to discretize the time derivatives of both field and particle equations [@Parker-jcp-93-bounded-multiscale-pic]. This is so because solving low-frequency field equations demands using large timesteps, but if particle orbits are computed with such timesteps, large plasma response errors result [@langdon-jcp-79-pic_ts]. In Ref. [@chen-jcp-11-ipic], a self-adaptive, charge-and-energy-conserving particle mover was developed that provided simultaneously accuracy and efficiency. Falgorithmic elements: 1. Estimate the sub-timestep $\Delta\tau$ using a second order estimator [@chen2013analytical]. 2. Integrate the orbit over $\Delta\tau$ using a Crank-Nicolson scheme. 3. If a particle orbit crosses a cell boundary, make it land at the first encountered boundary. 4. Accumulate the particle moments to the grid points. In the last step, the current density is orbit-averaged (over $\Delta t=\sum\Delta\tau$) to ensure global energy conservation. Additionally, binomial smoothing can be introduced without breaking energy or charge conservation. This is done in the particle pusher by using the binomially smoothed electric field, and the binomially smoothed orbit averaged current density in Ampere’s equation. The resulting Ampere’s equation reads: $$\epsilon_{0}\frac{E_{i}^{n+1}-E_{i}^{n}}{\Delta t}+SM(\bar{j})_{i}^{n+1/2}=\left\langle \overline{j}\right\rangle ^{n+1/2},\label{eq:bi-amperelaw}$$ where the orbit averaged current density is: $$\overline{j}_{i}^{n+1/2}=\frac{1}{\Delta t\Delta x}\sum_{p}\sum_{\nu=1}^{N_{\nu}}q_{p}S(x_{i}-x_{p}^{\nu+1/2})v_{p}^{\nu+1/2}\Delta\tau^{\nu}.\label{eq:javerage}$$ The binomial operator $SM$ is defined as $SM(Q)_{i}=\frac{Q_{i-1}+2Q_{i}+Q_{i+1}}{4}.$ A detailed description of the algorithm can be found in Ref. [@chen-jcp-11-ipic; @chen-jcp-12-ipic_gpu]. The kinetically enslaved JFNK residual is defined from Eq. \[eq:bi-amperelaw\] as: $$G_{i}(E^{n+1})=E_{i}^{n+1}-E_{i}^{n}+\frac{\Delta t}{\epsilon_{0}}\left(SM(\bar{j}[E^{n+1}])_{i}^{n+1/2}-\left\langle \overline{j}\right\rangle ^{n+1/2}\right).\label{eq:bi-amperelaw-residual}$$ The functional dependence of $\bar{j}$ with respect to $E^{n+1}$ has been made explicit. Evaluation of $\bar{j}[E^{n+1}]$ requires one particle integration step, and each linear and nonlinear iteration of the JFNK method requires one residual evaluation . We summarize the main elements of the JFNK nonlinear solver next. The JFNK solver\[sec:JFNK-method\] ================================== In its outer loop, JFNK employs Newton-Raphson’s method to solve a nonlinear system $\mathbf{G}(\mathbf{x})=0$, where $\mathbf{x}$ is the unknown, by linearizing the residual and inverting linear systems of the form: $$\left.\frac{\partial\mathbf{G}}{\partial\mathbf{x}}\right|^{(k)}\delta\mathbf{x}^{(k)}=-\mathbf{G}(\mathbf{x}^{(k)}),\label{eq:Newton-Raphson step}$$ with $\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\delta\mathbf{x}^{(k)}$, and $(k)$ denotes the nonlinear iteration number. Nonlinear convergence is reached when: $$\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}<\epsilon_{t}=\epsilon_{a}+\epsilon_{r}\left\Vert \mathbf{G}(\mathbf{x}^{(0)})\right\Vert _{2},\label{eq-Newton-conv-tol}$$ where $\left\Vert \cdot\right\Vert _{2}$ is the Euclidean norm, $\epsilon_{t}$ is the total tolerance, $\epsilon_{a}$ is an absolute tolerance, $\epsilon_{r}$ is the Newton relative convergence tolerance, and $\mathbf{G}(\mathbf{x}^{(0)})$ is the initial residual. Such linear systems are solved iteratively with a Krylov subspace method (e.g. GMRES), which only requires matrix-vector products to proceed. Because the linear system matrix is a Jacobian matrix, matrix-vector products can be implemented Jacobian-free using the Gateaux derivative: $$\left.\frac{\partial\mathbf{G}}{\partial\mathbf{x}}\right|^{(k)}\mathbf{v}=\lim_{\epsilon\rightarrow0}\frac{\mathbf{G}(\mathbf{x}^{(k)}+\epsilon\mathbf{v})-\mathbf{G}(\mathbf{x}^{(k)})}{\epsilon},\label{eq:gateaux}$$ where $\mathbf{v}$ is a Krylov vector, and $\epsilon$ is in practice a small but finite number (p. 79 in [@kelley1987iterative]). Thus, the evaluation of the Jacobian-vector product only requires the function evaluation $\mathbf{G}(\mathbf{x}^{(k)}+\epsilon\mathbf{v})$, and there is no need to form or store the Jacobian matrix. This, in turn, allows for a memory-efficient implementation. An inexact Newton method [@inexact-newton] is used to adjust the convergence tolerance of the Krylov method at every Newton iteration according to the size of the current Newton residual, as follows: $$\left\Vert J^{(k)}\delta\mathbf{x}^{(k)}+\mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}<\zeta^{(k)}\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}\label{eq-inexact-newton}$$ where $\zeta^{(k)}$ is the inexact Newton parameter and $J^{(k)}=\left.\frac{\partial\mathbf{G}}{\partial\mathbf{x}}\right|^{(k)}$ is the Jacobian matrix. Thus, the convergence tolerance of the Krylov method is loose when the Newton state vector $\mathbf{x}^{(k)}$ is far from the nonlinear solution, and tightens as $\mathbf{x}^{(k)}$ approaches the solution. Superlinear convergence rates of the inexact Newton method are possible if the sequence of $\zeta^{(k)}$ is chosen properly (p. 105 in [@kelley1987iterative]). Here, we employ the prescription: $$\begin{aligned} \zeta^{A(k)} & = & \gamma\left(\frac{\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}}{\left\Vert \mathbf{G}(\mathbf{x}^{(k-1)})\right\Vert _{2}}\right)^{\alpha},\\ \zeta^{B(k)} & = & \min[\zeta_{max},\max(\zeta^{A(k)},\gamma\zeta^{\alpha(k-1)})],\\ \zeta^{(k)} & = & \min[\zeta_{max},\max(\zeta^{B(k)},\gamma\frac{\epsilon_{t}}{\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}})],\end{aligned}$$ with $\alpha=1.5$ , $\gamma=0.9$, and $\zeta_{max}=0.2$. The convergence tolerance $\epsilon_{t}$ is defined in Eq. \[eq-Newton-conv-tol\]. In this prescription, the first step ensures superlinear convergence (for $\alpha>1$), the second avoids volatile decreases in $\zeta_{k}$, and the last avoids oversolving in the last Newton iteration. The Jacobian system Eq. \[eq:Newton-Raphson step\] must be preconditioned for efficiency. Here, we employ right preconditioning, which transforms the original system into the equivalent one: $$JP^{-1}\mathbf{y}=-\mathbf{G}(\mathbf{x})$$ where $J=\partial\mathbf{G}/\partial\mathbf{x}$ is the Jacobian matrix, $P$ is a preconditioner, and $\delta\mathbf{x}=P^{-1}\mathbf{y}$. The Jacobian-free preconditioned system employs $$\mathit{J}P^{-1}\mathbf{v}=\lim_{\epsilon\rightarrow0}\frac{\mathbf{G}(\mathbf{x}+\epsilon P^{-1}\mathbf{v})-\mathbf{G}(\mathbf{x})}{\epsilon}\label{eq:gateaux-prec}$$ for each Jacobian-vector product. An important feature of preconditioning is that, while it may substantially improve the convergence properties of the Krylov iteration (when $P$ approximates $J$ and is relatively easy to invert), it does not alter the solution of the system upon convergence. The purpose of this study is to formulate an effective, fast preconditioner $P$ for the implicit PIC kinetic system. Before deriving the preconditioner, however, we review the fundamental CPU speedup limits of implicit vs. explicit PIC. Performance limits of implicit PIC ================================== As mentioned earlier, the ability of implicit PIC to take large timesteps without numerical instabilities does not necessarily translate into performance gains of implicit PIC over its explicit counterpart [@chen-jcp-11-ipic]. In this section, we summarize the back-of-envelope estimate for the CPU speedup introduced in the reference that supports this statement. We begin by estimating the CPU cost for a given PIC solver to advance the solution for a given time span $\Delta T$ as: $$CPU=\frac{\Delta T}{\Delta t}N_{pc}\left(\frac{L}{\Delta x}\right)^{d}C,\label{eq:CPU-estimation-1}$$ where $N_{pc}$ is the number of particles per cell, ($L/\Delta x$) is the number of cells per dimension, $d$ is the number of physical dimensions, and $C$ is the computational complexity of the solver employed, measured in units of a standard explicit PIC Vlasov-Poisson leap-frogd timestep. Accordingly, the implicit-to-explicit speedup is given by: $$\frac{CPU_{ex}}{CPU_{im}}\sim\left(\frac{\Delta x_{im}}{\Delta x_{ex}}\right)^{d}\left(\frac{\Delta t}{\Delta t_{ex}}\right)\frac{1}{C_{im}},$$ where we denote $\Delta t$ to be the implicit timestep. Assuming that all particles take a fixed sub-timestep $\Delta\tau$ in the implicit scheme, and that the cost of one timestep with the explicit PIC solver is comparable to that of a single implicit sub-step, it follows that $C_{im}\sim N_{FE}\left(\Delta t/\Delta\tau_{im}\right)$, i.e., the cost of the implicit solver exceeds that of the explicit solver by the number of function evaluations ($N_{FE}$) per $\Delta t$ multiplied by the number of particle sub-steps $\left(\Delta t/\Delta\tau_{im}\right)$. Assuming typical values for $\Delta\tau_{im}\sim\min[0.1\Delta x/v_{th},\Delta t_{imp}]$, $\Delta t_{ex}\sim0.1\omega_{pe}^{-1}$, $\Delta x_{im}\sim0.2/k$, and $\Delta x_{ex}\sim\lambda_{D}$, we find that the CPU speedup scales as: $$\frac{CPU_{ex}}{CPU_{imp}}\sim\frac{0.2}{(5k\lambda_{D})^{d}}\min\left[\frac{1}{k\lambda_{D}},\sqrt{\frac{m_{i}}{m_{e}}}\right]\frac{1}{N_{FE}}.\label{eq:CPU-ex-im-1}$$ This result supports two important conclusions. Firstly, it predicts that the CPU speedup is asymptotically independent of the implicit time step $\Delta t$ for $\Delta t\gg\Delta\tau_{im}$. The effect of the implicit time step is captured in the extra power of one in the $(k\lambda_{D})$ term, once one accounts for sub-stepping, but that effect disappears when the mesh becomes coarse enough (i.e., $k\lambda_{D}<\sqrt{m_{e}/m_{i}}$). Also, it predicts that the speedup improves with larger ion-to-electron mass ratio, indicating that the approach is more advantageous when one employs realistic mass ratios. Because the CPU speedup is asymptotically independent of $\Delta t$, algorithmically it will be advantageous to use a time step that is large enough to be in the asymptotic regime, but no larger. This will motivate the choice in the preconditioner to include only electron stiff physics. Secondly, Eq. \[eq:CPU-ex-im-1\] indicates that large CPU speedups are possible when $k\lambda_{D}\ll1$, particularly in multiple dimensions, but only if $N_{FE}$ is kept small and bounded. The latter point motivates the development of suitable preconditioning strategies. We focus on this in the next section. Fluid preconditioning the electrostatic implicit PIC kinetic system\[sec:formulation-pc\] ========================================================================================= The preconditioner of the nonlinear kinetic JFNK solver needs to return an approximation for the $E$-field update only. The approximate $E$-field update will be found from a linearized fluid model, consistently closed with particle moments. As will be shown, the fluid model provides an inexpensive approximation to the kinetic Jacobian. We demonstrate the concept in the 1D electrostatic, multispecies PIC model. Formulation of the fluid preconditioner --------------------------------------- Following standard procedure [@knoll2004jacobian], we work with the linearized form of the governing equations to derive a suitable preconditioner. The linearized, orbit-averaged, binomially smoothed 1D Ampere’s residual equation (Eq. \[eq:discete-Ampere\] with $E=E_{0}+\delta E$, and $\delta\bar{j}\equiv\int_{0}^{\Delta t}\delta jdt/\Delta t$) reads: $$\delta E=-\Delta t\left(G(E_{0})+\frac{1}{\varepsilon_{0}}SM(\delta\bar{j})\right),\label{eq:delta-Ampere-disc}$$ where $G(E_{0})=E_{0}-E^{n}+\frac{\Delta t}{\varepsilon_{0}}(SM(\bar{j}_{0}^{n+\nicefrac{1}{2}})-\left\langle \bar{j}_{0}\right\rangle )$ is the residual of Ampere’s law, the superscript $n$ denotes last timestep, and the subscript 0 of the $E$-field denotes the current Newton state. From the discussion in the previous section, for the purpose of preconditioning we consider only the linear response of electron contribution to the current ($\delta\bar{j}\simeq-e\delta\bar{\Gamma}$ where $\Gamma$ is the electron flux). Thus, the electric field update in the preconditioner will be found from: $$\delta E\approx-\Delta t\left(G(E_{0})-\frac{e}{\varepsilon_{0}}SM(\delta\bar{\Gamma})\right),\label{eq:delta-Ampere-e_only}$$ where $\delta\bar{\Gamma}=\frac{1}{\Delta t}\int_{0}^{\Delta t}dt\delta\Gamma(t)$, a time-average between timestep $n$ and $n+1$.. We approximate the linear response of the electron current via the continuity and momentum equations of electrons, closed with moments from particles (as in the implicit moment method [@mason-jcp-81-im_pic]). The continuity equation for electrons is $$\frac{\partial n}{\partial t}+\frac{\partial\Gamma}{\partial x}=0,\label{eq:continuity}$$ where where $n$ is electron number density. Linearizing, we obtain: $$\frac{\partial\delta n}{\partial t}=-\frac{\partial\delta\Gamma}{\partial x},\label{eq:delta-continuity}$$ where we have used particle conservation ($\partial n_{0}/\partial t+\partial\Gamma_{0}/\partial x=0$), which is satisfied at all iteration levels owing to exact charge conservation [@chen-jcp-11-ipic]. We then take the time-average ($\frac{1}{\Delta t}\int_{0}^{\Delta t}dt$, equivalently to the orbit average in Eq. \[eq:javerage\]) of Eq. \[eq:delta-continuity\] to obtain $$\delta n=-\Delta t\frac{\partial\delta\bar{\Gamma}}{\partial x}.\label{eq:delta-continuity-disc}$$ The update equation for $\delta\bar{\Gamma}$ is found from the electron momentum equation, which in conservative form reads $$m\left[\frac{\partial\Gamma}{\partial t}+\frac{\partial}{\partial x}\left(\frac{\Gamma\Gamma}{n}\right)\right]=-enE-\frac{\partial P}{\partial x}\label{eq:momentum}$$ where $m$ is the electron mass, $P\equiv nT$ is the electron pressure, and $T$ is the electron temperature. Linearizing it, we obtain: $$m\left[\frac{\partial\delta\Gamma}{\partial t}+\frac{\partial}{\partial x}\left(\frac{2\Gamma_{0}\delta\Gamma}{n_{0}}-\frac{\Gamma_{0}\Gamma_{0}}{n_{0}^{2}}\delta n\right)\right]+e(n_{0}\delta E+\delta nE_{0})+\frac{\partial(\delta nT_{0})}{\partial x}=0,\label{eq:delta-momentum}$$ where $T_{0}\equiv\int f(v)m(v-u)(v-u)dv/n_{0}$ is the current temperature (or normalized pressure). Closures for $\Gamma_{0}$, $n_{0}$ and $T_{0}$ are obtained from current particle information. In Eq. \[eq:delta-momentum\], we take $m\left[\partial\Gamma_{0}/\partial t+\partial(\Gamma_{0}\Gamma_{0}/n_{0})/\partial x\right]+en_{0}E_{0}+\partial(n_{0}T_{0})/\partial x=0$ by ansatz. To close the fluid model, we have neglected the linear temperature response $\delta T$. To cast Eq. \[eq:delta-momentum\] in a useful form, we take its time-derivative to get (assuming that $n_{0}$, $E_{0}$, and $T_{0}$ do not vary with time): $$m\frac{\partial^{2}\delta\Gamma}{\partial t^{2}}+e(n_{0}\frac{\partial\delta E}{\partial t}+\frac{\partial\delta n}{\partial t}E_{0})+\frac{\partial}{\partial x}(T_{0}\frac{\partial\delta n}{\partial t})=0,\label{eq:delta-momentum-dt}$$ and then time-average the result to find (substituting Eqs. \[eq:delta-Ampere-disc\] and \[eq:delta-continuity\]): $$\frac{2m\delta\bar{\Gamma}}{\Delta t^{2}}+e^{2}n_{0}\delta\bar{\Gamma}-eE_{0}\frac{\partial\delta\bar{\Gamma}}{\partial x}-\frac{\partial}{\partial x}(T_{0}\frac{\partial\delta\bar{\Gamma}}{\partial x})=-n_{0}G(E_{0}).\label{eq:delta-momentum-avg}$$ Here, we have neglected the convective term for simplicity, and approximated the first time-derivative term as: $$\frac{\partial\delta\Gamma}{\partial t}\simeq\frac{2\delta\bar{\Gamma}}{\Delta t}\label{eq:ddGamma-dt}$$ (which is exact if $\delta\Gamma(t)$ is linear with $t$). We discretize Eq. \[eq:delta-momentum-avg\] with space-centered finite differences, resulting in a tridiagonal system, which we invert for $\delta\bar{\Gamma}$ using a direct solver. Finally, we substitute the solution of $\delta\bar{\Gamma}$ in Eq. \[eq:delta-Ampere-e\_only\] to find the $E$-field update. Extension to curvilinear meshes ------------------------------- The fully implicit PIC algorithm has been recently extended to curvilinear meshes [@chacon-jcp-13-curvpic]. In this section, we rewrite the above fluid model on a 1D non-uniform mesh using a map $x=x(\xi)$. In 1D, the curvilinear form of in Eqs. \[eq:continuity\] and \[eq:momentum\] can be derived straightforwardly by replacing every $dx$ with $\mathcal{J}d\xi$, where $\mathcal{J}\equiv dx/d\xi$ is the Jacobian. It follows that the continuity equation in logical space is written as: $$\frac{\partial n}{\partial t}+\frac{1}{\mathcal{J}}\frac{\partial\Gamma}{\partial\xi}=0.\label{eq:continuity-curv}$$ The transformed momentum equation is $$m\left[\frac{\partial\Gamma}{\partial t}+\frac{1}{\mathcal{J}}\frac{\partial}{\partial\xi}\left(\frac{\Gamma\Gamma}{n}\right)\right]=qnE-\frac{1}{\mathcal{J}}\frac{\partial P}{\partial\xi}.\label{eq:momentum-curv}$$ Similar to the procedure described above, linearizing and discretizing Eqs. \[eq:continuity-curv\] and \[eq:momentum-curv\] again results in a tridiagonal system. Electrostatic wave dispersion relations --------------------------------------- It is instructive to look at the dispersion relation of Eq. \[eq:delta-Ampere-disc\], \[eq:delta-continuity\] and \[eq:delta-momentum\], for both electrons and ions. Figure \[fig:eiwave-disper\] shows the dispersion relation of electron plasma waves and ion acoustic waves [@FChenbook], from which we make the following observations. The stiffest wave is the electron plasma wave, whose frequency $\omega_{pe}$ is essentially insensitive to the wave number $k$ for $k\lambda_{D}<1$. The wave frequency increases for $k\lambda_{D}>1$, but in that range the plasma wave is highly Landau-damped [@jackson1960longitudinal]. In contrast to the electron wave, the ion wave frequency increases with $k$ for $k\lambda_{D}<1$, but saturates at $\sim\omega_{pi}$ for $k\lambda_{D}>1$. In a propagating ion acoustic wave (IAW), nonlinear effects lead to wave steepening. Because of the wave dispersion, the IAW steepening stops when the high frequency waves propagate slower than the low frequency ones [@krall1997we]. Those high frequency ion waves are physically important, and therefore need to be resolved. For this reason, in our numerical experiments, we limit the implicit time step to $\Delta t\sim0.1\omega_{pi}^{-1}$. The frequency gap between the electron and ion waves is about a factor of $\sqrt{m_{i}/m_{e}}$, which provides enough room to place the algorithm in the large timestep asymptotic regime (Eq. \[eq:CPU-ex-im-1\]). ![\[fig:eiwave-disper\]Dispersion relations of electron and ion waves in an electrostatic plasma. The dispersions can be obtained by Fourier analysis of the fluid model of Eq. \[eq:delta-Ampere-disc\], \[eq:delta-continuity\] and \[eq:delta-momentum\], for both electrons and ions, assuming that $E_{0},\Gamma_{0}$, $n_{0},T_{0}=$const.](iaw-disper) Asymptotic behavior of the implicit PIC formulation in the quasineutral limit.\[sub:Asymptotic-preservation-in\] ---------------------------------------------------------------------------------------------------------------- Since the implicit scheme is able to use large grid sizes and timesteps stably, it is important to ensure that the fluid preconditioner be able to capture relevant asymptotic regimes correctly [@degond-jcp-10-ap_pic]. In the context of electrostatic PIC, the relevant asymptotic regime is the quasineutral limit, which manifests when the domain length is much larger than the Debye length ($L\gg\lambda_{D}$) and when $m_{e}\ll m_{i}$. In this limit, the electric field must be found from the fluid equations [@fernsler2005quasineutral], and leads to the well know ambipolar electric field, $E=-\frac{1}{en}\partial_{x}P$. In our context, the algorithm must be well behaved when $L$ varies from $\sim\lambda_{D}$ to $\gg\lambda_{D}$, and for arbitrary mass ratios. In particular, the fluid preconditioner must feature these properties to successfully accelerate the kinetic algorithm. To confirm that this is the case, following Ref. [@degond-jcp-10-ap_pic] we normalize the electron fluid equations to the following reference quantities: $$\hat{x}=\frac{x}{x_{0}},\:\hat{v}=\frac{v}{v_{0}},\:\hat{t}=\frac{tv_{0}}{x_{0}},\:\hat{n}=\frac{n}{n_{0}},\:\hat{q}=\frac{q}{q_{0}},\:\hat{m}=\frac{m}{m_{0}},\:\hat{E}=\frac{Eq_{0}x_{0}}{k_{B}T_{0}}.$$ We choose $x_{0}=L$, $v_{0}=\sqrt{k_{B}T_{0}/m_{0}}$, $q_{0}=e$, $m_{0}=m_{i}$. For electrons, $q=-e$, and hence $\hat{q}=-1$. The normalized preconditioning equations become: $$\begin{aligned} \hat{\lambda}_{D}^{2}\frac{\partial\hat{E}}{\partial\hat{t}}-\hat{\Gamma} & = & 0,\label{eq:Ampere-law-norm}\\ \frac{\partial\hat{n}}{\partial\hat{t}}+\frac{\partial\hat{\Gamma}}{\partial\hat{x}} & = & 0,\label{eq:continuity-norm}\\ \hat{m}\frac{\partial\hat{\Gamma}}{\partial\hat{t}}+\hat{n}\hat{E}+\hat{T}\frac{\partial\hat{n}}{\partial\hat{x}} & = & 0,\label{eq:momentum-norm}\end{aligned}$$ where in Eq. \[eq:momentum-norm\] we have neglected the convective term. Substituting Eq. \[eq:Ampere-law-norm\] into Eq. \[eq:momentum-norm\], we find the equation for the electric field: $$\hat{m}\frac{\partial}{\partial\hat{t}}\left(\hat{\lambda}_{D}^{2}\frac{\partial\hat{E}}{\partial\hat{t}}\right)+\hat{n}\hat{E}+\hat{T}\frac{\partial\hat{n}}{\partial\hat{x}}=0,\label{eq:ap-momentum}$$ where $\lambda_{D}$ may change in time and space. The solution of $\hat{E}$ is well behaved as $\hat{m}\hat{\lambda}_{D}^{2}\rightarrow0$, where we indeed find that $\hat{n}\hat{E}=-\hat{T}\frac{\partial\hat{n}}{\partial\hat{x}}$, which is the correct (ambipolar) $E$-field. Our fluid preconditioner is based on the linearization of Eqs. \[eq:Ampere-law-norm\]-\[eq:momentum-norm\], and therefore inherits this asymptotic property. In what follows, we will demonstrate among other things the effectiveness of the preconditioner as we vary the domain size and the mass ratio. Numerical experiments\[sec:Numerical-experiments\] ================================================== We use the IAW problem for testing the performance of the fluid-based preconditioner. IAW propagation is a multi-scale problem determined by the coupling between electrons and ions. The 1D case used in Ref. [@chen-jcp-11-ipic] features large-amplitude IAWs in an unmagnetized, collisionless plasma without significant damping. We initialize the calculation with the following ion distribution function: $$f(x,v,t=0)=f_{M}(v)\left[1+a\cos\left(\frac{2\pi}{L}x\right)\right]\label{eq:initf}$$ where $f_{M}(v)$ is the Maxwellian distribution, $a$ is the perturbation level, $L$ is the domain size. The spatial distribution is approximated by first putting ions randomly with a constant distribution, e.g. $x^{0}\in[0,L]$. The electrons are distributed in pairs with ions according to the Debye distribution[@williamson1971initial]. Specifically, in each $e$-$i$ pair, the electron is situated away from the ion by a small distance, $dx=\mathrm{ln}(R)$ where $R\in(0,1)$ is a uniform random number (note that we normalize all lengths with the electron Debye length). We then shift the particle position by a small amount such that $x=x^{0}+a\cos\left(\frac{2\pi}{L}x^{0}\right)$, with $a=0.2$. For testing the solver performance with non-uniform meshes, the mesh adaptation in the periodic domain is provided by the map [@chacon-jcp-13-curvpic]: $$x(\xi)=\xi+\frac{L}{2\pi}(1-\frac{N\Delta x_{\nicefrac{L}{2}}}{L})\sin\left(\frac{2\pi\xi}{L}\right),\label{eq:map-xofxi}$$ which has the property that the Jacobian $J$ is also periodic. Here, $N$ is the number of mesh points, and $\Delta x_{\nicefrac{L}{2}}$ is the physical mesh resolution at $x=\xi=L/2$. Before we begin the convergence studies, it is informative to look at the condition number of the Jacobian system, which can be estimated as the number of times we step over the explicit CFL: $$\sigma\propto\omega_{pe}\Delta t=0.1\frac{\omega_{pe}}{\omega_{pi}}=0.1\sqrt{\frac{m_{i}}{m_{e}}},\label{eq:condition_number}$$ where we have used that $\Delta t\sim0.1\omega_{pi}^{-1}$, and we have assumed $k\lambda_{D}<1$. The first important observation is that, as expected, the Jacobian system will become harder to solve as we increase the ion-to-electron mass ratio. Secondly, the condition number does not depend on $k\lambda_{D}$. The latter, while surprising, is a consequence of our chosen implicit time step upper bound. Dependence of $\sigma$ with $k\lambda_{D}$ is recovered for $k\lambda_{D}>1$, but in this regime Langmuir waves are highly Landau-damped [@jackson1960longitudinal], and do not survive in the system. We demonstrate the performance of the fluid preconditioner by varying several relevant parameters, namely, the implicit timestep $\Delta t$, the mass ratio $m_{i}/m_{e}$, the domain length $L$, the mesh size $N_{x}$, and the number of particles per cell $N_{pc}$. We begin with the implicit timestep, which we vary from $0.01\omega_{pi}^{-1}$ to $0.25\omega_{pi}^{-1}$. For this test, we choose $L=100$, $N_{x}=128$, $N_{pc}=1000$, and $m_{i}/m_{e}=1836$. As shown in Figure \[fig:FE-dt\], the performance for preconditioned and unpreconditioned solvers is about the same for small time steps, where the Jacobian system is not stiff. However, significant differences in performance develop for larger timesteps, reaching a factor of 2 to 3 as the timestep approaches $0.2\omega_{pi}^{-1}$. Overall, the preconditioner is able to keep the linear and nonlinear iteration count fairly well bounded as the timestep increases. ![\[fig:FE-dt\]The performance of the JFNK solver against the timestep, with $L=100$, $N_{x}=128$, $N_{pc}=1000$, and $m_{i}/m_{e}=1836$. The number of function evaluations are well controlled by the preconditioner over a large range of $\Delta t$. ](./FE_dt-iaw) For bounded $N_{FE}$, Eq. \[eq:CPU-ex-im-1\] predicts that the actual CPU time should be largely insensitive to the timestep size. This is confirmed in Fig. \[fig:CPU-dt\], which shows the CPU performance of a series of computations with a fixed simulation time-span. Clearly, the total CPU time is essentially independent of $\Delta t$ with preconditioning (but not without). Also, both with and without preconditioning, the average particle pushing time, which is the average CPU time used for all particle pushes during the simulation time-span, saturates for large enough time steps (e.g. $v_{the}\Delta t>1\sim10\Delta x$), indicating that we have reached an asymptotically large time step. Even though the CPU performance of the preconditioned solver is independent of $\Delta t$, the use of larger timesteps is beneficial for the following reasons. Firstly, the orbit-averaging performed to obtain the plasma current density helps with noise reduction, as it provides the time average of many samplings per particle [@cohen-jcp-82-orbit_averaging]. Secondly, the operational intensity (computations per memory operation) per particle orbit increases with the timestep, which helps enhance the computing performance (or efficiency) and offset communication latencies in the simulation [@chen-jcp-12-ipic_gpu]. ![\[fig:CPU-dt\]Overall CPU performance as a function of timestep, comparing the unpreconditioned and preconditioned solvers in terms of the average particle pushing time (obtained by the total CPU time divided by the average number of iterations) (left) and wall clock CPU time (right). $L=100$, $N_{x}=128$, $m_{i}/m_{e}=1836$, and the time-span is fixed at 4.67$\omega_{pi}^{-1}$ for all computations.](./CPU_dt-iaw) The performance of the preconditioner vs. the electron-ion mass ratio for both uniform and non-uniform meshes is shown in Tables \[tab:ES-prec-performance-iaw-uniform\] and \[tab:ES-prec-performance-IAW-non-uniform\]. To make a fair comparison, both uniform and non-uniform meshes have the same finest mesh resolution, which locally resolves the Debye length. From the tables it is clear that similar performance gains of the preconditioned solver vs. the unpreconditioned one are obtained for both uniform and non-uniform meshes. The dependence of the GMRES performance on the mass ratio is much weaker with the preconditioner: as the mass ratio increases by a factor of 100, the GMRES iteration count increases by a factor of 5 without the preconditioner, vs. a factor of 2 with the preconditioner. Although not completely independent of the mass ratio, the solver behavior is consistent with the asymptotic analysis made in Sec. \[sub:Asymptotic-preservation-in\]. ------- -------- ------- -------- ------- Newton GMRES Newton GMRES 100 4 8 4 7 1600 5 21.2 4 10.1 10000 5.8 50.1 5.5 13.5 ------- -------- ------- -------- ------- : \[tab:ES-prec-performance-iaw-uniform\]Solver performance with and without the fluid preconditioner for the IAW case with $L=100$, $N_{x}=512$, and $N_{pc}=1000$ on a uniform mesh. For all the test cases, $\Delta t=0.1\omega_{pi}^{-1}$. The Newton and GMRES iteration numbers are obtained by an average over 20 timesteps. For all the runs, we have kept the ion and electron temperature constant. ------- -------- ------- -------- ------- Newton GMRES Newton GMRES 100 4 7.6 4 7 1600 5 21.3 5.1 12.1 10000 5.8 48.6 5.3 16.5 ------- -------- ------- -------- ------- : \[tab:ES-prec-performance-IAW-non-uniform\]Solver performance with and without the fluid preconditioner for the IAW case with the non-uniform mesh ($N_{x}=64$ and the smallest mesh size 0.2). The impact of the domain length in the solver performance is shown in Fig. \[fig:FE\_L\_iaw\]. Clearly, the solver performance remains fairly insensitive to the domain length both with and without the preconditioner, even though the domain length varies from 10 to 1000 Debye lengths. This is consistent with the condition number analysis in Eq. \[eq:condition\_number\]. The impact of the preconditioner in the number of GMRES iterations is expected for the time step chosen. ![\[fig:FE\_L\_iaw\]Solver performance as a function of the domain size, with $N_{x}=64$, $N_{pc}=1000$, $\Delta t=0.1\omega_{pi}^{-1}$.](./FE_L-iaw) ![\[fig:FE-particle\]NFE of GMRES and Newton iterations as a function of average number of particles per cell for a domain size $L=100$ with $N_{x}=128$ uniformly distributed cells. ](./FE_np-iaw) ![\[fig:FE-particle\]NFE of GMRES and Newton iterations as a function of average number of particles per cell for a domain size $L=100$ with $N_{x}=128$ uniformly distributed cells. ](./FE_np-iaw-Newton) The impact of the number of particles in the performance of the solver is shown in Fig. \[fig:FE-particle\], which depicts the iteration count of both Newton and GMRES vs. the number of particles. The timestep is varied by a factor of two, corresponding to about one-tenth and one-fifth of the inverse ion plasma frequency ($\omega_{pi}^{-1}$). As expected, the solver performs better with smaller timesteps and with larger number of particles. The number of linear and nonlinear iterations increases as the number of particles decreases. This behavior is likely caused by the increased interpolation noise associated with fewer particles: the noise in charge density results in fluctuations in the self-consistent electric field, making the Jacobian-related calculations less accurate, thus delaying convergence. The preconditioner seems to ameliorate the impact of having too few particles on the performance of the algorithm, thus robustifying the nonlinear solver. ![\[fig:FE-dx\]Solver performance vs. the number of grid-points $N_{x}$ for $\omega_{pi}\Delta t=0.093$, $L=100\lambda_{D}$, and $N_{pc}=1000$.](./FE_nx-iaw) The impact of the number of grid points on the solver performance is shown in Fig. \[fig:FE-dx\] for $\omega_{pi}\Delta t=0.093$, $L=100\lambda_{D}$, and $N_{pc}=1000$. We see that the linear and nonlinear iteration count remains fairly constant with respect to $N_{x}$, with and without preconditioning. This is consistent with the condition number result in Eq. \[eq:condition\_number\] (which is independent of the wavenumber). However, despite the fact that the number of iterations is virtually independent of the number of grid points, the CPU time grows significantly with it. Figure \[fig:CPU-nx\] shows that the computational cost scales as $N_{x}^{2}$ for $N_{x}$ large enough. The reason is two-fold. On the one hand, since we keep the number of particles per cell fixed, the computational cost of pushing particles increases proportionally with the number of grid points. On the other hand, as we refine the grid, the cost per particle increases because particles have to cross more cells (for a given timestep). In multiple dimensions, the particle orbit will sample $N^{1/d}$ cells on average, for large enough $N$ (or $\Delta t$), with $N$ and $d$ denoting the total number of grid points and dimensions, respectively. Hence, the cost of particle crossing will scale as $N^{1/d}$, and the computational cost will scale as $N^{1+1/d}$. In this sense, the 1D configuration is the least favorable. ![\[fig:CPU-nx\]The performance of the JFNK solver against the number of grid-points, with $N_{pc}=1000$. The average particle pushing time is shown on the left and total CPU time for a total time-span $80$ is shown on the right. Both particle pushing time and total CPU time scale as $N_{x}^{2}$ for large enough $N_{x}$.](./CPU_nx-iaw) The performance of the implicit PIC solver vs. the explicit PIC one is compared in Fig. \[fig:CPU-scaling\], which depicts the CPU speedup vs. $k\lambda_{D}$. For this test, we choose $m_{i}/m_{e}=1836$, $\Delta t=0.1\omega_{pi}^{-1}$, and $N_{pc}=1000$. In the implicit tests, the number of grid-points is kept fixed at $N_{x}=32$ as $L$ increases with $k=2\pi/L$. In the explicit computations, $\Delta x\simeq0.3\lambda_{D}$ is kept constant for stability, and therefore the number of grid-points increases with $L$. Both implicit and explicit tests employ a uniform mesh. We monitor the scaling power index of Eq. \[eq:CPU-ex-im-1\] with and without the preconditioner. We test the performance with a large implicit timestep (about 40 times larger than the explicit timestep). The scaling index is found to be $\sim1.86$ for small domain sizes, close to the expected value of 2. As $L$ increases, the scaling index becomes $\sim1$. The scaling index turns at $k\lambda_{D}\sim\sqrt{m_{e}/m_{i}}\sim0.025$, as predicted by Eq. \[eq:CPU-ex-im-1\]. The estimated scaling index of 2 would be recovered if one increased the timestep proportionally to $L$, but this would result in timesteps too large with respect to $\omega_{pi}^{-1}$. Overall, these results are in very good agreement with our simple estimates. The preconditioned solver gains about a factor of two compared to the un-preconditioned one, insensitively to $k\lambda_{D}$, which is consistent with the results depicted in Fig. \[fig:FE\_L\_iaw\]. We see that for $k\lambda_{D}<10^{-3}$, the implicit scheme delivers speedups of about three orders of magnitude vs. the explicit approach, while remaining exactly energy- and charge-conserving. The setup in Fig. \[fig:CPU-scaling\] employs a uniform mesh. However, sometimes it is necessary to resolve the Debye length locally, e.g. at a shock front or a boundary layer near a wall. In this case, using a non-uniform mesh is advantageous [@chacon-jcp-13-curvpic]. We test the performance of the preconditioner on a non-uniform mesh for a nonlinear ion acoustic shock wave, as setup in Ref. [@chacon-jcp-13-curvpic]. Specifically, we use $L=100\lambda_{D}$, $N_{pc}=2000$, $N_{x}=64$, and $\Delta t=0.1\omega_{pi}^{-1}$ for 20 timesteps. The minimum resolution is $\Delta x=0.5\lambda_{D}$ at the shock location (as in [@chacon-jcp-13-curvpic], we perform the simulation in the reference frame of the shock). With a nonlinear tolerance $\epsilon_{r}=2\times10^{-4}$, we have found that, with preconditioning, the average number of Newton and GMRES iterations is 3 and 10.1, respectively, compared to 3.6 and 23 without preconditioning. The performance gain in the linear solve is about factor of two, comparable to that obtained for a uniform mesh with similar problem parameters. Similar performance gains are found with tighter nonlinear tolerances: for $\epsilon_{r}=10^{-8}$, we find 5.1 Newton and 46.6 GMRES iterations without preconditioning, vs. 5 and 20.5 with it. ![\[fig:CPU-scaling\]The implicit PIC solver performance compared with the explicit scheme. The performance gain increases with the domain size. For the parameters used, the performance gain of the implicit solver is enhanced by the preconditioner by about a factor of 2.](./scaling) Conclusions\[sec:Conclusion\] ============================= This study has focused on the development of a preconditioner for a recently proposed fully implicit, JFNK-based, charge- and energy-conserving particle-in-cell electrostatic kinetic model [@chen-jcp-11-ipic]. In the reference, it was found that, for large enough implicit time steps $\Delta t$, the potential implicit-to-explicit CPU speedup scaled as $\frac{1}{N_{FE}(k\lambda_{D})^{d}}$, with $N_{FE}$ the number of function evaluations per time step, and $k\lambda_{D}\propto\lambda_{D}/L$. Thus, large speedups are expected when $k\lambda_{D}\ll1$ provided that $N_{FE}$ is kept bounded. While the CPU speedup does not scale directly with $\Delta t$, the use of large $\Delta t$ is advantageous to maximize operational intensity [@chen-jcp-12-ipic_gpu] (i.e., to maximize floating point operations per byte communicated), and to control numerical noise via orbit averaging [@cohen-jcp-82-orbit_averaging]. We have targeted a preconditioner based on an electron fluid model, which is sufficient to capture the stiffest time scales, and thus enable the use of large implicit time steps while keeping the number of function evaluations bounded. The performance of the preconditioned kinetic JFNK solver has been analyzed with various parametric studies, including time step, mass ratio, domain length, number of particles, and mesh size. The number of function evaluations is found to be insensitive against changes in all of these, delivering a robust nonlinear solver. The CPU time of the implicit PIC solver is found to be insensitive to the time step (as expected), but to scale with the square of the number of mesh points in 1D. This scaling is due to the number of particles per cell being kept constant, and to the number of particle crossings increasing linearly with the mesh resolution. The latter scaling will be more benign in multiple dimensions, as particle orbits remain one-dimensional. Speedups of about three orders of magnitude vs. explicit PIC are demonstrated when $\lambda_{D}\ll L$ (i.e., in the quasineutral regime). Based on the speedup prediction in [@chen-jcp-11-ipic], more dramatic speedups are expected in multiple dimensions. We conclude that the proposed algorithm shows much promise for extension to multiple dimensions and to electromagnetic simulations. This will be the subject of future work. #### Acknowledgments {#acknowledgments .unnumbered} This work was partially sponsored by the Office of Fusion Energy Sciences at Oak Ridge National Laboratory, and by the Los Alamos National Laboratory (LANL) Directed Research and Development Program. This work was performed under the auspices of the US Department of Energy at Oak Ridge National Laboratory, managed by UT-Battelle, LLC under contract DE-AC05-00OR22725, and the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory, managed by LANS, LLC under contract DE-AC52-06NA25396.

--- abstract: 'Autonomous navigation is a key skill for assistive and service robots. To be successful, robots have to navigate avoiding going through the personal spaces of the people surrounding them. Complying with social rules such as not getting in the middle of human-to-human and human-to-object interactions is also important. This paper suggests using Graph Neural Networks to model how inconvenient the presence of a robot would be in a particular scenario according to learned human conventions so that it can be used by path planning algorithms. To do so, we propose two ways of modelling social interactions using graphs and benchmark them with different Graph Neural Networks using the SocNav1 dataset. We achieve close-to-human performance in the dataset and argue that, in addition to promising results, the main advantage of the approach is its scalability in terms of the number of social factors that can be considered and easily embedded in code, in comparison with model-based approaches. The code used to train and test the resulting graph neural network is available in a public repository.' author: - 'Luis J. Manso$^{1}$, Ronit R. Jorvekar$^{2}$, Diego R. Faria$^{1}$, Pablo Bustos$^{3}$ and Pilar Bachiller$^{3}$[^1][^2][^3]' bibliography: - 'IEEEabrv.bib' - 'bib.bib' title: 'Graph Neural Networks for Human-aware Social Navigation' --- Deep Learning in Robotics and Automation; Domestic Robots; Social Human-Robot Interaction Introduction {#intro} ============ robot navigation deals with the challenge of endowing mobile social robots with the capability of considering the emotions and safety of people nearby while moving around their surroundings. There is a wide range of works studying human-aware navigation from a considerably diverse set of perspectives. Pioneering works such as [@Pacchierotti2005] started taking into account the personal spaces of the people surrounding the robots, often referred to as proxemics. In addition to proxemics, human motion patterns were analysed in [@Hansen2009] to estimate whether humans are willing to interact. Semantic properties were also considered in [@Cosley2009]. Although not directly applied to navigation, the relationships between humans and objects were used in the context of ambient intelligence in [@Bhatt2010]. Proxemics and object affordances were jointly considered in [@Vega2019] for navigation purposes. Two extensive surveys on human-aware navigation can be found in [@Rios-Martinez2015] and [@Charalampous2017]. Despite the previously mentioned approaches being built on well-studied psychological models, they have limitations. Considering new factors programmatically (*i.e.*, writing additional code) involves a potentially high number of coding hours, makes systems more complex, and increases the chances of including bugs. Additionally, with every new aspect to be considered for navigation, the decisions made become less *explainable*, which is precisely one of the main advantages of model-based approaches over data-driven ones. Besides the mentioned model scalability and explainability issues, model-based approaches have the intrinsic and rather obvious limitation that they only account for what the model explicitly considers. Given that these models are manually written by humans, they cannot account for aspects that the designers are not aware of. Approaches leveraging machine learning have also been published. The parameters of a social force model (see [@Helbing1995]) are learned in [@Ferrer2013] and [@Patompak2019] to navigate in human-populated environments. Inverse reinforcement learning is used in [@Ramon-Vigo2014] and [@Vasquez2014] to plan navigation routes based on a list of humans in a radius. Social norms are implemented using deep reinforcement learning in [@Chen2017], again, considering a set of humans. An approach modelling crowd-robot interaction and navigation control is presented in [@Chen2019]. It features a two-module architecture where single interactions are modelled and then aggregated. Although its authors reported good qualitative results, the approach does not contemplate integrating additional information (*e.g.*, relations between humans and objects, structure and size of the room). The work in [@martins2019clusternav] tackles the same problem using Gaussian Mixture Models. It has the advantage of requiring less training data, but the approach is also limited in terms of the input information used. All the previous works and many others not mentioned have achieved outstanding results. Some model-based approaches such as [@Cosley2009] or [@Vega2019] can leverage structured information to take into account space affordances. Still, the data considered to make such decisions are often handcrafted features based on an arbitrary subset of the data that a robot would be able work with. There are many reasons motivating seeking for a learning-based approach not requiring feature handcrafting or manual selection. Their design is time-consuming and often requires a deep understanding of the particular domain (see discussion in [@Lecun2015]). Additionally, there is generally no guarantee that a particular hand-engineered set of features is close to being the best possible one. On the other hand, most end-to-end deep learning approaches have important limitations too. They require a big amount of data and computational resources that are often scarce and expensive, and they are hard to explain and manually fine-tune. Somewhere in the middle of the spectrum, we have proposals advocating not to choose between hand-engineered features or end-to-end learning. In particular, [@Battaglia2018] proposes Graph Neural Networks (GNNs) as a means to perform learning that allows combining raw data with hand-engineered features, and most importantly, learn from structured information. The relational inductive bias of GNNs is specially well-suited to learn about structured data and the relations between different types of entities, often requiring less training data than other approaches. In this line, we argue that using GNNs for human-aware navigation makes possible integrating new social cues in a straightforward fashion, by including more data in the the graphs that they are fed. In this paper we use different GNN models to estimate social navigation compliance, *i.e.*, given a robot pose and a scenario where humans and objects can be interacting, estimating to what extent a robot would be disturbing the humans if it was located in such a pose. GNNs are proposed because the information that social robots can work with is not just a map and a list of people, but a more sophisticated data structure where the entities represented have different relations among them. For example, social robots can have information about who a human is talking to, where people are looking at, who is friends with who, or who is the owner of an object in the scene. Regardless of how this information is acquired, it can be naturally represented using a graph, and GNNs are a particularly well-suited and scalable machine learning approach to work with these graphs. Graph Neural Networks ===================== Graph Neural Networks (GNNs) are a family of machine learning approaches based on neural networks that take graph-structured data as input. They allow classifying and making regressions on graphs, nodes, edges, as well as predicting link existence when working with partially observable phenomena. Except for few exceptions (*e.g.*, [@ying2018hierarchical]) GNNs are composed by similar stacked blocks/layers operating on a graph whose structure remains static but the features associated to its nodes are updated in every layer of the network (see Fig. \[fig:gnnlayer\]). {width="60.00000%"} As a consequence, the features associated to the nodes of the graph in each layer become more abstract and are influenced by a wider context as layers go deeper. The features in the nodes of the last layer are frequently used to perform the final classification or regression. The first published efforts on applying neural networks to graphs date back to [@Sperduti1998]. GNNs were further studied and formalised in [@Gori2005] and [@F.2009]. However, it was with the appearance of Gated Graph Neural Networks (GG-NNs, [@Li2015]) and especially Graph Convolutional Networks (GCNs, [@Kipf2016a]) that GNNs gained traction. The work presented in [@Battaglia2018] reviewed and unified the notation used in the GNNs existing to the date. Graph Convolutional Networks (GCN) [@Kipf2016a] are one of the most common GNN blocks. Because of its simplicity, we build on the GCN block to provide the reader with an intuition of how GNNs work in general. Following the notation proposed in [@Battaglia2018], GCN blocks operate over a graph $G=(V,E)$, where $V=\{v_i\}_{i=1:N^v}$ is a set of nodes, being $v_i$ the feature vector of node $i$ and $N^v$ the number of vertices in the graph. $E$ is a set of edges $E=\{(s_k,r_k)\}_{k=1:N^e}$, where $s_k$ and $d_k$ are the source and destination indices of edge $k$ and $N^e$ is the number of edges in the graph. Each GCN layer generates an updated representation $v_i'$ for each node $v_i$ using two functions: $$\overline{e}_i = \displaystyle\rho^{e \rightarrow v}(E) = \sum_{\{k:r_k=i\}}e_k,$$ $$v_i' = \phi^{v}(\overline{e}_i, v_i)=NN_v([\overline{e}_i,v_i]).$$ For every node $v_i$, the first function ($\displaystyle\rho^{e \rightarrow v}(E)$) aggregates the feature vectors of other nodes with an edge towards it and generates a temporary aggregated feature $\overline{e}_i$ which is used by the second function. In a second pass, the function $\phi^{v}(\overline{e}_i, v_i)$ is used to generate updated $v_i'$ feature vectors from the aggregated feature vectors using a neural network (usually a multi-layer perceptron, but the framework does not make any assumption on this). Such a learnable function is the same for all the nodes. By stacking several blocks where features are aggregated and updated, the feature vectors can carry information from nodes far away in the graph and convey higher level features that can be finally used for classification or regressions. Several means of improving GCNs have been proposed. Relational Graph Convolutional Networks (RGCNs [@Schlichtkrull2018]) extends GCNs by considering different types of edges separately and applies the resulting model to vertex classification and link prediction. Graph Attention Networks (GATs [@Velickovic2018]) extend GCNs by adding self-attention mechanisms (see [@vaswani2017attention]) and applies the resulting model to vertex classification. For a more detailed review of GNNs and the generalised framework, please refer to [@Battaglia2018]. Formalisation of the problem {#problem} ============================ The aim of this work is to analyse the scope of GNNs in the field of human-aware social navigation. Our study has been set up using the *SocNav1* dataset (see [@manso2019socnav]). It contains scenes with a robot in a room, a number of objects and a number of people that can potentially be interacting with other objects or people. Each scene is labelled with a value from $0$ to $100$ depending on to what extent the subjects that labelled the scenarios considered that the robot is disturbing the people in the scene. The dataset provides 16336 labelled samples to be used for training purposes, 556 scenarios as the development dataset and additional 556 for final testing purposes. As previously noted, GNNs are a flexible framework that allows working somewhere in the middle of end-to-end and feature engineered learning. Developers can use as many data features as desired and are free to structure the input graph data as they please. The only limitations are those of the particular GNN layer blocks used. In particular, while GCN and GAT do not support labelled edges, RGCN and GG-NN do. To account for this limitation, two representations were used in the experiments, depending on the GNN block to be tested: one without edge labels and one with them. The first version of the scene-to-graph transformation used to represent the scenarios does not use labelled edges. It uses 6 node types (the features associated to each of the types are detailed later in this section): - **robot:** The dataset only includes one robot in each scene, so there is just one robot symbol in each of the graphs. However, GNNs do not have such restriction. - **wall:** A node for each of the segments defining the room. They are connected to the room node. - **room:** Used to represent the room where the robot is located. It is connected to the robot. - **object:** A node for each object in the scene. - **human:** A node for each human. Humans might be interacting with objects or other humans. - **interaction:** An interaction node is created for every human-to-human or human-to-object interaction. Figure \[fig:scenarios\] depicts two areas of a scenario where four humans are shown in a room with several objects. Two of the humans are interacting with each other, another human is interacting with an object, and the remaining human is not engaging in interaction with any human or object. The structure of the resulting non-labelled graph is shown in Fig.\[fig:graph4\]. \ The features used for **human** and **object** nodes are: distance, the relative angle from the robot’s point of view, and its orientation, from the robot’s point of view too. For **room** symbols the features are: the distance to the closest human and the number of humans. For the **wall** segments and the **interaction** symbols, the features are the distance and orientation from the robot’s frame of reference. For wall segments, the position is the centre of the segment and the orientation is the tangent. For interactions, the position is the midpoint between the interacting symbols, and the orientation is the tangent of the line connecting the endpoints. Features related to distances are expressed in meters, whereas those related to angles are actually expressed as two different numbers, $sin(\alpha)$ and $cos(\alpha)$. The final features of the nodes are built by concatenating the one-hot encoding that determines the type of the symbol and the features for the different node types. It is worth noting that by building feature vectors this way, their size increases with every new type. This limitation is currently being studied by the GNN scientific community. For the GNN blocks that can work with labelled edges, a slightly different version of the scene-to-graph transformation is used. The first difference is that in this version of the scenario-to-graph model there are no interaction nodes. The elements interacting are linked to each other directly. Robot, room, wall, human and object nodes are attributed with the same features as in the previous model. The second difference is related to the labelling of the edges. In this domain, the semantics of the edges can be inferred from the types of the nodes being connected. For example, *wall* and *room* nodes are always connected by the same kind of relation “composes”. Similarly, *humans* and *object* nodes are always connected by the relation “interacts\_with\_human”. The same holds the other way around: “composes” relations only occur between *wall* and *room* nodes, and “interacts\_with\_human” relations only occur with *humans* and *object* nodes. Therefore, for simplicity, the label used for the edges is the concatenation of the types involved. The structure of the resulting labelled graph for the scenario depicted in Fig.\[fig:scenarios\] is shown in Fig.\[fig:graphR\]. Because all nodes are connected to the robot, the GNNs were trained to perform the regression on the feature vector of the *robot* node in the last layer. Using the previously mentioned dataset and the two proposed scenario-to-graph transformations, different architectures using different GNN blocks were compared. Experimental results ==================== The four GNN blocks used in the experiments were: - Graph Convolutional Networks (GCN) [@Kipf2016a]. - Gated Graph Neural Networks (GG-NN) [@Li2015]. - Relational Graph Convolutional Networks (RGCNs) [@Schlichtkrull2018]. - Graph Attention Networks(GAT) [@Velickovic2018]. Using 1 and 2 layers for the neural network implementing $v_{i}'$ (GAT and GAT2 respectively in table \[tbl:experiment\_results\]). If available, each of these GNN blocks was benchmarked using Deep Graph Library (DGL) [@wang2019dgl] and PyTorch-Geometric (PyG) [@fey2019pyg]. In addition to using several layers of the same GNN building blocks, alternative architectures combining RGCN and GAT layers were also tested: 1. A sequential combination of $n$ RGCN layers followed by $n$ GAT layers with the same number of hidden units (alternative 8 in table \[tbl:experiment\_results\]). 2. An interleaved combination of $n$ RGCN and $n$ GAT layers with the same number of hidden units (alternative 9 in table \[tbl:experiment\_results\]). 3. A sequential combination of $n$ RGCN layers followed by $n$ GAT layers with a linearly decreasing number of hidden units (alternative 10 in table \[tbl:experiment\_results\]). As a result, 10 framework-architecture combinations were benchmarked. Table \[tbl:experiment\_results\] describes them and provides their corresponding performance on the development dataset. To benchmark the different architectures, 5000 training sessions were launched using the SocNav1 training dataset and evaluated using the SocNav1 development dataset. The hyperparameters were randomly sampled from the range values shown in Table \[tbl:experiment\_hp\]. ---- --------- --------------- ------------- Network Loss architecture (MSE) 1 DGL GCN 0.02289 2 **DGL** **GAT** **0.01701** 3 DGL GAT2 0.01740 4 PyG GCN 0.02243 5 PyG GAT 0.01731 6 PyG RGCN 0.01871 7 PyG GG-NN 0.02700 8 PyG RGCN$||$GAT 1 0.02089 9 PyG RGCN$||$GAT 2 0.02103 10 PyG RGCN$||$GAT 3 0.01995 ---- --------- --------------- ------------- : A description of the different framework/architecture combinations and the experimental results obtained from their benchmark for the SocNav1 dataset.[]{data-label="tbl:experiment_results"} Hyperparameter Min Max ----------------- ------ ------ epochs patience batch size 100 1500 hidden units 50 320 attention heads 2 9 learning rate 1e-6 1e-4 weight decay 0.0 1e-6 layers 2 8 dropout 0.0 1e-6 alpha 0.1 0.3 : Ranges of hyperparameter values sampled. The ’attention heads’ parameter is only applicable to Graph Attention Network blocks.[]{data-label="tbl:experiment_hp"} The results obtained (see table \[tbl:experiment\_results\]) suggest that, for the dataset and the framework/architecture combinations benchmarked, DGL/GAT delivered the best results, with a loss of 0.01701 for the development dataset. The parameters used by the DGL/GAT combination were: batch size: 273, number of hidden units: 129, number of attention heads: 2, number of attention heads in the last layer: 3, learning rate: 5e-05, weight decay regularisation: 1e-05, number of layers: 4, no dropout, alpha parameter of the ReLU non-linearity 0.2114. After selecting the best set of hyperparameters, the network was compared with a third *test* dataset, obtaining an MSE of 0.03173. Figures \[fig:scenario1\_hm\] and \[fig:scenario2\_hm\] provide an intuition of the output of the network for the scenarios depicted in figures \[fig:scenario1\] and \[fig:scenario2\] considering all the different positions of the robot in the environment when looking along the $y$ axis. It is worth noting that, due to the subjective nature of the labels in the dataset (human feelings are utterly subjective), there is some level of disagreement even among humans. To compare the performance of the network with human performance, we asked 5 subjects to label all the scenarios of the development dataset. The mean MSE obtained for the different subjects was 0.02929. The subjects achieved an MSE of 0.01878 if their decisions were averaged before computing the error. Overall, the results suggest that the network performs very close to human accuracy in the test set (0.03173 versus 0.02929). Figure \[fig:histogram\] shows an histogram comparing the error made by the GNN-based regression in the test set. ![Histogram of the absolute error in the test dataset for the network performing best in the development dataset.[]{data-label="fig:histogram"}](histogram){width="\columnwidth"} Most algorithms presented in section \[intro\] deal with modelling human intimate, personal, social and interaction spaces instead of social inconvenience, which seems to be a more general term. Keeping that in mind, the algorithm proposed in [@Vega2019] was tested against the test dataset and got a MSE of 0.12965. The relative bad performance can be explained by the fact that other algorithms do not take into account walls and that their actual goal is to model personal spaces instead of feelings in general. Regarding the effect of the presence of humans, we can see from Fig. \[fig:scenario2\_hm\] that the learnt function is slightly skewed to the front of the humans, but not as much as modelled in other works such as [@kirby2009companion] or [@Vega2019]. One of the possible reasons why the “personal space” is close to being circular is the fact that, in the dataset, humans appear to be standing still. It is still yet to be studied, probably using more detailed and realistic datasets, how would the personal space look like if humans were moving. The results obtained using GNN blocks supporting edge labels were inferior to those obtained using GAT, which does not support edge labels. Two reasons might be the cause of this phenomena: **a)** as mentioned in section \[problem\] the labels edges can be inferred from the types of the nodes, so that information is to some extent redundant; **b)** the inductive bias of GATs is strong and appropriate for the problem at hand. This does not mean that the same results would be obtained in other problems where the label of the edges cannot be inferred. Conclusions =========== To our knowledge, this paper presented the first graph neural network for human-aware navigation. The scene-to-graph transformation model and the graph neural network developed as a result of the work presented in this paper achieved a performance comparable to that of humans. Even though the results achieved are remarkable, the key fact is that this approach allows to include more relational information. This will allow to include more sources of information in our decisions without a big impact in the development. There is room for improvement, particularly related to: **a)** personalisation (different people generally feel different about robots), and **b)** movement (the inconvenience of the presence of a robot is probably influenced by the movement of the people and the robot). Still, we include interactions and walls, features which are seldom considered in other works. As far as we know, interactions have only been considered in [@Vega2019] and [@Cruz-maya2019]. The code to test the resulting GNN has been published in a public repository as open-source software[^4], as well as the code implementing the scene-to-graph transformation and the code train the models suggested. [^1]: $^{1}$Luis J. Manso and Diego R. Faria are with School of Engineering and Applied Science, Computer Science Department, Aston University, United Kingdom. [l.manso@aston.ac.uk]{} [^2]: $^{2} $Ronit R. Jorvekar is with Department of Computer Engineering, Pune Institute of Computer Technology, India. [^3]: $^{3} $Pablo Bustos and Pilar Bachiller are with Robotics and Artificial Vision Laboratory, Caceres School of Technology, Universidad de Extremadura, Extremadura. [^4]: https://github.com/robocomp/sngnn

--- abstract: | Improving previous calculations, we compute the $J/\psi~\pi\rightarrow \mbox{charmed mesons}$ cross section using QCD sum rules. Our sum rules for the $J/\psi~\pi\rightarrow \bar{D}~D^*$, $D~\bar{D}^*$, ${\bar D}^*~D^*$ and ${\bar D}~D$ hadronic matrix elements are constructed by using vaccum-pion correlation functions, and we work up to twist-4 in the soft-pion limit. Our results suggest that, using meson exchange models is perfectly acceptable, provided that they include form factors and that they respect chiral symmetry. After doing a thermal average we get $\langle\sigma^{\pi J/\psi} v\rangle\sim 0.3$ mb at $T=150\MeV$. author: - 'Francisco O. Durães$^1$, Hungchong Kim$^2$, Su Houng Lee$^{2,3}$, Fernando S. Navarra$^1$ and Marina Nielsen$^1$' title: 'Progress in the determination of the $J/\psi-\pi$ cross section' --- Introduction ============ For a long time charmonium suppression has been considered as one of the best signatures of quark gluon plasma (QGP) formation [@ma86]. Recently this belief was questioned by some works. Detailed simulations [@thews] of a population of $c - \overline{c}$ pairs traversing the plasma suggested that, given the large number of such pairs, the recombination effect of the pairs into charmonium Coulomb bound states is non-negligible and can even lead to an enhancement of $J/\psi$ production. This conclusion received support from the calculations of [@rapp]. Taking the existing calculations seriously, it is no longer clear that an overall suppression of the number of $J/\psi$’s will be a signature of QGP. A more complex pattern can emerge, with suppression in some regions of the phase space and enhancement in others [@huf; @dnn]. Whatever the new QGP signature (involving charm) turns out to be, it is necessary to understand better the $J/\psi$ dissociation mechanism by collisions with comoving hadrons. Since there is no direct experimental information on $J/\psi$ absorption cross sections by hadrons, several theoretical approaches have been proposed to estimate their values. In order to elaborate a theoretical description of the phenomenon, we have first to choose the relevant degrees of freedom. Already at this point no consensus has been found. Some approaches were based on charm quark-antiquark dipoles interacting with the gluons of a larger (hadron target) dipole [@bhp; @kha2; @lo] or quark exchange between two (hadronic) bags [@wongs; @mbq], whereas other works used the meson exchange mechanism [@mamu98; @osl; @haglin; @linko; @ikbb; @nnr]. In this case it is not easy to decide in favor of quarks or hadrons because we are dealing with charm quark bound states, which are small and massive enough to make perturbation theory meaningful, but not small enough to make non-perturbative effects negligible. Charmonium is “the borderguard of the mysterious border of perturbative world of quarks and gluons and the non-perturbative world of hadrons” [@kardok]. In principle, different approaches apply to different energy regimes and we might think that at lower energies we can use quark-interchange models [@wongs; @mbq] or meson exchange models [@mamu98; @osl; @haglin; @linko; @ikbb; @nnr] and, at higher energies we can use perturbative QCD [@bhp; @kha2; @lo]. However, even at low energies, the short distance aspects may become dominant and spoil a non-perturbative description. In a similar way, non-perturbative effects may be important even at very high energies [@hdnnr]. Inspite of the difficulties, some progress has been achieved. This can be best realized if we compare our knowledge on the subject today with what we knew a few years ago, described by B. Mueller (in 1999) [@muel] as “...the state of the theory of interactions between $J/\psi$ and light hadrons is embarrassing. Only three serious calculations exist (after more than 10 years of intense discussion about this issue!) and their results differ by at least two orders of magnitude in the relevant energy range. There is a lot to do for those who would like to make a serious contribution to an important topic”. In the subsequent three years about 30 papers on this subject appeared and now the situation is much better, at least in what concerns the determination of the order of magnitude, which, as it will be discussed below, in the case of the $J/\psi$ pion interaction, is determined to be $ 1 \, < \, \sigma_{J/\psi - \pi} \, < \, 10 $ mb in the energy region close to the open charm production threshold. One of the main things that we have learned is the near-threshold behavior of $\sigma_{J/\psi-\pi}$. This is quite relevant because in a hadron gas at temperatures of $100 - 300$ MeV most of the $J/\psi-\pi$ interactions occur at relatively low energies, barely sufficient to dissociate the charmonium state. In some calculations a rapid growth of the cross sections with the energy was found [@haglin; @linko; @osl]. This behavior was criticized and considered to be incompatible with empirical information extracted from $J/\psi$ photoproduction [@zinovjev]. This criticism, however, made use of the vector meson dominance hypothesis (VDM), which, in the case of charm, is rather questionable [@hufkop]. The introduction of form factors in the effective Lagrangian approach, while reducing the order of magnitude of the cross section, did not change this fast growing trend around the threshold. Later, again in the context of meson exchange models, it was established [@nnr] that the correct implementation of chiral symmetry prevents the cross section from rising steeply around the threshold. In a different approach, with QCD sum rules (QCDSR) [@nnmk], the behavior found in [@nnr] was confirmed and in the present work, with improved QCDSR, we confirm again the smooth threshold behavior. We, thus, believe that this question has been answered. Another, phenomenologically less important, but conceptually interesting issue is the energy dependence in the region far from threshold. Results obtained with the non-relativistic quark model [@wongs] indicated a rapidly falling cross section. This behavior is due to the gaussian tail of the quark wave functions used in the quark exchange model. This result of the quark model approach could be mimicked within chiral meson Lagrangian approaches with the introduction of $\sqrt{s}$ dependent form factors [@oslw; @ikbb]. For $J/\psi-N$ interactions, it was found in ref. [@lo] that this behaviour depends ultimately on the gluon distribution in the proton at low $x$. In the case of $J/\psi-N$, for certain parametrizations of the gluon density one could find a falling trend for the cross section [@lo], but no definite conclusion could yet be drawn. If the $J/\psi$ is treated as an ordinary hadron, its cross section for interaction with any other ordinary hadron must increase smoothly at higher energies, in much the same way as the proton-proton or pion-proton cross sections. The underlying reason is the increasing role played by perturbative QCD dynamics and the manifestation of the partonic nature of all hadrons. Among the existing calculations, no one is strictly valid at $\sqrt{s} \simeq 20$ GeV, except the one of ref. [@hdnnr], which is designed to work at very high energies and which gives, for the $J/\psi$-nucleon cross section the value $\sigma_{J/\psi -N} \simeq 5$ mb. This number can be considered as a guide for $J/\psi - \pi$ cross section in the high energy regime. It should, however, be pointed out, that the calculation of ref. [@hdnnr] is based on a purely nonperturbative QCD approach. The inclusion of a perturbative contribution will add to the quoted value and will have a larger weight at higher energies. A similar conclusion was reached in [@gerland]. In the traditional short distance QCD approach the cross section grows monotonically [@bhp; @kha2; @lo]. As a side-product, the theoretical effort to estimate the charmonium-hadron cross section motivated a series of calculations [@nosform; @nos; @bnn], within the framework of QCD sum rules, of form factors and coupling constants involving charmed hadrons, that may be relevant also to other problems in hadron physics. In this work we improve the calculation done in ref. [@nnmk] by considering sum rules based on a three-point function with a pion. We work up to twist-4, which allows us to study the convergence of the OPE expansion. Since the method of the QCDSR uses QCD explicitly, we believe that our work brings a significant progress to this important topic. The paper is organized as follows: in the next section we review the method of QCD sum rules, giving special emphasis to the QCD side. In section III we present some formulas for the computation of open charm production amplitudes and in section IV we give our numerical results. Finally some concluding remarks are given in section V. The Method ========== In the QCDSR approach [@svz; @rry], the short range perturbative QCD is extended by an OPE expansion of the correlator, giving a series in inverse powers of the squared momentum with Wilson coefficients. The convergence at low momentum is improved by using a Borel transform. The coefficients involve universal quark and gluon condensates. The quark-based calculation of a given correlator is equated to the same correlator, calculated using hadronic degress of freedom via a dispersion relation, giving sum rules from which a hadronic quantity can be estimated. Let us start with the general vaccum-pion correlation function: \_[34]{} = d\^4x d\^4y e\^[-ip\_2.y]{} e\^[ip\_3.x]{}  0|T{j\_3(x)j\_4(0)j\_\^(y)}|(p\_1), \[cor\] with the currents given by $j_\mu^\psi=\overline{c} \gamma_\mu c$, $j_3=\overline{u} \Gamma_3 c $ and $j_4=\overline{c} \Gamma_4d$. $p_1$, $p_2$, $p_3$ and $p_4$ are the four-momenta of the mesons $\pi$, $J/\psi$, $M_3$ and $M_4$ respectively, and $\Gamma_3$ and $\Gamma_4$ denote specific gamma matrices corresponding to the process envolving the mesons $M_3$ and $M_4$. For instance, for the process $J/\psi~\pi\rightarrow \bar{D}~D^*$ we will have $\Gamma_3=\gamma_\nu$ and $\Gamma_4=i\gamma_5$. The advantage of this approach as compared with the 4-point calculation in ref. [@nnmk], is that we can consider more terms in the OPE expansion of the correlation function in Eq. (\[cor\]) and, therefore, check the “convergence” of the OPE expansion. Following ref. [@kl], we can rewrite Eq. (\[cor\]) as: \_[34]{}=- Tr\[S\_[ac]{}(p\_3-k)\_ S\_[cb]{}(p\_3-p\_2-k)\_4 D\_[ab]{}(k,p\_1)\_3\], \[corqq\] where S\_[ab]{}(p)=i[+m\_cp\^2-m\_c\^2]{}\_[ab]{} is the free $c$-quark propagator, and $D_{ab}(k,p)$ denotes the quark-antiquark component with a pion, which can be separated into three pieces depending on the Dirac matrices involved [@kl; @klo]: D\_[ab]{} (k,p) = \_[ab]{}  , with $a,\,b$ and $c$ being color indices. The three invariant functions of $k,p$ are defined by A(k,p)=[112]{}d\^4x e\^[ik.x]{}0||[d]{}(x)i\_5 u(0) |(p),\ B\^(k,p)=[112]{}d\^4x e\^[ik.x]{}0||[d]{}(x)\^\_5 u(0)|(p),\ C\^ (k,p)=-[124]{}d\^4x e\^[ik.x]{}0||[d]{}(x) \^\_5 u(0)|(p). Using the soft-pion theorem, PCAC and working at the order ${\cal O}(p_{\mu} p_{\nu})$ we get up to twist-4 [@kl; @bbk]: A(k,p)&=&[(2)\^412]{}[ q f\_]{}\^[(4)]{}(k),\ B\_(k,p)&=&[(2)\^412]{}f\_\^[(4)]{}(k),\ C\_(k,p)&=&-[(2)\^424]{}[ q 3f\_]{}(p\_g\_[\_1]{}-p\_g\_[\_1]{}) \^[(4)]{}(k), \[ABC\] where $\delta^2$ is defined by the matrix element $\langle 0| {\bar d} g_s {\tilde {\cal G}}^{\alpha\beta} \gamma_\beta u | \pi(p) \rangle = i \delta^2 f_\pi p^\alpha$, where ${\tilde {\cal G}}_{\alpha\beta}=\epsilon_{\alpha\beta\sigma\tau} {\cal G}^{\sigma \tau}/2$ and ${\cal G}_{\alpha \beta} = t^A G_{\alpha \beta}$. The additional contributions to the OPE comes from the diagrams where one gluon, emitted from the $c$-quark propagator, is combined with the quark-antiquark component. Specifically, the $c$-quark propagator with one gluon being attached is given by [@rry] $$\begin{aligned} -{g_s {\cal G}_{\alpha \beta} \over 2 (k^2 -m^2_c)^2 } \left [ k_\alpha \gamma_\beta - k_\beta \gamma_\alpha + ({\ooalign{\hfil/\hfil\crcr$k$}} + m_c) i \sigma_{\alpha \beta} \right ]\ ,\end{aligned}$$ Taking the gluon stress tensor into the quark-antiquark component, one can write down the correlation function into the form $$\begin{aligned} \Pi_{\mu34} = -4 \int {d^4 k \over (2\pi)^4} Tr \bigg[ \bigg(S_{\alpha\beta}(p_3-k)\gamma_\mu S(p_3-p_2-k) \nonumber \\ +S(p_3-k)\gamma_\mu S_{\alpha\beta}(p_3-p_2-k)\bigg)\Gamma_4 D^{\alpha \beta}(k,p_1)\Gamma_3\bigg]\;, \label{corqqg}\end{aligned}$$ where we have already contracted the color indices, and where we have defined S\_(k)=-[12 (k\^2 -m\^2\_c)\^2 ]{}  , and D\^(k,p)= \_5 \_ E\^(k,p) +\^\^ F\_(k,p), with $$\begin{aligned} E^{\rho\lambda\alpha \beta}(k,p) &=& -{1\over32}\int d^4x~e^{ik.x} \langle 0 | {\bar d}(x)\gamma_5\sigma^{\rho\lambda} g_s {G}^ {\alpha\beta} u| \pi (p) \rangle \,, \nonumber \\ F_{\tau\theta\delta}(k,p) &=& {1 \over 32} \int d^4x~e^{ik.x} \langle 0 | {\bar d}(x) \gamma_\tau g_s {\tilde G}_{\theta\delta}u | \pi (p) \rangle \,.\end{aligned}$$ Up to twist-4 and at order ${\cal O} (p_\mu p_\nu)$, the two functions appearing above are given by [@kl; @bbk] $$\begin{aligned} E^{\rho\lambda\alpha\beta} &=& {i\over32} f_{3\pi} \left(p^\alpha p^\rho g^{\lambda\beta} -p^\beta p^\rho g^{\lambda\alpha}-p^\alpha p^\lambda g^{\rho\beta} +p^\beta p^\lambda g^{\rho\alpha}\right) (2\pi)^4\delta^{(4)}(k) \nonumber\\ F_{\tau\theta\delta} &=& -{i \delta^2 f_\pi \over 3 \times 32} (p_\theta g_{\tau\delta} - p_\delta g_{\tau \theta}) (2\pi)^4 \delta^{(4)}(k) \ , \label{EF}\end{aligned}$$ where $f_{3\pi}$ is defined by the vacuum-pion matrix element $\langle 0| {\bar d} g_s \sigma_{\alpha\beta}\gamma_5{\tilde {\cal G}}^{\alpha \beta}u | \pi(p) \rangle$ [@bbk]. The phenomenological side of the correlation function, $\Pi_{\mu34}$, is obtained by the consideration of $J/\psi$, $M_3$ and $M_4$ state contribution to the matrix element in Eq. (\[cor\]). The hadronic amplitudes are defined by the matrix element: i&=&(p\_2,)| M\_3(-p\_3,) M\_4(-p\_4,) (p\_1)\ &=&i \_[34]{}(p\_1,p\_2,p\_3,p\_4) \_2\^ f\_3\^[\*]{}f\_4\^[\*]{}, where $f_i^{*\alpha}=\epsilon_i^{*\alpha}$ for the $D^*$ meson and $f_i^ {*\alpha}=1$ for the $D$ meson. The phenomenological side of the sum rule can be written as (for the part of the hadronic amplitude that will contribute to the cross section) [@nnmk]: \_[34]{}\^[phen]{}=-[f\_\_3\_4  \_[34]{}(p\_2\^2-\^2)(p\_3\^2-m\_3\^2)(p\_4\^2-m\_4\^2)]{} + , \[phendds\] where h. r. means higher resonances, and where $\lambda_i$ is related with the corresponding meson decay constant: $\langle D|j_D|0\rangle=-\lambda_D={\md^2 f_D/ m_c}$ and $\langle 0|j_\alpha|D^*\rangle=\lambda_{D^*}\epsilon_\alpha= \mds f_{D^*}\epsilon_\alpha$. Hadronic Amplitudes for $J/\psi~\pi\rightarrow$ Open Charm ========================================================== The hadronic amplitudes can be written in terms of many different structures. In terms of the structures that will contribute to the cross section we can write - for the process $J/\psi~\pi\rightarrow \bar{D}~D^*$: \_=\_1\^[DD\^\*]{}p\_[1]{}p\_[1]{} + \_2\^[DD\^\*]{} p\_[1]{}p\_[2]{} +\_3\^[DD\^\*]{}p\_[1]{}p\_[3]{} + \_4\^[DD\^\*]{} g\_+ \_5\^[DD\^\*]{}p\_[2]{}p\_[3]{}, \[strudds\] - for the process $J/\psi~\pi\rightarrow \bar{D}~D$: \_=\_[DD]{} \_p\_[1]{}\^ p\_3\^p\_4\^, \[strudd\] - for the process $J/\psi~\pi\rightarrow \bar{D}^*~D^*$: &&\_= \_1\^[D\^\*D\^\*]{} H\_+ \_2\^[D\^\*D\^\*]{} J\_ +\_3\^[D\^\*D\^\*]{}g\_\_p\_[1]{}\^ p\_2\^p\_3\^+ \_4\^[D\^\*D\^\*]{}\_ p\_[3]{} p\_[1]{}\^p\_3\^\ &+& \_5\^[D\^\*D\^\*]{} \_p\_[3]{} p\_[1]{}\^p\_2\^ +\_6\^[D\^\*D\^\*]{}\_p\_[3]{}p\_[1]{}\^ p\_2\^ + \_7\^[D\^\*D\^\*]{}\_ p\_[1]{} p\_[1]{}\^ p\_2\^\ &+& \_8\^[D\^\*D\^\*]{}\_p\_[1]{} p\_[1]{}\^p\_4\^ +\_9\^[D\^\*D\^\*]{}\_p\_[1]{}\^ + \_[10]{}\^[D\^\*D\^\*]{}\_p\_[1]{} p\_[1]{}\^p\_3\^ +\_[11]{}\^[D\^\*D\^\*]{}\_ p\_[2]{}\^\ &+&\_[12]{}\^[D\^\*D\^\*]{}\_ p\_[1]{} p\_[1]{}\^p\_3\^+ \_[13]{}\^[D\^\*D\^\*]{} \_p\_[3]{} p\_[1]{}\^p\_3\^. \[strudsds\] with $H_{\mu\nu\rho}=(\epsilon_{\nu\alpha\beta\gamma}g_{\mu\rho}-\epsilon_{\rho \alpha\beta\gamma}g_{\mu\nu})p_{1}^\alpha p_2^\beta p_3^\gamma + \epsilon_{\mu\rho\alpha\beta}p_{2\nu} p_1^\alpha p_2^\beta$ and $J_{\mu\nu\rho}=(\epsilon_{\nu\rho\alpha\beta}p_{1\mu}+\epsilon_{\mu\rho \alpha\beta}p_{1\nu} +\epsilon_{\mu\nu\alpha\beta}p_{1\rho})p_{2}^\alpha p_3^\beta + \epsilon_{\mu\nu\alpha\beta}p_{2\rho}p_{1}^\alpha p_3^\beta$. In Eqs. (\[strudds\]), (\[strudd\]) and (\[strudsds\]), $\Lambda_i$ are the parameters that we will evaluate from the sum rules. In principle all the independent structures appearing in $H_{\mu\nu\rho}$ and $J_{\mu\nu\rho}$ would have independent parameters $\Lambda_i$. However, since in our approach we get exactly the same sum rules for all of them, we decided to group them with the same parameters. Inserting the results in Eqs. (\[ABC\]) and (\[EF\]) into Eqs. (\[corqq\]) and (\[corqqg\]) we can write a sum rule for each of the structures appearing in Eqs. (\[strudds\]), (\[strudd\]) and (\[strudsds\]). To improve the matching between the phenomenological and theoretical sides we follow the usual procedure and make a single Borel transformation to all the external momenta taken to be equal: $-p_2^2=-p_3^2=-p_4^2=P^2\rightarrow M^2$. The problem of doing a single Borel transformation is the fact that terms associated with the pole-continuum transitions are not suppressed [@io2]. In the present case we have two kinds of these transitions: double pole-continuum and single pole-continuum. In the limit of similar meson masses it is easy to show that the Borel behaviour of the three-pole, double pole-continuum and single pole-continuum contributions are $e^{-m_M^2/M^2}/M^4,\,e^{-m_M^2/M^2}/M^2$ and $e^{-m_M^2/M^2}$ respectively. Therefore, we can single out the three-pole contribution from the others by introducing two parameters in the phenomenological side of the sum rule, which will account for the double pole-continuum and single pole-continuum contributions. The expressions for all 19 sum rules are given in the Appendices A, B and C. Results and Discussion ====================== The parameter values used in all calculations are $m_c=1.37\,\GeV$, $m_\pi=140\,\MeV$, $m_D=1.87\,\GeV$, $m_{D^*}=2.01\, \GeV$, $\mpsi=3.097\,\GeV$, $f_\pi=131.5\,\MeV$, $\langle\overline{q}q\rangle=-(0.23)^3\,\GeV^3$, $m_0^2=0.8\,\GeV^2$, $\delta^2=0.2\,\GeV^2$, $f_{3\pi}=0.0035\,\GeV^2$ [@bbk]. For the charmed mesons decay constants we use the values from [@bbk] for $f_D$ and $f_{D^*}$ and the experimental value for $f_\psi$: f\_=270, f\_D=170,f\_[D\^\*]{}=240. \[num\] In ref. [@dlnn] we have analyzed the sum rule for the process $J/\psi~\pi\rightarrow \bar{D}~D$. Here we choose to show the sum rule for $\Lambda_1^{D*D*}$ in Eq. (\[c1\]), as an example of the sum rules for the process $J/\psi~\pi\rightarrow \bar{D}^*~D^*$. In Fig. 1 we show the QCD sum rule results for $\Lambda_1^{D^*D^*}+A_1^{D^*D^*}M^2+B_1^{D^*D^*}M^4$ as a function of $M^2$. The dots, squares and diamonds give the twist-2, 3 and 4 contributions respectively. The triangles give the final QCDSR results. We see that the twist-3 and 4 contributions are small as compared with the twist-2 contribution, following the same behaviour as the sum rule for the process $J/\psi~\pi\rightarrow \bar{D}~D$ given in [@dlnn]. In general all the other sum rules are similar and contain twist-2, twist-3 and twist-4 contributions corresponding to the first, second, and third terms inside the brackets in the right hand side of Eq. (\[c1\]). Only the sum rules for $\Lambda_{10}^{D^*D^*}$ up to $\Lambda_{13}^{D^*D^*}$, $\Lambda_{4}^{DD^*}$ and $\Lambda_{5}^{DD^*}$ do not get the leading twist contribution, and will be neglected in the evaluation of the cross section. It is also interesting to notice that if we consider only the leading twist contributions we recover the sum rules obtained in ref. [@nnmk]. The triangles in Fig. 1 follow almost a straight line in the Borel region $6\leq M^2\leq16\,\GeV^2$. This show that the single pole-continuum transitions contribution is small. The value of the amplitude $\Lambda_1^{D^*D^*}$ is obtained by the extrapolation of the fit to $M^2=0$ [@bnn; @nos; @io2]. Fitting the QCD sum rule results to a quadratic form we get \_1\^[D\^\*D\^\*]{}10.5\^[-3]{}. \[ampli\] Since we worked in the soft pion limit, $\Lambda_1^{D^*D^*}$, as well as all other $\Lambda$, is just a number. All particle momenta dependence of the amplitudes is contained in the Dirac structure. In obtaining the results shown in Fig. 1 we have used the numerical values for the meson decay constants given in Eq. (\[num\]). However, it is also possible to use the respective sum rules, as done in [@nnmk]. The two-point sum rules for the meson decay constants are given in the appendix D. The behaviour of the results for the hadronic amplitudes does not change significantly if we use the two-point sum rules for the meson decay constants instead of the numerical values, leading only to a small change in the value of the amplitudes. In Fig. 2 we show, for a comparison, both results in the case of $\Lambda_1^{D^*D^*}$. Using the respective sum rules for the meson decay constants we get \_1\^[D\^\*D\^\*]{}13.9\^[-3]{}. \[ampli2\] We will use these two procedures to estimate the errors in our calculation. It is important to mention that our results agree completely with the value obtained in [@nnmk]. The results for all other sum rules show a similar behaviour and the amplitudes can be extracted by the extrapolation of the fit to $M^2=0$. The QCDSR results, evaluated using the numerical values for the meson decay constants, as well as the quadratic fits for the amplitudes associated with the process $J/\psi~\pi\rightarrow\bar{D}~D^*$ are shown in Fig. 3. The values for the parameters associated with the process $J/\psi~\pi \rightarrow\bar{D}~D^*$ are given in Table I. $\Lambda_1^{DD^*}$ $\Lambda_2^{DD^*}$ $\Lambda_3^{DD^*}$ $\Lambda_4^{DD^*}$ $\Lambda_5^{DD^*}$ -------------------- ------------------------- --------------------- -------------------- ------------------------- $14\pm2 \GeV^{-2}$ $-7.2\pm0.9$ GeV$^{-2}$ $-58\pm8 \GeV^{-2}$ $14.6\pm2.2$ $-15.6\pm2.2 \GeV^{-2}$ : \[tab:table1\]The best fitted values for the parameters associated with the process $J/\psi~\pi\rightarrow\bar{D}~D^*$. For the process $J/\psi~\pi\rightarrow\bar{D}~D$ we have only one parameter which is given by [@dlnn] \_[DD]{}=13.21.8\^[-3]{}, \[amplidd\] and the 13 parameters associated with the process $J/\psi~\pi \rightarrow\bar{D}^*~D^*$ are given in Table II. $\Lambda_1^{D^*D^*}$ $\Lambda_2^{D^*D^*}$ $\Lambda_3^{D^*D^*}$ $\Lambda_4^{D^*D^*}$ $\Lambda_5^{D^*D^*}$ ------------------------------------ ----------------------------- ------------------------- ------------------------- ------------------------- $12.2\pm1.7 \GeV^{-3}$ $-12.8\pm1.8$ GeV$^{-3}$ $12.5\pm1.7 \GeV^{-3}$ $-24.6\pm3.4 \GeV^{-3}$ $9.8\pm1.6 \GeV^{-3}$ $\Lambda_6^{D^*D^*}$ $\Lambda_7^{D^*D^*}$ $\Lambda_8^{D^*D^*}$ $\Lambda_9^{D^*D^*}$ $\Lambda_{10}^{D^*D^*}$ $9.7\pm1.6 \GeV^{-3}$ $-13.0\pm1.8$ GeV$^{-3}$ $-13.8\pm1.8 \GeV^{-3}$ $-5.4\pm0.9 \GeV^{-1}$ $2.5\pm0.2 \GeV^{-3}$ $\Lambda_{11}^{D^*D^*}(\GeV^{-1})$ $\Lambda_{12}^{D^*D^*}$ $\Lambda_{13}^{D^*D^*}$ $(-5.5\pm0.5)10^{-3}$ $-0.022\pm0.002$ GeV$^{-3}$ $0.53\pm0.03 \GeV^{-3}$ : \[tab:table2\]The best fitted values for the parameters associated with the process $J/\psi~\pi\rightarrow\bar{D}^*~D^*$. The errors in all parameters were estimated by the evaluation of the sum rules using the numerical values and the two-point QCDSR for the meson decay constants. Having the QCD sum rule results for the amplitudes of the three processes $J/\psi~\pi\rightarrow \bar{D}~D^*,~\bar{D}~D,~\bar{D}^*~D^*$, given in Eqs. (\[strudds\]), (\[strudd\]) and (\[strudsds\]) we can evaluate the cross sections. After including isospin factors, the differential cross section for the $J/\psi-\pi$ dissociation is given by =[196s[**p**]{}\_[i,cm]{}\^2]{} \_[spin]{}|[M]{}|\^2 , \[sig\] where ${\bf p}_{i,cm}$ is the three-momentum of $p_1$ (or $p_2$) in the center of mass frame (with $p_1~(p_2)$ being the four-momentum of the $\pi~(J/\psi)$): \_[i,cm]{}\^2=[(s,\^2,\^2)4s]{} , with $\lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz$, $s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$. In Eq. (\[sig\]), the sum over the spins of the amplitude squared is given by \_[spin]{}|[M]{}|\^2 =  [M]{}\_[M]{}\_[’’]{}\^\*(g\^[’]{}-[p\_2\^p\_2\^[’]{}\^2]{})(g\^[’]{}-[p\_3\^p\_3\^[’]{}\^2]{}), \[m2ds\] for $J/\psi~\pi\rightarrow \bar{D}~D^*$, with $p_3~(p_4)$ being the four-momentum of $D^*~(D)$. \_[spin]{}|[M]{}|\^2 =  [M]{}\_[M]{}\_[’]{}\^\*(g\^[’]{}-[p\_2\^p\_2\^[’]{}\^2]{}), \[m2dd\] for $J/\psi~\pi\rightarrow \bar{D}~D$, and \_[spin]{}|[M]{}|\^2 =  [M]{}\_[M]{}\_[’’’]{}\^\*(g\^[’]{}-[p\_2\^p\_2\^[’]{}\^2]{})(g\^[’]{}-[p\_3\^p\_3\^[’]{}\^2]{})(g\^[’]{}-[p\_4\^p\_4\^[’]{}\^2]{}), \[m2ss\] for $J/\psi~\pi\rightarrow \bar{D}^*~D^*$. The structures multiplying $\Lambda_4^{DD^*}$ and $\Lambda_5^{DD^*}$ in Eq. (\[strudds\]), and $\Lambda_{11}^{D^*D^*}$ in Eq. (\[strudsds\]) break chiral symmetry [@nnr]. To evaluate the effect of breaking chiral symmetry in the process $J/\psi~\pi\rightarrow \bar{D}~D^* +\bar{D}^*~D$ we show, in Fig. 4, the cross section calculated using all structures in Eq. (\[strudds\]) (dashed line) and neglecting $\Lambda_4^{DD^*}$ and $\Lambda_5^{DD^*}$ (solid line). From Fig. 4 we see that the cross section obtained with the amplitude that breaks chiral symmetry grows very fast near the threshold. Since this is the energy region where this kind of process is probable more likely to happen, it is very important to use models that respect chiral symmetry when evaluating the $J/\psi-\pi$ cross section. As mentioned before, the sum rules for $\Lambda_4^{DD^*}$, $\Lambda_5^{DD^*}$ and $\Lambda_{10}^{D^*D^*}$ up to $\Lambda_{13}^{D^*D^*}$, do not get the leading order contribution and will be neglected when evaluating the cross sections. It is important to keep in mind that, since our sum rules were derived in the limit $p_{1}\rightarrow0$, we can not extend our results to large values of $\sqrt{s}$. For this reason we will limit our calculation to $\sqrt{s}\leq4.5 \GeV$. In Fig. 5 we show separately the contributions for each one of the process. Our first conclusion is that our results show that, for values of $\sqrt{s}$ far from the $J/\psi~\pi\rightarrow \bar{D}^{*}~{D}^*$ threshold, $\sigma_{J/\psi\pi\rightarrow \bar{D}^{*}{D}^*} \, \geq \, \sigma_{J/\psi\pi\rightarrow \bar{D}{D}^*+D\bar{D}^*} \, \geq \, \sigma_{J/\psi\pi\rightarrow \bar{D}{D}}$, in agreement with the model calculations presented in [@osl] but in disagreement with the results obtained with the nonrelativistic quark model of [@wongs], which show that the state $\bar{D}^*D$ has a larger production cross section than $\bar{D}^{*}{D}^*$. Furthermore, our curves indicate that the cross section grows monotonically with the c.m.s. energy but not as fast, near the thresholds, as it does in the calculations in Refs. [@osl; @haglin; @linko; @ikbb; @nnr]. Again, this behavior is in opposition to [@wongs], where a peak just after the threshold followed by continuous decrease in the cross section was found. In Fig. 6 we show, for comparison, our result for the total cross section for the $J/\psi~\pi$ dissociation (solid lines) and the results from meson exchange model [@osl] obtained with a cut-off $\Lambda=1\GeV$ (dot-dashed line), quark exchange model [@wongs] (dashed line) and short distance QCD [@kha2; @lo] (dotted line). The shaded area in our results give an evaluation of the uncertainties in our calculation obtained with the two procedures described above. It is very interesting to notice that bellow the $DD^*$ threshold, our result and the results from meson exchange and quark exchange models are in a very good agreement. However, as soon as the $DD^*$ channel is open the cross section obtained with the meson exchange and quark exchange models show a very fast grown, as a function of $\sqrt{s}$, as compared with our result. As discussed above, this is due to the fact that chiral symmetry is broken in these two model calculations. The momentum distribution of thermal pions in a hadron gas depends on the effective temperature $T$ with an approximate Bose-Einsten distribution. Therefore, in a hadron gas, pions collide with the $J/\psi$ at different energies, and the relevant quantity is not the value of the cross section at a given energy, but the thermal average of the cross section. The thermal average of the cross section is defined by the product of the dissociation cross section and the relative velocity of initial state particles, averaged over the energies of the pions: $\langle\sigma^{\pi J/\psi} v\rangle$, and is given by [@linko] \^[J/]{} v=[\_[z\_0]{}\^dz\[z\^2-(\_1+ \_2)\^2\]\[z\^2-(\_1-\_2)\^2\]K\_1(z)\^[J/]{}(s=z\^2T\^2)4\_1\^2K\_2(\_1)\_2\^2K\_2(\_2)]{}, where $\alpha_i=m_i/T~(i=1$ to $4)$, $z_0=$ max$(\alpha_1+\alpha_2,\alpha_3+ \alpha_4)$ and $K_i$ is the modified Bessel function. As shown in Fig. 7, $\langle\sigma^{\pi J/\psi} v\rangle$ increases with the temperature. Since the $J/\psi$ dissociation by a pion requires energetic pions to overcome the energy threshold, it has a small thermal average at low temperatures. The magnitude of our thermal average cross section is of the same order as the meson exchange model calculation in ref. [@linko] with a cut-off $\Lambda=1\GeV$. The shaded area in Fig. 7 give an evaluation of the uncertainties in our calculation due to the two procedures used to extract the hadronic amplitudes. Summary and Conclusions ======================= We have evaluated the hadronic amplitudes for the $J/\psi$ dissociation by pions using the QCD sum rules based on a three-point function using vaccum-pion correlation functions. We have considered the OPE expansion up to twist-4 and we have worked in the soft-pion limit. Our work improves the former QCDSR calculation, done with a four-point function at the pion pole [@nnmk], since we have included more terms in the OPE expansion. We have shown that the twist-3 and twist-4 contributions to the sum rules are small when compared with the leading order contribution, showing a good “convergence” of the OPE expansion. We have checked that, taking the appropriated limit, we recover the previous result of [@nnmk]. From the theoretical point of view, the use of QCDSR in this problem was responsible for real progress, being a step beyond models and beyond the previous leading twist calculations [@bhp; @kha2; @lo; @arleo; @nnmk]. This is specially true in the low energy region, close to the open charm production threshold. At higher energies our treatment is less reliable due to the approximations employed. Although a more sophisticated analysis of our uncertainties is still to be done, the shaded area in Fig. 6 shows that we can make some unambiguous statement concerning the behavior of $\sigma_{\pi J/\psi}$ with the energy $\sqrt{s}$. Our cross section grows monotonically with the c.m.s. energy but not as fast, near the thresholds, as it does in the calculations using meson exchange models [@osl; @haglin; @linko; @ikbb; @nnr]. We have also shown the importance of respecting chiral symmetry, since the increase of the cross section near the threshold is strongly intensified when chiral symmetry is broken. In other words, our results suggest that, [*using meson exchange models is perfectly acceptable, provided that they include form factors and that they respect chiral symmetry*]{}. With these precautions, they can be a good tool to make predictions at somewhat higher energies. We have also evaluated the thermal average of the $J/\psi-\pi$ dissociation cross section. It increases with the temperature and at $T=150\,\MeV$ we get $\langle\sigma^{\pi J/\psi} v\rangle\sim0.2-0.4$ mb which is compatible with the values presented in Fig. 5 of ref. [@linko], i.e., in a meson exchange model with monopole form factor with cut-off $\Lambda = 1 $ GeV. The use of this information will reduce the uncertainties in the calculations of the hadronic lifetimes of $J/\psi$, which are needed in simulations like those of ref. [@rapp]. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to J. Hüfner for fruitful discussions. We thank Yongseok Oh for providing the data to construct fig. 6. M.N. would like to thank the hospitality and financial support from the Yonsei University during her stay in Korea. S.H.L and H.K. were supported by Korean Research Foundation Grant (KRF-2002-015-CP0074). This work was supported by CNPq and FAPESP-Brazil. Sum rules for the process $J/\psi~\pi\rightarrow \bar{D}~D^*$ ================================================= Using $\Gamma_3=\gamma_\nu$ and $\Gamma_4=i\gamma_5$ in Eqs. (\[corqq\]) and (\[corqqg\]), we obtain the following sum rules for the structures in Eq. (\[strudds\]): \_1\^[DD\^\*]{}+A\_1\^[DD\^\*]{}M\^2+B\_1\^[DD\^\*]{}M\^4&=&[1 C\_[DD\^\*]{} f\_[DD\^\*]{}(M\^2)]{} , \_2\^[DD\^\*]{}+A\_2\^[DD\^\*]{}M\^2+B\_2\^[DD\^\*]{}M\^4&=&[1 C\_[DD\^\*]{} f\_[DD\^\*]{}(M\^2)]{} , \_3\^[DD\^\*]{}+A\_3\^[DD\^\*]{}M\^2+B\_3\^[DD\^\*]{}M\^4&=&[1 C\_[DD\^\*]{} f\_[DD\^\*]{}(M\^2)]{} , \_4\^[DD\^\*]{}+A\_4\^[DD\^\*]{}M\^2+B\_4\^[DD\^\*]{}M\^4&=&[m\_c\^2 C\_[DD\^\*]{} f\_[DD\^\*]{} (M\^2)]{} [|[q]{}qf\_]{}[e\^[-m\_c\^2/M\^2]{}M\^2]{}, \_5\^[DD\^\*]{}+A\_5\^[DD\^\*]{}M\^2+B\_5\^[DD\^\*]{}M\^4&=&[-2 C\_[DD\^\*]{} f\_[DD\^\*]{}(M\^2)]{} [|[q]{}qf\_]{}[e\^[-m\_c\^2/M\^2]{}M\^2]{}, where f\_[DD\^\*]{}(M\^2)= [1\^2-\^2]{}, and C\_[DD\^\*]{}=[m\_c\^2f\_Df\_[D\^\*]{}f\_]{}. $A_i^{DD^*}$ and $B_i^{DD^*}$ are the parameters introduced to account for double pole-continuum and single pole-continuum transitions respectively. Sum rules for the process $J/\psi~\pi\rightarrow \bar{D}~D$ =========================================================== Using $\Gamma_3=i\gamma_5$ and $\Gamma_4=i\gamma_5$ in Eqs. (\[corqq\]) and (\[corqqg\]), we obtain the following sum rule for the structure in Eq. (\[strudd\]) [@dlnn]:\ = [m\_c\^2\^4f\_[D]{}\^2f\_]{} [e\^[-m\_c\^2/M\^2]{}M\^2]{} . \[sr\] Sum rules for the process $J/\psi~\pi\rightarrow \bar{D}^*~D^*$ ================================================= Using $\Gamma_3=\gamma_\nu$ and $\Gamma_4=\gamma_\rho$ in Eqs. (\[corqq\]) and (\[corqqg\]), we obtain the following sum rules for the structures in Eq. (\[strudsds\]): \_1\^[D\^\*D\^\*]{}+A\_1\^[D\^\*D\^\*]{}M\^2+B\_1\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \[c1\] \_2\^[D\^\*D\^\*]{}+A\_2\^[D\^\*D\^\*]{}M\^2+B\_2\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_3\^[D\^\*D\^\*]{}+A\_3\^[D\^\*D\^\*]{}M\^2+B\_3\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_4\^[D\^\*D\^\*]{}+A\_4\^[D\^\*D\^\*]{}M\^2+B\_4\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_5\^[D\^\*D\^\*]{}+A\_5\^[D\^\*D\^\*]{}M\^2+B\_5\^[D\^\*D\^\*]{}M\^4=[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_6\^[D\^\*D\^\*]{}+A\_6\^[D\^\*D\^\*]{}M\^2+B\_6\^[D\^\*D\^\*]{}M\^4=[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_7\^[D\^\*D\^\*]{}+A\_7\^[D\^\*D\^\*]{}M\^2+B\_7\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_8\^[D\^\*D\^\*]{}+A\_8\^[D\^\*D\^\*]{}M\^2+B\_8\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_9\^[D\^\*D\^\*]{}+A\_9\^[D\^\*D\^\*]{}M\^2+B\_9\^[D\^\*D\^\*]{}M\^4&=&[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_[10]{}\^[D\^\*D\^\*]{}+A\_[10]{}\^[D\^\*D\^\*]{}M\^2+B\_[10]{}\^[D\^\*D\^\*]{}M\^4=[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_[11]{}\^[D\^\*D\^\*]{}+A\_[11]{}\^[D\^\*D\^\*]{}M\^2+B\_[11]{}\^[D\^\*D\^\*]{}M\^4&=&[2C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} [m\_c|[q]{}qf\_]{}[e\^[-m\_c\^2/M\^2]{}M\^2]{}, \_[12]{}\^[D\^\*D\^\*]{}+A\_[12]{}\^[D\^\*D\^\*]{}M\^2+B\_[12]{}\^[D\^\*D\^\*]{}M\^4=[1C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{} , \_[13]{}\^[D\^\*D\^\*]{}+A\_[13]{}\^[D\^\*D\^\*]{}M\^2+B\_[13]{}\^[D\^\*D\^\*]{}M\^4=-[2C\_[D\^\*D\^\*]{} f\_[D\^\*D\^\*]{}(M\^2)]{}[f\_\^23]{} [e\^[-m\_c\^2/M\^2]{}M\^4]{}, where f\_[D\^\*D\^\*]{}(M\^2)= [1\^2-\^2]{}and C\_[D\^\*D\^\*]{}=[1\^2f\_[D\^\*]{}\^2f\_]{}. Sum rules for the meson decay constants ======================================= For consistency we use in our analysis the QCDSR expressions for the decay constants of the $J/\psi,~D^*$ and $D$ mesons up to dimension four in lowest order of $\alpha_s$: &&f\_D\^2 = [3m\_c\^28\^2m\_D\^4]{}\_[m\_c\^2]{}\^[s\_D]{}ds  [(s-m\_c\^2)\^2s]{}e\^[(m\_D\^2-s)/M\^2]{}\ &&- [m\_c\^3m\_D\^4]{} |[q]{}qe\^[(m\_D\^2-m\_c\^2)/M\^2]{} ,\[fhr\]\ &&f\_[D\^\*]{}\^2 = [18\^2\^2]{}\_[m\_c\^2]{}\^[s\_[D\^\*]{}]{}ds  -[ m\_c\^2]{}|[q]{}qe\^[(\^2-m\_c\^2)/M\^2]{}, \[fhs\]\ &&f\_\^2 = [14\^2]{}\_[4m\_c\^2]{}\^[s\_]{}ds [(s+2m\_c\^2) s\^[3/2]{}]{}e\^[(\^2-s)/M\^2]{},\[fpsi\] where $s_M$ stands for the continuum threshold of the meson $M$, which we parametrize as $s_M=(m_{M}+\Delta_s)^2$. The values of $s_M$ are, in general, extracted from the two-point function sum rules for $f_D$ and $f_{D^*}$ and $f_\psi$ in Eqs. (\[fhr\]), (\[fhs\]) and (\[fpsi\]). Using the Borel region $3 \leq M_M^2\leq 6 \GeV^2$ for the $D^*$ and $D$ mesons and $6 \leq M_M^2 \leq 12 \GeV^2$ for the $J/\psi$, we found good stability for $f_D$, $f_{D*}$ and $f_\psi$ with $\Delta_s\sim0.6\GeV$. We obtained $f_D=160\pm5\MeV$, $f_{D^*}=220\pm10\MeV$ and $f_\psi=280\pm10\MeV$, which are compatible with the numerical values in Eq. (\[num\]). T. Matsui and H. Satz, Phys. Lett. [**B178**]{}, 416 (1986). R. L. Thews, hep-ph/0206179; R. L. Thews and J. Rafelski, Nucl. Phys. [**A698**]{}, 575 (2002); R. L. Thews, M. Schroedter, J. Rafelski Phys. Rev. [**C63**]{}, 054905 (2001). L. Grandchamp and R. Rapp, hep-ph/0209141; Nucl. Phys. [**A709**]{}, 415 (2002); Phys. Lett. [**B523**]{}, 60 (2001). B.Z. Kopeliovich, A. Polleri and J. Hüfner, Phys. Rev. Lett. [**87**]{}, 112302 (2001), and references therein. F.O. Durães, F.S. Navarra and M. Nielsen, nucl-th/0210043. G. Bhanot and M.E. Peskin, Nucl. Phys. [**B156**]{}, 391 (1979); M.E. Peskin, Nucl. Phys. 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--- abstract: 'Energy-transport equations for the transport of fermions in optical lattices are formally derived from a Boltzmann transport equation with a periodic lattice potential in the diffusive limit. The limit model possesses a formal gradient-flow structure like in the case of the energy-transport equations for semiconductors. At the zeroth-order high temperature limit, the energy-transport equations reduce to the whole-space logarithmic diffusion equation which has some unphysical properties. Therefore, the first-order expansion is derived and analyzed. The existence of weak solutions to the time-discretized system for the particle and energy densities with periodic boundary conditions is proved. The difficulties are the nonstandard degeneracy and the quadratic gradient term. The main tool of the proof is a result on the strong convergence of the gradients of the approximate solutions. Numerical simulations in one space dimension show that the particle density converges to a constant steady state if the initial energy density is sufficiently large, otherwise the particle density converges to a nonconstant steady state.' address: - 'Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany' - 'Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria' author: - Marcel Braukhoff - Ansgar Jüngel title: 'Energy-transport systems for optical lattices: derivation, analysis, simulation' --- [^1] Introduction ============ An optical lattice is a spatially periodic structure that is formed by interfering optical laser beams. The interference produces an optical standing wave that may trap neutral atoms [@Blo05]. The lattice potential mimics the crystal lattice in a solid, while the trapped atoms mimic the valance electrons in a solid state crystal. In contrast to solid materials, it is easily possible to adjust the geometry and depth of the potential of an optical lattice. Another advantage is that the dynamics of the atoms, if cooled down to a few nanokelvin, can be followed on the time scale of milliseconds. Therefore, optical lattices are ideal systems to study physical phenomena that are difficult to observe in solid crystals. Moreover, they are promising candidates to realize quantum information processors [@Jak04] and extremely precise atomic clocks [@ADH15]. The dynamics of ultracold fermionic clouds in an optical lattice can be modeled by the Fermi-Hubbard model with a Hamiltonian that is a result of the lattice potential created by interfering laser beams and short-ranged collisions [@DGH15]. In the semi-classical picture, the interplay between diffusive and ballistic regimes can be described by a Boltzmann transport equation [@GNZ09], which is able to model qualitatively the observed cloud shapes [@SHR12]. In this paper, we investigate moment equations which are formally derived from a Boltzmann equation in the diffusive regime. The motivation of our work is the observation that in the (relative) high-temperature limit, the lowest-order diffusion approximation of the Boltzmann equation leads to a logarithmic diffusion equation [@MRR11] which has nonphysical properties in the whole space (for instance, it loses mass). Our aim is to derive the next-order approximation, leading to energy-transport equations for the particle and energy densities, and to analyze and simulate the resulting system of degenerate parabolic equations under periodic boundary conditions [@MRR11]. The starting point is the (scaled) Boltzmann equation for the distribution function $f(x,p,t)$, $$\label{1.be} \pa_t f + u\cdot\na_x f + \na_x V\cdot\na_p f = Q(f),$$ where $x\in{{\mathbb R}}^d$ is the spatial variable, $p$ is the crystal momentum defined on the $d$-dimensional torus ${{\mathbb T}}^d$ with unit measure, and $t>0$ is the time. The velocity $u$ is defined by $u(p)=\na_p\eps(p)$ with the energy $\eps(p)$, $V(x,t)$ is the potential, and $Q(f)$ is the collision operator. Compared to the semiconductor Boltzmann equation, there are two major differences. First, the band energy $\eps(p)$ is given by the periodic dispersion relation $$\label{1.eps} \eps(p) = -2\eps_0\sum_{i=1}^d \cos(2\pi p_i), \quad p\in{{\mathbb T}}^d.$$ The constant $\eps_0$ is a measure for the tunneling rate of a particle from one lattice site to a neighboring one. In semiconductor physics, usually a parabolic band structure is assumed, $\eps(p)=\frac12|p|^2$ [@Jue09]. This formula also appears in kinetic gas theory as the (microscopic) kinetic energy. The band energy in optical lattice is bounded, while it is unbounded when $\eps(p)=\frac12|p|^2$. This has important consequences regarding the integrability of the equilibrium distribution (see below). Second, the potential $V$ is given by $V=-U_0n$, where $n=\int_{{{\mathbb T}}^d}fdp$ is the particle density and $U_0>0$ models the strength of the on-site interaction between spin-up and spin-down components [@SHR12]. In semiconductor physics, $V$ is the electric potential which is a given function or determined self-consistently from the (scaled) Poisson equation [@Jue09]. The definition $V=-U_0 n$ leads to unexpected “degeneracies” in the moment equations, see the discussion below. The collision operator is given as in [@SHR12] by the relaxation-time approximation $$Q(f) = \frac{1}{\tau}({{\mathcal F}}_f-f),$$ where $\tau>0$ is the relaxation time and ${{\mathcal F}}_f$ is determined by minimizing the free energy for fermions associated to under the constraints of mass and energy conservation (see Section \[sec.deriv\] for details), leading to $${{\mathcal F}}_f(x,p,t) = \big(\eta + \exp(-\lambda_0(x,t)-\lambda_1(x,t)\eps(p))\big)^{-1}, \quad x\in{{\mathbb R}}^d,\ p\in{{\mathbb T}}^d,\ t>0,$$ where $(\lambda_0,\lambda_1)$ are the Lagrange multipliers resulting from the mass and energy constraints. For $\eta=1$, we obtain the Fermi-Dirac distribution, while for $\eta=0$, ${{\mathcal F}}_f$ equals the Maxwell-Boltzmann distribution. We may consider ${{\mathcal F}}_f$ as a function of $(\lambda_0,\lambda_1)$ and write ${{\mathcal F}}(\lambda_0,\lambda_1;p)=[\eta+\exp((-\lambda_0-\lambda_1\eps(p))]^{-1}$. The variable $\lambda_1$ can be interpreted as the negative inverse (absolute) temperature, while $\lambda_0$ is related to the so-called chemical potential [@Jue09]. Since the band energy is bounded, the equilibrium ${{\mathcal F}}_f$ is integrable even when $\lambda_1>0$, which means that the absolute temperature may be negative. In fact, negative absolute temperatures can be realized in experiments with cold atoms [@RMR10]. Negative temperatures occur in equilibrated (quantum) systems that are characterized by an inverted population of energy states. The thermodynamical implications of negative temperatures are discussed in [@Ram56]. In the following, we detail the main results of the paper. Formal derivation and entropy structure {#formal-derivation-and-entropy-structure .unnumbered} --------------------------------------- Starting from the Boltzmann equation , we derive formally moment equations in the limit of large times and dominant collisions. More precisely, the particle density $n=\int_{{{\mathbb T}}^d}{{\mathcal F}}dp$ and energy density $E=\int_{{{\mathbb T}}^d}{{\mathcal F}}\eps dp$ solve the energy-transport equations $$\label{1.et1} \pa_t n + {\operatorname{div}}J_n = 0, \quad \pa_t E + {\operatorname{div}}J_E - J_n\cdot\na V = 0$$ for $x\in{{\mathbb R}}^d$, $t>0$, where the particle and energy current densities are given by $$\label{1.et2} J_n = -\sum_{i=0}^1 D_{0i}\na\lambda_i - \lambda_1 D_{00}\na V,\quad J_E = -\sum_{i=0}^1 D_{1i}\na\lambda_i - \lambda_1 D_{10}\na V,$$ and the diffusion coefficients depend nonlocally on ${{\mathcal F}}$ and hence on $(\lambda_0,\lambda_1)$; see Proposition \[prop.deriv\]. The structure of system - is similar to the semiconductor case [@JKP11] but the diffusion coefficients $D_{j}$ are different. For $V=-U_0n$, the Joule heating term $J_n\cdot\na V$ contains the squared gradient $|\na n|^2$, while in the semiconductor case it contains $|\na V|^2$ which is generally smoother than $|\na n|^2$. System - possesses a formal gradient-flow or entropy structure. Indeed, the entropy $H$, defined in Section \[sec.ent\], is nonincreasing in time, $$\frac{dH}{dt} = -\int_{{{\mathbb R}}^d}\sum_{i,j=0}^1 \na\mu_i^\top L_{ij}\na\mu_j dx\le 0;$$ see Proposition \[prop.ent\]. Here, the functions $\mu_0=\lambda_0+\lambda_1 V$ and $\mu_1=\lambda_1$ are called the dual entropy variables, and the coefficients $L_{ij}$ are defined in . In the dual entropy variables, the potential terms are eliminated, leading to the “symmetric” problem $$\pa_t\begin{pmatrix} n \\E \end{pmatrix} = {\operatorname{div}}\bigg(\begin{pmatrix} L_{00} & L_{01} \\ L_{10} & L_{11} \end{pmatrix} \na\begin{pmatrix} \mu_0 \\ \mu_1\end{pmatrix}\bigg),$$ where the matrix $(L_{ij})$ is symmetric and positive definite. This formal gradient-flow structure allows for the development of an existence theory but only for uniformly positive definite diffusion matrices [@DGJ97]. A general existence result (including electric potentials) is still missing. A further major difficulty comes from the fact that the system possesses certain “degeneracies” in the mapping $(n,E)\mapsto \mu=(\mu_0,\mu_1)$ and the entropy production $-dH/dt$. For instance, the determinant of the Jacobi matrix $\pa(n,E)/\pa\mu$ may vanish at certain points. Such a situation also occurs for the semiconductor energy-transport equations but only at the [*boundary*]{} of the domain of definition (namely at $E=0$). In the present situation, the degeneracy may occur at points in the [*interior*]{} of the domain of definition. In view of these difficulties, an analysis of the general energy-transport model - is currently out of reach. This motivates our approach to introduce a simplified model. Analysis of high-temperature energy-transport models {#analysis-of-high-temperature-energy-transport-models .unnumbered} ---------------------------------------------------- We show the existence of weak solutions to a simplified energy-transport model. It is argued in [@MRR11] that the temperature is large (relative to the nanokelvin scale) in the center of the atomic cloud for long times. Therefore, we simplify - by performing the high-temperature limit. For high temperatures, the relaxation time may be approximated by $\tau(n)=\tau_0/(n(1-n))$ [@SHR12 Suppl.]. As $\theta=-1/\lambda_1$ can be interpreted as the temperature, the high-temperature limit corresponds to the limit $\lambda_1\to 0$. Expanding ${{\mathcal F}}(\lambda_0,\lambda_1)$ around $(\lambda_0,0)$ up to zeroth order leads to the diffusion equation (see Section \[sec.high\]) $$\label{1.log} \pa_t n = {\operatorname{div}}\bigg(\frac{\tau_0\na n}{n(1-\eta n)}\bigg) \quad\mbox{in }{{\mathbb R}}^d.$$ In the case $\eta=0$, we obtain the logarithmic diffusion equation $\pa_t n = \tau_0\Delta\log n$ which predicts a nonphysical behavior. Indeed, in two space dimensions, it can be shown that the particle number is not conserved and the unique smooth solution exists for finite time only; see, e.g., [@DaDe99; @Vaz06]. We expect a similar behavior when $\eta>0$. This motivates us to compute the next-order expansion. It turns out that at first order and with $V=-U_0n$, $(n,E)$ is solving the (rescaled) energy-transport equations $$\begin{aligned} \label{1.n} \pa_t n &= {\operatorname{div}}\bigg(\frac{W\na n}{n(1-\eta n)}\bigg), \\ \pa_t W &= \frac{2d-1}{2d}{\operatorname{div}}\bigg(\frac{\na W}{n(1-\eta n)}\bigg) - U\frac{W|\na n|^2}{n(1-\eta n)}. \label{1.w}\end{aligned}$$ where $U=U_0/(2d\eps_0^2)$ and $W=1-UE$ is the “reverted” energy. The case $W=0$ corresponds to the maximal energy $E=1/U_0$. Taking into account the periodic lattice structure, we solve - on the torus ${{\mathbb T}}^d$, together with the initial conditions $n(0)=n^0$, $W(0)=W^0$ in ${{\mathbb T}}^d$. The structure of the diffusion equation is similar to , but the diffusion coefficient contains $W$ as a factor, adding a degeneracy to the singular logarithmic diffusion equation. It is an open problem whether this factor removes the unphysical behavior of the solution to in ${{\mathbb R}}^d$. We avoid this problem by solving - in a bounded domain and by looking for strictly positive particle densities. Is is another open problem to prove the existence of solutions to - in the whole space. Because of the squared gradient term in , the energy $E$ (or $W$) is not conserved but the total energy $W_{\rm tot}=W-(U/2)n^2$. In fact, in terms of $W_{\rm tot}$, the squared gradient term is eliminated, $$\label{1.wtot} \pa_t W_{\rm tot} = {\operatorname{div}}\bigg(\frac{\na W}{2n(1-\eta n)} + \frac{UW}{1-\eta n}\na n\bigg).$$ Unfortunately, this formulation does not help for the analysis since the treatment of $\pa_t(n^2)=2n\pa_t n$ is delicate as $\pa_t n$ lies in the dual space $H^1({{\mathbb T}}^d)'$ but $n$ is generally not an element of $H^1({{\mathbb T}}^d)$ because of the degeneracy (we have only $W^{1/2}\na n\in L^2({{\mathbb T}}^d)$). The analysis of system - is very challenging since the first equation is degenerate in $W$, and the second equation contains a quadratic gradient term. In the literature, there exist existence results for degenerate equations with quadratic gradient terms [@DGLS06; @GiMa08], but the degeneracy is of porous-medium type. A more complex degeneracy was investigated in [@Cro12]. In our case, the degeneracy comes from another variable, which is much more delicate to analyze. Related problems appear in semiconductor energy-transport theory, but only partial results have been obtained so far. Let us review these results. The existence of stationary solutions to with the current densities $$J_n = -\na(n\theta) + n\na V, \quad J_E = -\kappa_0\na \theta+\frac52\theta J_n, \quad E = \frac32 n\theta,$$ close to the constant equilibrium has been shown in [@AlRo17]. The idea is that in such a situation, the temperature $\theta$ is strictly positive which removes the degeneracy in the term $\na(n\theta)$. The parabolic system was investigated in [@JPR13; @LLS16], and the global existence of weak solutions was shown without any smallness condition but for a simplifed energy equation. Again, the idea was to prove a uniform positivity bound for the temperature, which removes the degeneracy. A more general result (but without electric potential) was achieved in [@ZaJu15] for the system $$\pa_t n = \Delta (\theta^\alpha n), \quad \pa_t(\theta n) = \Delta(\theta^{\alpha+1}n) + \frac{n}{\tau}(1-\theta)$$ in a bounded domain, where $0<\alpha<1$. The global existence of weak solutions to the corresponding initial-boundary-value problem was proved. Again, the idea is a positivity bound for $\theta$ but this bound required a nontrivial cut-off procedure and several entropy estimates. In this paper, we make a step forward in the analysis of nonlinear parabolic systems with nonstandard degeneracies by solving - without any positive lower bound for $W$. Since $W$ may vanish, we can expect a gradient estimate for $n$ only on $\{W>0\}$. Although the quadratic gradient term also possesses $W$ as a factor, the treatment of this term is highly delicate, because of low time regularity. Therefore, we present a result only for a time-discrete version of -, namely for its implicit Euler approximation $$\begin{aligned} \frac{1}{{\triangle t}}(n^k-n^{k-1}) &= {\operatorname{div}}\bigg(\frac{W^k\na n^k}{n(1-\eta n^k)}\bigg), \label{1.appn} \\ \frac{1}{{\triangle t}}(W^k-W^{k-1}) &= \frac{2d-1}{2d}{\operatorname{div}}\bigg(\frac{\na W^k}{n^k (1-\eta n^k)}\bigg) - U\frac{W^k|\na n^k|^2}{n^k(1-\eta n^k)} \label{1.appw}\end{aligned}$$ for $x\in{{\mathbb T}}^d$, where ${\triangle t}>0$ and $(n^{k-1},W^{k-1})$ are given functions. We show the existence of a weak solution $(n^k,W^k)$ satisfying $n^k\ge 0$, $W^k\ge 0$ and $W^kn^k$, $W^k\in H^1({{\mathbb T}}^d)$; see Theorem \[thm.ex\]. In one space dimension and under a smallness assumption on the variance of $W^{k-1}$ and $n^{k-1}$, the strict positivity of $W^k$ can be proved; see Theorem \[thm.ex2\]. The existence proof is based on the solution of a regularized and truncated problem by means of the Leray-Schauder fixed-point theorem. Standard elliptic estimates provide bounds uniform in the approximation parameters. The key step is the proof of the strong convergence of the gradient of the particle density. For this, we show a general result for degenerate elliptic problems; see Proposition \[prop.conv\]. This result seems to be new. Standard results in the literature need the ellipticity of the differential operator [@BoMu92]. Unfortunately, we are not able to perform the limit $\triangle t\to 0$ since some estimates in the proof of Proposition \[prop.conv\] are not uniform in $\triangle t$; also see Remark \[rem.comm\] for a discussion. Numerical simulations {#numerical-simulations .unnumbered} --------------------- The time-discrete system - is discretized by finite differences in one space dimension and solved in an semi-implicit way. The large-time behavior exhibits an interesting phenomenon. If the initial energy $W^0$ is constant and sufficiently large, the solution $(n(t),W(t))$ converges to a constant steady state. However, if the constant $W^0$ is too small, the stationary particle density is nonconstant. In both cases, the time decay is exponential fast, but the decay rate becomes smaller for smaller constants $W^0$ since the diffusion coefficient in is smaller too. The paper is organized as follows. Section \[sec.deriv\] is devoted to the formal derivation of the general energy-transport model and its entropy structure, similar to the semiconductor case [@BeDe96]. The high-temperature expansion is performed in Section \[sec.high\], leading to the energy-transport system -. The strong convergence of the gradients is shown in Section \[sec.conv\]. In Section \[sec.ex\] the existence result is stated and proved. The numerical simulations are presented in Section \[sec.num\], and the Appendix is concerned with the calculation of some integrals involving the velocity $u(p)$ and energy $\eps(p)$. Formal derivation and entropy structure {#sec.deriv} ======================================= Derivation from a Boltzmann equation ------------------------------------ We consider the following semiclassical Boltzmann transport equation for the distribution function $f(x,p,t)$ in the diffusive scaling: $$\label{2.be} \alpha\pa_t f_\alpha + u\cdot\na_x f_\alpha + \na V_\alpha\cdot \na_p f_\alpha = \frac{1}{\alpha}Q_\alpha(f_\alpha), \quad (x,p)\in{{\mathbb R}}^{d}\times{{\mathbb T}}^d,\ t>0,$$ where $\alpha>0$ is the Knudsen number [@BeDe96], $(x,p)$ are the phase-space variables (space and crystal momentum), and $t>0$ is the time. We recall that the velocity equals $u(p)=\na_p \eps(p)$, where the energy $\eps(p)$ is given by . The potential $V_\alpha$ is defined by $V_\alpha=-U_0 n_\alpha$. In the physical literature [@SHR12], the collision operator $Q_\alpha$ is given by the relaxation-time approximation $$Q_\alpha(f) = \frac{1}{\tau_\alpha}({{\mathcal F}}_f-f),$$ where the function ${{\mathcal F}}_f$ is determined by maximizing the free energy associated to under the constraints $$\label{2.mec} \int_{{{\mathbb T}}^d}({{\mathcal F}}_f-f)dp = 0, \quad \int_{{{\mathbb T}}^d}({{\mathcal F}}_f-f)\eps(p)dp = 0,$$ which express mass and energy conservation during scattering events. The solution of this problem is given by $${{\mathcal F}}_f(x,p,t) = \frac{1}{\eta + \exp(-\lambda_0(x,t)-\lambda_1(x,t)\eps(p))},$$ where $\lambda_0$ and $\lambda_1$ are the Lagrange multipliers and $\eta\ge 0$ is a parameter which may take the values $\eta=0$ (Maxwell-Boltzmann statistics) or $\eta=1$ (Fermi-Dirac statistics). The relaxation time $\tau_\alpha\ge 0$ generally depends on the particle density but at this point we do not need to specifiy the dependence. We show the following result. \[prop.deriv\] Let $f_\alpha$ be a (smooth) solution to the Boltzmann equation . We assume that the formal limits $f=\lim_{\alpha\to 0}f_\alpha$, $g=\lim_{\alpha\to 0}(f_\alpha-{{\mathcal F}}_{f_\alpha})/\alpha$, and $\tau=\lim_{\alpha\to 0}\tau_\alpha$ exist. Then the particle and energy densities $$n = n[{{\mathcal F}}_f] = \int_{{{\mathbb T}}^d}{{\mathcal F}}_f dp, \quad E = E[{{\mathcal F}}_f] = \int_{{{\mathbb T}}^d}{{\mathcal F}}_f\eps(p)dp$$ are solutions to -, and the diffusion coefficients $D_{ij}=(D_{ij}^{k\ell})\in{{\mathbb R}}^{d\times d}$ are defined by $$D_{ij}^{k\ell} = \tau\int_{{{\mathbb T}}^d}u_k u_\ell{{\mathcal F}}_f(1-\eta{{\mathcal F}}_f)\eps(p)^{i+j}dp, \quad i,j=0,1,\ k,\ell=1,\ldots,d.$$ The proof of the proposition is similar to those of Propositions 1 and 2 in [@JKP11]. For the convenience of the reader, we present the (short) proof. To derive the balance equations, we multiply the Boltzmann equation by $1$ and $\eps$, respectively, and integrate over ${{\mathbb T}}^d$: $$\begin{aligned} & \pa_t n[f_\alpha] + \frac{1}{\alpha}\mathrm{div}_x\int_{{{\mathbb T}}^d}uf_\alpha dp = 0, \label{2.m1} \\ & \pa_t E[f_\alpha] + \frac{1}{\alpha}\mathrm{div}_x\int_{{{\mathbb T}}^d} \eps u f_\alpha dp - \frac{1}{\alpha}\na_x V_\alpha\cdot\int_{{{\mathbb T}}^d}uf_\alpha dp = 0. \label{2.m2}\end{aligned}$$ The integrals involving the collision operator vanish in view of mass and energy conservation; see . We have integrated by parts in the last integral on the left-hand side of . Next, we insert the Chapman-Enskog expansion $f_\alpha={{\mathcal F}}_{f_\alpha}+\alpha g_\alpha$ (which in fact defines $g_\alpha$) in - and observe that the function $p\mapsto u(p)\eps(p)^j{{\mathcal F}}_{f_\alpha}(p)$ is odd for any $j\in{{\mathbb N}}_0$ such that its integral over ${{\mathbb T}}^d$ vanishes. This leads to $$\begin{aligned} & \pa_t n[{{\mathcal F}}_{f_\alpha}] + \alpha\pa_t n[g_\alpha] + \mathrm{div}_x\int_{{{\mathbb T}}^d}ug_\alpha dp = 0, \\ & \pa_t E[{{\mathcal F}}_{f_\alpha}] + \alpha\pa_t E[g_\alpha] + \mathrm{div}_x\int_{{{\mathbb T}}^d}u\eps g_\alpha dp - \na_x V\cdot\int_{{{\mathbb T}}^d}ug_\alpha dp = 0.\end{aligned}$$ Passing to the formal limit $\alpha\to 0$ gives the balance equations with $$\label{2.auxJ} J_n = \int_{{{\mathbb T}}^d} ugdp, \quad J_E = \int_{{{\mathbb T}}^d} u\eps gdp.$$ To specify the current densities, we insert the Chapman-Enskog expansion in , $$\alpha\pa_t({{\mathcal F}}_{f_\alpha}+\alpha g_\alpha) + u\cdot\na_x({{\mathcal F}}_{f_\alpha}+\alpha g_\alpha) + \na_x V\cdot\na_p({{\mathcal F}}_{f_\alpha}+\alpha g_\alpha) = -\frac{g_\alpha}{\tau_\alpha},$$ and perform the formal limit $\alpha\to 0$, $$\label{2.auxg} u\cdot\na_x{{\mathcal F}}_f + \na_x V\cdot\na_p {{\mathcal F}}_f = -\frac{g}{\tau}.$$ A straightforward computation shows that $$\na_x{{\mathcal F}}_f = {{\mathcal F}}_f(1-\eta{{\mathcal F}}_f)(\na_x\lambda_0 + \eps\na_x\lambda_1), \quad \na_p{{\mathcal F}}_f = {{\mathcal F}}_f(1-\eta{{\mathcal F}}_f)u\lambda_1,$$ and inserting this into gives an explicit expression for $g$: $$g = -\tau {{\mathcal F}}_f(1-\eta{{\mathcal F}}_f)\big(u\cdot\na_x\lambda_0 + \eps u\cdot\na_x\lambda_1 + \lambda_1\na_x V\cdot u\big).$$ Therefore, the current densities lead to . This finishes the proof. In the following we write ${{\mathcal F}}_f={{\mathcal F}}(\lambda)$, where $$\label{2.F} {{\mathcal F}}(\lambda) = \frac{1}{\eta + \exp(-\lambda_0-\lambda_1\eps(p))}, \quad \lambda=(\lambda_0,\lambda_1)\in{{\mathbb R}}^2,\ p\in{{\mathbb T}}^d.$$ \[prop.D\] The diffusion matrix ${{\mathcal D}}=(D_{ij})\in{{\mathbb R}}^{2d\times 2d}$ is symmetric and positive definite. The proof is similar to Proposition 3 in [@JKP11]. Let $z=(w,y)\in{{\mathbb R}}^{2d}$ with $w$, $y\in{{\mathbb R}}^d$. Then $$\begin{aligned} z^\top{{\mathcal D}}z &= w^\top D_{00}w + 2w^\top D_{01}y + y^\top D_{11}y \\ &= \int_{{{\mathbb T}}^d}{{\mathcal F}}(1-\eta{{\mathcal F}})\sum_{i=1}^d(u_iw_i+\eps u_i y_i)\sum_{j=1}^d (u_jw_j+\eps u_j y_j)dp \\ &= \int_{{{\mathbb T}}^d}{{\mathcal F}}(1-\eta{{\mathcal F}})\sum_{i=1}^d\big|u_i(w_i+\eps y_i)\big|^2 dp \ge 0.\end{aligned}$$ Since $D_{ij}^{k\ell}$ is symmetric in $(i,j)$ and $(k,\ell)$, the symmetry of ${{\mathcal D}}$ is clear. Entropy structure {#sec.ent} ----------------- The entropy structure of - follows from the abstract framework presented in [@JKP11]. In the following, we make this framework explicit. First, we introduce the entropy $$\begin{aligned} H(t) &= \int_{{{\mathbb R}}^d} h(\lambda)dx, \quad\mbox{where} \\ h(\lambda) &= \int_{{{\mathbb T}}^d}\big( {{\mathcal F}}\log{{\mathcal F}}+ \eta^{-1}(1-\eta{{\mathcal F}})\log(1-\eta{{\mathcal F}})\big)dp.\end{aligned}$$ The entropy density $h$ can be reformulated as $$\begin{aligned} h(\lambda) &= \int_{{{\mathbb T}}^d}\bigg({{\mathcal F}}\log\frac{{{\mathcal F}}}{1-\eta{{\mathcal F}}} - \frac{1}{\eta}\log\frac{1}{1-\eta{{\mathcal F}}}\bigg)dp \nonumber \\ &= \int_{{{\mathbb T}}^d}\big({{\mathcal F}}(\lambda_0+\lambda_1\eps) - \frac{1}{\eta}\log (1+\eta e^{\lambda_0+\lambda_1\eps})\big)dp \nonumber \\ &= n\lambda_0 + E\lambda_1 - \frac{1}{\eta}\int_{{{\mathbb T}}^d}\log (1+\eta e^{\lambda_0+\lambda_1\eps})dp. \label{2.h}\end{aligned}$$ The following result shows that the entropy is nonincreasing in time. \[prop.ent\] It holds that $$\label{2.dHdt} \frac{dH}{dt} = -\int_{{{\mathbb R}}^d}\sum_{i,j=0}^1\na\mu_i^\top L_{ij}\na\mu_j dx \le 0,$$ where $\mu_0=\lambda_0+\lambda_1 V$ and $\mu_1=\lambda_1$ are the so-called dual entropy variables and $$\label{2.L} L_{00} = D_{00}, \quad L_{01} = L_{10} = D_{01}-D_{00}V, \quad L_{11} = D_{11} - 2D_{01}V + D_{00}V^2.$$ Identity implies that $$\frac{\pa h}{\pa\lambda_i} = \frac{\pa n}{\pa\lambda_i}\lambda_0 + \frac{\pa E}{\pa\lambda_i}\lambda_1, \quad i=0,1,$$ and consequently, $$\begin{aligned} \pa_t h(\lambda) &= \frac{\pa h}{\pa\lambda_0}\pa_t\lambda_0 + \frac{\pa h}{\pa\lambda_1}\pa_t\lambda_1 \\ &= \lambda_0\bigg(\frac{\pa n}{\pa\lambda_0}\pa_t\lambda_0 + \frac{\pa n}{\pa\lambda_1}\pa_t\lambda_1\bigg) + \lambda_1\bigg(\frac{\pa E}{\pa\lambda_0}\pa_t\lambda_0 + \frac{\pa E}{\pa\lambda_1}\pa_t\lambda_1\bigg) \\ &= \lambda_0\pa_t n+\lambda_1\pa_t E.\end{aligned}$$ Therefore, using - and integration by parts, $$\begin{aligned} \frac{dH}{dt} &= \int_{{{\mathbb R}}^d}\pa_t h(\lambda) dx = \int_{{{\mathbb R}}^d}\big(J_n\cdot\na\lambda_0 + J_E\cdot\na\lambda_1 + \na V\cdot J_n \lambda_1\big)dx \\ &= -\int_{{{\mathbb R}}^d}\big(D_{00}|\na\mu_0|^2 + 2(D_{01}-D_{00}V)\na\mu_0\cdot\na\mu_1 \\ &\phantom{xx}{}+ (D_{11}-2D_{01}V+D_{00}V^2)|\na\mu_1|^2\big)dx,\end{aligned}$$ which proves the identity in . Using the positive definiteness of ${{\mathcal D}}$, a computation shows that $(L_{ij})$ is positive definite too, and the inequality in follows. Singularities and degeneracies in the energy-transport system ------------------------------------------------------------- We denote by $ n$ and $ E$ the particle and energy densities depending on the dual entropy variable $\mu=(\mu_0,\mu_1) =(\lambda_0+V\lambda_1,\lambda_1)$. We have the (implicit) formulation $$\begin{aligned} n(\mu) &= \int_{{{\mathbb T}}^d}\frac{dp}{\eta + \exp(-\mu_0+U_0 n(\mu) - \mu_1\eps(p))}, \\ E(\mu) &= \int_{{{\mathbb T}}^d}\frac{\eps(p)dp}{\eta + \exp(-\mu_0+U_0 n(\mu) - \mu_1\eps(p))}.\end{aligned}$$ \[lem.det\] Let $\omega_i(\mu):=\int_{{{\mathbb T}}^d}{{\mathcal F}}(1-\eta{{\mathcal F}})\eps(p)^i dp$, $i\in{{\mathbb N}}_0$. Then $$\det\frac{\pa(n, E)}{\pa\mu} = \frac{\omega_0\omega_2-\omega_1^2}{1-U_0\mu_1\omega_0}.$$ We differentiate $$\frac{\pa n}{\pa\mu_0} = \bigg(1+U_0\mu_0\frac{\pa n}{\pa\mu_0}\bigg)\omega_0, \quad \frac{\pa n}{\pa\mu_1} = U n + U_0\mu_1\omega_0\frac{\pa n}{\pa\mu_0} + \omega_1.$$ This gives after a rearrangement $$\frac{\pa n}{\pa\mu_0} = \frac{\omega_0}{1-U_0\mu_1\omega_0}, \quad \frac{\pa n}{\pa\mu_1} = \frac{U n\omega_0+\omega_1}{1-U_0\mu_1\omega_0}.$$ In a similar way, we obtain $$\frac{\pa E}{\pa\mu_0} = \frac{\omega_1}{1-U_0\mu_1\omega_0}, \quad\frac{\pa E}{\pa\mu_1} = \frac{U n\omega_1 + U_0\mu_1(\omega_1 -\omega_0\omega_2)+\omega_2}{1-U_0\mu_1\omega_0},$$ and with $$\det\frac{\pa(n, E)}{\pa\mu} = \frac{\pa n}{\pa\mu_0} \frac{\pa E}{\pa\mu_1} - \frac{\pa n}{\pa\mu_1} \frac{\pa E}{\pa\mu_0} = \frac{(\omega_0\omega_2-\omega_1^2)(1-U_0\mu_1\omega_0)}{(1-U_0\mu_1\omega_0)^2},$$ the conclusion follows. Since $\mu_1$ can be positive and $\omega_0>0$, the expression $1-U_0\mu_1\omega_0$ may vanish, so the determinant of $\pa(n,E)/\pa\mu$ may be not finite. Moreover, the numerator of the determinant may vanish, and the function $\mu\mapsto(n,E)$ may be not invertible. This is made more explicit in the following remark. In the Maxwell-Boltzmann case, we can make the numerator of the determinant in Lemma \[lem.det\] explicit. Indeed, it is clear that $\omega_0= n$ and $\omega_1= E$. For the computation of $\omega_2$, we observe first that $$\begin{aligned} n &= \int_{{{\mathbb T}}^d}{{\mathcal F}}dp = \exp(\mu_0-U_0 n\mu_1)\prod_{k=1}^d\int_{{{\mathbb T}}} \exp\big(-2\eps_0\cos(2\pi p_k)\big)dp_k \nonumber \\ &= \exp(\mu_0-U_0 n\mu_1)I_0^d, \label{2.nI} \\ E &= \int_{{{\mathbb T}}^d}{{\mathcal F}}\eps dp = -2\eps_0\exp(\mu_0-U_0 n\mu_1)\sum_{i=1}^d\prod_{k\neq i}^d \int_{{{\mathbb T}}}\exp\big(-2\eps_0\cos(2\pi p_k)v)dp_k \nonumber \\ &\phantom{xx}{}\times \int_{{{\mathbb T}}} \exp\big(-2\eps_0\cos(2\pi p_i)\big)\cos(2\pi p_i)dp_i \nonumber \\ &= -2d\eps_0\exp(\mu_0-U_0 n\mu_1)I_0^{d-1}I_1, \label{2.EI}\end{aligned}$$ where, by symmetry, $$I_0 := \int_{{{\mathbb T}}}\exp\big(-2\eps_0\cos(2\pi p_1)\big)dp_1, \quad I_1 := \int_{{{\mathbb T}}}\exp\big(-2\eps_0\cos(2\pi p_1)\big)\cos(2\pi p_1)dp_1.$$ Now, we have $$\omega_2 = 4\eps_0^2\sum_{i=1}^d\int_{{{\mathbb T}}^d}\cos^2(2\pi p_i){{\mathcal F}}dp + 8\eps_0^2\sum_{i=1}^d\sum_{j=1,\,j\neq i}^d \int_{{{\mathbb T}}^d}\cos(2\pi p_i)\cos(2\pi p_j){{\mathcal F}}dp.$$ Expanding the exponentials in ${{\mathcal F}}$ and using -, we find that $$\int_{{{\mathbb T}}^d}\cos(2\pi p_i)\cos(2\pi p_j){{\mathcal F}}dp = \exp(\mu_0-U_0 n\mu_1) I_0^{d-2}I_1^2 = \frac{E^2}{(2d\eps_0)^2n},$$ and this expression is independent of $i\neq j$. For $i=j$, we use the identity $\sin(2\pi p_i){{\mathcal F}}=(\pa{{\mathcal F}}/\pa p_i)/(4\pi\eps_0\mu_1)$ and integration by parts: $$\begin{aligned} \int_{{{\mathbb T}}^d}\cos^2(2\pi p_i){{\mathcal F}}dp &= \int_{{{\mathbb T}}^d}(1-\sin^2(2\pi p_i)){{\mathcal F}}dp = n - \int_{{{\mathbb T}}^d}\frac{\sin(2\pi p_i)}{4\pi\eps_0\mu_1} \frac{\pa{{\mathcal F}}}{\pa p_i}dp \\ &= n + \int_{{{\mathbb T}}^d}\frac{\cos(2\pi p_i)}{2\eps_0\mu_1}{{\mathcal F}}dp. \end{aligned}$$ Summing over $i=1,\ldots,n$ gives $$4\eps_0^2\sum_{i=1}^d \int_{{{\mathbb T}}^d}\cos^2(2\pi p_i){{\mathcal F}}dp = 4\eps_0^2 dn - \frac{E}{\mu_1}.$$ We conclude that $\omega_2 = 4\eps_0^2dn - E/\mu_1 + (d-1)E^2/(dn)$ and consequently, $$\omega_0\omega_2-\omega_1^2 = 4\eps_0^2 dn - \frac{En}{\mu_1} - \frac{E^2}{d}.$$ For certain values of $\mu_1$ or $(n,E)$, this expression may vanish such that $\det\pa(n,E)/\pa\mu=0$ at these values. This shows that the relation between $(n,E)$ and $\mu$ needs to treated with care. A tedious computation, detailed in [@Bra17 Chapter 8.4], shows that the entropy production can be written as $$\sum_{i,j=0}^1\int_{{{\mathbb R}}^d}\na\mu_i^\top L_{ij}\na\mu_j dx = g_1(\lambda)(1-U_0\mu_1\omega_0)|\na n|^2 + g_2(\lambda)|g_3(\lambda)\na n-\na E|^2,$$ where $g_i(\lambda)$, $i=1,2,3$, are functions depending on $\omega_i$, defined in Lemma \[lem.det\], and on $$\Gamma_i = \int_{{{\mathbb T}}^d}\eps(p)^i|\na\eps|^2{{\mathcal F}}(1-\eta{{\mathcal F}})dp, \quad i=0,1,2.$$ The above formula shows that we lose the gradient estimate if $1-U_0\mu_1\omega_0=0$. For the semiconductor energy-transport equations in the parabolic band approximation, we do not face the singularities and degeneracies occuring in the model for optical lattices. Indeed, let the potential $V$ be given (to simplify). According to Example 6.8 in [@Jue09], we have $$n = \mu_1^{-3/2}\exp(\mu_0+\mu_1 V), \quad E = \frac32\mu_1^{-5/2}\exp(\mu_0+\mu_1 V).$$ Then $$\det\frac{\pa(n,E)}{\pa\mu} = \det\begin{pmatrix} n & nV-E \\ E & -5E/(2\mu_1)+EV \end{pmatrix} = -\frac23E^2,$$ which is nonzero as long as $E>0$. Furthermore, by Remark 8.12 in [@Jue09], it holds that $\omega_0=n$, $\omega_1=E$, and $\omega_2=5E^2/(3n)$, and so $$\omega_0\omega_2-\omega_1^2 = \frac23 E^2.$$ This expression is degenerate only at the boundary of the domain of definition (i.e. at $E=0$). Often, such kind of degeneracies may be handled; an important example is the porous-medium equation. In the case of optical lattices, the degeneracy may occur in the interior of the domain of definition, which is much more delicate. High-temperature expansion {#sec.high} ========================== The Lagrange multiplier $\lambda_1$ is interpreted as the negative inverse temperature, so high temperatures correspond to small values of $|\lambda_1|$. In this section, we perform a high-temperature expansion of -, i.e., we expand ${{\mathcal F}}(\lambda)$ around $(\lambda_0,0)$ for small $|\lambda_1|$ up to first order. Our ansatz is $$\begin{aligned} {{\mathcal F}}(\lambda) &= {{\mathcal F}}(\lambda_0,0) + \frac{\pa {{\mathcal F}}}{\pa\lambda_1}(\lambda_0,0)\lambda_1 + O(\lambda_1^2) \nonumber \\ &= {{\mathcal F}}(\lambda_0,0) + \eps{{\mathcal F}}(\lambda_0,0)\big(1-\eta{{\mathcal F}}(\lambda_0,0)\big)\lambda_1 + O(\lambda_1^2). \label{3.F}\end{aligned}$$ Zeroth-order expansion ---------------------- At zeroth-order, we have by , , and using formula from the appendix, $$\begin{aligned} n &= \int_{{{\mathbb T}}^d}{{\mathcal F}}(\lambda_0,0)dp + O(\lambda_1) = {{\mathcal F}}(\lambda_0,0) + O(\lambda_1), \\ J_n &= -\tau\int_{{{\mathbb T}}^d} \big(u\otimes u\na_x{{\mathcal F}}(\lambda_0,0) + u\na_x V\cdot\na_p{{\mathcal F}}(\lambda_0,0)\big)dp \\ &= -\tau\int_{{{\mathbb T}}^d}(u\otimes u)n dp + O(\lambda_1) = -\frac{\tau}{2}(4\pi\eps_0)^2\na n + O(\lambda_1).\end{aligned}$$ Therefore, up to order $O(\lambda_1)$, we infer that $$\pa_t n = \frac12(4\pi\eps_0)^2{\operatorname{div}}(\tau\na n), \quad x\in{{\mathbb R}}^d,\ t>0,$$ At high temperature, the relaxation time depends on the particle density in a nonlinear way, $\tau=\tau_0/(n(1-\eta n))$ [@MRR11]. At low densities, i.e.$\eta=0$, we obtain the logarithmic diffusion equation $$\pa_t n = \eps_1\Delta\log n, \quad t>0, \quad n(0,\cdot)=n_0\quad\mbox{in }{{\mathbb R}}^d,$$ where $\eps_1=\frac12\tau_0(4\pi\eps_0)^2$. We already mentioned in the introduction that the (smooth) solution to this equation in two space dimensions loses mass, which is unphysical. Therefore, we compute the next-order expansion. First-order expansion --------------------- We calculate, using , $$\begin{aligned} n &= \int_{{{\mathbb T}}^d}\big({{\mathcal F}}(\lambda_0,0)+\eps({{\mathcal F}}(1-\eta{{\mathcal F}}))(\lambda_0,0) \lambda_1\big)dp + O(\lambda_1^2) \\ &= {{\mathcal F}}(\lambda_0,0) + ({{\mathcal F}}(1-\eta{{\mathcal F}}))(\lambda_0,0)\lambda_1\int_{{{\mathbb T}}^d}\eps(p)dp + O(\lambda_1^2) = {{\mathcal F}}(\lambda_0,0) + O(\lambda_1^2), \\ E &= {{\mathcal F}}(\lambda_0,0)\int_{{{\mathbb T}}^d}\eps(p)dp + ({{\mathcal F}}(1-\eta{{\mathcal F}}))(\lambda_0,0)\lambda_1 \int_{{{\mathbb T}}^d}\eps(p)^2dp + O(\lambda_1^2) \\ &= 2d\eps_0^2({{\mathcal F}}(1-\eta{{\mathcal F}}))(\lambda_0,0)\lambda_1 + O(\lambda_1^2) = 2d\eps_0^2({{\mathcal F}}(1-\eta{{\mathcal F}}))(\lambda_0,0)\lambda_1 + O(\lambda_1^2).\end{aligned}$$ Therefore, by , $${{\mathcal F}}(\lambda) = n + \eps\frac{E}{2d\eps_0^2} + O(\lambda_1^2),$$ and and $\na_p\eps=u$ give, up to order $O(\lambda_1^2)$, $$\begin{aligned} g &= -\tau\big(u\cdot\na_x{{\mathcal F}}(\lambda) + \na_x V\cdot\na_p{{\mathcal F}}(\lambda)\big) \\ &= -\tau u\cdot\bigg(\na_x n + \frac{E}{2d\eps_0^2}\na_x V\bigg) - \tau\eps u\cdot\frac{\na_x E}{2d\eps_0^2}.\end{aligned}$$ Then, by - and taking into account –, we infer that, again up to first order, $$\begin{aligned} J_n &= -\tau\int_{{{\mathbb T}}^d} u\otimes udp\na_x n - \tau\int_{{{\mathbb T}}^d}u\otimes udp\na_x V\frac{E}{d\eps_1} = -8\pi^2\tau\eps_0^2\na_x n - \frac{4\pi^2}{d}\tau E\na_x V, \\ J_E &= -\frac{\tau}{2d\eps_0^2}\int_{{{\mathbb T}}^d}\eps^2(u\otimes u)dp \na_x E = -4\pi^2\frac{2d-1}{d}\eps_0^2\na_x E.\end{aligned}$$ Therefore, the first-order expansion leads to $$\begin{aligned} & \pa_t n = 8\pi^2\eps_0^2{\operatorname{div}}\bigg(\tau\na n + \frac{\tau}{2d\eps_0^2}E\na V\bigg), \label{2.et1} \\ & \pa_t E = 8\pi^2\eps_0^2\frac{2d-1}{2d}{\operatorname{div}}(\tau\na E) - 8\pi^2\eps_0^2\tau\na V\cdot\bigg(\na n+\frac{E}{2d\eps_0^2}\na V\bigg). \label{2.et2}\end{aligned}$$ We rescale the time by $t_s=(8\pi^2\eps_0^2\tau_0)t$ and introduce $U=U_0/(2d\eps_0^2)$ and $W=1-UE$. Then, writing again $t$ instead of $t_s$, system - becomes $$\label{2.eq} \pa_t n = {\operatorname{div}}\bigg(\frac{W\na n}{n(1-\eta n)}\bigg), \quad \pa_t W = \frac{2d-1}{2d}{\operatorname{div}}\bigg(\frac{\na W}{n(1-\eta n)}\bigg) - U\frac{W|\na n|^2}{n(1-\eta n)}.$$ The existence of weak solutions to a time-discrete version of , together with periodic boundary conditions, is shown in Section \[sec.ex\]. A strong convergence result for the gradient {#sec.conv} ============================================ The key tool of the existence analysis of Section \[sec.ex\] is the following result on the strong convergence of the gradients of certain approximate solutions for the following equation. Let ${\triangle t}>0$, $\overline{n}\in L^\infty(\Omega)$, $y\in L^\infty(\Omega)\cap H^1(\Omega)$, and $\Psi\in C^1({{\mathbb R}})$. We consider the equation $$\label{auxeq2} \frac{1}{{\triangle t}}(n-\overline{n}) = {\operatorname{div}}\big(\na(y\Psi(n))-\Psi(n)\na y\big)$$ for $n\in L^\infty(\Omega)$ such that $y\Psi(n)\in H^1(\Omega)$. \[prop.conv\] Let $\Omega\subset{{\mathbb R}}^d$ be a bounded domain, $\eps>0$, $\triangle t>0$, let $\overline{n}\in L^\infty(\Omega)$ be such that $\overline{n}\ge 0$ in $\Omega$, and let $\Psi\in C^1({{\mathbb R}})$ satisfy $\Psi'>0$. Let $(y_\eps)$ be a bounded sequence in $H^1(\Omega)$ satisfying $y_\eps\ge C(\eps)>0$ for some $C(\eps)>0$, $y_\eps\to y$ strongly in $L^2(\Omega)$ and weakly in $H^1(\Omega)$ as $\eps\to 0$. Furthermore, let $n_\eps\in L^2(\Omega)$ with $\Psi(n_\eps)\in H^1(\Omega)$ be a weak solution to $$\label{auxeq} \frac{1}{{\triangle t}}\int_\Omega(n_\eps-\overline{n})\phi dx + \int_\Omega y_\eps\na \Psi(n_\eps)\cdot\na\phi dx = 0$$ for all $\phi\in H^1(\Omega)$. Then there exist a function $n\in L^\infty(\Omega)$ such that $y\Psi(n)\in H^1(\Omega)$, being a weak solution of , and a subsequence of $(n_\eps)$, which is not relabeled, such that, as $\eps\to 0$, $$\begin{aligned} y^{1/2}y_\eps^{1/2}\na \Psi(n_\eps) \to \na(y\Psi(n))-\Psi(n)\na y & \quad\mbox{strongly in }L^2(\Omega), \\ \mathrm{1}_{\{y>0\}}(n_\eps-n)\to 0 &\quad\mbox{strongly in }L^2(\Omega).\end{aligned}$$ [*Step 1.*]{} First, we derive some uniform bounds. Set $M=\|\overline{n}\|_{L^\infty(\Omega)}$. Taking $(\Psi(n_\eps)-\Psi(M))_+=\max\{0,\Psi(n_\eps)-\Psi(M)\}$ as a test function in , we find that $$\begin{aligned} \frac{1}{{\triangle t}}\int_\Omega & \big((n_\eps-M)-(\overline{n}-M)\big) (\Psi(n_\eps)-\Psi(M))_+ dx \\ &{}+ \int_\Omega y_\eps\na \Psi(n_\eps)\cdot\na(\Psi(n_\eps)-\Psi(M))_+ = 0.\end{aligned}$$ Since $-(\overline{n}-M)(\Psi(n_\eps)-\Psi(M))_+\ge 0$, it follows that $$\frac{1}{{\triangle t}}\int_\Omega(n_\eps-M)_+(\Psi(n_\eps)-\Psi(M))_+ dx + \int_\Omega y_\eps|\na(\Psi(n_\eps)-\Psi(M))_+|^2 dx \le 0$$ and hence, $n_\eps\le M$ in $\Omega$. In a similar way, using $\Psi(n_\eps)_-=\min\{0,\Psi(n_\eps)\}$ as a test function and using $\overline{n}\ge 0$, we infer that $n_\eps\ge 0$. This shows that $(n_\eps)$ is bounded in $L^\infty(\Omega)$. Hence, there exists a subsequence which is not relabeled such that, as $\eps\to 0$, $$\label{auxn} n_\eps\rightharpoonup^* n\quad \mbox{weakly* in }L^\infty(\Omega)$$ for some function $n\in L^\infty(\Omega)$. Since $\Phi'$ is positive and continuous on ${{\mathbb R}}$, the boundedness of $(n_\eps)$ in $L^\infty(\Omega)$ implies that there exists a constant $C>0$ such that $1/C\leq \Psi'(n_\eps)\leq C$ for all $\eps>0$. Next, we choose the test function $\phi=\Psi(n_\eps)$ in and use $\Psi(n_\eps)\leq \Psi(M)$ to find that $$\frac{1}{{\triangle t}}\int_\Omega n_\eps\Psi(n_\eps) dx + \int_\Omega y_\eps|\na\Psi(n_\eps)|^2 dx \le \frac{\Psi(M)}{{\triangle t}}\int_\Omega \overline{n} dx.$$ We deduce that $(y_\eps^{1/2}\na\Psi(n_\eps))$ is bounded in $L^2(\Omega)$ and, for a subsequence, $$\label{auxna} y_\eps^{1/2}\na\Psi(n_\eps) \rightharpoonup \xi\quad\mbox{weakly in }L^2(\Omega)$$ for some function $\xi\in L^2(\Omega)$. Now, let $\phi$ be a smooth test function. Then we can take the limit, for a subsequence, in and obtain $$\label{auxeq_im_Beweis} \frac{1}{{\triangle t}}\int_\Omega(n-\overline{n})\phi dx + \int_\Omega y^{1/2}\xi\cdot\na\phi dx = 0.$$ Note that this equation holds also for all $\phi\in H^1(\Omega)$. [*Step 2.*]{} As $(y_\eps)$ is strongly converging in $L^2(\Omega)$, we deduce from that $y_\eps^{1/2}(n_\eps-n)\rightharpoonup 0$ weakly in $L^2(\Omega)$. We claim that this convergence is even strong. Indeed, the sequence $$\na(y_\eps n_\eps^2) = 2\frac{y_\eps^{1/2}n_\eps}{\Psi'(n_\eps)} y_\eps^{1/2}\na \Psi(n_\eps) + n_\eps^2\na y_\eps$$ is uniformly bounded in $L^2(\Omega)$. Thus, $(y_\eps n_\eps^2)$ is bounded in $H^1(\Omega)$ and by compactness, for a subsequence, $y_\eps n_\eps^2\to \zeta\ge 0$ strongly in $L^2(\Omega)$ or $y_\eps^{1/2}n_\eps\to \zeta^{1/2}$ strongly in $L^4(\Omega)$. The strong convergence of $(y_\eps^{1/2})$ in $L^2(\Omega)$ and the weak\* convergence of $(n_\eps)$ in $L^\infty(\Omega)$ imply that $y_\eps^{1/2}n_\eps\rightharpoonup y^{1/2}n$ weakly in $L^2(\Omega)$. Therefore, $\zeta^{1/2}=y^{1/2}n$ and $$\label{auxyn} y_\eps^{1/2}(n_\eps-n)\to 0\quad\mbox{strongly in }L^2(\Omega).$$ This proves the claim. [*Step 3.*]{} The next goal is to show that $$\label{auxnn} \mathrm{1}_{\{y>0\}}(n_\eps-n)\to 0\quad\mbox{strongly in }L^2(\Omega).$$ Taking into account , the strong convergence of $(y_\eps^{1/2})$ in $L^2(\Omega)$, and the $L^\infty$ bound for $(n_\eps)$, it follows that $$\|y^{1/2}(n_\eps-n)\|_{L^2(\Omega)} \le \|y_\eps^{1/2}(n_\eps-n)\|_{L^2(\Omega)} + \|y^{1/2}-y_\eps^{1/2}\|_{L^2(\Omega)}\|n_\eps-n\|_{L^\infty(\Omega)}$$ converges to zero. Thus, for a subsequence, $y^{1/2}(n_\eps-n)\to 0$ a.e.in $\Omega$ and consequently, $n_\eps-n\to 0$ a.e. in $\{y>0\}$. Then the a.e. pointwise convergence $\mathrm{1}_{\{y>0\}}(n_\eps-n)\to 0$ and the dominated convergence theorem show . [*Step 4.*]{} We wish to identify $\xi$ in and . The bounds for $(\na y_\eps)$ and $(y_\eps\na\Psi(n_\eps))$ in $L^2(\Omega)$ show that $(\na(y_\eps\Psi(n_\eps)))$ is bounded in $L^2(\Omega)$ and so, $(y_\eps \Psi(n_\eps))$ is bounded in $H^1(\Omega)$. By compactness, for a subsequence, $\na(y_\eps\Psi(n_\eps))\rightharpoonup \na(y\Psi(n))$ weakly in $L^2(\Omega)$ and $y_\eps\Psi(n_\eps)\to y\Psi(n)$ strongly in $L^2(\Omega)$. We can identify the limit since $y_\eps\to y$ strongly in $L^2(\Omega)$ and $\Psi(n_\eps)\rightharpoonup^* \theta$ weakly\* in $L^\infty(\Omega)$ with $\theta=\Psi(n)$ in $\{y>0\}$ lead to $y_\eps \Psi(n_\eps)\rightharpoonup y\Psi(n)$ weakly in $L^2(\Omega)$. Moreover, since $\Psi(n_\eps)\to \Psi(n)$ strongly in $L^2(\Omega)$ and $\na y_\eps\to\na y$ weakly in $L^2(\Omega)$, we infer that $$\label{auxyy} y_\eps\na \Psi(n_\eps) = \na(y_\eps \Psi(n_\eps))-\Psi(n_\eps) \na y_\eps\rightharpoonup\na(y \Psi(n))-\Psi(n)\na y \quad\mbox{weakly in }L^2(\Omega).$$ Here, we have used additionally that $(y_\eps\na \Psi(n_\eps))$ is bounded in $L^2(\Omega)$ and that $L^1(\Omega)$ is dense in $L^2(\Omega)$. Similarly as above, we deduce that $$\label{auxabschaetzung_gut_fuer_spaeter} \|(y^{1/2}-y_\eps^{1/2})y_\eps^{1/2}\na \Psi(n_\eps)\|_{L^1(\Omega)} \leq \|(y^{1/2}-y_\eps^{1/2})\|_{L^2(\Omega)} \|y_\eps^{1/2}\na \Psi(n_\eps)\|_{L^2(\Omega)}$$ converges to zero. Therefore, $y^{1/2}\xi=\na(y \Psi(n))-\Psi(n)\na y$ in $\Omega$. [*Step 5.*]{} We obtain from that $$y^{1/2}\big(y_\eps\na \Psi(n_\eps)-y^{1/2}\xi\big)\rightharpoonup 0 \quad\mbox{weakly in }L^2(\Omega)$$ and consequently, $$\begin{aligned} \label{auxc1} \int_\Omega y_\eps\xi\cdot\big(y^{1/2}\na \Psi(n_\eps)-\xi\big)dx &= \int_\Omega \xi\cdot y^{1/2}\big(y_\eps\na \Psi(n_\eps)-y^{1/2}\xi\big)dx \\ &\phantom{xx}{}+\int_\Omega |\xi|^2\big(y-y_\eps\big)dx \to 0, \nonumber\end{aligned}$$ applying the dominated convergence theorem to the last integral. Furthermore, using the test function $\phi=y(\Psi(n_\eps)-\Psi(n))$ in , $$\begin{aligned} \bigg| & \int_\Omega y_\eps y^{1/2}\na \Psi(n_\eps)\cdot \big(y^{1/2}\na \Psi(n_\eps)-\xi\big)dx\bigg| \\ &= \bigg|\int_\Omega\Big(y_\eps y|\na\Psi(n_\eps)|^2 - y_\eps\na\Psi(n_\eps) \cdot\big(\na(y\Psi(n))-\Psi(n)\na y\big)\Big)dx\bigg|\\ &= \bigg|\int_\Omega y_\eps\na \Psi(n_\eps)\cdot\na(y(\Psi(n_\eps)-\Psi(n))) - \int_\Omega y_\eps^{1/2}\na\Psi(n_\eps) \cdot\na y\big(y_\eps^{1/2}(\Psi(n_\eps)-\Psi(n))\big)dx \bigg| \\ &= \bigg|\frac{1}{{\triangle t}}\int_\Omega(\overline{n}-n_\eps)y(\Psi(n_\eps)-\Psi(n))dx - \int_\Omega y_\eps^{1/2}\na\Psi(n_\eps)\cdot\na y\big(y_\eps^{1/2} (\Psi(n_\eps)-\Psi(n))\big)dx \bigg| \\ &\le \frac{1}{{\triangle t}}\|n_\eps-\overline{n}\|_{L^2(\Omega)} \|y(\Psi(n_\eps)-\Psi(n))\|_{L^2(\Omega)} \\ &\phantom{xx}{}+ \|y_\eps^{1/2}\na\Psi(n_\eps)\|_{L^2(\Omega)} \bigg(\int_\Omega y_\eps(\Psi(n_\eps)-\Psi(n))^2|\na y|^2 dx\bigg).\end{aligned}$$ By , we have $\|y(\Psi(n_\eps)-\Psi(n))\|_{L^2(\Omega)}\to 0$ and by , $y_\eps(\Psi(n_\eps)-\Psi(n))^2\to 0$ in $\Omega$ for a subsequence. Then, by dominated convergence, $\int_\Omega y_\eps(\Psi(n_\eps)-\Psi(n))^2|\na y|^2 dx\to 0$. We have proved that $$\label{auxc2} \int_\Omega y_\eps y^{1/2}\na \Psi(n_\eps)\cdot \big(y^{1/2}\na \Psi(n_\eps) - \xi\big)dx \to 0.$$ Subtracting from , we conclude that $$\int_\Omega y_\eps|y^{1/2}\na \Psi(n_\eps)-\xi|^2 dx \to 0.$$ Taking into account this convergence and , it follows again by the dominated convergence theorem that $$\begin{aligned} \int_\Omega yy_\eps|\na \Psi(n_\eps)|^2 dx &= \int_\Omega y_\eps|y^{1/2}\na \Psi(n_\eps)-\xi|^2dx \\ &\phantom{xx}{}+2\int_\Omega y_\eps\big(y^{1/2}\na \Psi(n_\eps)-\xi\big)\cdot \xi dx + \int_\Omega y_\eps|\xi|^2dx \to \int_\Omega y|\xi|^2 dx.\end{aligned}$$ This shows the first part of the proposition. [*Step 6.*]{} It remains to show that the limit $n$ solves . Let $\phi\in W^{1,\infty}(\Omega)$. Since (a subsequence of) $(n_\eps)$ converges weakly\* to $n$ in $L^\infty(\Omega)$, we have $$\frac{1}{{\triangle t}}\int_\Omega(n_\eps-\overline{n})\phi dx\to \frac{1}{{\triangle t}}\int_\Omega(n-\overline{n})\phi dx.$$ Furthermore, $$\begin{aligned} \int_\Omega y_\eps\na\Psi(n_\eps)\cdot\na\phi dx &= \int_\Omega y^{1/2}y_\eps^{1/2}\na\Psi(n_\eps)\cdot\na\phi dx \\ &\phantom{xx}{} + \int_\Omega (y^{1/2}-y_\eps^{1/2})y_\eps^{1/2}\na\Psi(n_\eps)\cdot\na\phi dx.\end{aligned}$$ By Step 5, the first integral converges to $$\int_\Omega\big(\na(y\Psi(n))-\Psi(n)\na y\big)\cdot\na\phi dx,$$ while the second integral converges to zero since $$\begin{aligned} \bigg|\int_\Omega & (y^{1/2}-y_\eps^{1/2})y_\eps^{1/2}\na\Psi(n_\eps)\cdot\na\phi dx \bigg| \\ &\le \|y_\eps^{1/2}\na\Psi(n_\eps)\|_{L^2(\Omega)} \|y^{1/2}-y_\eps^{1/2}\|_{L^2(\Omega)}\|\na\phi\|_{L^\infty(\Omega)}\to 0.\end{aligned}$$ We conclude that holds in the weak sense for test functions in $W^{1,\infty}(\Omega)$ but a density argument shows that it is sufficient to take test functions in $H^1(\Omega)$. This finishes the proof. Existence of solutions to the high-temperature model {#sec.ex} ==================================================== We prove the existence of weak solutions to in ${{\mathbb T}}^d$. We recall the definition of the total (“reverted”) energy $$\label{ex.wtot} W_{\rm tot}^k = W^k - \frac{U}{2}(n^k)^2$$ and introduce the total variance $$\label{var} V^k := \int_{{{\mathbb T}}^d}\bigg((W^k)^2 - \int_{{{\mathbb T}}^d}W^kdz\bigg)^2dx + U\int_{{{\mathbb T}}^d}W^{k-1}dx\int_{{{\mathbb T}}^d}\bigg(n^k - \int_{{{\mathbb T}}^d}n^kdz\bigg)^2 dx.$$ The main result is as follows. \[thm.ex\] Let ${\triangle t}>0$, $U>0$, $\eta\in(0,1]$, $0<\delta<1/(1+\eta)$ and let $$n^{k-1},\ W^{k-1}\in L^\infty({{\mathbb T}}^d), \quad \delta \le n^{k-1}\le\frac{1-\delta}{\eta}, \quad W^{k-1}\ge 0\mbox{ in }{{\mathbb T}}^d.$$ Then there exists a weak solution $(n^k,W^k)$ to - in the following sense: It holds $\delta\le n^k\le\|n^{k-1}\|_{L^\infty({{\mathbb T}}^d)} \le (1-\delta)/\eta$, $0\le W^k\le\|W^{k-1}\|_{L^\infty({{\mathbb T}}^d)}$ in ${{\mathbb T}}^d$, $W^kn^k$, $W^k\in H^1({{\mathbb T}}^d)$, as well as $$\begin{aligned} \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}(n^k-n^{k-1})\phi_0 dx &= -\int_\Omega\frac{\na(W^kn^k)-n^k\na W^k}{g(n^k)} \cdot\na\phi_0 dx, \label{1.ntau} \\ \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}(W^k-W^{k-1})W^k\phi_1 dx &= -\frac{2d-1}{2d}\int_{{{\mathbb T}}^d}\frac{\na W^k\cdot\na(W^k\phi_1)}{g(n^k)}dx \label{1.wtau} \\ &\phantom{xx}{} - U\int_{{{\mathbb T}}^d}\frac{|\na (W^kn^k)-n^k\na W^k|^2}{g(n^k)} \phi_1 dx \nonumber\end{aligned}$$ for all $\phi_0\in H^1({{\mathbb T}}^d)$ and $\phi_1\in H^1({{\mathbb T}}^d)\cap L^\infty({{\mathbb T}}^d)$, where $g(n^k)=n^k(1-\eta n^k)$. For this solution, the following monotonicity properties hold: $$\label{ex.mono} \int_{{{\mathbb T}}^d}W_{\rm tot}^{k}dx \ge \int_{{{\mathbb T}}^d}W_{\rm tot}^{k-1} dx, \quad V^k + {\triangle t}\frac{2d-1}{d}\int_{{{\mathbb T}}^d}\frac{|\na W^k|^2}{g(n^k)}dx \le V^{k-1},$$ where $W_{\rm tot}^k$ and $V^k$ are defined in and , respectively. Moreover, if $$\label{ex.assump} \frac{U}{2} \int_{{{\mathbb T}}^d}\bigg(n^{k-1}-\int_{{{\mathbb T}}^d}n^{k-1}dz\bigg)^2dx < \int_{{{\mathbb T}}^d}W^{k-1}dx$$ holds then $W^k\not\equiv 0$. \[rem.comm\]1. The existence result holds true for more general functions $g(n)$ under the assumption that $g(n)$ is strictly positive for $\delta\le n\le(1-\delta)/\eta$. 2. One may interpret $W^k$ as a “renormalized” solution since we need test functions of the form $W^k\phi_1$ in order to avoid vacuum sets $W^k=0$. Such an idea has been used, for instance, for the compressible quantum Navier-Stokes equations to avoid vacuum sets in the particle density [@Jue10]. Test functions of the type $W^k\phi$ allow for the trivial solution $n^k=n^{k-1}$ and $W^k=0$ but assumption excludes this situation. It means that no constant steady state with $W^k=0$ exists if the variance of $n^{k-1}$ is small compared to the energy $\int_{{{\mathbb T}}^d}W^{k-1}dx$. 3. The second inequality in involves $W^{k-2}$ which makes sense when the equations are solved iteratively, starting from $k=1$. Also can be iterated. Indeed, if holds for $(n^{k-1},W^{k-1})$, the monotonicity property and mass conservation $\int_{{{\mathbb T}}^d}n^k dx=\int_{{{\mathbb T}}^d}n^{k-1}dx$ imply that $$\begin{aligned} \frac{U}{2} & \int_{{{\mathbb T}}^d}\bigg(n^{k} - \int_{{{\mathbb T}}^d}n^{k}dz\bigg)^2 = \frac{U}{2}\int_{{{\mathbb T}}^d}(n^k)^2 dx - \frac{U}{2}\bigg(\int_{{{\mathbb T}}^d}n^k dx\bigg)^2 \\ &\le \int_{{{\mathbb T}}^d}(W^k-W^{k-1})dx + \frac{U}{2}\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx - \frac{U}{2}\bigg(\int_{{{\mathbb T}}^d}n^{k-1} dx\bigg)^2 < \int_{{{\mathbb T}}^d} W^k dx.\end{aligned}$$ 4. We are not able to perform the limit ${\triangle t}\to 0$. The reason is that we cannot perform the limit in the quadratic gradient term $|\na (W^kn^k)-n^k\na W^k|^2$, since we cannot prove the strong convergence of $\na (W^kn^k)-n^k\na W^k$. Proposition \[prop.conv\] provides such a result for the time-discrete elliptic case. The key step is to show that $$\begin{aligned} \int_\Omega \frac{\na(W^kn^k)-n^k\na W^k}{g(n^k)} &\cdot\na(Wn-W^kn^k) dx \\ &= \frac{1}{{\triangle t}}\big\langle n^k-n^{k-1},W^kn^k-Wn\big\rangle \to 0,\end{aligned}$$ where $n$, $W$ are the (weak) limits of $(n^k)$, $(W^k)$, respectively, and $\langle\cdot,\cdot\rangle$ is the dual product between $H^1({{\mathbb T}}^d)'$ and $H^1({{\mathbb T}}^d)$. It is possible to show that ${\triangle t}^{-1}(n^k-n^{k-1})$ is bounded in $H^1({{\mathbb T}}^d)'$, but the limit $W^kn^k-Wn\to 0$ strongly in $H^1({{\mathbb T}}^d)$ (more precisely: the limit of the piecewise constant in time construction of $W^kn^k$ in $L^2(0,T;H^1({{\mathbb T}}^d))$) cannot be expected. In the one-dimensional case and under the smallness condition below, we can show that $W^k$ is positive, which allows us to define the weak solution to - in the standard sense (with test functions $\phi_1$ instead of $W^k\phi_1$). We set $$\overline{W^{k-1}} = \int_{{{\mathbb T}}}W^{k-1}dx, \quad \overline{n^{k-1}} = \int_{{{\mathbb T}}}n^{k-1}dx.$$ \[thm.ex2\] Let the assumptions of Theorem \[thm.ex\] hold, let $d=1$, $G=\max_{\delta\le s\le \|n^{k-1}\|_{L^\infty({{\mathbb T}})}}g(s)$, and let $(n^k,W^k)$ for $k\ge 0$ be the solution given by Theorem \[thm.ex\]. We assume that $$\label{assump2} \frac{G}{{\triangle t}}\big\|W^{k-1}-\overline {W^{k-1}}\big\|_{L^2({{\mathbb T}})}^2 + U\bigg(\frac{G\overline {W^{k-2}}}{{\triangle t}}+\frac12\bigg) \big\|n^{k-1}-\overline {n^{k-1}}\big\|_{L^2({{\mathbb T}})}^2 < \overline {W^{k-1}},$$ Then $W^k$ is strictly positive, $n^k\in H^1({{\mathbb T}})$, and - hold in the sense of $H^1({{\mathbb T}})'$. We proceed to the proof of Theorems \[thm.ex\] and \[thm.ex2\]. In this section, $\eps$ denotes a positive parameter and not the band energy. Since we are not concerned with the kinetic equations, no notational confusion will occur. Let ${\triangle t}>0$, $\alpha$, $\gamma$, $\delta$, $\eps>0$ satisfying $\gamma<1$ and $\delta<1/(1+\eta)$. Define the truncations $$[n]_\delta = \max\big\{\delta,\min\{(1-\delta)/\eta,n\}\big\}, \quad [W]_\gamma = \max\big\{0,\min\{1/\gamma,W\}\big\},$$ and $g_\delta(s)=[s]_\delta(1-\eta[s]_\delta)$ for $s\in{{\mathbb R}}$. Then $g_\delta$ is continuous and strictly positive. Given $W^{k-1}$, $n^{k-1}\in L^\infty({{\mathbb T}}^d)$ satisfying $\delta\le n^{k-1}\le (1-\delta)/\eta$, we solve the regularized and truncated nonlinear problem in ${{\mathbb T}}^d$ $$\begin{aligned} \frac{1}{{\triangle t}}(n^k-n^{k-1}) &= {\operatorname{div}}\bigg(\frac{[W^k]_\gamma+\eps}{g_\delta(n^k)} \na n^k\bigg), \label{ex.appn} \\ \frac{1}{{\triangle t}}(W^k-W^{k-1}) &= \frac{2d-1}{2d}{\operatorname{div}}\bigg(\frac{\na W^k}{g_\delta(n^k)}\bigg) - U\frac{[W^k]_\gamma}{g_\delta(n^k)}\frac{|\na n^k|^2}{1+\alpha|\na n^k|^2}. \label{ex.appw}\end{aligned}$$ Let us explain the approximation -. The truncation of $W^k$ with parameter $\gamma$ ensures that the coefficients are bounded, while the truncation $g_\delta(n^k)$ with parameter $\delta>0$ guarantees that the denominator is always positive. The regularization parameter $\eps$ gives strict ellipticity for , since generally the first term on the right-hand side of without $\eps$ is degenerate. Finally, the approximation of the quadratic gradient term with parameter $\alpha$ avoids regularity issues since it holds $|\na n^k|^2\in L^1({{\mathbb T}}^d)$ only. Solution of an approximated problem ----------------------------------- First, we prove the existence of solutions to -. There exists a weak solution $(n^k,$ $W^k)\in H^1({{\mathbb T}}^d)^2$ to -. We define the fixed-point operator $S:L^2({{\mathbb T}}^d)^2\times[0,1]\to L^2({{\mathbb T}}^d)^2$ by $S(n^*,W^*;\theta)=(n,W)$, where $(n,W)\in H^1({{\mathbb T}}^d)^2$ is the unique solution to the linear problem $$\label{ex.LM} a_0(n,\phi_0) = F_0(\phi_0), \quad a_1(W,\phi_1)=F_1(\phi_1) \quad\mbox{for all }\phi_0,\,\phi_1\in H^1({{\mathbb T}}^d),$$ where $$\begin{aligned} a_0(n,\phi_0) &= \int_{{{\mathbb T}}^d}\frac{[W^*]_\gamma+\eps}{g_\delta(n^*)}s \na n\cdot\na\phi_0 dx + \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}n\phi_0 dx, \\ F_0(\phi_0) &= \frac{\theta}{{\triangle t}}\int_{{{\mathbb T}}^d} n^{k-1}\phi_0 dx, \\ a_1(W,\phi_1) &= \frac{2d-1}{2d}\int_{{{\mathbb T}}^d} \frac{\na W\cdot\na\phi_1}{g_\delta(n^*)}dx + \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}W\phi_1 dx, \\ F_1(\phi_1) &= \frac{\theta}{{\triangle t}}\int_{{{\mathbb T}}^d}W^{k-1}\phi_1 dx - \theta U\int_{{{\mathbb T}}^d}\frac{[W^*]_\gamma}{g_\delta(n^*)} \frac{|\na n|^2}{1+\alpha|\na n|^2}\phi_1 dx.\end{aligned}$$ The approximation and truncation ensure that these forms are bounded on $H^1({{\mathbb T}}^d)$. The bilinear forms $a_0$ and $a_1$ are coercive. By the Lax-Milgram lemma, there exists a unique solution $(n,W)\in H^1({{\mathbb T}}^d)^2$ to . Thus, the fixed-point operator is well defined (and has compact range). Furthermore, $S(n^*,W^*;0)=0$. Standard arguments show that $S$ is continuous. Let $(n,W)$ be a fixed point of $S(\cdot,\cdot;\theta)$, i.e., $(n,W)$ solves - with $(n^k,W^k)$ replaced by $(n,W)$. With the test functions $\phi_0=n$ and $\phi_1=W$ and the inequality $$\bigg(\frac{1}{{\triangle t}}n - \frac{\theta}{{\triangle t}}n^{k-1}\bigg)n \ge \frac{1}{2{\triangle t}}\big(n^2 - (n^{k-1})^2\big),$$ we find that $$\begin{aligned} \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}n^2 dx + \int_{{{\mathbb T}}^d}\frac{[W]_\gamma+\eps}{g_\delta(n)}|\na n|^2 dx &\le \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx, \\ \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}W^2 dx + \frac{2d-1}{2d}\int_{{{\mathbb T}}^d}\frac{|\na W|^2}{g_\delta(n)}dx &\le \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}(W^{k-1})^2 dx \\ &\phantom{xx}{} - \theta U\int_{{{\mathbb T}}^d}\frac{[W]_\gamma W}{g_\delta(n)} \frac{|\na n|^2}{1+\alpha|\na n|^2}dx.\end{aligned}$$ The last integral is nonnegative since $[W]_\gamma W\ge 0$. Therefore, $$\|n\|_{H^1({{\mathbb T}}^d)} \le C(\eps), \quad \|W\|_{H^1({{\mathbb T}}^d)} \le C(\delta),$$ where $C(\eps)$ and $C(\delta)$ are positive constants independent of $(n,W)$. This provides the necessary uniform bound for all fixed points of $S(\cdot,\cdot;\theta)$. We can apply the Leray-Schauder fixed-point theorem to infer the existence of a fixed point for $S(\cdot,\cdot;1)$, i.e. of a weak solution to -. Removing the truncation ----------------------- The following maximum principle holds. Let $(n^k,W^k)$ be a weak solution to -. Then $$\delta\le n^k\le \|n^{k-1}\|_{L^\infty({{\mathbb T}}^d)}\le \frac{1-\delta}{\eta}, \quad 0\le W^k\le \frac{1}{\gamma}\quad\mbox{in }{{\mathbb T}}^d,$$ where $\gamma\le 1/\|W^{k-1}\|_{L^\infty({{\mathbb T}}^d)}$. We choose $(n^k-\delta)_-=\min\{0,n^k-\delta\}$ as a test function in : $$\begin{aligned} \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d} &\big((n^k-\delta)-(n^{k-1}-\delta)\big)(n^k-\delta)_-dx \\ &{}+ \int_{{{\mathbb T}}^d}\frac{[W^k]_\gamma+\eps}{g_\delta(n^k)} \na n^k\cdot\na(n^k-\delta)_- dx = 0.\end{aligned}$$ Since $-(n^{k-1}-\delta)(n^k-\delta)_-\ge 0$, this gives $$\frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}(n^k-\delta)_-^2 dx \le -\int_{{{\mathbb T}}^d}\frac{[W^k]_\gamma+\eps}{g_\delta(n^k)}|\na(n^k-\delta)_-|^2 dx \le 0,$$ and hence, $n^k\ge\delta$ in ${{\mathbb T}}^d$. In a similar way, the test function $(n^k-N)_+=\max\{0,n^k-N\}$ with $N:=\|n^{k-1}\|_{L^\infty({{\mathbb T}}^d)}$ leads to $n^k-N\le 0$ in ${{\mathbb T}}^d$. Next, we use $W^k_-\le 0$ as a test function in : $$\begin{aligned} \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d} & (W_-^k)^2 dx + \frac{2d-1}{2d}\int_{{{\mathbb T}}^d}\frac{|\na W_-|^2}{g_\delta(n^k)}dx \\ &= \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}W^{k-1}W_-^k dx - U\int_{{{\mathbb T}}^d}\frac{[W^k]_\gamma W_-^k}{g_\delta(n^k)} \frac{|\na n^k|^2}{1+\alpha|\na n^k|^2}dx \le 0.\end{aligned}$$ We deduce that $W^k\ge 0$. The proof of $W^k\le \|W^{k-1}\|_{L^\infty({{\mathbb T}}^d)} \le 1/\gamma$ is similar, using the test function $(W^k-\|W^{k-1}\|_{L^\infty({{\mathbb T}}^d)})_+$. We have shown that $(n^k,W^k)$ solves $$\begin{aligned} \frac{1}{{\triangle t}}(n^k-n^{k-1}) &= {\operatorname{div}}\bigg(\frac{W^k+\eps}{g(n^k)} \na n^k\bigg), \label{ex.aln} \\ \frac{1}{{\triangle t}}(W^k-W^{k-1}) &= \frac{2d-1}{2d}{\operatorname{div}}\bigg(\frac{\na W^k}{g(n^k)}\bigg) - U\frac{W^k}{g(n^k)}\frac{|\na n^k|^2}{1+\alpha|\na n^k|^2}, \label{ex.alw}\end{aligned}$$ where $g(n)=n(1-\eta n)$. The limit $\alpha\to 0$ ----------------------- Let $(n_\alpha^k,W_\alpha^k)$ be a weak solution to -. We use the test function $n^k_\alpha$ in , $$\label{ex.auxn} \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}(n^k_\alpha)^2 dx + \int_{{{\mathbb T}}^d}\frac{W^k_\alpha+\eps}{g(n^k_\alpha)}|\na n^k_\alpha|^2 dx \le \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx,$$ and the test function $W^k_\alpha$ in , $$\label{ex.auxw} \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}(W^k_\alpha)^2 dx + \frac{2d-1}{2d}\int_{{{\mathbb T}}^d}\frac{|\na W_\alpha^k|^2}{g(n^k_\alpha)} \le \frac{1}{2{\triangle t}}\int_{{{\mathbb T}}^d}(W^{k-1})^2 dx,$$ which provides immediately uniform $H^1({{\mathbb T}}^d)$ estimates since $g(n^k_\alpha)\ge C(\delta)>0$: $$\|n^k_\alpha\|_{H^1({{\mathbb T}}^d)} \le C(\delta,\eps,{\triangle t}), \quad \|W^k_\alpha\|_{H^1({{\mathbb T}}^d)} \le C(\delta,{\triangle t}),$$ where the constants are independent of $\alpha$. By compactness, this implies the existence of a subsequence which is not relabeled such that, as $\alpha\to 0$, $$\begin{aligned} n^k_\alpha\to n^k,\ W^k_\alpha\to W^k &\quad\mbox{strongly in }L^2({{\mathbb T}}^d), \\ n^k_\alpha\rightharpoonup n^k,\ W^k_\alpha\rightharpoonup W^k &\quad\mbox{weakly in }H^1({{\mathbb T}}^d).\end{aligned}$$ This shows that, maybe for a subsequence, $W^k_\alpha/g(n^k_\alpha)\to W^k/g(n^k)$ and $1/g(n^k_\alpha)\to 1/g(n^k)$ a.e. in ${{\mathbb T}}^d$, and by dominated convergence, strongly in $L^2({{\mathbb T}}^d)$. We claim that $n^k_\alpha\to n^k$ strongly in $H^1({{\mathbb T}}^d)$. Let $y_\alpha:=(W^k_\alpha+\eps)/g(n^k_\alpha)$. Then $y_\eps\ge\eps/\sup_{s\in[\delta,N]}g(s)>0$, where $N=\|n^{k-1}\|_{L^\infty({{\mathbb T}}^d)}$, and $y_\alpha\to y:=W^k/g(n^k)\ge 0$ strongly in $L^2(\Omega)$. Thus, $y_\alpha^{1/2}\na n^k_\alpha\rightharpoonup y^{1/2}\na n^k$ weakly in $L^2(\Omega)$, and it follows that $$\int_{{{\mathbb T}}^d}y_\alpha \na n^k\cdot\na(n^k_\alpha-n^k)dx \to 0.$$ Taking $n^k_\alpha-n^k$ as a test function in , we obtain $$\int_{{{\mathbb T}}^d}y_\alpha\na n^k_\alpha\cdot\na(n^k_\alpha-n^k)dx = -\frac{1}{{\triangle t}}\int_\Omega(n^k_\alpha-n^{k-1})(n^k_\alpha-n^k)dx \to 0.$$ Subtraction of these integrals leads to $$\int_{{{\mathbb T}}^d}y_\alpha|\na(n^k_\alpha-n^k)|^2 dx \to 0.$$ Since $y_\alpha\ge\eps/\sup_{s\in[\delta,N]}g(s)>0$, this proves the claim. In particular, $1/(1+\alpha|\na n^k_\alpha|^2)\to 1$ in $L^2({{\mathbb T}}^d)$. From this, we can directly deduce that $$\frac{|\na n^k_\alpha|^2}{1+\alpha|\na n^k_\alpha|^2} \to |\na n^k|^2 \quad \mbox{in }L^1({{\mathbb T}}^d).$$ The above convergence results are sufficient to pass to the limit $\alpha\to 0$ in -, showing that $(n^k,W^k)$ solves $$\begin{aligned} \frac{1}{{\triangle t}}(n^k-n^{k-1}) &= {\operatorname{div}}\bigg(\frac{W^k+\eps}{g(n^k)} \na n^k\bigg), \label{ex.epsn} \\ \frac{1}{{\triangle t}}(W^k-W^{k-1}) &= \frac{2d-1}{2d}{\operatorname{div}}\bigg(\frac{\na W^k}{g(n^k)}\bigg) - U\frac{W^k}{g(n^k)}|\na n^k|^2, \label{ex.epsw}\end{aligned}$$ The limit $\eps\to 0$ --------------------- This limit is the delicate part of the proof. We first state a lemma concerning weak and strong convergence. \[lem.weak.strong\] Let $(f_n)$ be a weakly and $(g_n)$ be a strongly converging sequence in $L^2(\Omega)$ which have the same limit. If $|f_n(x)|\leq |g_n(x)|$ for all $n\in \mathbb N$ and a.e. $x\in{{\mathbb T}}^d$, then $(f_n)$ converges strongly in $L^2(\Omega)$. Let $f$ denote the weak limit of $(f_n)$ and $(g_n)$. Due to the weak lower semi-continuity of the norm, $$\begin{aligned} \int_{{{\mathbb T}}^d}|f(x)|^2dx &\leq \liminf_{n\to\infty} \int_{{{\mathbb T}}^d}|f_n(x)|^2dx \\&\leq \limsup_{n\to\infty} \int_{{{\mathbb T}}^d}|f_n(x)|^2dx \leq \lim_{n\to\infty} \int_{{{\mathbb T}}^d}|g_n(x)|^2dx = \int_{{{\mathbb T}}^d}|f(x)|^2dx. \end{aligned}$$ Thus, the limes inferior and superior coincide and $\|f_n\|_{L^2(\Omega)}\to \|f\|_{L^2(\Omega)}$. Together with the weak convergence of $(f_n)$, we deduce the strong convergence. Let $(n^k_\eps,W^k_\eps)$ be a weak solution to -. Inequalities and show the following bounds uniform in $\eps$: $$\begin{aligned} \|n^k_\eps\|_{L^\infty({{\mathbb T}}^d)} + \|(W^k_\eps+\eps)^{1/2}\na n^k_\eps\|_{L^2({{\mathbb T}}^d)} &\le C(\delta,{\triangle t}), \label{ex.eps1} \\ \|W^k_\eps\|_{L^\infty(\Omega)} + \|W^k_\eps\|_{H^1({{\mathbb T}}^d)} &\le C(\delta,{\triangle t}). \label{ex.eps2}\end{aligned}$$ By compactness, there exists a subsequence (not relabeled) such that, as $\eps\to 0$, $$\begin{aligned} & n_\eps^k\rightharpoonup^* n^k \quad\mbox{weakly* in }L^\infty({{\mathbb T}}^d), \label{ex.neps} \\ & W^k_\eps\to W^k\quad\mbox{strongly in }L^2({{\mathbb T}}^d), \quad W^k_\eps\rightharpoonup W^k\quad\mbox{weakly in }H^1({{\mathbb T}}^d). \label{ex.weps}\end{aligned}$$ Again, we need strong convergence for $\na n^k_\eps$. Since equation is degenerate, we obtain a weaker result. For this, let $\Psi\in C^2({{\mathbb R}})$ be strictly monotonically increasing and satisfy $\Psi'(t)=1/g(t)$ for $\delta\leq t\leq (1-\delta)/\eta$. Thus, we can apply Proposition \[prop.conv\] for $y_\eps=W^k_\eps+\eps$ and $y=W^k$ to conclude that, up to a subsequence, $$\begin{aligned} \label{ex.prop} & (W^k)^{1/2}(W^k_\eps+\eps)^{1/2}\na \Psi(n_\eps^k) \to \na(W^k\Psi(n^k))-\Psi(n^k)\na W^k, \\ & \mathrm{1}_{\{W^k>0\}}(n_\eps^k-n^k) \to 0 \quad\mbox{strongly in } L^2(\Omega). \nonumber\end{aligned}$$ The latter convergence implies that $n^k_\eps-n^k\to 0$ a.e.in $\{W^k>0\}$ and, by dominated convergence, $$\label{ex.pos} \frac{\mathrm{1}_{\{W^k>0\}}}{g(n^k_\eps)} \to \frac{\mathrm{1}_{\{W^k>0\}}}{g(n^k)}\quad\mbox{strongly in }L^2({{\mathbb T}}^d).$$ Now, let $y_\eps=(W^k_\eps+\eps)/g(n^k_\eps)$ and $y=W^k/g(n^k)$. We know that $y_\eps$, $y\in L^\infty({{\mathbb T}}^d)\cap H^1({{\mathbb T}}^d)$ and $ y_\eps\to y$ strongly in $L^2({{\mathbb T}}^d)$. Thus, we can again apply Proposition \[prop.conv\] with $\Psi=\mathrm{Id}$ and infer that $W^kn^k\in H^1({{\mathbb T}}^d)$ as well as $$\frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}(n^k-n^{k-1})\phi_0 dx +\int_\Omega\frac{\na(W^kn^k)-n^k\na W^k}{g(n^k)} \cdot\na\phi_0 dx=0$$ for all $\phi_0\in H^1({{\mathbb T}}^d)$. Let $\phi_1$ be a smooth test function. We use the test function $W^k\phi_1$ in the weak formulation of : $$\begin{aligned} \label{approx.eq.Wk.eps} 0 &= \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}(W^k_\eps-W^{k-1})W^k\phi_1 dx + \frac{2d-1}{2d}\int_{{{\mathbb T}}^d} \frac{\na W^k_\eps\cdot\na(W^k\phi_1)}{g(n^k_\eps)}dx \\ &\phantom{xx}{}+ U\int_{{{\mathbb T}}^d}W^k\frac{W^k_\eps}{g(n^k_\eps)} |\na n^k_\eps|^2\phi_1 dx =: I^1_\eps+I^2_\eps+I^3_\eps. \nonumber\end{aligned}$$ We pass to the limit $\eps\to 0$ term by term. By , $$I^1_\eps \to \frac{1}{{\triangle t}}\int_{{{\mathbb T}}^d}(W^k-W^{k-1})W^k\phi_1 dx.$$ For the integral $I^2_\eps$, we use the strong convergence and the weak convergence of $(\na W^k_\eps)$ in $L^2({{\mathbb T}}^d)$ to infer that $$\begin{aligned} I^2_\eps &= \int_{{{\mathbb T}}^d}\frac{\mathrm{1}_{\{W^k>0\}}}{g(n^k_\eps)} W^k\na W^k_\eps\cdot\na\phi_1 dx + \int_{{{\mathbb T}}^d}\frac{\mathrm{1}_{\{W^k>0\}}}{g(n^k_\eps)}\na W^k_\eps\cdot\na W^k \phi_1 dx \\ &\to \int_{{{\mathbb T}}^d}\frac{\mathrm{1}_{\{W^k>0\}}}{g(n^k)}W^k\na W^k\cdot\na\phi_1 dx + \int_{{{\mathbb T}}^d}\frac{\mathrm{1}_{\{W^k>0\}}}{g(n^k)}|\na W^k|^2\phi_1 dx \\ &= \int_{{{\mathbb T}}^d}\frac{\na W^k\cdot\na(W^k\phi_1)}{g(n^k)}dx.\end{aligned}$$ The remaining integral $I^3_\eps$ requires some work. As a preparation, using Proposition \[prop.conv\], we infer similarly to that $$h_\eps :=(W^k)^{1/2}\left(\frac{W^k_\eps+\eps}{g(n^k_\eps)}\right)^{1/2}\na n_\eps^k \to \frac{\na(W^kn^k)-n^k\na W^k}{g(n^k)^{1/2}}=:h$$ strongly in $L^2({{\mathbb T}}^d)$. Let $$\xi_\eps :=\bigg(\frac{W^k_\eps}{g(n^k_\eps)}\bigg)^{1/2}\na n_\eps^k.$$ Then $(\xi_\eps)$ is bounded in $L^2(\Omega)$ and admits a weakly convergent subsequence, i.e. $\xi_\eps\rightharpoonup\xi$ for some $\xi\in L^2({{\mathbb T}}^d)$. Similarly as in the proof of Proposition \[prop.conv\], i.e. with an argument as in , we can find that, up to a subsequence, $(W^k_\eps+\eps)^{1/2}\xi_\eps\rightharpoonup h$ weakly in $L^2({{\mathbb T}}^d)$ implying $(W^k)^{1/2}\xi=h$. In particular, $$f_\eps:=(W^k)^{1/2}\xi_\eps\rightharpoonup(W^k)^{1/2}\xi = h \quad\mbox{ weakly in }L^2({{\mathbb T}}^d).$$ Since $|f_\eps(x)|\leq |h_\eps(x)|$ for a.e. $x\in{{\mathbb T}}^d$ and all $\eps>0$, we can apply Lemma \[lem.weak.strong\] and obtain that, up to a subsequence, $f_\eps$ converges strongly in $L^2(\Omega)$. Thus, $$I^3_\eps = U\int_{{{\mathbb T}}^d}f_\eps^2\phi_1 dx \to U\int_{{{\mathbb T}}^d}\frac{|\na(W^kn^k)-n^k\na W^k|^2}{g(n^k)}\phi_1 dx.$$ Hence, passing to the limit $\eps\to 0$ in , we infer that $(n^k,W^k)$ solves -. Energy estimate --------------- We claim that the total energy $\int_{{{\mathbb T}}^d}W_{\rm tot}^k dx$ is nondecreasing in $k$. Let $(n^k_\eps,W^k_\eps)$ be a weak solution to -. Then $$\begin{aligned} {\triangle t}\int_{{{\mathbb T}}^d} & \bigg(W^k_\eps-W^{k-1} - \frac{U}{2}\big((n_\eps^k)^2-(n^{k-1})^2\big)\bigg)dx \\ &\ge {\triangle t}\int_{{{\mathbb T}}^d}\big(W^k_\eps-W^{k-1} - U(n_\eps^k-n^{k-1})n_\eps^k\big)dx.\end{aligned}$$ Taking the test functions $\phi_1=U$ in and $\phi_0=n_\eps^k$ in and subtracting both equations, the above integral becomes $$\begin{aligned} {\triangle t}\int_{{{\mathbb T}}^d} & \bigg(W^k_\eps-W^{k-1} - \frac{U}{2}\big((n_\eps^k)^2-(n^{k-1})^2\big)\bigg)dx \\ &\ge -U\int_{{{\mathbb T}}^d}\frac{W_\eps^k}{g(n_\eps^k)}|\na n_\eps^k|^2 dx + U\int_{{{\mathbb T}}^d}\frac{W_\eps^k+\eps}{g(n_\eps^k)}|\na n_\eps^k|^2 dx \ge 0.\end{aligned}$$ Thus, with the lower semi-continuity of the norm, we have $$\begin{aligned} \int_{{{\mathbb T}}^d}\bigg(W^{k-1}-\frac{U}{2}(n^{k-1})^2\bigg)dx &\le \liminf_{\eps\to 0} \int_{{{\mathbb T}}^d}\bigg(W_\eps^k-\frac{U}{2}(n^k_\eps)^2\bigg) dx \\ &\le \int_{{{\mathbb T}}^d}\bigg(W^{k}-\frac{U}{2}(n^{k})^2\bigg)dx.\end{aligned}$$ In view of mass conservation $\int_{{{\mathbb T}}^d}n^kdx=\int_{{{\mathbb T}}^d}n^{k-1}dx$, it follows by Jensen’s inequality that $$\begin{aligned} \int_{{{\mathbb T}}^d}W^k dx &\ge \int_{{{\mathbb T}}^d}W^{k-1}dx + \frac{U}{2}\int_{{{\mathbb T}}^d}\big((n^k)^2-(n^{k-1})^2\big)dx \nonumber \\ &\ge \int_{{{\mathbb T}}^d}\bigg(W^{k-1}-\frac{U}{2}(n^{k-1})^2\bigg)dx + \frac{U}{2} \bigg(\int_{{{\mathbb T}}^d}n^{k-1}dx\bigg)^2. \label{energy.est}\end{aligned}$$ This shows the energy inequality in . Finally, assumption gives $\int_{{{\mathbb T}}^d}W^k dx>0$ and consequently $W^k\not\equiv 0$. An estimate for the variance ---------------------------- We claim that the total variance $$V^k := \int_{{{\mathbb T}}^d}\bigg(W^k-\int_{{{\mathbb T}}^d}W^k dz\bigg)^2dx + U\int_{{{\mathbb T}}^d}W^{k-1}dx\int_{{{\mathbb T}}^d}\bigg(n^k-\int_{{{\mathbb T}}^d} n^kdz\bigg)^2 dx$$ is nonincreasing in $k$. For the proof, we observe that, taking the test function $\phi_1=1$ in the weak formulation of and performing the limit $\eps\to 0$, $$\label{ex.monow} \int_{{{\mathbb T}}^d}W^k dx \le \int_{{{\mathbb T}}^d}W^{k-1}dx.$$ Thus, by the energy estimate , $$\begin{aligned} \frac{U}{2}\int_{{{\mathbb T}}^d}(n^k)^2dx &\le \frac{U}{2}\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx - \int_{{{\mathbb T}}^d}(W^{k-1}-W^k)dx \nonumber \\ &\le \frac{U}{2}\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx. \label{ex.monon}\end{aligned}$$ We employ again to find that $$\begin{aligned} \bigg(\int_{{{\mathbb T}}^d} & W^{k-1}dx\bigg)^2 - \bigg(\int_{{{\mathbb T}}^d}W^k dx\bigg)^2 \\ &= \bigg(\int_{{{\mathbb T}}^d} W^{k-1}dx + \int_{{{\mathbb T}}^d} W^k dx\bigg) \bigg(\int_{{{\mathbb T}}^d} W^{k-1}dx - \int_{{{\mathbb T}}^d} W^k dx\bigg) \\ &\le \bigg(\int_{{{\mathbb T}}^d} W^{k-1}dx + \int_{{{\mathbb T}}^d} W^k dx\bigg) \frac{U}{2}\bigg(\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx - \int_{{{\mathbb T}}^d}(n^k)^2 dx\bigg).\end{aligned}$$ In view of , the second bracket on the right-hand side is nonnegative, such that leads to $$\bigg(\int_{{{\mathbb T}}^d} W^{k-1}dx\bigg)^2 - \bigg(\int_{{{\mathbb T}}^d}W^k dx\bigg)^2 \le U\int_{{{\mathbb T}}^d}W^{k-1}dx \bigg(\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx - \int_{{{\mathbb T}}^d}(n^k)^2 dx\bigg).$$ We take the test function $\phi_1=2{\triangle t}$ in : $$\begin{aligned} 0 &\ge 2\int_{{{\mathbb T}}^d}(W^k - W^{k-1})W^k dx + {\triangle t}\frac{2d-1}{d}\int_{{{\mathbb T}}^d}\frac{|\na W^k|^2}{g(n^k)}dx \\ &\ge \int_{{{\mathbb T}}^d}(W^k)^2 dx - \int_{{{\mathbb T}}^d}(W^{k-1})^2 dx + {\triangle t}\frac{2d-1}{d}\int_{{{\mathbb T}}^d}\frac{|\na W^k|^2}{g(n^k)}dx.\end{aligned}$$ Combining the previous two inequalities, we arrive at $$\begin{aligned} \int_{{{\mathbb T}}^d} (W^k)^2 dx &- \bigg(\int_{{{\mathbb T}}^d}W^{k} dx\bigg)^2 + {\triangle t}\frac{2d-1}{d}\int_{{{\mathbb T}}^d}\frac{|\na W^k|^2}{g(n^k)}dx \\ &\le \int_{{{\mathbb T}}^d}(W^{k-1})^2 dx - \bigg(\int_{{{\mathbb T}}^d}W^{k-1}dx\bigg)^2 \\ &\phantom{xx}{}+ U\int_{{{\mathbb T}}^d}W^{k-1}dx \bigg(\int_{{{\mathbb T}}^d}(n^{k-1})^2 dx - \int_{{{\mathbb T}}^d}(n^k)^2 dx\bigg).\end{aligned}$$ Since the measure of ${{\mathbb T}}^d$ is one, we have $$\int_{{{\mathbb T}}^d} (W^k)^2 dx - \bigg(\int_{{{\mathbb T}}^d}W^{k} dx\bigg)^2 = \int_{{{\mathbb T}}^d}\bigg((W^k)^2 - \bigg(\int_{{{\mathbb T}}^d}W^k dz\bigg)\bigg)^2 dx.$$ Thus, taking into account mass conservation $\int_{{{\mathbb T}}^d}n^k dx=\int_{{{\mathbb T}}^d}n^{k-1}dx$ and $\int_{{{\mathbb T}}^d}W^{k-1}dx \le\int_{{{\mathbb T}}^d}W^{k-2}dx$ (see ), $$\begin{aligned} \int_{{{\mathbb T}}^d} \bigg((W^k)^2 &- \bigg(\int_{{{\mathbb T}}^d}W^k dz\bigg)\bigg)^2 dx + {\triangle t}\frac{2d-1}{d}\int_{{{\mathbb T}}^d}\frac{|\na W^k|^2}{g(n^k)}dx \\ &\le \int_{{{\mathbb T}}^d}\bigg((W^{k-1})^2 - \bigg(\int_{{{\mathbb T}}^d}W^{k-1} dz\bigg)\bigg)^2 dx \\ &\phantom{xx}{}+ U\int_{{{\mathbb T}}^d}W^{k-2}dx \int_{{{\mathbb T}}^d}\bigg((n^{k-1})^2 - \bigg(\int_{{{\mathbb T}}^d}n^{k-1} dz\bigg)\bigg)^2 dx \\ &\phantom{xx}{}- U\int_{{{\mathbb T}}^d}W^{k-1}dx \int_{{{\mathbb T}}^d}\bigg((n^k)^2 - \bigg(\int_{{{\mathbb T}}^d}n^k dz\bigg)\bigg)^2 dx,\end{aligned}$$ and the claim follows after using the lower-semicontinuity of the $L^2$-norm. Proof of Theorem \[thm.ex2\] ---------------------------- Let $d=1$. The second inequality in implies that $$\int_{{{\mathbb T}}^1}|\pa_x W^k|^2dx\leq \frac{G}{{\triangle t}}V^{k-1},$$ where $G=\max_{\delta\le s\le\{n^{k-1}\|_{L^\infty({{\mathbb T}})}\}}g(s) \ge\|g(n^k)\|_{L^\infty({{\mathbb T}})}$. By the mean-value theorem, there exists $x_0\in{{\mathbb T}}$ such that $$W^k(x) = W^k(x_0) + \int^x_{x_0}\pa_x W^k(z)dz \ge \int_{{{\mathbb T}}}W^k(z)dz - \int_{{{\mathbb T}}}|\pa_x W^k|dz.$$ Then, using Jensen’s inequality and the energy estimate in , $$\begin{aligned} W^k(x) &\ge \int_{{\mathbb T}}W^kdx - \int_{{\mathbb T}}|\pa_x W^k|^2 dx \\ &\ge \int_{{\mathbb T}}W^{k-1}dx - \frac{U}{2}\int_{{\mathbb T}}\bigg(n^{k-1}-\int_{{\mathbb T}}n^{k-1}dz\bigg)^2dx - \frac{G}{{\triangle t}}V^{k-1}.\end{aligned}$$ By definition of $V^k$, the right-hand side is positive if $$\begin{aligned} \overline{W^{k-1}} > \frac{G}{{\triangle t}}\int_{{\mathbb T}}(W^{k-1}-\overline{W^{k-1}})^2 dx + U\bigg(\frac{G}{{\triangle t}}\overline{W^{k-2}} + \frac12\bigg) \int_{{\mathbb T}}(n^{k-1}-\overline{n^{k-1}})^2 dx,\end{aligned}$$ which is our assumption. Since $W^{k}\in H^1({{\mathbb T}})\hookrightarrow C^0([0,1])$, we conclude that $W^k>0$ in $[0,1]$. Then we can use $\phi_1=\phi/W^k$ as a test function in and obtain the standard weak formulation of for test functions $\phi\in H^1({{\mathbb T}})$. Furthermore, for $\phi\in H^1({{\mathbb T}})$, $$\begin{aligned} \int_{{\mathbb T}}n^k\pa_x\phi dx &= \int_{{\mathbb T}}n^k\pa_x\bigg(W^k\pa_x\bigg(\frac{\phi}{W^k}\bigg) + \pa_x W^k\frac{\phi}{W^k}\bigg)dx \\ &= -\int_{{\mathbb T}}\pa_x(n^kW^k)\frac{\phi}{W^k}dx + \int_{{\mathbb T}}n^k\pa_xW^k\frac{\phi}{W^k}dx,\end{aligned}$$ showing that $n^k\in H^1({{\mathbb T}})$ and finishing the proof. Numerical simulations {#sec.num} ===================== We solve the one-dimensional equations and on the torus in conservative form, i.e. for the variables $n$ and $W_{\rm tot}$. The equations are discretized by the implicit Euler method and solved in a semi-implicit way: $$\begin{aligned} \frac{1}{{\triangle t}}(n^k-n^{k-1}) &= \pa_x\bigg(\frac{W^{k-1}\pa_x n^k}{n^{k-1}(1-\eta n^{k-1})} \bigg), \label{num.n} \\ \frac{1}{{\triangle t}}(W^k_{\rm tot}-W^{k-1}_{\rm tot}) &= \pa_x\bigg( \frac{\pa_x W^k}{2n^k(1-\eta n^k)} + \frac{UW^k}{1-\eta n^k}\pa_x n^k\bigg), \label{num.w}\end{aligned}$$ where $W^k_{\rm tot}=W^k-(U/2)(n^k)^2$, $x\in{{\mathbb T}}=(0,1)$. The spatial derivatives are discretized by centered finite differences with constant space step $\triangle x>0$. For given $(W^{k-1},n^{k-1})$, the first equation is solved for $n^k$. This solution is employed in the second equation which is solved for $W^k_{\rm tot}$. Finally, we define $W^k=W^k_{\rm tot}+(U/2)(n^k)^2$. We choose the parameters $U=10$, $\eta=1$, ${\triangle t}=10^{-5}$, and $\triangle x=10^{-2}$. The initial energy $W^0$ is constant and the initial density equals $$n^0(x) = \left\{\begin{array}{ll} 3/4 &\quad\mbox{for }1/4\le x\le 3/4, \\ 1/4 &\quad\mbox{else}, \end{array}\right. \quad x\in[0,1].$$ ![Evolution of $(n,W)$ with initial energy $W^0=1$.[]{data-label="fig.evol1"}](Bild1.pdf){width="140mm"} ![Evolution of $(n,W)$ with initial energy $W^0=1/4$.[]{data-label="fig.evol2"}](Bild2.pdf){width="140mm"} The time evolution of the particle density and energy is shown in Figure \[fig.evol1\] for initial energy $W^0=1$. The variables converge to the constant steady state $(n^\infty,W^\infty)$ as $t\to\infty$, which is almost reached after time $t=0.1$. Since equations - are conservative, the total particle number $\int_0^1 n(x,t)dx$ and the total energy $\int_0^1 W_{\rm tot}(x,t)dx$ are constant in time. Consequently, the values for the steady state can be computed explicitly. We obtain for $W^0=1$, $$\begin{aligned} n^\infty &= \int_0^1 n^0(x)dx = \frac12, \\ W^\infty &= W^\infty_{\rm tot} + \frac{U}{2}(n^\infty)^2 = \int_0^1\bigg(W^0(x)-\frac{U}{2}n^0(x)^2\bigg)dx + \frac{U}{2}(n^\infty)^2 = \frac{11}{16}.\end{aligned}$$ The energy stays positive for all times, so the high-temperature equations are strictly parabolic, and the convergence to the (constant) steady state is quite natural. The situation is different in Figure \[fig.evol2\], where the particle density converges to a [*nonconstant*]{} steady state $n^\infty$ (we have chosen $W^0=1/4$). This can be understood as follows. By contradiction, let both the particle density and energy be converging to a constant steady state. Then $n^\infty=1/2$ (see the above calculation) and $$W^\infty = \int_0^1\bigg(W^0(x)-\frac{U}{2}n^0(x)^2\bigg)dx + \frac{U}{2}(n^\infty)^2 = -\frac{1}{16}.$$ However, this contradicts the fact that the energy $W$ is nonnegative which follows from the maximum principle. Therefore, it is plausible that either $n$ or $W$ cannot converge to a constant. If $n^\infty$ is not constant, $W^\infty\na n^\infty$ is constant only if $W^\infty=0$. Thus, it is reasonable that the energy converges to zero, while $n^\infty$ is not constant. One may say that there is not sufficient initial “reverted” energy to level the particle density. ![Decay rates for various initial energies.[]{data-label="fig.decay"}](Bild3.pdf){width="140mm"} Another difference between Figure \[fig.evol1\] and Figure \[fig.evol2\] is the time scale. For larger initial energies, the convergence to equilibrium is faster. In fact, Figure \[fig.decay\] shows that the decay of the $\ell^2$ norm of $n(t)-n^\infty$ and $W(t)-W^\infty$ is exponential. Here, we have chosen the initial particle density $n^0(x)=\frac14$ for $0\le x<\frac12$, $n^0(x)=\frac34$ for $\frac12\le x<1$, and the initial energy $W^0\in\{\frac14,\frac12,1\}$. For $W^0\in\{1,\frac12\}$, we have $n^\infty=\frac12$ and $W^\infty=\max\{0,W^0-U/32\}$. For $W^0=\frac14$, it holds that $W^\infty=0$ and we have set $n^\infty(x)=n(x,2)$. Finally, we compute the numerical convergence rates for different space and time step sizes $\triangle x$ and ${\triangle t}$, respectively. Since there is no explicit solution available, we choose as reference solution the solution to - with $\triangle x=1/1680$ (for the computation of the spatial $\ell^2_x$ error) and ${\triangle t}=1/5040$ (for the computation of the $\ell^2_t\ell^2_x$ error). Figure \[fig.conv1\] shows that the temporal error is linear in ${\triangle t}$, and Figure \[fig.conv2\] indicates that the spatial error is quadratic in $\triangle x$. These values are expected in view of our finite-difference discretization and they confirm the validity of the numerical scheme. ![Numerical convergence in time.[]{data-label="fig.conv1"}](Bild4.pdf){width="110mm"} ![Numerical convergence in space.[]{data-label="fig.conv2"}](Bild5.pdf){width="110mm"} Calculation of some integrals ============================= We recall that $\eps(p)=-2\eps_0\sum_{k=1}^d\cos(2\pi p_i)$. Then $u_i(p)=(\pa\eps/\pa p_i)(p)=4\pi\eps_0\sin(2\pi p_i)$, and we calculate $$\begin{aligned} \int_{{{\mathbb T}}^2}\eps^2 dp &= 4\eps_0^2\int_0^1\sum_{k=1}^d\cos^2(2\pi p_k)dp_k = 2d\eps_0^2, \label{2.eps2} \\ \int_{{{\mathbb T}}^d} u_iu_j dp &= (4\pi\eps_0)^2\delta_{ij}\int_0^1\sin^2(2\pi p_i) dp_i = \frac12(4\pi\eps_0)^2\delta_{ij}, \label{2.uu} \\ \int_{{{\mathbb T}}^d} \eps u_iu_j dp &= -2\eps_0(4\pi\eps_0)^2\sum_{k=1}^d\int_{{{\mathbb T}}^d} \cos(2\pi p_k)\sin(2\pi p_i)\sin(2\pi p_j)dp \label{2.epsuu} \\ &= -2\eps_0(4\pi\eps_0)^2\delta_{ij}\int_0^1\cos(2\pi p_i)\sin^2(2\pi p_i)dp_i = 0, \nonumber \\ \int_{{{\mathbb T}}^d} \eps^2 u_iu_j dp &= \frac13\int_{{{\mathbb T}}^d}\frac{\pa}{\pa p_i}(\eps^3) \frac{\pa\eps}{\pa p_j}dp = -\frac13\int_{{{\mathbb T}}^d}\eps^3\frac{\pa^2\eps}{\pa p_i\pa p_j}dp = 0 \quad\mbox{if }i\neq j, \nonumber\end{aligned}$$ since $\pa^2\eps/\pa p_i\pa p_j=0$ for $i\neq j$. We compute the integral $\int_{{{\mathbb T}}^d}\eps^2 u_i^2dp$. First, let $d=1$. Then $$\int_{{{\mathbb T}}} \eps^2 u_1^2 dp_1 = 4\eps_0^2(4\pi\eps_0)^2\int_0^1\cos^2(2\pi p_1)\sin^2(2\pi p_1)dp_1 = \frac{\eps_0^2}{2}(4\pi\eps_0)^2 = 8\pi^2\eps_0^4.$$ Furthermore, for $d>1$, $$\begin{aligned} \int_{{{\mathbb T}}^d} & \eps^2 u_i^2 dp = 4\eps_0^2(4\pi\eps_0)^2\int_{{{\mathbb T}}^d}\bigg(\sum_{k=1,\,k\neq i}^d \cos(2\pi p_k) + \cos(2\pi p_i)\bigg)^2\sin^2(2\pi p_i)dp \\ &= 4\eps_0^2(4\pi\eps_0)^2\left(\int_{{{\mathbb T}}^{d}}\bigg(\sum_{k=1,\,k\neq i}^d \cos(2\pi p_k)\bigg)^2\sin^2(2\pi p_i)dp + \int_{{\mathbb T}}\cos^2(2\pi p_i)\sin^2(2\pi p_i)dp_i\right) \\ &= 4\eps_0^2(4\pi\eps_0)^2 \int_{T^{d}} \sum_{k=1,\,k\neq i}^d\cos^2(2\pi p_k)\sin^2(2\pi p_i)dp + 8\pi^2\eps_0^4 \\ &= 4\eps_0^2(4\pi\eps_0)^2\sum_{k=1,\,k\neq i}^d\int_0^1\cos^2(2\pi p_k)dp_k \int_0^1\sin^2(2\pi p_i)dp_i + 8\pi^2\eps_0^4 \\ &= (d-1)\eps_0^2(4\pi\eps_0)^2 + 8\pi^2\eps_0^4 = 8(2d-1)\pi^2\eps_0^4.\end{aligned}$$ We conclude that $$\label{2.eps2uu} \int_{{{\mathbb T}}^d}\eps^2 u_iu_j dp = 8(2d-1)\pi^2\eps_0^4\delta_{ij}.$$ [11]{} G. Alì and V. Romano. 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--- abstract: 'The degree-degree correlation is important in understanding the structural organization of a network and the dynamics upon a network. Such correlation is usually measured by the assortativity coefficient $r$, with natural bounds $r \in [-1,1]$. For scale-free networks with power-law degree distribution $p(k) \sim k^{-\gamma}$, we analytically obtain the lower bound of assortativity coefficient in the limit of large network size, which is not -1 but dependent on the power-law exponent $\gamma$. This work challenges the validation of assortativity coefficient in heterogeneous networks, suggesting that one cannot judge whether a network is positively or negatively correlated just by looking at its assortativity coefficient.' author: - 'Dan Yang$^{1}$' - 'Liming Pan$^{1}$' - 'Tao Zhou$^{1,2}$' title: 'Lower bound of assortativity coefficient in scale-free networks' --- Introduction ============ The last decade has witnessed a great change where the studies on networks, being of very limited interests mainly from mathematical society in the past, have received a huge amount of attention from many branches of sciences [@Newman2010; @Chen2012]. The fundamental reason is that the network structure could well describe the interacting pattern of individual elements, which leads to many complex phenomena in biological, social, economic, communication, transportation and physical systems. Characterizing the structural features [@Costa2007; @Barthelemy2011; @Holme2012] is the foundation of the correct understanding about the dynamics of networks [@Albert2002; @Dorogovtsev2002; @Newman2003; @Boccaletti2006], the dynamics on networks (e.g., epidemic spreading [@Zhou2006; @Pastor-Satorras2015], transportation [@Tadic2007; @Wang2007], evolutionary game [@Szabo2007; @Perc2010], synchronization [@Arenas2008], and other social and physical processes [@Dorogovtsev2008; @Castellano2009]), as well as the network-related applications [@Fortunato2010; @Costa2011; @Lu2011; @Lu2012]. Thus far, the characterization of networks is still a challenging task, because some seemingly standalone structural properties are indeed statistically dependent to each other, resulting in many non-trivial structural constrains that are rarely understood [@Zhou2007; @Lopez2008; @Baek2012; @Orsini2015]. Let’s look at the five fundamental structural features of general networks: (i) the density $\rho$, quantified by the ratio of the number of edges $M$ to the possibly maximum value $N(N-1)/2$, where $N$ is the number of nodes; (ii) the degree $k_i$ of a node $i$ (defined as the number of $i$’s associated edges) as well as the degree distribution $p(k)$ [@Barabasi1999]; (iii) the average distance $\langle d \rangle$ over all pairs of nodes in a connected network [@Watts1998]; (iv) the assortativity coefficient $r$ that quantifies the degree-degree correlation (with mathematical definition shown later) [@Newman2002; @Newman2003b]; (v) the clustering coefficient $c_i$ of a node (defined as the ratio of the number of edges between $i$’s neighbors to the possibly maximum value) and the average clustering coefficient $\langle c \rangle$ over all nodes with degree larger than 1 [@Watts1998]. Even though these measures are very simple compared with many other recently proposed network measures, there are complicated correlations between them. For example, scale-free networks are of very low density [@Genio2011] and usually small average distance [@Cohen2003], a node of larger degree is often with smaller clustering coefficient [@Ravasz2003; @Zhou2005], a very high-density network must be with high clustering coefficient and small-world property [@Markov2013; @Orsini2015], networks with high clustering coefficients tend to have high assortativity coefficients [@Foster2011], to name just a few. In this paper, we focus on the mathematical relationship between assortativity coefficient and degree distribution. In particular, as the majority of real-world networks are found to be very heterogeneous in degree, we consider a typical class of networks [@Caldarelli2007; @Barabasi2009]: the scale-free networks with power-law degree distribution $p(k)\sim k^{-\gamma}$, where $\gamma>0$ is called the power-law exponent. Both the degree distribution and assortativity coefficient are shown to be critical in understanding the structural organization of a network and the dynamics upon a network [@Dorogovtsev2002; @Albert2002; @Newman2003; @Boccaletti2006; @Caldarelli2007; @Barabasi2009; @Barrat2008; @Noldus2015], hence to uncover the correlation between these two fundamental measures could help us in clarifying whether a structural property is significant or just a statistical consequence of another property [@Orsini2015]. Menche *et al.* [@Menche2010] analyzed the maximally disassortative scale-free networks and found that the lower bound of assortativity coefficient, $r_{\min}$, will approach to zero when $2<\gamma<4$ as the increase of the network size $N$. Instead of an explicit value, they only provide the order of $r_{\min}$ in the large $N$ limit. Dorogovtsev *et al.* [@Dorogovtsev2010] considered a specific class of recursive trees with power-law degree distribution and found that the assortativity coefficient is always zero. Raschke *et al.* [@Raschke2010] showed both analytical and numerical results that the assortativity coefficient depends on the network size $N$. Litvak and Van Der Hofstad [@Litvak2013; @Hofstad2014] highlighted the problem that the assortativity coefficient in disassortative networks systematically decreases with the network size, and they provide some mathematical explanation on this phenomenon. In particular, they showed that in the large $N$ limit, the assortativity coefficient is no less than zero [@Hofstad2014]. The above analyses, either on network configuration model with given degree distribution or on some specific scale-free network models, showed non-trivial dependencies between $r$ and $N$, as well as between $r$ and $\gamma$. However, to our best knowledge, an explicit lower bound of $r$ in scale-free networks has not been reported in the literatures. In this paper, we will derive the lower bound of $r$ in scale-free networks for $N\rightarrow \infty$, and then most of the above-mentioned conclusions can be considered as direct deductions from our results. This paper is organized as follows. In the next section, we will analytically derive the lower bound of assortativity coefficient. In section III and section IV, we will validate the theoretic bound via simulation and empirical analysis, respectively. Lastly, we will draw the conclusion and discuss the theoretical significance and practical relevance of our contribution in section V. Theoretic Bound =============== A network is called assortative if nodes tend to connect with other nodes with similar degrees, and disassortative if high-degree nodes tend to connect with low-degree nodes. In this paper, we focus on undirected simple networks, of which the degree correlation is usually measured by the assortativity coefficient, defined as [@Newman2002; @Newman2003b]: $$\label{eq:pearson} r=\frac{{{M}^{-1}}\sum\nolimits_{i}{{{j}_{i}}{{k}_{i}}}-{{[{{M}^{-1}}\sum\nolimits_{i}{\frac{1}{2}({{j}_{i}}+{{k}_{i}})}]}^{2}}}{{{M}^{-1}}\sum\nolimits_{i}{\frac{1}{2}({{j}_{i}}^{2}+{{k}_{i}}^{2})}-{{[{{M}^{-1}}\sum\nolimits_{i}{\frac{1}{2}({{j}_{i}}+{{k}_{i}})}]}^{2}}},$$ where $M$ is the number of edges and ${{j}_{i}}$, ${{k}_{i}}$ are the degrees of the nodes at the ends of the $i$th edge, with $i=1,2,...,M$. The assortativity coefficient $r$ is actually the Pearson correlation coefficient between the degrees of neighboring nodes, which is supposed to have natural bounds $r \in [-1,1]$, where $r=-1$ indicates the completely negative correlation, while $r=+1$ suggests the perfectly positive correlation. It is straightforward to accept that a network is assortative when $r>0$ and disassortative when $r<0$ [@Newman2002; @Newman2003b; @Newman2003c]. However, such affirm could be wrong since the natural bounds do not imply a uniform distribution of $r$ in $[-1,1]$ and furthermore, the bounds of $r$ in a given network ensemble can be different from $[-1,1]$. For example, if for a given network ensemble, $r$ has non-trivial bounds $[0,1]$, claiming a network with a small positive $r$ to be assortative can be unreliable since it’s quite close to the lower bound of all possible values. In such case, a network with zero or very small assortativity coefficient could be disassortative or assortative. In fact, Dorogovtsev *et al.* [@Dorogovtsev2010] found a specific class of growing trees, which are strongly correlated but the assortativity coefficients are always zero. ![(Color online) Generating networks from a given degree sequence $\underline{k_{0}}=\{k_{1},k_{2},k_{3},k_{4}\}=\{1,2,2,3\}$. (a) Each stub is labeled by the degree of the node it is attached, and thus the stub sequence is obtained as $\underline{x_{0}}=\{x_{1},x_{2},\cdot\cdot\cdot,x_{7},x_{8}\}=\{1,2,2,2,2,3,3,3\}$. Realizations from the network ensemble defined by the degree sequence $\underline{k_{0}}$ can be represented by the corresponding $\underline{x_{0}}$, such as (b) $g_{\theta_{0}}=\{(x_{1},x_{6}),(x_{2},x_{7}),(x_{3},x_{5}),(x_{4},x_{8})\}$, one possible realization of simple networks without self-loops or multiple edges; and (c) $g_{\theta_{1}}=\{(x_{1},x_{6}),(x_{2},x_{7}),(x_{3},x_{8}),(x_{4},x_{5})\}$, one possible realization of networks with self-loops and multiple edges. []{data-label="illustrate"}](figure1.eps){width="48.00000%"} To uncover the non-trivial correlation between assortativity coefficient and degree distribution, we consider the most widely studied network ensemble, where the degree sequence is fixed and networks satisfying the degree constrain appear with equal probabilities [@JPark2004; @Bianconi2008]. Networks in such ensemble can be sampled by the uniform configuration model (UCM) [@Newman2001]. Initially, each node $i$ is assigned a given degree $k_i$ according to the degree sequence and thus attached by $k_i$ half-edges (‘stubs’). Then, random pairs of stubs are connected to form edges, without any self-loops and multiple edges. Considering a degree sequence $\underline{k}=\{k_{1},k_{2},k_{3},\cdot\cdot\cdot,k_{N}\}$ with $k_i$ being arranged in a non-decreasing order as $k_{1}\leq k_{2}\leq \cdot\cdot\cdot \leq k_{N}$. Obviously, there are $\sum^N_i k_i = 2M$ stubs. By labeling each stub with the degree of the attached node, one can obtain the stub sequence $\underline{x}=\{x_{1},x_{2},x_{3},\cdot\cdot\cdot,x_{2M}\}$, which is also arranged in a non-decreasing order as $x_{1}\leq x_{2}\leq \cdot\cdot\cdot \leq x_{2M}$. This procedure is illustrated in Fig. \[illustrate\](a). The four circles denote four nodes with degrees $\underline{k_{0}}=\{k_{1},k_{2},k_{3},k_{4}\}=\{1,2,2,3\}$. The stubs are labeled with the degrees of nodes they are attached to, as $\underline{x_{0}}=\{x_{1},x_{2},\cdot\cdot\cdot,x_{7},x_{8}\}=\{1,2,2,2,2,3,3,3\}$. It is clear that a possible realization of networks from the degree sequence $\underline{k}$ can also be represented by the corresponding stub sequence $\underline{x}$ as: $g_{\theta}=\{(x_{\theta_{1}},x_{\theta_{2}}),(x_{\theta_{3}},x_{\theta_{4}}),(x_{\theta_{5}},x_{\theta_{6}}),\cdot\cdot\cdot,(x_{\theta_{2M-1}},x_{\theta_{2M}})\}$, where $\theta$ is a rearrangement of the sequence $\{1,2,3,\cdot\cdot\cdot,2M\}$, and $\theta_{i}$ is the $i$-th element in the rearranged sequence. Consider then the degree sequence $\underline{k_{0}}$ in Fig. \[illustrate\](a), by rearranging $\underline{x_{0}}$, we can get all possible networks generated from $\underline{k_{0}}$. As an illustration, Fig. \[illustrate\](b) shows a possible realization of the network ensemble by $\underline{k_{0}}$: $g_{\theta_{0}}=\{(x_{1},x_{6}),(x_{2},x_{7}),(x_{3},x_{5}),(x_{4},x_{8})\}$. For each $g_{\theta}$, the corresponding assortativity coefficient $r_{\theta}$ can be rewritten via the $\underline{x}$ sequence as: $$\begin{aligned} \label{eq:rtheta} r_\theta = \frac{{M^{-1}\sum_{i=1}^M x_{\theta_{2i-1}}x_{\theta_{2i}} - {\langle x \rangle}^{2}}}{\langle x^{2}\rangle-\langle x\rangle^{2}},\end{aligned}$$ where $$\langle x \rangle =\frac{1}{M}\sum_{i=1}^M(x_{\theta_{2i-1}}\!\!\!+\!\!x_{\theta_{2i}})$$ and $$\langle x^{2} \rangle = \frac{1}{M}\sum_{i=1}^M(x^{2}_{\theta_{2i-1}}\!\!\!+\!\!x^{2}_{\theta_{2i}}).$$ From Eq. (\[eq:rtheta\]), it is observed that the first term in the dominator $$S_{\theta}=\frac{1}{M}\sum_{i=1}^M x_{\theta_{2i-1}}x_{\theta_{2i}}$$ is the only term that the rearrangement $g_{\theta}$ affects, while other terms are fixed given $\underline{x}$. That is to say, $g_{\theta_{\mathrm{min}}}$ which minimize $S_{\theta}$ will also minimize the assortativity coefficient $r$. Actually $g_{\theta_{\mathrm{min}}}$ can be determined based on the branch-and-bound idea [@Hallin1992; @Guo2015], which is $g_{\theta_{\mathrm{min}}}=\{(x_{1},x_{2M}),(x_{2},x_{2M-1}),(x_{3},x_{2M-2}),\cdot\cdot\cdot,(x_{M},x_{M+1})\}$ (see *Appendix A* for the proof of the above proposition). Notice that, in general $g_{\theta}$ can not guarantee the absence of self-loops and multiple edges. For example, as showed in Fig. \[illustrate\](c), $g_{\theta_{1}}=\{(x_{1},x_{6}),(x_{2},x_{7}),(x_{3},x_{8}),(x_{4},x_{5})\}$ is one possible realization in the network ensemble $g_\theta$, which has both one self-loop and one multiple edge. However we can prove that if $p(k)\sim k^{-\gamma}$ with $\gamma >2$, the self-loops and multiple edges will vanish in the thermodynamical limit (i.e., $N\rightarrow \infty$) for the specific network generated by $g_{\theta_{\mathrm{min}}}$ (see *Appendix B* for the proof). Taking $g_{\theta_{\mathrm{min}}}$ into Eq. (\[eq:rtheta\]), the lower bound of the assortativity coefficient $r$ is: $$\label{eq:rmin} r_{{\mathrm{min}}}=\frac{{M^{-1}\sum_{i=1}^M x_{i}x_{2M+1-i} - {\langle x\rangle}^{2}}}{\langle x^{2}\rangle-\langle x\rangle^{2}},$$ which holds for any fixed degree sequence that allows $g_{\theta_{\mathrm{min}}}$ to be a simple network without any self-loops or multiple edges. Now we consider the expected value $\mathrm{E}[r_{\mathrm{min}}]$ when the degree sequence is drawn from a power-law distribution $p(k)\sim k^{-\gamma}$. Del Genio *et al.* [@Genio2011] showed a fundamental mathematical constraint that when $0 \leq \gamma \leq 2$, the graphical fraction (i.e., the ratio of the number of graphical sequences that can be realized as simple networks to the total number of degree sequences with an even degree sum in a given ensemble) approaches zero, and in fact the majority of real scale-free networks are of power-law exponents $\gamma>2$ [@Caldarelli2007; @Clauset2009]. Therefore, in this work, we focus on scale-free networks with $\gamma>2$. Obviously, if the degrees $\underline{k}$ are sampled from a power-law distribution $p(k)\sim k^{-\gamma}$, the corresponding stubs $\underline{x}$ also follow a power-law distribution $f(x)\sim x^{-\beta}$ with $\beta=\gamma-1$. ![(Color online) Simulated results of $r_{\mathrm{min}}$ versus $k_{\mathrm{min}}$ for different degree exponents $\gamma$. The five curve from up to bottom denote the sample average with $\gamma =2.5$, $\gamma =4.5$, $\gamma =5.5$, $\gamma =7.5$ and $\gamma =9.5$, respectively. The network size is fixed as $N=10^6$. Each data point is obtained by averaging over 2000 independent runs. It is observed that $r_{\mathrm{min}}$ converges as the increase of $k_{\mathrm{min}}$. []{data-label="RminKm"}](figure2.eps){width="48.00000%"} As $\underline{x}$ is arranged in non-decreasing order, we have $x_i\leq x_M, \forall i \leq M$. Thus the following inequality always holds: $$\label{eq:ineq1} \begin{split} {{S}_{\theta}}=&{{M}^{-1}}\sum\nolimits_{i=1}^{M}{{{x}_{i}}{{x}_{2M-1+i}}} \\ \le & {{x}_{M}}{{M}^{-1}}\sum\nolimits_{i=1}^{M}{{{x}_{2M-1+i}}}. \end{split}$$ When $N\rightarrow\infty$, the right hand side of Eq. (\[eq:ineq1\]) can be approximated by continuous variables as: $$\begin{aligned} \label{eq:eqnar1} {{x}_{M}}{{M}^{-1}}\sum\nolimits_{i=1}^{M}{{{x}_{2M+1-i}}}\approx 2{{x}_{M}}\int_{{{x}_{M}}}^{N-1}{xf(x)dx}. %&=&2{{x}_{M}}\frac{\beta -1}{{{x}_{min}}^{1-\beta }}\left( \frac{1}{2-\beta }{{k}_{c}}^{2-\beta }-\frac{1}{2-\beta }{{x}_{M}}^{2-\beta } \right)\\\end{aligned}$$ Doing the integral in Eq. (\[eq:eqnar1\]), and taking into account the fact that $x_{M}\sim M^{0}$ [@Baek2012], we have: $$\begin{aligned} \label{eq:eqnar2} &S_{\theta}=&\left\{ \begin{matrix} O\left( {{N}^{3-\gamma }} \right)~~~~2<\gamma <3 \\ O\left( \ln N \right)~~~~~~~~~~~~\gamma=3 \\ O\left( 1 \right)~~~~~~~~~~3<\gamma <4. \\ \end{matrix} \right .\end{aligned}$$ Similarly, we also have: $$\label{eq:x} \langle x\rangle=\left\{ \begin{matrix} O\left( {{N}^{3-\gamma }} \right)~~~~2<\gamma <3 \\ O\left( \ln N \right)~~~~~~~~~~~~\gamma=3 \\ O\left( 1 \right)~~~~~~~~~~3<\gamma <4 \\ \end{matrix} \right .$$ and $$\label{eq:x} \langle x^2 \rangle=\left\{ \begin{matrix} O\left( {{N}^{4-\gamma }} \right)~~~~2<\gamma <4 \\ O\left( \ln N \right)~~~~~~~~~~~~\gamma=4. \\ \end{matrix} \right .$$ Evidently, the denominator has the strict larger order when $2<\gamma \leq 4$, hence $$\label{eq:Rmin0} r_{\mathrm{min}}=0$$ in the thermodynamic limit $N\rightarrow \infty$. In the case $\gamma>4$, we show numerically that $r_{\mathrm{min}}$ decreases with $k_{\mathrm{min}}$ and eventually approaches to a steady value as $k_{\mathrm{min}}$ becomes large, as shown in Fig. \[RminKm\]. Therefore we set a relatively large $k_{\mathrm{min}}$, which also allows us to take variables in $\underline{x}$ as continuous. Below, we carry out our derivation based on continuous variables, thus the distribution function of $\underline{x}$ can be normalized as $$\tilde{f}(x)=\frac{\beta -1}{{{k}_{\mathrm{min}}}}{{(\frac{x}{{{k}_{\mathrm{min}}}})}^{-\beta }}$$ For $\gamma>4$, both of the first or second moment of $\underline{x}$ exsit, and they can be calculated as follows, $$\langle x\rangle =\int_{{{k}_{\mathrm{min}}}}^{N-1}{x{\tilde{f}}\,}(x)dx=\frac{\gamma -2}{\gamma -3}{{k}_{\mathrm{min}}}$$ and $$\langle x^2\rangle =\int_{{{k}_{\mathrm{min}}}}^{N-1}{x^{2}{\tilde{f}}\,}(x)dx=\frac{\gamma -2}{\gamma -4}{k}_{\mathrm{min}}^{2}$$ when $N\rightarrow\infty$. ![(Color online) Simulated results of $r_{\mathrm{min}}$ versus the theoretical prediction by Eq. (6). The blue hexagons represent the theoretical values of the lower bound of assortativity coefficients of a series of randomly generated scale-free networks with different degree exponents. The red squares denote the simulated values, obtained by the degree-preserving edge-rewiring procedure. The network size is set to be $N=50000$ and the minimum degree is set as $k_{\min}=50$. Each data point is averaged over 500 independent runs. The inset shows the results of a single run, where the simulation result is still perfectly in accordance with the theoretical prediction.[]{data-label="fig2"}](figure3.eps){width="48.00000%"} We divide each element in $\underline{x}$ by ${k}_{\mathrm{min}}$, then we get a new sequence $\underline{\hat{x}}$, which certainly follows the power-law distribution with the same exponent $\beta$, but the minimum value is rescaled to be $1$, i.e., $\hat{f}(x)=(\beta-1)x^{-\beta}$. Obviously $$\label{SS} {S}_{\theta_{\mathrm{min}}}={k}_{\mathrm{min}}^{2}\hat{S}_{\theta_{\mathrm{min}}}.$$ To get the expected value of $\hat{S}_{\theta_{\mathrm{min}}}$, we introduce the joint distribution of order statistics $p\left( {{\hat{x}}_{m}}={{t}_{1}},{{\hat{x}}_{n}}={{t}_{2}} \right)$, which is the probability that the $m$th element in the $\underline{\hat{x}}$ takes value $t_1$, while the $n$th takes value $t_2$. It reads [@David2003] $$\begin{split} p&\left( {{\hat{x}}_{m}}={{t}_{1}},{{\hat{x}}_{n}}={{t}_{2}} \right)=(2M)!\frac{{{\left[ F\left( {{t}_{1}} \right) \right]}^{m-1}}}{\left( m-1 \right)!}\\ \times& \frac{{{\left[ F\left( {{t}_{2}} \right)-F\left( {{t}_{1}} \right) \right]}^{n-m-1}}}{\left( n-m-1 \right)!}\\ \times& \frac{{{\left[ 1-F\left( {{t}_{2}} \right) \right]}^{2M-n}}}{\left( 2M-n \right)!}\hat{f}\left( {{t}_{1}} \right)\hat{f}\left( {{t}_{2}} \right), \end{split}$$ where $F(t)$ is the cumulative distribution function $$%F(t)=\int_{{{k}_{m}}}^{t}{\widetilde{f}(x)dx}=1-{{(\frac{t}{{{k}_{m}}})}^{1-\beta }} F(t)=\int_{1}^{t}{\hat{f}(x)dx}=1-t^{1-\beta}.$$ Using a shorthand $c=\frac{1}{\beta-1}$, we express $\mathrm{E}\left[ {{\hat{x}}_{i}}{{\hat{x}}_{2M+1-i}} \right]$ in terms of $p\left( {{\hat{x}}_{m}}={{t}_{1}},{{\hat{x}}_{n}}={{t}_{2}} \right)$ as: $$\begin{split} \mathrm{E}&\left[ {{\hat{x}}_{i}}{{\hat{x}}_{2M+1-i}} \right]\!\!\\ =&\!\!\!\int\!\!\!\!\int_{1\leq {{t}_{1}}\leq {{t}_{2}}}^{{\infty }}\!\!\!\!\,{{t}_{1}}{{t}_{2}}p\left( {{\hat{x}}_{i}}={{t}_{1}},{{\hat{x}}_{2M+1-i}}={{t}_{2}} \right)d{{t}_{1}}d{{t}_{2}} \\ =&\frac{\Gamma\left( 2M+1 \right)}{\Gamma\left( 2M+1-2c \right)}\frac{\Gamma\left( i+2-c \right)}{\Gamma\left( i+2 \right)}\frac{\Gamma\left( 2M-i+1-2c \right)}{\Gamma\left( 2M-i+1-c \right)} \\ =&{{\left( 2M+1 \right)}^{2c}}{{\left( i+2-c \right)}^{-c}}{{\left( 2M-i+1-2c \right)}^{-c}}, \end{split}$$ where $\Gamma(x)=\int_0^{\infty} t^{x-1}e^{-t}dt$ is the well-known Gamma function and $\Gamma(n)=(n-1)!$ for any positive integer $n$. Obviously, $M\rightarrow \infty$ in the thermodynamic limit $N \rightarrow \infty$, and then $$\begin{split} \hat{S}_{\theta_{\mathrm{min}}}=&\underset{M\to \infty }{\mathop{\lim }}\,{{M}^{-1}}\sum\nolimits_{i=1}^{M}\mathrm{E}\left[ {{\hat{x}}_{i}}{{\hat{x}}_{2M+1-i}} \right] \\ =&2\underset{M\to \infty }{\mathop{\text{lim}}}\,\frac{\sum\nolimits_{i=1}^{M}\,{{\left( i+2-c \right)}^{-c}}{{\left( 2M-i+1-2c \right)}^{-c}}}{{{\left( 2M+1 \right)}^{1-2c}}{}} \\ =&2\underset{M\to \infty }{\mathop{\text{lim}}}\,\frac{\sum\nolimits_{i=1}^{M}\,{{\left( \frac{i}{2M+1} \right)}^{-c}}{{\left( 1-\frac{i}{2M+1} \right)}^{-c}}}{{{\left( 2M+1 \right)}}{}} \\ =&2\int_{1}^{\frac{1}{2}}{{u}^{-c}}{{\left( 1-u \right)}^{-c}}du \\ =&2B\left( \frac{1}{2};\frac{\beta -2}{\beta -1},\frac{\beta -2}{\beta -1} \right), \end{split}$$ where $u=\frac{i}{2M+1}$ and $B(\cdot)$ is the *incomplete beta function* $$B(x; a,b) = \int_{0}^{x} t^{a-1} (1-t)^{b-1} dt.$$ According to Eq. (\[SS\]), we thus get $$\label{eq:S} {S}_{\theta_{\mathrm{min}}}=2{k}_{\mathrm{min}}^{2}B\left( \frac{1}{2};\frac{\beta -2}{\beta -1},\frac{\beta -2}{\beta -1} \right).$$ When $\gamma>4$, substituting Eq. (\[eq:S\]) into Eq. (\[eq:rmin\]), we have $$\label{eq:gamma4} \begin{split} {{r}_{\mathrm{min}}}=&\frac{{{M}^{-1}}\mathop{\sum }_{i=1}^{M}{{x}_{i}}{{x}_{2M+1-i}}-{{\left\langle x \right\rangle }^{2}}}{\left\langle {{x}^{2}} \right\rangle -{{\left\langle x \right\rangle }^{2}}} \\ =&\frac{2B\left( \frac{1}{2};\frac{\beta -2}{\beta -1},\frac{\beta -2}{\beta -1} \right)-{{\left( \frac{\beta -1}{\beta -2} \right)}^{2}}}{\frac{\beta -1}{\beta -3}-{{\left( \frac{\beta -1}{\beta -2} \right)}^{2}}}\\ =&\frac{2B\left( \frac{1}{2};\frac{\text{ }\!\!\gamma\!\!\text{ }-3}{\text{ }\!\!\gamma\!\!\text{ }-2},\frac{\text{ }\!\!\gamma\!\!\text{ }-3}{\text{ }\!\!\gamma\!\!\text{ }-2} \right)-{{\left( \frac{\text{ }\!\!\gamma\!\!\text{ }-2}{\text{ }\!\!\gamma\!\!\text{ }-3} \right)}^{2}}}{\frac{\text{ }\!\!\gamma\!\!\text{ }-2}{\text{ }\!\!\gamma\!\!\text{ }-4}-{{\left( \frac{\text{ }\!\!\gamma\!\!\text{ }-2}{\text{ }\!\!\gamma\!\!\text{ }-3} \right)}^{2}}}. \end{split}$$ Combining Eq. (\[eq:gamma4\]) and Eq. (\[eq:Rmin0\]), the lower bound of scale-free networks is thus obtained as: $${{r}_{\mathrm{min} }}=\left\{ \begin{matrix} 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2<\gamma \le 4,~ \\ \frac{2B\left( \frac{1}{2};\frac{\text{ }\!\!\gamma\!\!\text{ }-3}{\text{ }\!\!\gamma\!\!\text{ }-2},\frac{\text{ }\!\!\gamma\!\!\text{ }-3}{\text{ }\!\!\gamma\!\!\text{ }-2} \right)-{{\left( \frac{\text{ }\!\!\gamma\!\!\text{ }-2}{\text{ }\!\!\gamma\!\!\text{ }-3} \right)}^{2}}}{\frac{\text{ }\!\!\gamma\!\!\text{ }-2}{\text{ }\!\!\gamma\!\!\text{ }-4}-{{\left( \frac{\text{ }\!\!\gamma\!\!\text{ }-2}{\text{ }\!\!\gamma\!\!\text{ }-3} \right)}^{2}}}~~~~~~~~~~\gamma >4. \\ \end{matrix}\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ } \right.$$ ![(Color online) The simulation result of the expected value of $r_{\mathrm{min}}$ for different $\gamma$, compared with the analytical result in Eq. (24). The red line represents the analytical result Eq. (24), and the squares, origin circle and blue up-triangles denote the simulation results for $N=10^{5}$, $N=10^{6}$, $N=10^{8}$, respectively. The minimum degree is fixed as $k_{\min}=150$. Each data point is averaged over 500 independent runs.[]{data-label="fig3"}](figure4.eps){width="48.00000%"} Simulations =========== We verify the analytical results by extensive numerical simulations. Given the degree sequence, we search for the network with minimum assortativity (i.e., the most disassortative network) via a degree-preserving edge-rewiring procedure [@Maslov2002; @Kim2004; @Zhao2006]. Specifically, to find the most disassortative network [@Xulvi2004], at each step two edges with four nodes at their ends are chosen at random. Now, we label these four nodes with $a$, $b$, $c$, and $d$ such that their degrees $k_a$, $k_b$, $k_c$ and $k_d$ are ordered as $$k_a\geq k_b \geq k_c \geq k_d.$$ Considering the operation to break the two edges and connect $a$ with $d$ and $b$ with $c$, if such operation will not generate multiple edges, we implement such operation, otherwise we do nothing. Since this edge-rewiring procedure has been proven to be ergodic [@Maslov2004], and $r$ is convex in the configuration space, the procedure will eventually converge to the network with $r_{\mathrm{min}}$ (see more details in [@Xulvi2004]). In the simulation, we first generate the degree sequence by the distribution $p(k)\sim k^{-\gamma}$. Given $N$ and $k_{\min}$, we will sample $N$ integers between $k_{\min}$ and $N-1$ and check if these $N$ degrees obey the graphical condition according to the Erdös-Gallai theorem [@Erdos1960]. If the degree sequence is graphical, it will be accepted and a scale-free network will be generated by the uniform configuration model [@Newman2001], otherwise it will be rejected. Notice that, when $\gamma>2$, almost every degree sequence is graphical [@Genio2010], and in fact we have not found any non-graphical degree sequence when $N\geq 5\times 10^4$ and $\gamma>2$ in the simualtion. First of all, we compare the simulation results with Eq. (\[eq:rmin\]), which is the starting point of our analysis. As shown in figure 3, for both averaged results and single-run result, the theoretical prediction perfectly agrees with the simulation, demonstrating the validation of Eq. (\[eq:rmin\]). Figure 4 shows the expected value $r_{\min}$ for the scale-free networks with degree distribution $p(k)\sim k^{-\gamma}$, with $\gamma$ varying from 2 to 10. Overall speaking, the simulation results agree well with the analytical results, and with the increasing of the network size $N$, the simulation results are getting close to the theoretical bound. Notice that, nearing $\gamma=2$ and $\gamma=4$, the deviation between analytical and simulation results is larger, which is resulted from the fact that the order of divergence in the numerator and dominator of Eq. (\[eq:rmin\]) becomes close (see also Eqs. (9)-(11)). Empirical Evidence ================== In this section, we test our theory on three large-scale real networks. (1) AS-Skitter [@Leskovec2005]: Autonomous systems topology graph of the Internet. (2) YouTube [@Leskovec2012]: The YouTube social network, in which users are connected with their friends in YouTube. (3) Web-Google [@Leskovec2009]: Nodes represent web pages and edges represent hyperlinks between them, disgarding the direction of links. All the three networks are undirected simple networks without any self-loops or multiple edges. As shown in figure 5, these networks are all scale-free networks with $\gamma>2$. ![(Color online) Degree distribution of the three real networks: (a) AS-Skitter, (b) YouTube, and (c) Web-Google. All the three networks are scale-free networks with power-law exponents being 2.3, 2.1 and 2.7, respectively. The power-law exponents are estimated by using the maximum likelihood method [@Clauset2009]. []{data-label="fig1"}](figure5.eps){width="48.00000%"} As shown in table 1, the assortativity coefficients of the three real networks are all very close to zero, in accordance with the theoretical prediction in Eq. (24). Considering the finite-size effects, as observed in figure 4, our theory agrees well with the empirical results. Furthermore, by using the degree-preserving edge-rewiring method, we can obtain the minimum and maximum assortativity coefficients. One can see from the last two columns in table 1 that the interval between $r_{\mathrm{min}}$ and $r_{\mathrm{max}}$ is quite narrow, similar to the phenomenon reported in Ref. [@Zhou2007]. This phenomenon strongly suggests that the widely applied assortativity coefficient is not a suitable measure for the degree-degree correlation at least for scale-free networks, since to know the coefficient, we still cannot make sure whether this network is positively correlated or not. Network N M $\gamma$ $r$ $r_{\mathrm{min}}$ $r_{\mathrm{max}}$ ------------ --------- ---------- ---------- ---------- -------------------- -------------------- As-Skitter 1965206 11095298 2.3 $-0.081$ $-0.086$ $-0.045$ YouTube 1134890 2987624 2.1 $-0.037$ $-0.044$ $-0.004$ Web-Google 875731 4322051 2.7 $-0.055$ $-0.065$ $0.108$ : The basic statistics and assortativity coefficients of the three real networks. $N$ and $M$ are the number of nodes and the number of edges, and $\gamma$ is the estimated power-law exponents. $r$ is the assortativity coefficient, while $r_{\mathrm{min}}$ and $r_{\mathrm{max}}$ denote the minimum and maximum assortativity coefficients obtained by the degree-preserving edge-rewiring procedure. The method to obtain $r_{\mathrm{max}}$ is similar to that for $r_{\mathrm{min}}$. []{data-label="table1"} Conclusion and Discussion ========================= This paper argued the invalidation of the well-known assortativity coefficient $r$ in highly heterogeneous networks, and in particular analytically obtained the lower bound of $r$ for scale-free networks with power-law exponent $\gamma >2$. According to the main result Eq. (24), when $2<\gamma \leq 4$, $r_{\min}$ will approach zero in the large $N$ limit and when $\gamma>4$, $r_{\min}$ will monotonously decrease as the increase of $\gamma$. In addition, as indicated by the simulation result in Figure 4, as the increase of network size, the lower bound $r_{\min}$ will also increase. The above results are in accordance with previous studies [@Menche2010; @Dorogovtsev2010; @Raschke2010; @Litvak2013; @Hofstad2014], meanwhile the advantage of the present study is that it considered a more general case instead of specific network models and derived the explicit lower bound of $r$. At the same time, there are still some disadvantages in the present work. For example, we have not obtained the upper bound of $r$ or the analytical relation between $r_{\min}$ and $\gamma$ for finite-size networks. These problems may be solved in the future studies, but we do not know whether they can be solved under the present framework. In addition, the in-depth analyses on the degree-degree correlation in directed networks are very interesting and challenging, as such kind of correlation in directed networks is much more complicated [@Williams2014; @Hoom2015]. In addition to the technical skills, this paper has demonstrated an important point of view that the assortativity coefficient is not a suitable measure for degree-degree correlation in heterogenous networks, since the possible range of $r$ in heterogeneous networks is very narrow [@Zhou2007], and to know the value of $r$ is usually not enough to draw a conclusion whether the target network is assortative or disassortative. Some scientists have suggested other measures for heterogeneous networks, most of which are rank-based coefficients, such as the Kendall-Gibbons’ Tau [@Kendall1990] suggested by Raschke *et al.* [@Raschke2010] and the Spearman’s Rho [@Spearman1904] suggested by Litvak and Van Der Hofstad [@Litvak2013; @Hofstad2014]. However, compared with the extensive studies on assortativity coefficient [@Noldus2015], the studies on rank-based coefficients in complex networks are very limited. Although we still do not know whether a rank-based coefficient is the best candidate in properly measuring degree-degree correlation in heterogeneous networks, to uncover the statistical properties of rank-based coefficients and to explore other possible candidates based on the more in-depth understanding of the network ensemble are significant in the current stage. The authors acknowledge Ya-Jun Mao, Zhi-Hai Rong, Wei Wang, Yifan Wu and Zhi-Dan Zhao for valuable discussion and research assistance, and the Stanford Large Network Dataset Collection (SNAP Datasets) for the real data. This work is partially supported by National Natural Science Foundation of China under Grants Nos. 61433014 and 11222543. Determining $g_{\theta_{\mathrm{min}}}$ that minimizes $r$ ========================================================== Inspired by the branch-and-bound idea [@Hallin1992; @Guo2015], in this section we will prove that $g_{\theta_{\mathrm{min}}}$ minimizes $r$, where $g_{\theta_{\mathrm{min}}}=\{(x_{1},x_{2M}),(x_{2},x_{2M-1}),(x_{3},x_{2M-2}),\cdot\cdot\cdot,(x_{M},x_{M+1})\}$. Given the degree sequence and the corresponding ensemble, we define ${{\xi }^{\left( 0 \right)}}$ as the set of all possible realizations $g_{\theta}$, and at each step $\epsilon \geq 1$ we subdivide ${{\xi }^{\left( \epsilon-1 \right)}}$ into two nonempty subsets ${{\xi }^{\left( \epsilon \right)}}$ and ${{\xi }^{\left( \epsilon-1 \right)}}\backslash {{\xi }^{\left( \epsilon \right)}}$ according to the following rule, which guarantees that ${{\xi }^{\left( \epsilon \right)}}$ contains at least one realization $g_{\theta}$ that minimizes $r$. In step one, we consider the stub of largest value and the stub of smallest value, which are $x_{2M}$ and $x_{1}$ respectively. If they are not connected, there must be two links $\left( {{x}_{1}},{{x}_{i}} \right)$ and $\left( {{x}_{j}},{{x}_{2M}} \right)$, with $1<i<2M$, $1<j<2M$ and $i\ne j$. Since $\underline{x}$ is arranged in a non-decreasing order, it is obvious that ${{x}_{1}}{{x}_{2M}}+{{x}_{i}}{{x}_{j}}\le {{x}_{1}}{{x}_{i}}+{{x}_{j}}{{x}_{2M}}$. Therefore, keeping other links the same, according to Eq. (2), the network with edges $\left( {{x}_{i}},{{x}_{j}} \right)$ and $\left( {{x}_{1}},{{x}_{2M}} \right)$ has no larger assortativity coefficient than the one with edges $\left( {{x}_{1}},{{x}_{i}} \right)$ and $\left( {{x}_{j}},{{x}_{2M}} \right)$. Then we define ${{\xi }^{\left( 1 \right)}}$ as the set of all realizations containing the edge $\left( {{x}_{1}},{{x}_{2M}} \right)$. Analogously, we further consider whether $x_{2M-1}$ and $x_{2}$ are connected in all realizations $g_{\theta} \in {{\xi }^{\left( 1 \right)}}$. Under the same arguments, ${{\xi }^{\left( 2\right)}}$ can be defined as the set of all realizations containing both edges $\left( {{x}_{1}},{{x}_{2M}} \right)$ and $\left( {{x}_{2}},{{x}_{2M-1}} \right)$. After $M$ steps, the above process ends up with ${{\xi }^{\left( M \right)}}=\{{g_{\theta_{\mathrm{min}}}}\}$, that is $$\label{} \begin{split} &{{\xi }^{\left( M\right)}}=\{{g_{\theta_{\mathrm{min}}}}\} \\ &=\{(x_{1},x_{2M}),(x_{2},x_{2M-1}),\cdot\cdot\cdot,(x_{M},x_{M+1})\}. \end{split}$$ Self-loops and multiple edges in $g_{\theta_{\mathrm{min}}}$ vanish in the thermodynamical limit ================================================================================================ The $g_{\theta_{\mathrm{min}}}$ derivated in *Appendix A* can not guarantee the absence of self-loops or multiple edges. Here we show that actually self-loops and multiple edges vanish in the thermodynamical limit for the specific structure of $g_{\theta_{\mathrm{min}}}$, given $p(k)\sim k^{-\gamma}$ with $\gamma >2$. First let’s consider self-loops. Clearly self-loops may appear only when two stubs of the same value are connected, since they might come from the same node. Under the specific arrangement of $g_{\theta_{\mathrm{min}}}$, stubs of values $\geq x_M$ always connect with stubs of values $\leq x_M$, thus connections among stubs with the same value only occur when their values are equal to $x_M$. Denote the number of stubs of value $x_M$ by $N(x_M)$ and the number of nodes with degree $x_M$ by $\widehat{N}(x_M)$, then it is obvious that $$N(x_M)=x_M\times \widehat{N}(x_M).$$ There are in total $\frac{1}{2} \widehat{N}(x_M) (\widehat{N}(x_M)-1)$ potential node pairs and in the worst case, $N(x_M)/2$ connections among these stubs will be generated. To avoid self-loops, there must be no less potential node pairs than the demanded connections, namely $$\label{eq:apB} \frac{1}{2} \widehat{N}(x_M) (\widehat{N}(x_M)-1) \geq \frac{N(x_M)}{2}.$$ Combining Eq. (B1) and Eq. (B2), the condition is reduced to $\widehat{N}(x_M) \geq x_M +1$. Actually for $x_M$ we have $x_M \sim {M^0}$, meanwhile $\widehat{N}(x_M)\sim N$ [@Baek2012; @Genio2011]. Thus in the thermodynamical limit (i.e., $N\rightarrow \infty$), $\widehat{N}(x_M) \geq x_M +1$ holds. That is to say, the self-loops vanish in the thermodynamical limit. Secondly, we consider the multiple edges. For an arbitrary node with degree $k$, which will draw $n (n \leq k)$ edges to the nodes with degree $k'$ according to the realization $g_{\theta_{\mathrm{min}}}$. Obviously, multiple edges can be avoided if $n \leq \widehat{N}(k')$, where $\widehat{N}(k')$ is the number of nodes with degree $k'$ in the given degree sequence. Thus if $n\leq \widehat{N}(k')$ holds for all possible values of $n$ and $k'$ in $g_{\theta_{\mathrm{min}}}$, we can conclude that multiple edges can be excluded for the certain realization $g_{\theta_{\mathrm{min}}}$. Notice that, in $g_{\theta_{\mathrm{min}}}$, two nodes are connected only if one is of degree $\geq x_M$ and the other is of degree $\leq x_M$. Therefore, we can only consider the case $k \geq x_M$ and $k' \leq x_M$, since if for all nodes with degree $k \geq x_M$, the multiple edges can be avoided, then for nodes with degree $k<x_M$, the multiple edges can also be avoided as these small-degree nodes cannot connect to each other in $g_{\theta_{\mathrm{min}}}$. In the thermodynamical limit of scale-free networks with degree distribution $p(k)\sim k^{-\gamma}$, the maximum degree scales in the order $k_{\mathrm{max}} \sim N^{\frac{1}{\gamma-1}}$ , hence when $\gamma >2$, we have $$n\leq k \leq k_{\mathrm{max}} \sim N^{\frac{1}{\gamma-1}} < N$$ for $N\rightarrow \infty$. At the same time, since $k' \leq x_M$, we have $\widehat{N}(k') \geq \widehat{N}(x_M)$, meanwhile $\widehat{N}(x_M)\sim N$ in the thermodynamical limit, that is $$\widehat{N}(k') \geq \widehat{N}(x_M) \sim N.$$ Combining Eq. (B3) and Eq. (B4), for all possible values of $n$ and $k'$, $n\leq \widehat{N}(k')$ in the limit $N\rightarrow \infty$. 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--- abstract: 'Dynamic dark energy (DDE) models are often designed to solve the cosmic coincidence (why, just now, is the dark energy density $\rho_{de}$, the same order of magnitude as the matter density $\rho_m$?) by guaranteeing $\rho_{de} \sim \rho_m$ for significant fractions of the age of the universe. However, such behaviour is neither sufficient nor necessary to solve the coincidence problem. Cosmological processes constrain the epochs during which observers can exist. Therefore, what must be shown is that a significant fraction of observers see $\rho_{de} \sim \rho_m$. Precisely when, and for how long, must a DDE model have $\rho_{de} \sim \rho_{m}$ in order to solve the coincidence? We explore the coincidence problem in dynamic dark energy models using the temporal distribution of terrestrial-planet-bound observers. We find that any realistic DDE model which can be parameterized as $w=w_0+w_a(1-a)$ over a few e-folds, has $\rho_{de} \sim \rho_m$ for a significant fraction of observers in the universe. This demotivates DDE models specifically designed to solve the coincidence using long or repeated periods of $\rho_{de} \sim \rho_m$.' author: - 'Chas A. Egan' - 'Charles H. Lineweaver' bibliography: - 'coincidenceII.bib' title: 'Dark-Energy Dynamics Required to Solve the Cosmic Coincidence' --- Introduction ============ In 1998, using supernovae Ia as standard candles, @Riess1998 and @Perlmutter1999 revealed a recent and continuing epoch of cosmic acceleration - strong evidence that Einstein’s cosmological constant $\Lambda$, or something else with comparable negative pressure $p_{de} \sim -\rho_{de}$, currently dominates the energy density of the universe [@Lineweaver1998]. $\Lambda$ is usually interpreted as the energy of zero-point quantum fluctuations in the vacuum [@ZelDovich1967; @Durrer2007] with a constant equation of state $w \equiv p_{de}/\rho_{de} = -1$. This necessary additional energy component, construed as $\Lambda$ or otherwise, has become generically known as “dark energy” (DE). A plethora of observations have been used to constrain the free parameters of the new standard cosmological model, , in which $\Lambda$ *does* play the role of the dark energy. Hinshaw et al. [@Hinshaw2006] find that the universe is expanding at a rate of $H_0 = 71 \pm 4\ km/s/Mpc$; that it is spatially flat and therefore critically dense ($\Omega_{tot 0} = \frac{\rho_{tot 0}}{\rho_{crit0}} = \frac{8 \pi G}{3 H_0^2} \rho_{tot 0} = 1.01 \pm 0.01$); and that the total density is comprised of contributions from vacuum energy ($\Omega_{\Lambda 0} = 0.74 \pm 0.02$), cold dark matter (CDM; $\Omega_{CDM 0} = 0.22 \pm 0.02$), baryonic matter ($\Omega_{b 0} = 0.044 \pm 0.003$) and radiation ($\Omega_{r 0} = 4.5 \pm 0.2 \times 10^{-5}$). Henceforth we will assume that the universe is flat ($\Omega_{tot 0} = 1$) as predicted by inflation and supported by observations. Two problems have been influential in moulding ideas about dark energy, specifically in driving interest in alternatives to . The first of these problems is concerned with the smallness of the dark energy density [@ZelDovich1967; @Weinberg1989; @Cohn1998]. Despite representing more than $70\%$ of the total energy of the universe, the current dark energy density is $\sim 120$ orders of magnitude smaller than energy scales at the end of inflation (or $\sim 80$ orders of magnitude smaller than energy scales at the end of inflation if this occurred at the GUT rather than Planck scale) [@Weinberg1989]. Dark energy candidates are thus challenged to explain why the observed DE density is so small. The standard idea, that the dark energy is the energy of zero-point quantum fluctuations in the true vacuum, seems to offer no solution to this problem. The second cosmological constant problem [@Weinberg2000b; @Carroll2001b; @Steinhardt2003] is concerned with the near coincidence between the current cosmological matter density ($\rho_{m0} \approx 0.26 \times \rho_{crit0}$) and the dark energy density ($\rho_{de0} \approx 0.74 \times \rho_{crit0}$). In the standard  model, the cosmological window during which these components have comparable density is short (just $1.5$ e-folds of the cosmological scalefactor $a$) since matter density dilutes as $\rho_{m} \propto a^{-3}$ while vacuum density $\rho_{de}$ is constant [@Lineweaver2007]. Thus, even if one explains why the DE density is much less than the Planck density (the smallness problem) one must explain why we happen to live during the time when $\rho_{de} \sim \rho_m$. The likelihood of this coincidence depends on the range of times during which we suppose we might have lived. In works addressing the smallness problem, @Weinberg1987 [@Weinberg1989; @Weinberg2000] considered a multiverse consisting of a large number of big bangs, each with a different value of $\rho_{de}$. There he asked, suppose that we could have arisen in any one of these universes; What value of $\rho_{de}$ should we expect our universe to have? While Weinberg supposed we could have arisen in another universe, we are simply supposing that we could have arisen in another time *in our universe*. We ask, what time $t_{obs}$, and corresponding densities $\rho_{de}(t_{obs})$ and $\rho_{m}(t_{obs})$ should we expect to observe? Weinberg’s key realization was that not every universe was equally probable: those with smaller $\rho_{de}$ contain more Milky-Way-like galaxies and are therefore more hospitable [@Weinberg1987; @Weinberg1989]. Subsequently, he, and other authors used the relative number of Milky-Way-like galaxies to estimate the distribution of observers as a function of $\rho_{de}$, and determined that our value of $\rho_{de}$ was indeed likely [@Efstathiou1995; @Martel1998; @Pogosian2007]. Our value of $\rho_{de}$ could have been found to be unlikely and this would have ruled out the type of multiverse being considered. Here we apply analogous reasoning to the cosmic coincidence problem. Our observerhood could not have happened at any time with equal probability [@Lineweaver2007]. By estimating the temporal distribution of observers we can determine whether the observation of $\rho_{de} \sim \rho_{m}$ was likely. If we find $\rho_{de} \sim \rho_{m}$ to be unlikely while considering a particular DE model, that will enable us to rule out that DE model. In a previous paper [@Lineweaver2007], we tested  in this way and found that $\rho_{de} \sim \rho_{m}$ is expected. In the present paper we apply this test to dynamic dark energy models to see what dynamics is required to solve the coincidence problem when the temporal distribution of observers is being considered. The smallness of the dark energy density has been anthropically explained in multiverse models with the argument that in universes with much larger DE components, DE driven acceleration starts earlier and precludes the formation of galaxies and large scale structure. Such universes are probably devoid of observers [@Weinberg1987; @Martel1998; @Pogosian2007]. A solution to the coincidence problem in this scenario was outlined by @Garriga1999 who showed that if $\rho_{de}$ is low enough to allow galaxies to form, then observers in those galaxies will observe $r \sim 1$. To quantify the time-dependent proximity of $\rho_m$ and $\rho_{de}$, we define a proximity parameter, $$\label{eq:definer} r \equiv \min \left[ \frac{\rho_{de}}{\rho_m}, \frac{\rho_m}{\rho_{de}} \right],$$ which ranges from $r \approx 0$, when many orders of magnitude separate the two densities, to $r = 1$, when the two densities are equal. The presently observed value of this parameter is $r_0 = \frac{\rho_{m 0}}{\rho_{de 0}} \approx 0.35$. In terms of $r$, the coincidence problem is as follows. If we naively presume that the time of our observation $t_{obs}$ has been drawn from a distribution of times $P_t(t)$ spanning many decades of cosmic scalefactor, we find that the expected proximity parameter is $r \approx 0 \ll 0.35$. In the top panel of Fig. \[fig:illustrated\_coincidence\] we use a naive distribution for $t_{obs}$ that is constant in $\log(a)$ to illustrate how observing $r$ as large as $r_0 \approx 0.35$ seems unexpected. In @Lineweaver2007 we showed how the apparent severity of the coincidence problem strongly depends upon the distribution $P_t(t)$ from which $t_{obs}$ is hypothesized to have been drawn. Naive priors for $t_{obs}$, such as the one illustrated in the top panel of Fig. \[fig:illustrated\_coincidence\], lead to naive conclusions. Following the reasoning of @Weinberg1987 [@Weinberg1989; @Weinberg2000] we interpret $P_t(t)$ as the temporal distribution of observers. The temporal and spatial distribution of observers has been estimated using large ($10^{11} M_{\sol}$) galaxies [@Weinberg1987; @Efstathiou1995; @Martel1998; @Garriga1999] and terrestrial planets [@Lineweaver2007] as tracers. The top panel of Fig. \[fig:illustrated\_coincidence\] shows the temporal distribution of observers $P_t(t)$ from @Lineweaver2007. ![[**(Top)**]{} The history of the energy density of the universe according to standard . The dotted line shows the energy density in radiation (photons, neutrinos and other relativistic modes). The radiation density dilutes as $a^{-4}$ as the universe expands. The dashed line shows the density in ordinary non-relativistic matter, which dilutes as $a^{-3}$. The thick solid line shows the energy of the vacuum (the cosmological constant) which has remained constant since the end of inflation. The thin solid peaked curve shows the proximity $r$ of the matter density to the vacuum energy density (see Eq. \[eq:definer\]). The proximity $r$ is only $\sim 1$ for a brief period in the $log(a)$ history of the cosmos. Whether or not there is a coincidence problem depends on the distribution $P_t(t)$ for $t_{obs}$. If one naively assumes that we could have observed any epoch with equal probability (the light grey shade) then we should not expect to observe $r$ as large as we do. If, however, $P_t(t)$ is based on an estimate of the temporal distribution of observers (the dark grey shade) then $r_0 \approx 0.35$ is not surprising, and the coincidence problem is solved under  [@Lineweaver2007]. [**(Bottom)**]{} The dark energy density history is modified in DDE models. Observational constraints on the dark energy density history are represented by the light grey shade (details in Section \[constraints\]).[]{data-label="fig:illustrated_coincidence"}](fig1.ps) A possible extension of the concordance cosmological model that may explain the observed smallness of $\rho_{de}$ is the generalization of dark energy candidates to include dynamic dark energy (DDE) models such as quintessence, phantom dark energy, k-essence and Chaplygin gas. In these models the dark energy is treated as a new matter field which is approximately homogenous, and evolves as the universe expands. DDE evolution offers a mechanism for the decay of $\rho_{de}(t)$ from the expected Planck scales ($10^{93}$ g/cm$^3$) in the early universe ($10^{-44}$ s) to the small value we observe today ($10^{-30}$ g/cm$^3$). The light grey shade in the bottom panel of Fig. \[fig:illustrated\_coincidence\] represents contemporary observational constraints on the DDE density history. Many DDE models are designed to solve the coincidence problem by having $\rho_{de}(t) \sim \rho_{m}(t)$ for a large fraction of the history/future of the universe [@Amendola2000; @Dodelson2000; @Sahni2000; @Chimento2000; @Zimdahl2001; @Sahni2002; @Chimento2003; @Ahmed2004; @Franca2004; @Mbonye2004; @delCampo2005; @Guo2005; @Olivares2005; @Pavon2005; @Scherrer2005; @Zhang2005; @delCampo2006; @Franca2006; @Feng2006; @Nojiri2006; @Amendola2006; @Amendola2007; @Olivares2007; @Sassi2007]. With $\rho_{de} \sim \rho_{m}$ for extended or repeated periods the hope is to ensure that $r \sim 1$ is expected. Our main goal in this paper is to take into account the temporal distribution of observers to determine when, and for how long, a DDE model must have $\rho_{de} \sim \rho_{m}$ in order to solve the coincidence problem? Specifically, we extend the work of @Lineweaver2007 to find out for which cosmologies (in addition to ) the coincidence problem is solved when the temporal distribution of observers is considered. In doing this we answer the question, Does a dark energy model fitting contemporary constraints on the density $\rho_{de}$ and the equation of state parameters, necessarily solve the cosmic coincidence? Both positive and negative answers have interesting consequences. An answer in the affirmative will simplify considerations that go into DDE modeling: any DDE model in agreement with current cosmological constraints has $\rho_{de} \sim \rho_m$ for a significant fraction of observers. An answer in the negative would yield a new opportunity to constrain the DE equation of state parameters *more strongly* than contemporary cosmological surveys. A different coincidence problem arises when the time of observation is conditioned on and the parameters of a model are allowed to slide. The tuning of parameters and the necessity to include ad-hoc physics are large problems for many current dark energy models. This paper does not address such issues, and the interested reader is referred to @Hebecker2001, @Bludman2004 and @Linder2006. In the coincidence problem addressed here we let the time of observation vary to see if $r(t_{obs}) \ge 0.35$ is unlikely according to the model. In Section \[solving\] we present several examples of DDE models used to solve the coincidence problem. An overview of observational constraints on DDE is given in Section \[constraints\]. In Section \[observers\] we estimate the temporal distribution of observers. Our main analysis is presented in Section \[results\]. Our main result - that the coincidence problem is solved for all DDE models fitting observational constraints - is illustrated in Fig. \[fig:severities\]. Finally, in Section \[conclusion\], we end with a discussion of our results, their implications and potential caveats. Dynamic Dark Energy Models in the Face of the Cosmic Coincidence {#solving} ================================================================ Though it is beyond the scope of this article to provide a complete review of DDE (see @Copeland2006 [@Szydlowski2006]), here we give a few representative examples in order to set the context and motivation of our work. Fig. \[fig:quintessence\_energy\] illustrates density histories typical of tracker quintessence, tracking oscillating energy, interacting quintessence, phantom dark energy, k-essence, and Chaplygin gas. They are discussed in turn below.  Quintessence ------------ In quintessence models the dark energy is interpreted as a homogenous scalar field with Lagrangian density ${\mathcal L}(\phi,X) = \frac{1}{2} \dot{\phi}^2 - V(\phi)$ [@Ozer1987; @Ratra1988; @Ferreira1998; @Caldwell1998; @Steinhardt1999; @Zlatev1999; @Dalal2001]. The evolution of the quintessence field and of the cosmos depends on the postulated potential $V(\phi)$ of the field and on any postulated interactions. In general, quintessence has a time-varying equation of state $w = \frac{p_{de}}{\rho_{de}} = \frac{\dot{\phi}^2 / 2 - V(\phi)}{\dot{\phi}^2 / 2 + V(\phi)}$. Since the kinetic term $\dot{\phi}^2 / 2$ cannot be negative, the equation of state is restricted to values $w \ge -1$. Moreover, if the potential $V(\phi)$ is non-negative then $w$ is also restricted to values $w \le +1$. If the quintessence field only interacts gravitationally then energy density evolves as $\frac{\delta \rho_{de}}{\rho_{de}} = -3(w+1) \frac{\delta a}{a}$ and the restrictions $-1 \le w \le +1$ mean $\rho_{de}$ decays (but never faster than $a^{-6}$) or remains constant (but never increases). ### Tracker Quintessence Particular choices for $V(\phi)$ lead to interesting attractor solutions which can be exploited to make $\rho_{de}$ scale (“track”) sub-dominantly with $\rho_r + \rho_m$. The DE can be forced to transit to a $\Lambda$-like ($w \approx -1$) state at any time by fine-tuning $V(\phi)$. In the $\Lambda$-like state $\rho_{de}$ overtakes $\rho_m$ and dominates the recent and future energy density of the universe. We illustrate tracker quintessence in Fig.  \[fig:quintessence\_energy\] using a power law potential $V(\phi) = M \phi^{-\alpha}$ (panel b) [@Ratra1988; @Caldwell1998; @Zlatev1999] and an exponential potential $V(\phi) = M \exp(1/\phi)$ (panel c) [@Dodelson2000]. The tracker paths are attractor solutions of the equations governing the evolution of the field. If the tracker quintessence field is initially endowed with a density off the tracker path (e.g.  an equipartition of the energy available at reheating) its density quickly approaches and joins the tracker solution. ### Oscillating Dark Energy @Dodelson2000 explored a quintessence potential with oscillatory perturbations $V(\phi) = M \exp(- \lambda \phi) \left[1 + A \sin(\nu \phi)\right]$. They refer to models of this type as tracking oscillating energy. Without the perturbations (setting $A=0$) this potential causes exact tracker behaviour: the quintessence energy decays as $\rho_r + \rho_m$ and never dominates. With the perturbations the quintessence energy density oscillates about $\rho_r + \rho_m$ as it decays (Fig.  \[fig:quintessence\_energy\]d). The quintessence energy dominates on multiple occasions and its equation of state varies continuously between positive and negative values. One of the main motivations for tracking oscillating energy is to solve the coincidence problem by ensuring that $\rho_{de} \sim \rho_{m}$ or $\rho_{de} \sim \rho_{r}$ at many times in the past or future. It has yet to be seen how such a potential might arise from particle physics. Phenomenologically similar cosmologies have been discussed in @Ahmed2004 [@Yang2005; @Feng2006]. ### Interacting Quintessence Non-gravitational interactions between the quintessence field and matter fields might allow energy to transfer between these components. Such interactions are not forbidden by any known symmetry [@Amendola2000b]. The primary motivation for the exploration of interacting dark energy models is to solve the coincidence problem. In these models the present matter/dark energy density proximity $r$ may be constant [@Amendola2000; @Zimdahl2001; @Amendola2003; @Franca2004; @Guo2005; @Olivares2005; @Pavon2005; @Zhang2005; @Franca2006; @Amendola2006; @Amendola2007; @Olivares2007] or slowly varying [@delCampo2005; @delCampo2006]. We plot a density history of the interacting quintessence model of @Amendola2000 in Fig.  \[fig:quintessence\_energy\]e. This model is characterized by a DE potential $V(\phi)=A \exp[B \phi]$ and DE-matter interaction term $Q = -C \rho_m \dot{\phi}$, specifying the rate at which energy is transferred to the matter fields. The free parameters were tuned such that radiation domination ends at $a=10^{-5}$ and that $r_{t \rightarrow \infty} = 0.35$. Phantom Dark Energy ------------------- The analyses of @Riess2004 and @Wood-Vasey2007 have mildly ($\sim 1 \sigma$) favored a dark energy equation of state $w_{de} < -1$. These values are unattainable by standard quintessence models but can occur in phantom dark energy models [@Caldwell2002], in which kinetic energies are negative. The energy density in the phantom field *increases* with scalefactor, typically leading to a future “big rip” singularity where the scalefactor becomes infinite in finite time. Fig. \[fig:quintessence\_energy\]f shows the density history of a simple phantom model with a constant equation of state $w=-1.25$. The big rip ($a=\infty$ at $t=57.5$ Gyrs) is not seen in $\log(a)$-space. @Caldwell2003 and @Scherrer2005 have explored how phantom models may solve the coincidence problem: since the big rip is triggered by the onset of DE domination, such cosmologies spend a significant fraction of their total time with $r$ large. For the phantom model with $w=-1.25$ (Fig. \[fig:quintessence\_energy\]f) @Scherrer2005 finds $r > 0.1$ for $12\%$ of the total lifetime of the universe. Whether this solves the coincidence or not depends upon the prior probability distribution $P_t(t)$ for the time of observation. @Caldwell2003 and @Scherrer2005 implicitly assume that the temporal distribution of observers is constant in time (i.e. $P_t(t)=\textrm{constant}$). For this prior the coincidence problem *is* solved because the chance of observing $r \ge 0.1$ is large ($12\%$). Note that for the “naive $P_t(t)$” prior shown in Fig. \[fig:illustrated\_coincidence\], the solution of @Caldwell2003 and @Scherrer2005 fails because $r > 0.1$ is brief in $\log(a)$-space. It fails in this way for many other choices of $P_t(t)$ including, for example, distributions constant in $a$ or $\log(t)$. K-Essence --------- In k-essence the DE is modeled as a scalar field with non-canonical kinetic energy [@Chiba2000; @Armendariz-Picon2000; @Armendariz-Picon2001; @Malquarti2003]. Non-canonical kinetic terms can arise in the effective action of fields in string and supergravity theories. Fig. \[fig:quintessence\_energy\]g shows a density history typical of k-essence models. This particular model is from @Armendariz-Picon2001 and @Steinhardt2003. During radiation domination the k-essence field tracks radiation sub-dominantly (with $w_{de}=w_r=1/3$) as do some of the other models in Fig. \[fig:quintessence\_energy\]. However, no stable tracker solution exists for $w_{de}=w_m (=0)$. Thus after radiation-matter equality, the field is unable to continue tracking the dominant component, and is driven to another attractor solution (which is generically $\Lambda$-like with $w_{de} \approx -1$). The onset of DE domination was recent in k-essence models because matter-radiation equality prompts the transition to a $\Lambda$-like state. K-essence thereby avoids fine-tuning in any particular numerical parameters, but the Lagrangian has been constructed ad-hoc. Chaplygin Gas ------------- A special fluid known as Chaplygin gas motivated by braneworld cosmology may be able to play the role of dark matter *and* the dark energy [@Bento2002; @Kamenshchik2001]. Generalized Chaplygin gas has the equation of state $p_{de}=-A \rho_{de}^{-\alpha}$ which behaves like pressureless dark matter at early times ($w_{de} \approx 0$ when $\rho_{de}$ is large), and like vacuum energy at late times ($w_{de} \approx -1$ when $\rho_{de}$ is small). In Fig. \[fig:quintessence\_energy\]h we show an example with $\alpha=1$. Summary of DDE Models --------------------- Two broad classes of DDE models emerge from our comparison: 1. In , tracker quintessence and k-essence models, the dark energy density is vastly different from the matter density for most of the lifetime of the universe (panels a, b, c, g of Fig. \[fig:quintessence\_energy\]). The coincidence problem can only be solved if the probability distribution $P_t(t)$ for the time of observation is narrow, and overlaps significantly with an $r \sim 1$ peak. If $P_t(t)$ is wide, e.g. constant over the life of the universe in $t$ or $\log(t)$, then observing $r \sim 1$ would be unlikely in these models and the coincidence problem *is not* resolved. 2. Tracking oscillating energy, interacting quintessence, phantom models and Chaplygin gas models (panels d, e, f, h of Fig. \[fig:quintessence\_energy\]) employ mechanisms to ensure that $r \sim 1$ for large fractions of the life of the universe. In these models the coincidence problem may be solved for a wider range of $P_t(t)$ including, depending on the DE model, distributions that are constant over the whole life of the universe in $t$, $\log(t)$, $a$ or $\log(a)$. The importance of an estimate of the distribution $P_t(t)$ is highlighted: such an estimate will either rule out models of the first category because they do not solve the coincidence problem, or demotivate models of the second because their mechanisms are unnecessary to solve the coincidence problem. This analysis does not addressed the problems associated with fine-tuning, initial conditions or ad hoc mechanisms of many DDE models [@Hebecker2001; @Bludman2004; @Linder2006]. We leave this line of enquiry temporarily to discuss contemporary *observational* constraints on the dark energy density history, because we wish to test what DE dynamics are required to solve the coincidence, beyond those which models must exhibit to satisfy standard cosmological observations. Current Observational Constraints on Dynamic Dark Energy {#constraints} ========================================================  Supernovae Ia ------------- Observationally, possible dark energy dynamics is explored almost solely using measurements of the cosmic expansion history. Recent cosmic expansion is directly probed by using type Ia supernova (SNIa) as standard candles [@Riess1998; @Perlmutter1999]. Each observed SNIa provides an independent measurement of the luminosity distance $d_l$ to the redshift of the supernova $z_{SN}$. The luminosity distance to $z_{SN}$ is given by \[eq:lumdist\] d\_l(z\_[SN]{}) = (1+z\_[SN]{}) \_[z=0]{}\^[z\_[SN]{}]{} where E(z) & = & \[eq:e\]\ & = & \^ and thus depends on $H_0$, $\Omega_{m 0}$, and the evolution of the dark energy $\rho_{de}(z) / \rho_{de 0}$. The radiation term, irrelevant at low redshifts, can be dropped from Equation \[eq:e\]. $\Omega_{de 0}$ is a dependent parameter due to flatness ($\Omega_{de 0} = 1-\Omega_{m 0}$). Contemporary datasets include $\sim 200$ supernovae at redshifts $z_{SN} \le 2.16$ ($a \ge 0.316$) [@Riess2007; @Wood-Vasey2007] and provide an effective continuum of constraints on the expansion history over that range [@Wang2005; @Wang2006]. The redshift range probed by SNIa is indicated in both panels of Fig. \[fig:SN1aConstraints\]. Cosmic Microwave Background --------------------------- The first peak in the cosmic microwave background (CMB) temperature power spectrum corresponds to density fluctuations on the scale of the sound horizon at the time of recombination. Subsequent peaks correspond to higher-frequency harmonics. The locations of these peaks in $l$-space depend on the comoving scale of the sound horizon at recombination, and the angular distance to recombination. This is summarized by the so-called CMB shift parameter $R$ [@Efstathiou1999; @Elgaroy2007] which is related to the cosmology by \[eq:cmbshift\] R = \_[z=0]{}\^[z\_[rec]{}]{} where $z_{rec} \approx 1089$ [@Spergel2006] is the redshift of recombination. The 3-year WMAP data gives a shift parameter $R=1.71 \pm 0.03$ [@Davis2007; @Spergel2006]. Since the dependence of Equation \[eq:cmbshift\] on $H_0$ and $\Omega_{m 0}$ differs from that of Equation \[eq:lumdist\], measurements of the CMB shift parameter can be used to break degeneracies between $H_0$, $\Omega_{m 0}$ and DE evolution in the analysis of SNIa. In the top panel of Fig. \[fig:SN1aConstraints\] we represent the CMB observations using a bar from $z=0$ to $z_{rec}$. Baryonic Acoustic Oscillations and Large Scale Structure -------------------------------------------------------- As they imprinted acoustic peaks in the CMB, the baryonic oscillations at recombination were expected to leave signature wiggles - baryonic acoustic oscillations (BAO) - in the power spectrum of galaxies [@EisensteinHu1998]. These were detected with significant confidence in the SDSS luminous red galaxy power spectrum [@Eisenstein2005]. The expected BAO scale depends on the scale of the sound horizon at recombination, and on transverse and radial scales at the mean redshift $z_{BAO}$, of galaxies in the survey. @Eisenstein2005 measured the quantity A(z\_[BAO]{}) = \^ to have a value $A(z_{BAO}=0.35) = 0.469 \pm 0.017$, thus constraining the matter density and the dark energy evolution parameters in a configuration which is complomentary to the CMB shift parameter and the SNIa luminosity distance relation. Ongoing BAO projects have been designed specifically to produce stronger constraints on the dark energy equation of state parameter $w$. For example, WIGGLEZ [@Glazebrook2007] will use a sample of high-redshift galaxies to measure the BAO scale at $z_{BAO} \approx 0.75$. As well as reducing the effects of non-linear clustering, this redshift is at a larger angular distance, making the observed scale more sensitive to $w$. Constraints from the BAO scale depend on the evolution of the universe from $z_{rec}$ to $z_{BAO}$ to set the physical scale of the oscillations. They also depend on the evolution of the universe from $z_{BAO}$ to $z=0$, since the observed angular extent of the oscillations depends on this evolution. The bar representing BAO scale observations in the top panel of Fig.  \[fig:SN1aConstraints\] indicates both these regimes. The amplitude of the BAOs - the amplitude of the large scale structure (LSS) power spectrum - is determined by the amplitude of the power spectrum at recombination, and how much those fluctuations have grown (the transfer function) between $z_{rec}$ and $z_{BAO}$. By comparing the recombination power spectrum (from CMB) with the galaxy power spectrum, the LSS linear growth factor can be measured and used to constrain the expansion history of the universe (independently of the BAO scale) over this redshift range. In practice, biases hinder precise normalization of the galaxy power spectrum, weakening this technique. The range over which this technique probes the DE is indicated in Fig.  \[fig:SN1aConstraints\]. Ages ---- Cosmological parameters from SN1a, CMB, LSS, BAO and other probes allow us to calculate the current age of the universe to be $13.8 \pm 0.1$ [@Hinshaw2006] assuming . Uncertainties on the age calculated in this way grow dramatically if we drop the assumption that the DE is vacuum energy ($w=-1$). An independent lower limit on the current age of the universe is found by estimating the ages of the oldest known globular clusters [@Hansen2004]. These observations rule out models which predict the universe to be younger than $12.7 \pm 0.7 \; \textrm{Gyrs}$ ($2 \sigma$ confidence): t\_0 & = & H\_0\^[-1]{} \_[z=0]{}\^\ & [ ]{}& 12.7 0.7 . Other objects can also be used to set this age limit @Lineweaver1999, but generally less successfully due to uncertainties in dating techniques. Assuming , an age of $12.7$ Gyrs corresponds to a redshift of $z \approx 5.5$. Contemporary age measurements are sensitive to the dark energy content from $z \approx 5.5$ to $z=0$. In the top panel of Fig. \[fig:SN1aConstraints\] we show this redshift interval. The evolution and energy content of the universe before $12.7$ Gyrs ago is not probed by these age constraints. Nucleosynthesis --------------- In addition to the constraints on the expansion history (SN1a, CMB, BAO and $t_0$) we know that $\rho_{de}/\rho_{tot} < 0.045$ (at $2 \sigma$ confidence) during Big Bang Nucleosynthesis (BBN) [@Bean2001b]. Larger dark energy densities imply a higher expansion rate at that epoch ($z \sim 6 \times 10^{8}$) which would result in a lower neutron to proton ratio, conflicting with the measured helium abundance, $Y_{\textrm{He}}$. Dark Energy Parameterization ---------------------------- Because of the variety of proposed dark energy models, it has become usual to summarize observations by constraining a parameterized time-varying equation of state. Dark energy models are then confronted with observations in this parameter space. The unique zeroth order parameterization of $w$ is $w=w_0$ (a constant), with $w=-1$ characterizing the cosmological constant model. The observational data can be used to constrain the first derivative of $w$. This additional dimension in the DE parameter space may be useful in distinguishing models which have the same $w_0$. From an observational standpoint, the obvious choice of 1st order parameterization is $w(z)=w_0+\frac{dw}{dz}z$ [@diPietro2003]. This is rarely used today since currently considered DDE models are poorly portrayed by this functional form. The most popular parameterization is $w(a)=w_0+w_a(1-a)$ [@Albrecht2006; @Linder2006b], which does not diverge at high redshift. @Linder2005 have argued that the extension of this approach to second order, e.g.  $w(a)=w_0+w_a(1-a)+w_{aa}(1-a)^2$, is not motivated by current DDE models. Moreover, they have shown that next generation observations are unlikely to be able to distinguish the quadratic from a linear expansion of $w$. @Riess2007 have illustrated this recently using new SN1a. An alternative technique for exploring the history of dark energy is to constrain $w(z)$ or $\rho_{de}(z)$ in independent redshift bins. This technique makes fewer assumptions about the specific shape of $w(z)$. In the absence of any strongly motivated parameterization of $w(z)$ this bin-wise method serves as a good reminder of how little we actually know from observation. Using luminosity distance measurements from SNIa, DE evolution has been constrained in this way in $\Delta z \sim 0.5$ bins out to redshift $z_{SN} \sim 2$ [@Wang2004b; @Huterer2005; @Riess2007]. In the future, BAO measurements at various redshifts may contribute to these constraints, however $z_{BAO}$ will probably never be larger than $z_{SN}$. Moreover, because the recombination redshift $z_{rec} \approx 1089$ is fixed, only the cumulative effect (from $z=z_{rec}$ to $z=0$) of the DE can be measured with the CMB and LSS linear growth factor. With only this single data point above $z_{SN}$, the bin-wise technique effectively degenerates to a parameterized analysis at $z > z_{SN}$. Summary of Current DDE Constraints ---------------------------------- If one assumes the popular $w_0 - w_a$ parameterization until last scattering, then all cosmological probes can be combined to constrain $w_0$ and $w_a$. In a recent analysis of SN1a, CMB and BAO observations, @Davis2007 found $w_0 = -1.0 \pm 0.4$ and $w_a = -0.4 \pm 1.8$ at $2 \sigma$ confidence (the contour is shown in Fig. \[fig:severities\]). Using the same observations, @Wood-Vasey2007 assumed $w_a = 0$ and found $w = w_0 = -1.09 \pm 0.16$ ($2 \sigma$). The evolution of $\rho_{de}$ is related to $w$ by covariant energy conservation [@Carroll2004book] = - 3 ( w(a)+1 ) . The dark energy density corresponding to the $w_0 - w_a$ parameterization of $w$ is thus given by \_[de]{}(z) = \_[de 0]{} e\^[3 w\_a (a - 1)]{} a\^[-3 (1 + w\_0 + w\_a)]{}. The cosmic energy density history is illustrated in Fig. \[fig:SN1aConstraints\]. Radiation and matter densities steadily decline as the dotted and dashed lines. With the DE equation of state parameterized as $w(a)=w_0+w_a(1-a)$, its density history is constrained to the light-grey area [@Davis2007]. If the evolution of $w$ is negligible, i.e. we condition on $w_a \approx 0$, then $w(a) \approx w_0$ and the DE density history lies within the dark-grey band [@Wood-Vasey2007]. If the dark energy is pure vacuum energy (or Einstein’s cosmological constant) then $w=-1$ and its density history is given by the horizontal solid black line. The Temporal Distribution of Observers {#observers} ====================================== The energy densities $\rho_r$, $\rho_m$ and $\rho_{de}$, and the proximity parameter $r$ we imagine we might have observed, depend on the distribution $P_t(t)$ from which we imagine our time of observation $t_{obs}$ has been drawn. What we can expect to observe must be restricted by the conditions necessary for our presence as observers [@Carter1974]. Thus, for example, it is meaningless to suppose we might have lived during inflation, or during radiation domination, or before the first atoms [@Dicke1961]. We can, however, suppose that we are randomly selected cosmology-discovering observers, and we can expect our observations of $\rho_m$ and $\rho_{de}$ to be typical of observations made by such observers. This is Vilenkin’s principle of mediocrity [@Vilenkin1995a]. Accordingly, the distribution $P_t(t)$ for the time of observation $t_{obs}$ is proportional to the temporal distribution of cosmology-discovering observers (referred to henceforth as simply “observers”). Thus to solve the coincidence problem one must show that the proximity parameter we measure, $r_0$, is typical of those measured by other observers. The most abundant elements in the cosmos are hydrogen, helium, oxygen and carbon [@Pagel1997]. In the past decade $>200$ extra solar planets have been observed via doppler, transit or microlensing methods. Extrapolation of current patterns in planet mass and orbital period are consistent with the idea that planetary systems like our own are common in the universe [@Lineweaver2003b]. All this does not necessarily imply that observers are common, but it does support the idea that terrestrial-planet-bound carbon-based observers, even if rare, may be the *most common* observers. In the following estimation of $P_t(t)$ we consider only observers bound to terrestrial planets. First the Planets... -------------------- @Lineweaver2001 estimated the terrestrial planet formation rate (PFR) by making a compilation of measurements of the cosmic star formation rate (SFR) and suppressing a fraction of the early stars $f(t)$ to correct for the fact that the metallicity was too low for those early stars to host terrestrial planetary systems, $$PFR(t) = const \times SFR(t) \times f(t).$$ In Fig. \[fig:tpfhz\] we plot the PFR reported by @Lineweaver2001 as a function of redshift, $z=\frac{1}{a}-1$. As illustrated in the figure, there is large uncertainty in the normalization of the formation history. The number of stars orbited by terrestrial planets normalizes the distribution of observers but, importantly, does not shift the distribution in time. Thus our analysis will not depend on the normalization of this function and this uncertainty *will not* propagate into our analysis. There are also uncertainties in the location of the turnover at high redshift, and in the slope of the formation history at low redshift - both of these *will* affect our results. ![The terrestrial planet formation rate as estimated by @Lineweaver2001. It is based on a compilation of SFR measurements and has been corrected for the low metallicity of the early universe, which prevents the terrestrial planet formation rate from rising as quickly as the stellar formation rate at $z {\mbox{$\:\stackrel{>}{_{\sim}}\:$} }4$.[]{data-label="fig:tpfhz"}](fig4.ps) The conversion from redshift to time depends on the particular cosmology, through the Friedmann equation, $$\begin{aligned} \left( \frac{da}{dt} \right)^2 & = & H(a)^2 a^2 \\ & = & H_0^2 \bigg[ \Omega_{r 0}a^{-2} + \Omega_{m 0}a^{-1} + {} \biggr. \nonumber \\ & & \biggl. \Omega_{de 0}\ \exp [3 w_a (a-1)]\ a^{-3 w_0 -3 w_a -1} \biggr]. \nonumber \end{aligned}$$ In Fig. \[fig:tpfht\] we plot the PFR from Fig. \[fig:tpfhz\] as a function of time assuming the best fit parameterized DDE cosmology. ![The terrestrial planet formation from Fig. \[fig:tpfhz\] is shown here as a function of time. The transformation from redshift to time is cosmology dependent. To create this figure we have used best-fit values for the DDE parameters, $w_0=-1.0$ and $w_a=-0.4$ [@Davis2007]. The y-axis is linear (c.f. the logarithmic axis in Fig. \[fig:tpfhz\]) and the family of curves have been re-normalized to highlight the sources of uncertainty important for this analysis: uncertainty in the width of the function, and in the location of its peak. The observer formation rate (OFR) is calculated by shifting the planet formation rate by some amount $\Delta t_{obs}$ ($=4\ \textrm{Gyrs}$) to allow the planet to cool, and the possible emergence of observers. These distributions are closed by extrapolating exponentially in $t$.[]{data-label="fig:tpfht"}](fig5.ps) ... then First Observers ------------------------ After a star has formed, some non-trivial amount of time $\Delta t_{obs}$ will pass before observers, if they arise at all, arise on an orbiting rocky planet. This time allows planets to form and cool and, possibly, biogenesis and the emergence observers. $\Delta t_{obs}$ is constrained to be shorter than the life of the host star. If we consider that our $\Delta t_{obs}$ has been drawn from a probability distribution $P_{\Delta t_{obs}}(t)$. The observer formation rate (OFR) would then be given by the convolution $$\label{eq:OFR} OFR(t) = const \times \int_{0}^{\infty} PFR(\tau) P_{\Delta t_{obs}}(t - \tau) d\tau.$$ In practice we know very little about $P_{\Delta t_{obs}}(t)$. It must be very nearly zero below about $\Delta t_{obs} \sim 0.5\ \textrm{Gyrs}$ - this is the amount of time it takes for terrestrial planets to cool and the bombardment rate to slow down. Also, it is expected to be near zero above the lifetimes of sun-like stars (much above $\sim 10\ \textrm{Gyrs}$). If we assume that our $\Delta t_{obs}$ is typical, then $P_{\Delta t_{obs}}(t)$ has significant weight around $\Delta t_{obs}=4\ \textrm{Gyrs}$ - the amount of time it has taken for us to evolve here on Earth. A fiducial choice, where *all* observers emerge $4\ \textrm{Gyrs}$ after the formation of their host planet, is $P_{\Delta t_{obs}}(t)=\delta(t-4\ \textrm{Gyrs})$. This choice results in an OFR whose shape is the same as the PFR, but is shifted $4\ \textrm{Gyrs}$ into the future, $$OFR(t) = const \times PFR(t - 4\ \textrm{Gyrs})$$ (see the lower panel of Fig. \[fig:tpfht\]). Even for non-standard $w_0$ and $w_a$ values, this fiducial OFR aligns closely with the $r(t)$ peak and the effect of a wider $P_{\Delta t_{obs}}$ is generally to increase the severity of the coincidence problem by spreading observers outside the $r(t)$ peak. Hence using our fiducial $P_{\Delta t_{obs}}$ (which is the narrowest possibility) will lead to conclusions which are conservative in that they underestimate the severity of the cosmic coincidence. If another choice for $P_{\Delta t_{obs}}$ could be justified, the cosmic coincidence would be more severe than estimated here. We will discuss this choice in Section \[conclusion\]. The OFR is then extrapolated into the future using a decaying exponential with respect to $t$ (the dashed segment in the lower panel of Fig. \[fig:tpfht\]). The observed SFH is consistent with a decaying exponential. We have tested other choices (linear & polynomial decay) and our results do not depend strongly on the shape of the extrapolating function used. The temporal distribution of observers $P_t(t)$ is proportional to the observer formation rate, $$P_t(t) = const \times OFR(t).$$ This observer distribution is similar to the one used by @Garriga1999 to treat the coincidence problem in a multiverse scenario. By comparison, our $OFR(t)$ distribution starts later because we have considered the time required for the build up of metallicity, and because we have included an evolution stage of $4\ \textrm{Gyrs}$. Our distribution also decays more quickly than theirs does. Some of our cosmologies suffer big-rip singularities in the future. In these cases we truncate $P_t(t)$ at the big-rip. Analysis and Results: Does fitting contemporary constraints necessarily solve the cosmic coincidence? {#results} ===================================================================================================== For a given model the proximity parameter observed by a typical observer is described by a probability distribution $P_r(r)$ calculated as $$P_r(r) = \sum \frac{dt}{dr} P_t(t(r)).$$ The summation is over contributions from all solutions of $t(r)$ (typically, any given value of $r$ occurs at multiple times during the lifetime of the Universe). In Fig. \[fig:pofr\] we plot $P_r(r)$ for the $w_0=-1.0$, $w_a=-0.4$ cosmology. In this case, observers are distributed over a wide range of $r$ values, with $71\%$ seeing $r>r_0$, and $29\%$ seeing $r<r_0$. ![The predicted distribution of observations of $r$ is plotted for the parameterized DDE model which best-fits cosmological observations: $w_0=-1.0$ and $w_a=-0.4$. The proximity parameter we observe $r_0 =\frac{\rho_{m 0}}{\rho_{de 0}} \approx 0.35$ is typical in this cosmology since only $29 \%$ of observers (vertical striped area) observe $r<0.35$. The upper and lower limits on this value resulting from uncertainties in the SFR are $38 \%$ and $20 \%$ respectively. Thus the severity of the cosmic coincidence in this model is $S=0.29\pm0.09$. This model does not suffer a coincidence problem.[]{data-label="fig:pofr"}](fig6.ps) We define the severity $S$ of the cosmic coincidence problem as the probability that a randomly selected observer measures a proximity parameter $r$ lower than we do: $$S = P(r < r_0) = 1 - P(r > r_0) = \int_{r=0}^{r_0} P_r(r) dr.$$ For the $w_0=-1.0$, $w_a=-0.4$ cosmology of Figs. $\ref{fig:tpfht}$ and $\ref{fig:pofr}$, the severity is $S=0.29 \pm 0.09$. This model does not suffer a coincidence problem since $29\%$ of observers would see $r$ lower than we do. If the severity of the cosmic coincidence would be near $0.95$ ($0.997$) in a particular model, then that model would suffer a $2 \sigma$ ($3 \sigma$) coincidence problem and the value of $r$ we observe really would be unexpectedly high. We calculated the severities $S$ for cosmologies spanning a large region of the $w_0-w_a$ plane and show our results in Fig. \[fig:severities\] using contours of equal $S$. The severity of the coincidence problem is low (e.g. $S {\mbox{$\:\stackrel{<}{_{\sim}}\:$} }0.7$) for most combinations of $w_0$ and $w_a$ shown. There is a coincidence problem, where the severity is high ($S {\mbox{$\:\stackrel{>}{_{\sim}}\:$} }0.8$), in two regions of this parameter space. These are indicated in Fig. \[fig:severities\]. Some features in Fig. \[fig:severities\] are worth noting: - Dominating the left of the plot, the severity of the coincidence increases towards the bottom left-hand corner. This is because as $w_0$ and $w_a$ become more negative, the $r$ peak becomes narrower, and is observed by fewer observers. - There is a strong vertical dipole of coincidence severity centered at $(w_0=0, w_a=0)$. For $(w_0 \approx 0, w_a > 0)$ there is a large coincidence problem because in such models we would be currently witnessing the very closest approach between DE and matter, with $\rho_{de} \gg \rhom$ for all earlier and later times (see Fig. \[fig:severityexamples\]c). For $(w_0 \approx 0, w_a < 0)$ there is an anti-coincidence problem because in those models we would be currently witnessing the DDE’s furthest excursion from the matter density, with $\rho_{de}$ and $\rhom$ in closer proximity for all relevant earlier and later times, i.e., all times when $P_t(t)$ is non-negligible. - There is a discontinuity in the contours running along $w_a = 0$ for phantom models ($w_0 < -1$). The distribution $P_t(t)$ is truncated by big-rip singularities in strongly phantom models (provided they remain phantom; $w_a > 0$). This truncation of late-time observers means that early observers who witness large values of $r$ represent a greater fraction of the total population. To illustrate these features, Fig. \[fig:severityexamples\] shows the density histories and observer distributions for four specific examples selected from the $w_0-w_a$ plane of Fig. \[fig:severities\]. We find that *all* observationally allowed combinations of $w_0$ and $w_a$ result in low severities ($S < 0.4$), i.e., there are large ($> 60\%$) probabilities of observing the matter and vacuum density to be at least as close to each other as we observe them to be. It is not suggested that for arbitrary models $P_{obs}$ should be large when and only when $\rho_{de} \sim \rho_{m}$. Indeed, it is easy to imagine $\rho_{de} \sim \rho_{m}$ when there are not observers. Moreover, in some non-standard cosmological models it is possible to contrive $\rho_{de} \ll \rho_{m}$ during times when observers do exist. What our results suggest is that, for models parameterized with $w_0$ and $w_a$ satisfying current constraints, most observers ($> 60\%$) will see $\rho_{de}$ and $\rho_m$ nearly equal ($r > 0.35$). Discussion {#conclusion} ==========   Particle-theoretic approaches to the cosmic coincidence problem have focussed on the generation of a constant or slowly varying density ratio $r$. However it has not been made clear precisely how slowly the ratio $r$ must evolve in order to solve the coincidence problem. In other words, the question “What DDE dynamics are required to solve the coincidence problem?” has not been addressed. In the present work we adopt the principle of mediocrity: that we should be typical observers, to try to answer this question. We estimate the temporal distribution of observers and devise a scheme for quantifying how unlikely the observation $r \ge 0.35$ is for an arbitrary DDE model. This scheme is applied to $w_0-w_a$ parametric models, and we identify regions of the $w_0-w_a$ parameter space in which the coincidence problem is most severe, however these are already strongly excluded by observations (see Fig. \[fig:severities\]). Thus the main result of our analysis is that any realistic DDE model which can be parameterized as $w=w_0+w_a(1-a)$ over a few e-folds, has $\rho_{de} \sim \rho_m$ for a significant fraction of observers. Central to our approach is the temporal distribution of observers as estimated using the distribution of terrestrial planets. Such observer selection effects are operating. Thus, while they may be difficult to quantify, they need to be considered whenever the cosmic coincidence is used to motivate new physics. These anthropic considerations operate in conjunction with (not in place of) fundamental explanations of the dark energy. Interacting quintessence models in which the proximity parameter asymptotes to a constant at late times [@Amendola2000; @Zimdahl2001; @Amendola2003; @Franca2004; @Guo2005; @Olivares2005; @Pavon2005; @Zhang2005; @Franca2006; @Amendola2006; @Amendola2007; @Olivares2007] have been proposed as a solution to the coincidence problem. More recently, @delCampo2005 [@delCampo2006] have argued for a broader class of interacting quintessence models that “soften” the coincidence problem by predicting a very slowly varying (though not constant) proximity parameter. Our analysis finds that $r$ need not asymptote to a constant, nor evolve particularly slowly, partially undermining the motivations for these interacting quintessence models. @Caldwell2003 and @Scherrer2005 have proposed that the coincidence problem may be solved by phantom models in which there is a future big-rip singularity because such cosmologies spend a significant fraction of their lifetimes in $r \sim 1$ states. In our work $P_t(t)$ is terminated by big-rip singularities in ripping models. In non-ripping models, however, the distribution is effectively terminated by the declining star formation rate. Therefore the big-rip gives phantom models only a marginal advantage over other models. This marginal advantage manifests as the discontinuity along $w_a=0$ on the left side of Fig. \[fig:severities\]. A running cosmological constant $\Lambda(t)$ could arise from the renormalization group (RG) in quantum field theory [@Shapiro2000; @Babic2002]. The running lambda term can mimic quintessence or phantom behaviour and transit smoothly between the two [@Sola2006]. RG models represent interesting alternatives to scalar-field models of dark energy. In some variants [@Grande2006b] additional fields are introduced to address the cosmic coincidence problem by predicting a slowly varying density ratio $r$. Our results demotivate such additions and favor simplistic RG models. How strongly do these results depend on the assumed time it takes for observers to arise, $\Delta t_{obs}$? In @Lineweaver2007, where we performed an anlysis similar to the present one (but limited to $w=-1$), we demonstrated that the results were robust to any choice $\Delta t_{obs} \sim [0,11]\ \textrm{Gyrs}$. However, for $\Delta t_{obs} {\mbox{$\:\stackrel{>}{_{\sim}}\:$} }12\ \textrm{Gyrs}$ that analysis resulted in an unavoidable coincidence problem because most observers would arise late (during DE domination) and would observe $\rho_{de} \gg \rho_{m}$. The validity of the results of the present analysis are similarly limited. We could improve our analysis, in the sense of getting tighter coincidence constraints (larger severities), if we used a less conservative $P_{\Delta t_{obs}}$. We used the most conservative choice - a delta function - because the present understanding of the time it takes to evolve into observers is too poorly developed to motivate any other form of $P_{\Delta t_{obs}}$. Another possible improvement is the DE equation of state parameterization. We used the current standard, $w=w_0+w_a(1-a)$, which may not parameterize some models well for very small or very large values of $a$. We conclude that DDE models need not be fitted with exact tracking or oscillatory behaviors specifically to solve the coincidence by generating long or repeated periods of $\rho_{de} \sim \rho_{m}$. Also, particular interactions guaranteeing $\rho_{de} \sim \rho_m$ for long periods are not well motivated. Moreover phantom models have no significant advantage over other DDE models with respect to the coincidence problem discussed here. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== CE acknowledges a UNSW School of Physics postgraduate fellowship. CE thanks the ANU’s RSAA for its kind hospitality, where this research was carried out. Numerical Values for Parameters of Models Illustrated in Fig. \[fig:quintessence\_energy\] {#paramvals} ========================================================================================== Model Parameter Value ---------------------------------- ----------- ------------------------ power law tracker quintessence $\alpha$ $2$ $M$ $1.4 \times 10^{-124}$ exponential tracker quintessence $M$ $1.3 \times 10^{-124}$ tracking oscillating energy $M$ $1.8 \times 10^{-126}$ $\lambda$ $4$ $A$ $0.99$ $\nu$ $2.7$ interacting quintessence $A$ $1.4 \times 10^{-119}$ $B$ $9.7$ $C$ $16$ Chaplygin gas $\alpha$ $1$ $A$ $2.8 \times 10^{-246}$ : Free parameters of the DDE models illustrated in Fig. \[fig:quintessence\_energy\]. These values were chosen such that observational constraints are crudely satisfied. These are by no means the only combinations fitting observations. These values are intended for the purposes of illustration in Fig. \[fig:quintessence\_energy\]. Units are Planck units. \[tab:paramvals\]

--- abstract: | Multi species turbulence in inhomogeneous magnetised plasmas is found to exhibit symmetry breaking in the dynamical alignment of a third species with the fluctuating electron density and vorticity with respect to the magnetic field direction and the species’ relative background gradients. The possibility of truly chiral aggregation of charged molecules in magnetized space plasma turbulence is discussed. This is a preprint version of a manuscript accepted 10/2012 for publication in [*Physics of Plasmas*]{}. author: - 'A. Kendl' title: Asymmetric chiral alignment in magnetized plasma turbulence --- Introduction: drift wave turbulence =================================== Drift wave turbulence [@Hor99] is composed of coupled nonlinear fluctuations of pressure $p$ and the electrostatic potential $\phi$ in an inhomogeneous plasma with a guiding magnetic field ${\bf B}$. The local electrostatic potential acts as a stream function for the dominant plasma flow with the drift velocity. The plasma is advected around the isocontours of a potential perturbation perpendicular to the magnetic field and forms a quasi two-dimensional vortex: an initial localized plasma density perturbation rapidly loses electrons along the magnetic field ${\bf B}$, leaving a positive electrostatic potential perturbation $\phi$ in its place. The associated electric field ${\bf E} = - {\boldsymbol \nabla}\phi$ is directed outward from the perturbation centre, and in combination with a magnetic field ${\bf B}$ causes an “E-cross-B” vortex drift motion of the whole plasma along isocontours of the potential with a velocity ${\bf v}_E = {\bf E} \times {\bf B}/B^2$. This gyration averaged drift motion is a result of the combined electric and magnetic forces acting on a charged particle with small gyration radius compared to background scales [@Hor99]. In the presence of a background density (or pressure) gradient ${\boldsymbol \nabla}n$ the vortex propagates in the direction perpendicular to both ${\boldsymbol \nabla}n$ and ${\bf B}$. Drift waves become unstable if the electron density can not rapidly adapt to the electrostatic potential along the direction parallel to ${\bf B}$, which can be caused by interaction of the electrons with other particles or waves. Nonlinear coupling of drift waves results in a self-sustained fully developed turbulent state [@Has83; @Sco90]. The requirements for the occurrence of drift waves are ubiquitously met in inhomogeneous magnetized plasmas when the drift scale $\rho_s = \sqrt{ (M_i T_e)/(eB)}$ is much smaller than the background pressure gradient length $L_p = ({\boldsymbol \nabla}\ln p)^{-1}$, or equivalently, when the fluctuation frequency $\omega$ is much smaller than the ion gyration frequency $\omega_{ci} = eB/M_i$. Here $M_i$ is the ion mass, $T_e$ the electron temperature (in eV), $e$ the elementary charge, $B$ the magnetic field strength, and $p = n_e T_e$ the electron pressure. Drift wave vortices are mainly excited in the size of a few drift scales $\rho_s$ and nonlinearly dually cascade to both smaller and larger scales. This perpendicular spatial scales of turbulent fluctuations are characteristically much smaller than any parallel fluctuation gradients and scales, so that wave vectors fulfill $k_{||} \ll k_{\perp}$. Particle aggregation and transport in turbulence is a widely studied subject in fluid dynamics [@Pro99]. In turbulent plasmas the aggregation of charged particles is in addition influenced by static and dynamic electric and magnetic fields. For gyrofluid drift wave turbulence in inhomogeneous magnetized plasmas it has been observed by Scott [@Sco05] that the gyrocenter density of light positive trace ions (in addition to electrons and a main ion species) in three-species fusion plasma drift wave turbulence tend to dynamically align with the fluctuating electron density, so that fluctuation amplitudes of the electron and trace ion densities are spatiotemporally closely correlated. Priego et al. [@Pri05] have found that the particle density of inertial impurties in a fluid drift wave turbulence model is also closely correlated with the ExB flow vorticity. The present analysis shows that the dynamical alignment in drift wave turbulence is not only sign selective with respect to the vorticity of trapping eddies, for any given trace species charge and background magnetic field direction, but also their large-scale background gradient. The resulting asymmetric effects on transport and aggregation of charged particles is specifically discussed for application to space plasmas, in particular for cases of a nearly collisionless plasma with cold ions and low parallel electron resistivity, where the electrostatic potential closely follows changes in the electron density. It is shown that in this case drift wave turbulence constitutes a unique mechanism for truly chiral vortical aggregation of charged molecules in space environments with a background magnetic field and plasma density gradients. This paper is organized as follows: in section II the cold-ion gyrofluid model equations for the present multi-species drift wave turbulence studies are introduced. Section III describes the numerical methods, and in section IV the computational results are presented. Section V discusses implications of the results for chiral particle aggregation in turbulent space plasmas, which poses a possible extraterrestrial physical mechanism for achieving enantiomeric excess in prebiotic molecular synthesis. The relation between the present gyrofluid model and the inertial fluid model by Priego et al. [@Pri05] is considered in the Appendix. Model: multi-species (cold ion) gyrofluid equations =================================================== Multi-species drift wave turbulence in the weakly collisional limit can be conveniently treated within a gyrofluid model. Here a strongly reduced variant of the gyrofluid electromagnetic model (“GEM”) by Scott [@Sco10] is adapted for numerical computation of three-species drift wave turbulence in a quasi-2D approximation in the electrostatic, isothermal, cold ion limit (neglecting finite Larmor radius effects) with a dissipative parallel coupling model. The gyrocenter particle densities $n_s$ for (electron, main ion, and trace ion) species $s \in (e,i,z)$ are evolved by nonlinear advection equations $$D_t n_s = (\partial_t + {\bf v}_E \cdot {\boldsymbol \nabla}) n_s = \partial_t n_s + [\phi,n_s] = C_s, \label{e:conti}$$ where all $C_s$ include hyperviscous dissipation terms $\nu_4 \nabla^4 n_s$ for numerical stability. The parallel gradient of the parallel fluid velocity components are for simplicity expressed by assuming a single parallel wave vector $k_{||}$ in the force balance equation along the magnetic field [@Has83]. Then $C_e$ in addition includes the coupling term $d(\phi - n_e)$, where $d$ is the parallel coupling coefficient proportional to $k_{||}^2/\nu_{ei}$. For weakly dissipative plasmas $d$ is well larger than unity, but for practical purposes $d = 2$ already sufficiently supports nearly adiabatic coupling between $n_e$ and $\phi$ while allowing reasonable time steps. The coupling can be regarded as non-adiabatic when $d <1$. The electrostatic potential is here derived from the local (linear) polarization equation $$\rho_m\nabla^2 \phi = n_e - n_i - a_z n_z, \label{e:locpol}$$ where $\rho_m = 1+ a_z \mu_z$ with $a_z = Z n_z/ n_e$ and $\mu_z = m_z/(Z m_i)$. In the local model only the density perturbations are evolved in the advection equations, which then gain an additional background advection term $g_s \partial_y \phi$ with $g_s = (L_{\perp}/L_{ns})$ on the right hand side, where $L_{\perp}$ is a normalising perpendicular scale (here set identical to $L_{ne}$ so that $g_e=g_i=1$) and $y$ is the coordinate perpendicular to both the magnetic field and the background gradient directions. In the following only trace ions with $a_z \ll 1$ are considered, and the parameter $a_z = 0.0001$ is going to be fixed, while other parameters are being varied. The 2D isothermal gyrofluid model and its relation to (inertial) fluid models is briefly discussed in the Appendix. Method: numerical simulation and analysis ========================================= The numerical scheme for solution of the advection and polarization equations is described (for a similar “Hasegawa Wakatani” type code where the ion density equation is replaced by a vorticity equation) in refs. [@Ken11; @Shu11]. The present computations use a $512 \times 512$ grid corresponding to $(64\;\rho_s)^2$ in units of the drift scale. This is a resolution well established for drift wave turbulence computations, resolving both the necessary small scales just below the gyro radius (or drift scale), and also allowing enough large scales for fully developed turbulent spectrum. The results converge for both higher resolution and same size in drift units, and for larger size with accordingly increased grid nodes. The computations are initialized with quasi-turbulent random density fields and run until a saturated state is achieved. The resulting turbulent advection dynamics is restrained to the 2-dimensional plane perpendicular to the (local) magnetic field [@Ken08]. Statistical analysis is performed on spatial and temporal fluctuation data in this saturated state. The turbulence characteristics of drift wave systems (like power spectra, probability distributions, etc.) in general have been extensively discussed before, so that here the focus is on analysis of the correlation of the fluctuating density of the additional species with the other fluctuating fields. Results: asymmetric dynamical alignment ======================================= In accordance with ref. [@Sco05] we also observe in our simulations that the gyrocenter density of a low-density massive trace ion species tends to dynamically align with the electron density $n_e$. The anomalous diffusion, clustering and pinch of impurities in plasma edge turbulence has also been studied previously by Priego et al. [@Pri05]. In this paper the impurities were modeled as a passive fluid advected by the electric and polarization drifts, while the ambient plasma turbulence was modeled using the 2D Hasegawa-Wakatani model. As a consequence of compressibility it has there also been found that the density of inertial impurities correlates with the vorticity of the ExB velocity. Trace impurities were observed to cluster in vortices of a precise orientation determined by the charge of the impurity particles [@Pri05]. The major difference to the present approach is that Priego et al. have considered only a constant impurity background (i.e. no impurity background gradient), but have in addition included additional nonlinear inertial effects through the impurity polarization drift. Linear inertial effects by polarization are already consistently included in the local gyrofluid model. The detailed correspondence between the two models is discussed in the Appendix. For an initial homogeneous distribution of impurities with no background gradient ($g_z=0$) and vanishing (or constant) initial perturbation $n_z(t=0)=0$ the perturbed gyrocenter density $n_z(t)$ remains zero for all times, which directly follows from the evolution equation $D_t n_z=0$. The perturbed fluid particle impurity density $N_z$ is related to the cold impurity gyrocenter density by the transformation $N_z = n_z + \mu_z \Omega$ (see Appendix). As a result the perturbed fluid particle impurity density evolves according to $D_t N_z = \mu_z D_t \Omega$ and directly follows the vorticity, which has also been observed in fluid simulations by Priego et al. [@Pri05]. In the following the focus is on additional effects of a finite impurity background density gradient specified by $g_z$. In the adiabatic (weakly collisional) limit the electrons can be assumed to follow a Boltzmann distribution with $$(n_0+n_e) = n_0 \exp[e\phi/T_e] \approx n_0 \; (1 + e\phi/T_e),$$ so that (for given $T_e$ and $n_0$) a positive $n_e$ perturbation corresponds to a positive localized potential perturbation $\phi$. The ${\bf v}_E \sim (-{\boldsymbol \nabla}\phi \times {\bf B})$ drift motion of a plasma vortex around a localized $\phi$ perturbation possesses a vorticity $${\bf \Omega} = {\boldsymbol \nabla}\times {\bf v}_E = ({\bf B}/|B|) \nabla^2 \phi,$$ related to the Laplacian of the electrostatic potential, and thus (depending on the sign of the fluctuating potential) a definite sense of rotation with respect to the background magnetic field ${\bf B}/|B|$. In ref. [@Sco05] the absolute correlation $|r(n_e,n_z)|$ between the (gyrocenter) density perturbations of electrons $n_e$ and trace ions $n_z$ has been determined. Our present analysis in addition shows that under specific conditions a definite sign relation appears for the sample correlation coefficient $$r(n_e,n_z) = { {\sum (n_e-\overline{n}_e)(n_z-\overline{n}_z) } \over \sqrt{ \sum(n_e-\overline{n}_e)^2 \sum(n_z-\overline{n}_z)^2} },$$ where the sum $\sum$ is taken over all grid points of the computational domain, and the bar $\overline{n}_s$ denotes the domain average of the specific particle density. In local computations with the present model, where the densities are split into a static spatially slowly varying background component $n_0$ with perpendicular gradient lengths $L_n = (\nabla \ln n_0)^{-1}$ and a fluctuating part with small amplitudes $n$ (typically in the range of a few percent), the sign and value of $r(n_e,n_z) \approx \pm (0.90 \pm 0.02)$ are observed to only depend on the relative sign but not the magnitude of the gradient lengths $L_{ne}$ and $L_{nz}$, for a given direction of the background magnetic field and fixed other parameters ($d=2$, $\mu_z = 10$). The result for $r(n_e,n_z)$ changes only marginally for most other parameter variations. In particular, the sign of the impurity charge $Z$ has no effect on the alignment property. Stronger adiabaticity leaves $r (d=10) \approx \pm (0.90 \pm 0.02)$ largely unchanged, while a smaller (strongly non-adiabatic) dissipative coupling coefficient results in $r (d=0.01) \approx \pm (0.79 \pm 0.02)$. In Fig. \[f:fig1\] the computed turbulent fields of vorticity (left figure) and a heavy (for example molecular) charged particle species density perturbation (right) in the 2D plane perpendicular to [**B**]{} are shown in a snapshot to be closely (negatively) spatially correlated for parameters $d=2$ (quasi-adiabatic) and $g_z = L_{\perp}/L_{nz} = + 0.001$ (co-aligned background density gradients). ![ *Left: Vorticity $\Omega(x,y)$; right: trace ion gyrocenter density perturbation $n_z(x,y)$. Normalized amplitudes are used and only a quarter of the complete computational domain is shown. $\Omega$ and $n_z$ here show close negative alignment for weak (quasi-adiabatic) coupling with $d=2$. The (co-aligned) impurity gradient length is here set to $L_{\perp}/L_{nz} = + 0.001$.*[]{data-label="f:fig1"}](kendl-fig-1-a.eps "fig:"){width="8.0cm"} ![ *Left: Vorticity $\Omega(x,y)$; right: trace ion gyrocenter density perturbation $n_z(x,y)$. Normalized amplitudes are used and only a quarter of the complete computational domain is shown. $\Omega$ and $n_z$ here show close negative alignment for weak (quasi-adiabatic) coupling with $d=2$. The (co-aligned) impurity gradient length is here set to $L_{\perp}/L_{nz} = + 0.001$.*[]{data-label="f:fig1"}](kendl-fig-1-b.eps "fig:"){width="8.0cm"} [Table 1: correlation coefficient $r(n_e,n_z)$]{}\ $g_z$ $d$ $r(n_e,n_z)$ ----------- -------- -------------------- $+ 0.001$ $2.00$ $ + 0.90 \pm 0.01$ $+ 0.100$ $2.00$ $ + 0.90 \pm 0.01$ $+ 1.000$ $2.00$ $ + 0.91 \pm 0.01$ $- 1.000$ $2.00$ $ - 0.90 \pm 0.01$ $- 0.100$ $2.00$ $ - 0.91 \pm 0.01$ $- 0.001$ $2.00$ $ - 0.90 \pm 0.01$ $+ 0.001$ $10.0$ $ + 0.90 \pm 0.02$ $+ 0.001$ $2.00$ $ + 0.90 \pm 0.01$ $+ 0.001$ $1.00$ $ + 0.92 \pm 0.01$ $+ 0.001$ $0.10$ $ + 0.87 \pm 0.01$ $+ 0.001$ $0.01$ $ + 0.79 \pm 0.02$ The observed sign-selective multi-species dynamical alignment effect is basically caused by the (linear) drive of density fluctuations $\partial_t n_z \sim g_z \partial_y \phi$ by this background advection term, where the potential $\phi$ is for each species just acting on the respective background gradients with length $L_{ns}$. The species gyrocenter densities are enhanced (positive partial time derivative) when the background advection is positive, and decreased when negative. Co-aligned background gradients of electrons and trace ions thus dynamically also imply co-alignment of the gyrocenter density perturbations. In case of adiabatic electrons this additionally implies negative alignment with vorticity, as can be seen in Fig. \[f:fig1\]. For counter-aligned background density gradients the plot for $n_z(x,y)$ in Fig. \[f:fig1\] has exactly the same topology only with signs reversed, i.e. blue and red in the colour scale exchanged. Strong non-adiabaticity ($d \ll 1$) only slightly reduces the electron-to-impurity density correlation coefficient $r(n_e,n_z)$ compared to adiabatic cases. Although the alignment between electron density and electrostatic potential is reduced for a non-adiabatic response, the alignment between the impurity gyrocenter density $n_z$ and vorticity $\Omega$ still remains. The results for $r(n_e,n_z)$ as a function of the dissipative coupling parameter $d$ and (co- or counter-aligned) impurity gradient length $g_z = L_{\perp}/L_{nz}$ are summarized in Table 1. ![ *A Gaussian trace ion puff placed in the turbulent plasma is initially localized around the center of the $x$ domain (and constant in $y$). Left: impurity density $n_z(x,y)$. Right: the product function $\Omega(x,y)\cdot n_z(x,y)$ shows in comparison with the top figure that impurity density and vorticity are preferentially negatively aligned in the right half of the domain (where the initial density gradients are co-aligned) and positively aligned on the left.* []{data-label="f:fig2"}](kendl-fig-2-a.eps "fig:"){width="8.0cm"} ![ *A Gaussian trace ion puff placed in the turbulent plasma is initially localized around the center of the $x$ domain (and constant in $y$). Left: impurity density $n_z(x,y)$. Right: the product function $\Omega(x,y)\cdot n_z(x,y)$ shows in comparison with the top figure that impurity density and vorticity are preferentially negatively aligned in the right half of the domain (where the initial density gradients are co-aligned) and positively aligned on the left.* []{data-label="f:fig2"}](kendl-fig-2-b.eps "fig:"){width="8.0cm"} ![ *A Gaussian trace ion puff placed in the turbulent plasma is initially localized around the center of the $y$ domain (and constant in $x$). Left: impurity density $n_z(x,y)$. Right: the product function $\Omega(x,y)\cdot n_z(x,y)$ shows that in this case no preferential sign of alignment between impurity density and vorticity is found.*[]{data-label="f:fig3"}](kendl-fig-3-a.eps "fig:"){width="8.0cm"} ![ *A Gaussian trace ion puff placed in the turbulent plasma is initially localized around the center of the $y$ domain (and constant in $x$). Left: impurity density $n_z(x,y)$. Right: the product function $\Omega(x,y)\cdot n_z(x,y)$ shows that in this case no preferential sign of alignment between impurity density and vorticity is found.*[]{data-label="f:fig3"}](kendl-fig-3-b.eps "fig:"){width="8.0cm"} The situation is different if there is no background distribution of the trace ion species, but only a smaller localized cloud (diffusing over time) with an initial spatial extension in the same order of magnitude as the turbulence scales. Then the cloud can have global gradients of its density in all directions with respect to the background electron (and primary ion) gradient, and the signs of $r$ approximately can cancel to zero by integration over the computational domain, while the absolute correlation coefficient $|r|$ remains near unity. The turbulent spreading of such a initially Gaussian trace ion puff (with zero background) strongly localized in $x$ or $y$ is shown in Figs. \[f:fig2\] and \[f:fig3\], respectively. In Fig. \[f:fig2\] a Gaussian trace ion puff placed in the turbulent plasma is initially localized around the center of the $x$ domain and constant in $y$. The product function $\Omega(x,y)\cdot n_z(x,y)$ shows (in the left part of the figure) in comparison with the impurity gyrocenter density $n_z(x,y)$ (right figure) that $n_z(x,y)$ and $\Omega(x,y)$ are preferentially negatively aligned in the right half of the domain, where the initial density gradients are co-aligned, and positively aligned on the left. This behaviour is in accordance with the previous result for a constant background gradient, as has been shown in Fig. \[f:fig1\] For the $x$-symmetric case in Fig. \[f:fig3\] no preferential sign of alignment between impurity gyrocenter density and vorticity is found. Also this case the perturbed fluid particle density $N_z$ is still correlated with vorticity according to the relation $D_t N_z = \mu_z D_t \Omega$. The preceding discussion of the present results was funded within the gyrocenter density $n_z$ representation of the gyrofluid model. It remains to be clarified when the advection by the background gradient or the inertial contribution is the dominant factor that determines alignment of the fluid particle density $N_z = n_z + \mu_z \Omega$ with vorticity. In Fig. \[f:fig4align\] the correlation coefficient $r(\Omega,N_z)$ is shown as a function of the impurity density gradient parameter $g_n$ at $d=2$ for $\mu_z=10$ (circles, bold solid line), $\mu_z=30$ (squares, thin dashed line) and $\mu_z=-10$ (diamonds, thin solid line). The alignment of perturbed impurity particle density with vorticity changes sign around $g_n \approx \mu_z/2$: for $g_n \ll \mu_z$ the alignment directly follows the vorticity with $r(\Omega,N_z)\approx 1$, and for $g_n \gg \mu_z$ the alignment is reversed towards negative vorticity with $r(\Omega,N_z) \approx -1$. The computations have been repeated for a non-adiabatic parallel coupling coefficient $d=0.1$, for which the results remain very similar. The overall impression given by Fig. \[f:fig4align\] does not change signficantly in this case, except for slight (order of per cent) differences within fluctuation averaging errors. The sign of alignment is thus determined by the strength of the impurity gradient in relation to the mass-to-charge ratio of the impurities. For example, for singly positively charged heavy molecules (like typical space biomolecules) with $\mu_z \gg 1$ and small impurity gradient $g_z \approx 1$ the alignment is tendentially towards positive vorticity. For vanishing impurity gradient ($g_z=0$) the correlation $r(\Omega,N_z)$ is always exactly $+1$ for positively charged impurities, and $-1$ for negative impurities. ![*Correlation coefficient $r(\Omega,N_z)$ as a function of the impurity density parameter $g_n$ for $\mu_z=10$ (circles, bold solid line), $\mu_z=30$ (squares, thin dashed line) and $\mu_z=-10$ (diamonds, thin solid line): the alignment of impurity particle density with vorticity changes sign around $g_n \approx \mu_z/2$.* []{data-label="f:fig4align"}](kendl-fig-4.eps){width="9.0cm"} The basic conclusion is that for given background gradients and magnetic field direction the perturbed trace ion species gyrocenter density dynamically aligns with a definite sign of the electron density fluctuations, and thus of the electrostatic potential fluctations, and consequently of vorticity ${\bf \Omega}$: for example, an excess of trace ions aggregates within vortices of clockwise direction, and a deficit is found in vortices with anti-clockwise direction (or vice versa, depending on global parameters). While, as usual in a fully developed turbulent state, vortices of both signs appear equally likely and evenly distributed over all turbulent scales, the trace particle aggregation on drift scales emerges with one prefered rotationality with respect to the background magnetic field and background particle gradients. Chiral molecular aggregation in drift wave turbulence ===================================================== Finally, a possible relevance of the asymmetric alignment effect on molecular chemistry in a magnetized space plasma environment is suggested. Homochirality of biomolecules – the fact that the essential chemical building blocks of life have a certain handedness while synthetic production leads to equal (racemic) distribution of left-handed and right-handed chiral structures – has ever since its discovery by Pasteur [@Pas48] posed a formidable puzzle [@Mas84; @Lou02]. Pasteur already had unsuccessfully tried to identify physical causes for this biological symmetry breaking, for example by imposing chirality through fluid vortices in a centrifuge, and by exposing chemical solutions to a magnetic field [@Pas84]. A number of theories and experiments on the origin of chirality have since been put forward, like effects of circularly polarized light on the molecular reactions [@Huc96; @Bai98; @Bow01] or a combination of magnetic field and non-polarized light [@Rik00], or possible electroweak effects on quantum chemistry [@Ber01]. All of these mechanisms could be active in interplanetary and interstellar space, and would hint on an extraterrestrial origin of early fundamental biomolecules. That chirality can in principle indeed be imposed by rotational forces has been confirmed experimentally for different situations [@Rib01; @Mic12], but what mechanism could invoke a specific directionality in turbulent (terrestrial or space) fluids or plasmas, where vortices of both senses of rotation generically occur mixed across all scales, has been left an open question. Here we argue that rotation asymmetric ion aggregation in drift vortices in magnetized space plasmas constitutes a mechanism for fostering a truly chiral environment for enantiomeric selective extraterrestrial formation of biomolecules. The drift scaling conditions and possibilities for occurrence of drift wave turbulence are well fulfilled for a number of typical space plasma parameters and magnetic field strengths, like in (warm ionized) clouds in the interstellar medium [@How06]. The background pressure or density gradient length of molecular ion species in the interstellar medium can take values across many orders of magnitude. While interstellar clouds range in size between a few and hundreds of parsecs (1 pc = $3.0857 \cdot 10^{16}$ m) and thus have global gradient lengths of the same order, the local gradients can be set by macroscopic turbulence (resulting from large scale magnetohydrodynamic motion or by chaotic external drive through winds and jets) and vary widely, from observable scales of 100 pc down to 1000 km [@Gae11]. Then again, the gradient of molecular ion density can also be set by a spatially varying degree of ionization, for example through inhomogeneous irradiation at the edge of molecular clouds. Magnetic field strengths in the interstellar medium can occur up to a few $\mu$G to mG, and temperatures range between 10 K in cold molecular clouds to 10000 K in warm ionized interstellar medium. The resulting drift (and vortex) scale $\rho_s$ is in the order between a few hundred meters and a few hundred kilometers. Typical vortex life times are in the order between seconds and hours. The mean free path (parallel to a magnetic field) between particle collisions is in the order of 1000 km, so that the plasma is only weakly collisional. The drift ratio is in the maximum order of $\delta = \rho_s / L_p = 0.1$ and smaller, thus well below unity. The relative fluctuation amplitudes (compared to background values of the plasma density) in drift wave turbulence are typically in the order of the drift ratio and thus here in the range of a few percent or below. Compared to macroscopic flow-driven or magnetohydrodynamic (MHD) turbulence, drift wave turbulence is most effective on smaller scales, in the order of $\rho_s$, and with relatively small amplitude. Drift wave vortices in the size between a few hundred meters and hundreds of kilometers are thus expected to be present in most interstellar plasmas. Next we consider, if such vortices truly can account for a chiral physical effect. Chiral synthesis usually requires a truly chiral influence [@Bar86; @Bar12]: the parity transformation $(P: x, v \rightarrow -x, -v)$ has to result in a mirror asymmetric state that needs to be different from simply a time reversal $(T: t \rightarrow -t)$ plus a subsequent rotation by $\pi$. A purely 2D vortex can not exert a truly chiral influence, while 3D funnel-like fluid vortices in principle do. While a bulk rotation per se can not cause any direct polarizing effect on the reaction path for molecular synthesis [@Fer99], particle aggregation on a supramolecular level in vortex motion has been experimentally shown to be able to lead to chiral selection [@Rib01]. To date it however remains unclear how chirality from a supramolecular aggregate (clusters or dust) could be transfered to single molecules [@Fer01]. In the case of drift wave turbulence the parallel wave-like dynamics imposes a wave vector $k_{||}$ that under $P$ changes direction with respect to the vorticity ${\bf \Omega}$ and ${\bf B}$ (which themselves are pseudovectors and remain invariant under $P$), equivalently to a 3D vortex tube. Drift wave turbulence is thus truly chiral, although it characteristically appears quasi-two-dimensional. In case of a large-scale density gradient $\nabla_{||} n$ along ${\bf B}$ the (small) $k_{||}$ is given by this gradient length, otherwise it is determined by local fluctuations. Molecular ions are advected by the drift vortex motion, and are subjected to chiral aggregation onto neutral molecules, clusters or dust particles that are not participating in the plasma rotation. If in addition (as to be rather expected in most cases) a trace ion background density gradient in the direction parallel to ${\bf B}$ is present, the resulting parallel wave vector $k_{||}$ prescribes in combination with vorticity ${\bf \Omega}$ a definite true chiral vortex effect. Chemical reactions in this system occur by collisions or aggregation of ions, that participate in the chiral vortex rotation, with neutral molecules or clusters of the quiescent (non rotating) background gas. The co- or counter alignment of ${\bf B}$ with ${\bf \Omega}$ can further act as chiral catalyst. The chiral influence however changes sign in different sides or regions of the (large-scale) molecular cloud, when the relative direction of background particle gradient and magnetic field direction reverses. This implies that different regions of the interstellar medium favour different tendencies for chiral selective aggregation, and thus for a potential enantiomeric excess in molecular syntheses. The excess rate could be expected in the order of the relative density fluctuations in drift wave turbulence, of a few percent. A possible problem that could not be addressed with the present electrostatic model is whether chiral alignment may be broken by strong Alfvènic activity. Electromagnetic computations for moderate beta values (a few percent as for fusion edge plasmas) in ref. [@Sco05] for absolute correlation have however shown no significant deviation of the alignment character compared to electrostatic computations, so that chiral alignment may be expected to survive for finite beta. Chiral aggregation in drift vortices should therefore be feasible locally at least in lower beta regions of the interstellar or interplanetary media. The necessary conditions for chiral aggregation thus appear rather restrictive and cosmologically rare. It may be concluded that either the suggested mechanism for chiral synthesis is subdominant (compared to any other of proposed or yet unknown mechanisms), or that a chiral excess of molecules should cosmologically be a rather rare phenomenon itself. Acknowledgements {#acknowledgements .unnumbered} ================ The author thanks B.D. Scott (IPP Garching) for valuable discussions on gyrofluid theory and computation. This work has been funded by the Austrian Science Fund (FWF) Y398. Appendix {#appendix .unnumbered} ======== The correspondence between the present cold-ion gyrofluid model with linear polarization and the HW model of Priego et al. [@Pri05] including inertial nonlinear polarization effects is in the following briefly lined out. For a general discussion on nonlinear polarization and dissipative correspondence between low-frequency fluid and gyrofluid equations we refer to ref. [@Sco07] by Scott. The Priego model (with notations and normalizations adopted to fit ours) consists of the 2D Hasegawa-Wakatani (HW) model $$\begin{aligned} & & D_t (N_e-x) = d (\phi -N_e) \\ & & D_t \Omega = d(\phi-N_e) \label{e:hw}\end{aligned}$$ where $\Omega = \nabla^2 \phi$ is the vorticity, $D_t = \partial_t + {\bf v}_E \cdot {\boldsymbol \nabla}$ is the convective derivative with ${\bf v}_E = ({\bf B}/|B|) \times {\boldsymbol \nabla}\phi$, and $D_t x = - \partial_y \phi$ in a local model where the background density gradient enters into the length scale normalisation. For simplicity we neglect in this presentation all dissipation terms (where Priego et al. have used normal second order dissipation, while we used fourth order hyperviscous dissipation terms). Here capital letters are used for the fluid particle densities $N_s$ for distinction to the gyrocenter densities $n_s$ of the gyrofluid model. The particle densities fulfill the direct quasi-neutrality condition $N_e = N_i$ for $N_z \ll N_{i0}$. The massive trace impurities are in Priego’s model assumed to be passively advected, but due to their inertia respond to the velocity ${\bf v}_z = {\bf v}_E + {\bf v}_{pz}$ with the additional polarization drift velocity ${\bf v}_{pz} = - \mu_z D_t \nabla \phi$ in the global compressional impurity continuity equation $$\partial_t N_z + {\boldsymbol \nabla}\cdot ( N_z {\bf v}_z) = 0$$ for the full impurity density $N_z$ (whereas only fluctuations of $N_e$ are evolved in the HW equation). For incompressional ExB velocity (when the magnetic field is homogeneous) this is equivalent to the full (global) impurity density equation used by Priego et al., which is (in our notation) given in ref. [@Pri05] as $$D_t N_z - {\boldsymbol \nabla}\cdot ( \zeta N_z D_t {\boldsymbol \nabla}\phi) = 0 \label{e:priego}$$ with $\zeta \equiv \mu_z \delta_0$. The factor $\delta_0 = \rho_s / L_{ne}$ stems from the normalization $\phi \rightarrow \delta_0^{-1} e \tilde \phi / T_{e0}$ of the local HW model. This introduces a nonlinear polarization term into the dynamics of massive impurities. In a local approximation (corresponding to the assumption of small fluctuations on a large static background), where only the ExB advection is kept as the sole nonlinearity, this equation reduces to: $$D_t N_z = \zeta N_{z0} D_t \Omega. \label{e:priegolin}$$ Now we derive a set of HW-like vorticity-density fluid equations from the (cold ion) gyrofluid model, and compare this with the Priego model. The present gyrofluid model consists of the local gyrocenter density equations $$\begin{aligned} D_t (n_e + g_e x) &=& d(\phi - n_e) \\ D_t (n_i + g_i x) &=& 0 \\ D_t (n_z + g_z x )&=& 0 \end{aligned}$$ and the local quasi-neutral polarization equation $$\sum_s \left[ {a_s \over \tau_s} (\Gamma_0 -1) \phi + a_s \Gamma_1 n_s \right] = 0.$$ where $\tau_s = T_{s0}/T_{e0}$, $a_s = Z n_s/ n_e$ and $\mu_s = m_s/(Z m_i)$. The gyrofluid advective derivative $D_t n_s = \partial_t n_s + [\psi,n_s]$ includes the gyro-averaged potential $\psi = \Gamma_1 \phi$ with $\Gamma_0 = [1+b]^{-1}$ and $\Gamma_1 = \Gamma_0^{1/2} = [1+(1/2)b]^{-1}$ in Pade approximation, where $b = -\tau_s \mu_s \nabla^2$. In Taylor approximation one would have $\Gamma_0 \approx [1-b]$ and $\Gamma_1 \approx [1-(1/2)b]$. The species coefficients are $\tau_e=1$, $a_e = -1$, $\mu_e=0$ for electrons, and $a_i=1$ for ions. The background gradient terms $g_s = \partial_x \ln n_{s0}$ fulfill (for zero background vorticity) the global quasi-neutrality condition $g_e = g_i + a_z g_z \approx g_i$ when $g_i \ll 1$ for trace impurities. In the normalization of perpendicular length scales by $L_{\perp} = L_n = |\partial_x \ln n_{e0}|^{-1}$ we then have $g_e=g_i=1$. The perturbed particle and gyrocenter densities are connected via the elements of the polarization equation as [@Sco07] $$a_s N_s = a_s \Gamma_1 n_s + \mu_s a_s \nabla^2 \phi$$ so that gyrofluid polarisation corresponds to the fluid particle quasi-neutrality condition $\sum_s a_s N_s = 0$. From now on again cold ions with $\tau_i=\tau_z=0$ are assumed, and thus $\psi=\phi$ and $\Gamma_0 = \Gamma_1 = 1$. The local polarization then reduces to $\rho_m \nabla^2 \phi = n_e - n_i -a_z n_z$ with $\rho_m = \sum_s a_s \mu_s = 1+ a_z \mu_z$. It can be immediately seen that the local cold-ion gyrofluid model already includes the (linear) polarization effect that has been added in the Priego model, when the gyrocenter density in the impurity continuity equation $D_t n_z =0$ is substituted by the fluid particle density: $$D_t N_z = D_t \alpha_z n_z + D_t \mu_z \alpha_z \Omega = \mu_z \alpha_z D_t \Omega$$ where $\alpha_s = a_s/Z = n_{s0}/n_{e0}$. This is identical to the linearized form eq. \[e:priegolin\] of the Priego model (where the full impurity density is evolved so that on the right hand side $\mu_z \alpha_z \rightarrow \mu_z \delta_0 N_{z0}$). Completely passive trace ions would be achieved in the gyrofluid model for $\mu_z=0$ so that $\rho_m=1$. A fully global set of nonlinear gyrofluid equations would include the nonlinear quasi-neutral polarization equation [@Str04], which (with $M_s=m_s/m_i$) is given as: $$\sum_s {\boldsymbol \nabla}\cdot (M_s n_s {\boldsymbol \nabla}\phi) + \sum_s q_s \Gamma_1 n_s = 0. \label{e:globalpol}$$ This equation is expensive to solve numerically and requires e.g. the use of a multi-grid solver. Many codes therefore apply the linearized form, which can be treated by standard fast Poisson solvers. This is usually regarded as a valid approximation as long as the turbulent density fluctuations are small compared to the background plasma density. The trace of the (cold ion) global polarization eq. \[e:globalpol\] again delivers the fluid-gyrofluid density relations: $$N_s = n_s + {\boldsymbol \nabla}\cdot (M_s n_s {\boldsymbol \nabla}\phi).$$ From this we get the fluid impurity continuity equation $$D_t N_z = D_t {\boldsymbol \nabla}\cdot (M_z n_z {\boldsymbol \nabla}\phi) = {\boldsymbol \nabla}\cdot (M_z n_z D_t {\boldsymbol \nabla}\phi)$$ where twice $D_t n_z = 0$ has been used. The term on the right hand side however still contains the gyrofluid density $n_z$. The approximation $$D_t N_z \approx {\boldsymbol \nabla}\cdot (\zeta N_z D_t {\boldsymbol \nabla}\phi) \label{e:result}$$ with the fluid particle density $N_z$ appears acceptable if either $\zeta \ll 1$ or if only small turbulent fluctuations on a static background can be assumed, so that the additional gyrocenter correction could be considered an order smaller than the other terms. 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--- abstract: 'In thermodynamics, quantum coherences—superpositions between energy eigenstates—behave in distinctly nonclassical ways. Here we describe how thermodynamic coherence splits into two kinds—“internal" coherence that admits an energetic value in terms of thermodynamic work, and “external" coherence that does not have energetic value, but instead corresponds to the functioning of the system as a quantum clock. For the latter form of coherence we provide dynamical constraints that relate to quantum metrology and macroscopicity, while for the former, we show that quantum states exist that have finite internal coherence yet with zero deterministic work value. Finally, under minimal thermodynamic assumptions, we establish a clock-work trade-off relation between these two types of coherences. This can be viewed as a form of time-energy conjugate relation within quantum thermodynamics that bounds the total maximum of clock and work resources for a given system.' author: - Hyukjoon Kwon - Hyunseok Jeong - David Jennings - Benjamin Yadin - 'M. S. Kim' title: 'Clock–Work Trade-Off Relation for Coherence in Quantum Thermodynamics' --- Classical thermodynamics describes the physical behavior of macroscopic systems composed of large numbers of particles. Thanks to its intimate relationship with statistics and information theory, the domain of thermodynamics has recently been extended to include small systems, and even quantum systems. One particularly pressing question is how the existence of quantum coherences, or superpositions of energy eigenstates, impacts the laws of thermodynamics [@Aberg14; @Skrzypczyk14; @Uzdin15; @Korzekwa16], in addition to quantum correlations [@Park13; @Reeb14; @Huber15; @Llobet15]. We now have a range of results for quantum thermodynamics [@Janzing2000; @Horodecki13; @Brandao13; @Brandao15; @Lostaglio15; @LostaglioX; @Cwiklinski15; @Goold16; @Bera16; @Wilming16; @LostaglioPRL; @Muller17; @Gour17] that have been developed within the resource-theoretic approach. A key advantage of the resource theory framework is that it avoids highly problematic concepts such as “heat" or “entropy" as its starting point. Its results have been shown to be consistent with traditional thermodynamics [@Weilmann16] while allowing for the inclusion of coherence as a resource [@Sterltsov17] in the quantum regime. Very recently, a framework for quantum thermodynamics with coherence was introduced in Ref. [@Gour17]. Remarkably, the thermodynamic structure (namely, which states $\hat\sigma$ are thermodynamically accessible from a given state $\hat\rho$) turns out to be fully describable in terms of a single family of entropies. This framework of “thermal processes” is defined by the following three minimal physical assumptions: (i) that energy is conserved microscopically, (ii) that an equilibrium state exists, and (iii) that quantum coherence has a thermodynamic value. These are described in more detail in [@Supplemental]. Note that thermal processes contain thermal operations (TOs) [@Janzing2000; @Horodecki13] as a subset, and coincide with TOs on incoherent states; however, in contrast to TOs they admit a straightforward description for the evolution of states with coherences between energy eigenspaces. In this Letter we work under the same thermodynamic assumptions (i)–(iii) as above and show that quantum coherence in thermodynamics splits into two distinct types: *internal* coherences between quantum states of the same energy, and *external* coherences between states of different energies. This terminology is used because the external coherences in a system are only defined relative to an external phase reference frame, while internal coherences are defined within the system as relational coherences between its subcomponents. We focus on the case of an $N$-partite system with noninteracting subsystems. The Hamiltonian is written as $\hat{H} = \sum_{i=1}^N \hat{H}_i$ and we assume that each $i$th local Hamiltonian $\hat{H}_i$ has an energy spectrum $\{ E_i \}$ with local energy eigenstates ${\left\vertE_i\right\rangle}$. Then a quantum state of this system may be represented as $$\hat\rho = \sum_{{\boldsymbol E}, {\boldsymbol E'} } \rho_{{\boldsymbol E}{\boldsymbol E'} } {\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E'\right\vert},$$ where ${\boldsymbol E} := (E_1, E_2, \cdots, E_N)$ and ${\left\vert\boldsymbol E\right\rangle} := {\left\vertE_1 E_2 \cdots E_N\right\rangle}$. We also define the total energy of the string ${\boldsymbol E}$ as ${\cal E}_{\boldsymbol E} := \sum_{i=1}^N E_i$. Classical thermodynamic properties are determined by the probability distribution of the local energies, including their correlations. This information is contained in the diagonal terms of density matrix $P(\boldsymbol E) := {\rm Tr} \left[ \hat\Pi_{\boldsymbol E} \hat\rho \right] = \rho_{\boldsymbol E \boldsymbol E}$, where $\hat\Pi_{\boldsymbol E} := {\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E\right\vert}$. Corresponding classical states could have degeneracies in energy, but we still have a distinguished orthonormal basis set $\{ {\left\vert\mathbf{E}\right\rangle} \}$. The probability distribution of the total energy is $\displaystyle p_{\cal E} := \sum_{\boldsymbol{E} \colon {\cal E}_{\boldsymbol E} = {\cal E}} P(\boldsymbol E)$. So every state has a corresponding classical state defined via the projection $\Pi(\hat\rho) := \sum_{\boldsymbol E} P(\boldsymbol E) {\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E\right\vert}$. However, a quantum system is defined by more than its classical energy distribution – it may have coherence in the energy eigenbasis. This coherence is associated with nonzero off-diagonal elements in the density matrix, namely ${\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E'\right\vert}$ for $\boldsymbol E \neq \boldsymbol E'$. The internal coherence corresponds to off-diagonal terms of the same total energy (where $\cal{E}_{\boldsymbol E} = \cal{E}_{\boldsymbol E'}$) and external coherence corresponds to terms with different energies ($\cal{E}_{\boldsymbol E} \neq \cal{E}_{\boldsymbol E'}$). For any state $\hat\rho$, we denote the corresponding state in which all external coherence is removed by $\mathcal{D}(\hat\rho) := \sum_{\mathcal E} \hat\Pi_{\mathcal E} \hat\rho \hat\Pi_{\mathcal E}$, where $\hat\Pi_{\mathcal E} := \sum_{\boldsymbol{E} \colon \mathcal{E}_{\boldsymbol{E}} = \mathcal{E}} \hat\Pi_{\boldsymbol E}$ is the projector onto the eigenspace of total energy $\mathcal{E}$. As illustrated in Fig. \[FIG1\], internal coherence may be used to extract work; however, it has been shown that external coherences obey a superselection rule (called “work locking") that forbids work extraction, and is unavoidable if one wishes to explicitly account for all sources of coherence in thermodynamics [@Lostaglio15]. We study this phenomenon by defining the process of extracting work *purely from the coherence*, without affecting the classical energy statistics. We find the conditions under which work can be deterministically extracted in this way from a pure state. Next, we show that external coherence is responsible for a system’s ability to act as a clock. The precision of the clock may be quantified by the quantum Fisher information (QFI) [@Braunstein94]; we show that the QFI satisfies a second-law-like condition, stating that it cannot increase under a thermal process. Finally, we derive a fundamental trade-off inequality between the QFI and the extractable work from coherence demonstrating how a system’s potential for producing work is limited by its ability to act as a clock and vice versa. [*Extractable work from coherence.—*]{} Here we demonstrate that, in a single-shot setting, work may be extracted from coherence without changing the classical energy distribution $P(\boldsymbol E)$ of the system. We consider the following type of work extraction process $$\hat\rho \otimes {\left\vert0\right\rangle} {\left\langle0\right\vert}_B \xrightarrow{\rm thermal~process} \Pi(\hat\rho) \otimes {\left\vertW\right\rangle}{\left\langleW\right\vert}_B,$$ in which the energy of a work qubit [@footnote] $B$ with Hamiltonian $\hat{H}_B = W {\left\vertW\right\rangle}{\left\langleW\right\vert}_B$ ($W \geq 0$) is raised from ${\left\vert0\right\rangle}_B$ to ${\left\vertW\right\rangle}_B$. For energy-block-diagonal states $\hat\rho = \mathcal{D}(\hat\rho)$ and $\hat\sigma = \mathcal{D}(\hat\sigma)$, the work distance is defined as $D_{\rm work} (\hat\rho \succ \hat\sigma) = \displaystyle \inf_{\alpha} \left[ F_\alpha(\hat\rho) - F_\alpha(\hat\sigma) \right] $ [@Brandao15], where $F_\alpha(\hat\rho) = k_B T S_\alpha(\hat\rho || \hat\gamma ) - k_B T \log Z$ is a generalized free energy based on the Rényi divergence $$S_\alpha({\hat{\rho}}||{\hat{\sigma}}) = \begin{cases} \frac{1}{\alpha-1} \log \Tr [ {\hat{\rho}}^\alpha {\hat{\sigma}}^{1-\alpha} ], & \alpha \in [0,1)\\ \frac{1}{\alpha-1} \log \Tr \left[ \left({\hat{\sigma}}^\frac{1-\alpha}{2\alpha} {\hat{\rho}} {\hat{\sigma}}^\frac{1-\alpha}{2\alpha}\right)^\alpha \right], & \alpha>1. \end{cases}$$ Here $Z = {\rm Tr} e^{-\hat{H}/(k_B T)}$ is a partition function of the system. The work distance is the maximum extractable work by a thermal process by taking $\hat\rho$ to $\hat\sigma$ [@Brandao15]. ![Thermodynamic resources for many-body quantum systems. Coherences between energy levels provide coherent oscillations and are resources for the composite system to act as a quantum clock. At the other extreme, projective energy measurements on the individual systems provide the classical energy statistics, which may display classical correlations. Intermediate between these two cases are quantum coherences that are internal to energy eigenspaces. Partial interconversions are possible between these three aspects.[]{data-label="FIG1"}](FIG1.eps){width="0.8\linewidth"} ![Thermomajorization graph for the energy-block-diagonal state ${\cal D}({\left\vert\psi\right\rangle}{\left\langle\psi\right\vert})$ and its projection to an incoherent state ${\Pi}({\left\vert\psi\right\rangle}{\left\langle\psi\right\vert})$ for a two-qubit state ${\left\vert\psi\right\rangle}$ studied in the text with coefficients $p_1$ and $p_2$. $Z$ represents the partition function of the system. (a) When $p_1 e^{\beta \omega_0}$ is the maximum among $p_i e^{\beta E_i}$, $W_{\rm coh}$ is positive, but (b) if another energy (e.g., $p_2 e^{2\beta\omega_0}$ in the plot) obtains the maximum, $W_{\rm coh}=0$. []{data-label="WcohQ"}](FIG2.eps){width="\linewidth"} Even when the initial state $\hat\rho$ is not block diagonal in the energy basis, the extractable work is still given by $D_{\rm work}(\mathcal{D}(\hat\rho) \succ \hat\sigma)$, so external coherence cannot be used to extract additional work [@Lostaglio15]. In order to exploit external coherence for work, one needs multiple copies of $\hat\rho$ [@Brandao13; @Korzekwa16] or ancilliary coherent resources [@Brandao13; @Aberg14; @Lostaglio15]. Thus, the single-shot extractable work purely from coherence is given by $$\label{Wcoh} W_{\rm coh} = \inf_\alpha \left[ F_\alpha({\cal D} (\hat\rho)) - F_\alpha(\Pi(\hat\rho)) \right].$$ For example, consider extracting work from coherence in the pure two-qubit state $${\left\vert\psi\right\rangle} = \sqrt{p_0} {\left\vert00\right\rangle} + \sqrt{p_1} \left( \frac{ {\left\vert01\right\rangle} + {\left\vert10\right\rangle}}{ \sqrt{2} } \right)+ \sqrt{p_2} {\left\vert11\right\rangle},$$ where each qubit has local Hamiltonain $H_i = \omega_0 {\left\vert1\right\rangle}{\left\langle1\right\vert}$. As shown in Fig. \[WcohQ\] using the concept of thermomajorization [@Horodecki13], we have $W_{\rm coh} >0$ only for sufficiently large $p_1$. In this case, we have the necessary condition $p_1 > (1+e^{\beta\omega_0} + e^{-\beta\omega_0})^{-1}$ and the sufficient condition $p_1 > e^{\beta\omega_0}/({1+e^{\beta\omega_0}})$ for $W_{\rm coh} > 0$, independent of $p_0$ and $p_2$ at an inverse temperature $\beta = (k_B T)^{-1}$. See Ref. [@Supplemental] for details. We generalize this statement to the internal coherence of an arbitrary pure state. Since external coherences cannot contribute to work, we need only consider the properties of the state dephased in the energy eigenbasis. \[PureWext\] Consider any pure state state ${\left\vert\psi\right\rangle}$ with $\mathcal{D}( {\left\vert\psi\right\rangle}{\left\langle\psi\right\vert}) = \sum_k p_\mathcal{E}|\psi_\mathcal{E}\rangle\langle \psi_\mathcal{E}|$, where $|\psi_\mathcal{E}\rangle$ is an energy $\mathcal{E}$ eigenstate. Nonzero work can be extracted deterministically from the internal coherence of $\hat\rho$ at an inverse temperature $\beta$ if and only if $\Pi\left({\left\vert\psi_{{\cal E}^*}\right\rangle}{\left\langle\psi_{{\cal E}^*}\right\vert}\right) \neq {\left\vert\psi_{{\cal E}^*}\right\rangle}{\left\langle\psi_{{\cal E}^*}\right\vert}$ for ${\cal E}^* = \underset{\cal E}{\rm argmax}~p_{\cal E}e^{\beta {\cal E}}$. Internal coherence has some overlap with nonclassical correlations, namely quantum discord [@Modi12]. Consider the following quantity which quantifies the sharing of free energy between subsystems: $$C_\alpha(\hat\rho_{1:2:\cdots:N}) := \beta \left[ F_\alpha(\hat\rho) - \sum_{i=1}^N F_\alpha(\hat\rho_i) \right],$$ where $\hat \rho_i$ is the local state of the $i$th subsystem. For nondegenerate local Hamiltonians, the extractable work from coherence can be written as $$W_{\rm coh} = k_B T \inf_{\alpha} \left[ C_\alpha( {\cal D}(\hat\rho)) - C_\alpha( \Pi(\hat\rho)) \right],$$ noting that the local free energies are the same for ${\cal D}(\hat\rho)$ and $\Pi(\hat\rho)$. This is of the same form as discord defined by Ollivier and Zurek [@Zurek01], expressed as a difference between total and classical correlations. Note that the classical correlations are defined here with respect to the energy basis, instead of the usual maximization over all local basis choices. This free-energy correlation is also related to “measurement-induced disturbance” [@Luo08] by considering $\Pi(\hat\rho)$ as a classical measurement with respect to the local energy bases. Unlike previous related studies [@Llobet15; @Korzekwa16], our result requires that only coherence is consumed in the work extraction processes, leaving all energy statistics unchanged. We may also consider the “incoherent" contribution to the extractable work, $W_{\rm incoh} := \inf_\alpha [F_\alpha(\Pi(\hat\rho)) - F_\alpha(\hat\gamma)] = F_{0}(\Pi(\hat\rho)) + k_B T \log Z$, which is the achievable work from an incoherent state $\Pi(\hat\rho)$ ending with a Gibbs state $\hat\gamma$. The sum of the coherent and incoherent terms cannot exceed the total extractable work from $\hat\rho$ to $\hat\gamma$, i.e., $W_{\rm coh} + W_{\rm incoh} \leq W_{\rm tot} = D_{\rm work}({\cal D}(\hat\rho) \succ \hat\gamma) $. The equality holds when $W_{\rm coh}$ in Eq. (\[Wcoh\]) is given at $\alpha = 0$. We also point out that this type of work extraction process operates without any measurement or information storage as in Maxwell’s demon [@Lloyd97; @Zurek03] or the Szilard engine [@Park13] in the quantum regime. Apart from the above example, a significant case is the so-called coherent Gibbs state [@Lostaglio15], defined for a single subsystem as ${\left\vert\gamma\right\rangle} := \sum_i \sqrt{\frac{e^{-\beta E_i}}{Z}} {\left\vertE_i\right\rangle}$. No work can be extracted from this state, as $\mathcal{D}({\left\vert\gamma\right\rangle}{\left\langle\gamma\right\vert}) = \hat\gamma$ – an instance of work locking. However, nonzero work can be unlocked [@Lostaglio15] from multiple copies ${\left\vert\gamma\right\rangle}^{\otimes N},\, N>1$. In fact, from Observation \[PureWext\], we see that $N=2$ is always sufficient to give $W_{\rm coh}>0$. This is because $p_{\mathcal{E}} e^{\beta \mathcal{E}}$ is proportional to the degeneracy of the $\mathcal{E}$ subspace – there always exists a degenerate subspace for $N \geq 2$, and this is guaranteed to have coherence. [*Coherence as a clock resource.—*]{} Having discussed the thermodynamical relevance of internal coherence, we now turn to external coherence. Suppose we have an initial state $\hat\rho_0 = \sum_{{\boldsymbol E}, {\boldsymbol E'}} \rho_{{\boldsymbol E}{\boldsymbol E'} } {\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E'\right\vert}$. After free unitary evolution for time $t$, this becomes $\hat\rho_t = \sum_{{\boldsymbol E}, {\boldsymbol E'} } \rho_{{\boldsymbol E}{\boldsymbol E'} } e^{- i \Delta \omega_{\boldsymbol E \boldsymbol E'} t} {\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E'\right\vert}$, where each off-diagonal component ${\left\vert\boldsymbol E\right\rangle} {\left\langle\boldsymbol E'\right\vert}$ rotates at frequency $\Delta \omega_{\boldsymbol E \boldsymbol E'} = ({\cal E}_{\boldsymbol E} - {\cal E}_{\boldsymbol E'})/\hbar$. Internal coherences do not evolve ($\Delta \omega_{\boldsymbol E \boldsymbol E'} = 0$), while external coherences with larger energy gaps, and hence higher frequencies, can be considered as providing more sensitive quantum clocks [@Chen10; @Komar14]. By comparing $\hat\rho_0$ with $\hat\rho_t$, one can estimate the elapsed time $t$. More precisely, the resolution of a quantum clock can be quantified by $(\Delta t)^2 = \langle (\hat{t} - t)^2 \rangle$, where $\hat{t}$ is the time estimator derived from some measurement on $\hat\rho_t$. The resolution is limited by the quantum Cramér-Rao bound [@Braunstein94], $(\Delta t)^2 \geq 1 / I_F(\hat{\rho}, {\hat{H})}$, where $I_F(\hat{\rho}, \hat{H}) = 2 \sum_{i,j} \frac{(\lambda_i - \lambda_j)^2}{\lambda_i + \lambda_j} |{\left\langlei\right\vert} \hat{H} {\left\vertj\right\rangle}|^2$ is the QFI, and $\lambda_i,\, {\left\verti\right\rangle}$ are the eigenvalues and eigenstates of $\hat\rho$, respectively. For the optimal time estimator $\hat{t}$ saturating the bound, the larger the QFI, the higher the clock resolution. The maximum value of QFI for a given Hamiltonian $\hat{H}$ is $(E_{\rm max} - E_{\rm min})^2$, which can be obtained by the equal superposition ${\left\vertE_{\rm min}\right\rangle} + {\left\vertE_{\rm max}\right\rangle}$ between the maximum ($E_{\rm max}$) and minimum ($E_{\rm min}$) energy eigenstates. The Greenberger-Horne-Zeilinger (GHZ) state in an $N$-particle two-level system is a state of this form. Another family of relevant measures of the clock resolution is the skew information $I_\alpha({\hat{\rho}}, \hat{H}) = \Tr({\hat{\rho}} \hat{H}) - \Tr({\hat{\rho}}^\alpha \hat{H} {\hat{\rho}}^{1-\alpha} \hat{H})$ for $0 \leq \alpha \leq 1$ [@Wigner63; @Braunstein94]. For pure states, both the QFI and skew information reduce to the variance: $\frac{1}{4} I_F({\left\vert\psi\right\rangle},\hat{H}) = I_\alpha({\left\vert\psi\right\rangle},\hat{H}) = V({\left\vert\psi\right\rangle},\hat{H}) := {\left\langle\psi\right\vert}{\hat{H}^2}{\left\vert\psi\right\rangle} - {\left\langle\psi\right\vert}\hat{H}{\left\vert\psi\right\rangle}^2$. In particular, the skew information of $\alpha=1/2$ has been studied in the context of quantifying coherence [@Luo17] and quantum macroscopicity [@Kwon18]. We also remark that a similar approach to “time references" in quantum thermodynamics has been recently suggested using an entropic clock performance quantifier [@Gour17]. We first note that, even though a quantum state might be very poor at providing work, it can still function as a good time reference. The coherent Gibbs state is a canonical example. As mentioned earlier, no work may be extracted from ${\left\vert\gamma\right\rangle}$; however, such a state does allow time measurements, since $I_F({\left\vert\gamma\right\rangle}, \hat{H}) = 4 \frac{\partial^2}{\partial \beta^2} \log Z$, which is proportional to the heat capacity $k_B \beta^2 \frac{\partial^2}{\partial \beta^2} \log Z$ [@Crooks12]. Furthermore, the QFI and skew information are based on monotone metrics [@Hansen08; @Petz11], and monotonically decrease under time-translation-covariant operations [@Marvian14; @Yadin16]. It follows that the resolution of a quantum clock gives an additional constraint of a second-law type. Under a thermal process, the quantum Fisher (skew) information $I_{F(\alpha)}$ of a quantum system cannot increase, i.e. $$\label{Imonotone} \Delta I_{F(\alpha)} \leq 0.$$ We highlight that this condition is independent from those obtained previously, based on a family of entropy asymmetry measures $ A_\alpha({\hat{\rho}}_S) = S_\alpha ({\hat{\rho}}_S || {\cal D} ({\hat{\rho}}_S))$ [@Lostaglio15] and modes of asymmetry $\sum_{{\cal E}_{\boldsymbol E} - {\cal E}_{\boldsymbol E'} = \omega} |\rho_{\boldsymbol E \boldsymbol E'}| $ [@LostaglioX; @MarvianPRA]. In Ref. [@Supplemental], we give an example of a state transformation that is forbidden by Eq. (\[Imonotone\]) but not by previous constraints. Importantly, condition Eq. (\[Imonotone\]) remains significant in the many-copy or independent and identically distributed limit. This follows from the additivity of the QFI and skew information, namely, $ I_{F(\alpha)} ({\hat{\rho}}^{\otimes N}, \hat{H})/ N = I_{F(\alpha)}({\hat{\rho}},\hat{H}_1)$ for all $N$, where $\hat{H}=\sum_{i=1}^N \hat{H}_i$. In contrast, the measure $A_\alpha$ is negligible in this limit: $\displaystyle \lim_{N\rightarrow \infty} A_\alpha({\hat{\rho}}^{\otimes N}) / N = 0$ for all $\alpha$ [@Lostaglio15]. The free energy $F_\alpha (\hat\rho)$ with $\alpha=1$ has been stated to be the unique monotone for asymptotic transformations [@Brandao13]. However, this is true *only* if one is allowed to use a catalyst containing external coherence between every energy level, ${\left\vertM\right\rangle} = |M|^{-1/2} \sum_{m \in M} {\left\vertm\right\rangle}$, where $M= \{ 0, \dots , 2N^{2/3} \}$, which contains a superlinear amount of clock resources $I_F= O(N^{4/3})$, so $I_F/N$ is unbounded. Thus, Eq. (\[Imonotone\]) is the first known nontrivial coherence constraint on asymptotic transformations under thermal processes without additional catalytic coherence resources. We can illustrate the physical implications of this condition in an $N$-particle two-level system with a local Hamiltonian $\hat{H}_i = 0 {\left\vert0\right\rangle}_i{\left\langle0\right\vert} + \omega_0 {\left\vert1\right\rangle}_i{\left\langle1\right\vert}$ for every $i$th particle. As noted above, for a product state ${\hat{\rho}}^{\otimes N}$, the QFI and skew information scale linearly with $N$. On the other hand, the GHZ state ${\left\vert\psi_{\rm GHZ}\right\rangle} = 2^{-1/2}({\left\vert0\right\rangle}^{\otimes N} + {\left\vert1\right\rangle}^{\otimes N})$ has quadratic scaling, $I_{F(\alpha)} ({\left\vert\psi_{\rm GHZ}\right\rangle}, \hat{H}) = O(N^2)$. Thus the restriction given by Eq. (\[Imonotone\]) indicates that a thermal process cannot transform a product state into a GHZ state. More generally, it is known that $I_F(\hat\rho, \hat{H}) \leq kN$ for $k$-producible states in $N$-qubit systems [@Toth12; @Hyllus12], so genuine multipartite entanglement is necessary to achieve a high clock precision of $I_F = O(N^2)$. Also note that the QFI has been used to quantify “macroscopicity", the degree to which a state displays quantum behavior on a large scale [@Frowis2012; @Yadin16]. [*Trade-off between work and clock resources.—*]{} Having examined the two types of thermodynamic coherence independently, it is natural to ask if there is a relation between them. Here, we demonstrate that there is always a trade-off between work and clock coherence resources. We first give the following bound in an $N$-particle two-level system: \[Clock/work trade-off for two-level subsystems\] For a system composed of N two-level particles with energy level difference $\omega_0$, the coherent work and clock resources satisfy $$\label{TradeOff} W_{\rm coh} \leq N k_B T (\log 2) H_b\left( \frac{1}{2} \left[1-\sqrt{\frac{I_F(\hat\rho, \hat{H})}{N^2 \omega_0^2}} \right] \right) ,$$ where $\hat{H}$ is the total Hamiltonian and $H_b(r) = -r \log_2 r -(1-r)\log_2(1-r) $ is the binary entropy. \[TradeOffTheorem\] This shows that a quantum state cannot simultaneously contain maximal work and clock resources. When the clock resource is maximal, $I_F = N^2 \omega_0^2$, no work can be extracted from coherence $W_{\rm coh} = 0$. Conversely, if the extractable work form coherence is maximal, $W_{\rm coh} = N k_B T \log 2$, the state cannot be utilized as a quantum clock as $I_F=0$. For $N=2$ we derive a tighter inequality: $$\label{QubitTradeOff} W_{\rm coh} + ( k_B T \log 2) \left( \frac{ I_F(\hat\rho, \hat{H})}{4\omega_0^2} \right) \leq k_B T \log 2 .$$ ![Trade-off between work and clock coherences. The solid line refers to Eq. (\[TradeOff\]), the dashed line refers to Eq. (\[QubitTradeOff\]), and the dotted line refers the tighter bound for $N=10$. []{data-label="TradeOffFig"}](FIG3.eps){width="0.9\linewidth"} We demonstrate that the GHZ state ${\left\vert\psi_{\rm GHZ}\right\rangle} = 2^{-1/2} ({\left\vert0\right\rangle}^{\otimes N} + {\left\vert1\right\rangle}^{\otimes N})$ and Dicke states ${\left\vertN,k\right\rangle} = \binom{N}{k}^{-1/2}\sum_{P} P ( {\left\vert1\right\rangle}^{k} {\left\vert0\right\rangle}^{N-k})$, summing over all permutations $P$ of subsystems, are limiting cases that saturate this trade-off relation. For a Dicke state, the extractable work from coherence is given by $W_{\rm coh} = k_B T \log \binom{N}{k} \approx N k_B T (\log 2) H_b(k/N)$. However, each Dicke state has $I_F = 0$ since it has support on a single energy eigenspace with ${\cal E} = k\omega_0$. In particular, when $k=N/2$, $W_{\rm coh} = N k_B T \log 2$, attaining the maximal value and saturating the bounds Eqs. (\[TradeOff\]) and (\[QubitTradeOff\]). The GHZ state behaves in the opposite way: ${\left\vert\psi_{\rm GHZ}\right\rangle}$ has maximal QFI $I_F({\left\vert\psi_{\rm GHZ}\right\rangle}, \hat{H}) = N^2 \omega_0^2$ while having no internal coherence; thus, $W_{\rm coh}=0$. In this case, we can see the saturation of both bounds Eqs. (\[TradeOff\]) and (\[QubitTradeOff\]). Furthermore, our two-level trade-off relation can be generalized for an arbitrary noninteracting $N$-particle system. \[TradeOffN\] Let $\hat H$ be a noninteracting Hamiltonian of $N$ subsystems, where the $i$th subsystem has an arbitrary (possibly degenerate) $d^{(i)}$-level spectrum $\{ E_1^{(i)} \leq E_2^{(i)} \leq \cdots \leq E_{d^{(i)}}^{(i)} \}$. Also define $\Delta_E^2 = \sum_{i=1}^N (\Delta_E^{(i)})^2 $ with $\Delta_E^{(i)} = E_{d^{(i)}}^{(i)} - E_1^{(i)}$. Then $$\label{GeneralTradeOff} W_{\rm coh} + k_B T \left( \frac{I_F(\hat\rho,\hat{H})}{2 \Delta_E^2} \right) \leq k_B T \sum_{n=1}^N \log d^{(n)}.$$ This is more generally applicable than Eq. (\[QubitTradeOff\]), but is weaker for two-level subsystems – maximal $I_F$ does not imply $W_{\rm coh}=0$ via Eq. (\[GeneralTradeOff\]). For systems with identical local $d$-level Hamiltonians, Eq. (\[GeneralTradeOff\]) reduces to $$\bar{w}_{\rm coh} + k_B T \left( \frac{I_F(\hat\rho,\hat{H})}{2N^2 \Delta_0^2} \right) \leq k_B T \log d ,$$ where $\bar{w}_{\rm coh} = W_{\rm coh} / N$ is extractable work per particle and $\Delta_E^2 = N \Delta_0^2$ where $\Delta_0$ is the maximum energy difference between the local energy eigenvalues. Our bounds do not limit the extractable work in the independent and identically distributed limit, since $I_F(\hat \rho^{\otimes N},\hat H) / N^2 \to 0$ for $N \gg 1$. We can also describe how one extends our analysis into the regime of weak interactions between local systems. We note that interactions can break degeneracies in energy eigenspaces, and energy eigenstates of the free Hamiltonian may not be expressed as a product of local states. However breaking of degeneracies in energy should only be treated above a finite experimental width $\epsilon$ in energy resolution. Weak perturbations admit a similar analysis in terms of work extraction up to $\epsilon$ fluctuations. This $\epsilon$-energy window allows us to use external coherences with energy gaps less than $\epsilon$ for work extraction by effectively treating them as internal coherences in the same energy levels. In this case, we can calculate how work and clock resources are perturbed and note that the trade-off relation Eq. (\[GeneralTradeOff\]) still holds with an ${\cal O}(\epsilon)$ correction. We also discuss an example for transverse Ising models in which the trade-off relation can be resolved using a quasiparticle picture (see Ref. [@Supplemental] for details). The more general interacting case is nontrivial and we leave this for future study. [*Remarks.—*]{} We have found that thermodynamic coherence in a many-body system can be decomposed into time- and energy-related components. 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Supplemental Material ===================== Physical assumptions for the analysis. ====================================== For the class of free operations that define the thermodynamic framework, we make the following three physical assumptions [@Gour17]. 1. *Energy is conserved microscopically.* We assume that any quantum operation $\mathcal{E}: \mathcal{B}(\mathcal{H}_A) \rightarrow \mathcal{B}(\mathcal{H}_{A'})$ that is thermodynamically free admits a Stinespring dilation of the form $$\label{micro} \mathcal{E}(\hat{\rho}_A) = {\rm tr}_C \hat{V} (\hat{\rho}_A \otimes \hat{\sigma}_B) \hat{V}^\dagger$$ where the isometry $\hat{V}$ conserves energy microscopically, namely we have $$\hat{V} (\hat{H}_A \otimes \mathbb{I}_B + \mathbb{I}_A \otimes \hat{H}_B) = (\hat{H}_{A'} \otimes \mathbb{I}_C + \mathbb{I}_{A'} \otimes \hat{H}_C) \hat{V}.$$ Here $H_S$ denotes the Hamiltonian for the system $S \in \{ A,A', B, C\}$. 2. *An equilibrium state exists.* We assume that for any systems $A$ and $A'$ there exist states $\hat{\gamma}_A$ and $\hat{\gamma}_{A'}$ such that $\mathcal{E} ( \hat{\gamma}_A) = \hat{\gamma}_{A'}$ for all thermodynamically free operations $\mathcal{E}$ between these two quantum systems. In the case where one admits an unbounded number of free states within the theory, this together with energy conservation essentially forces one to take $$\hat{\gamma}_A = \frac{1}{Z_A} e^{-\beta \hat{H}_A},$$ at some temperature $T = (k_B\beta)^{-1}$ and $Z_A = {\rm tr} \left[ e^{-\beta \hat{H}_A} \right]$. However one may also consider scenarios in which the thermodynamic equilibrium state deviates from being a Gibbs state. Here we restrict our analysis to thermal Gibbs states. 3. *Quantum coherences are not thermodynamically free.* Since one is interested in quantifying the effects of coherence in thermodynamics, one must not view coherence as a free resource that can be injected into a system without being included in the accounting. We therefore assume that the thermodynamically free operations do not smuggle in coherences in the following sense: if $\mathcal{E}$ has a microscopic description of the form \[micro\] then the same operation is possible with $\hat{\sigma}_B \rightarrow \mathcal{D} (\hat{\sigma}_B)$ and $\hat{V} \rightarrow \hat{W}$ being some other energy conserving isometry. In other words *no coherences in $\hat{\sigma}_B$ are exploited for free*. It can be shown this assumption has the mathematical consequence that $$\mathcal{E}( e^{-it \hat{H}_A} \hat{\rho}_A e^{it \hat{H}_A}) = e^{-i t \hat{H}_{A'}} \mathcal{E}(\hat{\rho}_A) e^{i t \hat{H}_{A'}},$$ for any translation through a time interval $ t \in \mathbb{R}$, namely covariance under time-translations. The first assumption is simply a statement of energy conservation, while the second assumption singles out a special state that is left invariant under the class of thermodynamic processes. Note that this equilibrium state could be a *non-Gibbsian* state and the formalism in [@Gour17] would still apply, however for our analysis here, we shall take the state to be a thermal Gibbs state, simply because this provides us with a notion of temperature and a comparision with traditional equilibrium thermodynamics. Finally the last assumption can be understood as a criterion for non-classicality for the framework, and the set of quantum operations defined by these three physical assumptions is called (generalized) *thermal processes* (TPs). By using these three physical assumptions (1–3), one can show that a set of TPs coincide with the set of time-translational covariant Gibbs-preserving maps [@Gour17]. We emphasize that TPs are different from thermal operations (TOs) [@Janzing2000; @Horodecki13]. A key difference is that TOs are defined by a energy conserved unitary operation between a system and a equilibrium bath $\hat\gamma_B = Z_B^{-1} e^{-\beta \hat{H}_B}$. $${\rm tr}_B \hat{V} (\hat\rho_A \otimes \hat\gamma_B) \hat{V}^\dagger,$$ satisfying $[\hat{V}, \hat{H}_A + \hat{H}_B] =0$. Crucially, TOs fix the auxiliary system to be in the Gibbs state. One might think that the assumptions (1–3) imply that TP coincides with TO but this is not the case. The operations in TO all obey (1–3) and so $TO \subseteq TP$, however there exist transformations in TP that are not in TO. To show this, consider the regime in which all Hamiltonians are trivial, namely $\hat{H}=0$. For this scenario the equilibrium Gibbs state becomes the maximally mixed state $\frac{1}{d} \mathbb{1}$ and conditions (1) and (3) are trivially true for any map. The set of TOs for this situation coincide with the set of *noisy operations*, while the set of TPs coincide with the set of *unital maps*. It is known that these two sets are not the same. However it is known that the *state interconversion structure* of noisy operations *coincides* with that of unital maps: we have $\hat\rho \longrightarrow \hat\sigma$ under a noisy operation if and only if it is possible under a unital operation. Therefore one might conjecture that these two classes of operations have essentially the same “power”, in the sense just described. It has been shown in [@Gour17] that when restricted to states block-diagonal in energy that TP and TO coincide exactly in terms of state interconversion (both are governed by thermo-majorization), however at present it is unclear how/whether the interconversion structure of these two classes differ for states with coherence, and there is no obvious physical principle to choose one over the other. However, one nice aspect of TPs is that they admit a complete description in terms a single family of entropies that have a natural interpretation, while at present no complete set exists for TOs. Note that the third assumption only accounts for *external* coherences within the framework. The way in which we account for internal coherences in our analysis is to demand that the diagonal components (in the basis $|\mathbf{E} \rangle$) of the state are left invariant by the evolutions considered. Also note that if assumptions (1) and (3) hold then the isometry $\hat{W}$ can be taken to be equal to $\hat{V}$. Work extraction from a pure state coherence =========================================== Proof of Observation 1 ---------------------- We find the conditions under which work can be deterministically extracted from coherence in a pure state. After energy block diagonalizing, the state can be written as ${\cal D}({\left\vert\Psi\right\rangle}{\left\langle\Psi\right\vert}) = \sum_{\cal E} p_{\cal E} {\left\vert\psi_{\cal E}\right\rangle} {\left\langle\psi_{\cal E}\right\vert}$, where ${\left\vert\psi_{\cal E}\right\rangle}$ are pure eigenstates of energy ${\cal E}$. Suppose $\cal E^*$ gives the maximum value of $\log p_{\cal E} e^{\beta \cal E}$. Then we have $$\begin{aligned} W_{\rm coh} &= \inf_\alpha \left[ F_\alpha({\cal D}(\hat\rho) ) - F_\alpha(\Pi(\hat\rho)) \right] \\ &= \inf_\alpha \left( \frac{1}{\alpha-1} \right) \log \left[ \frac{\sum_{\cal E} p_{{\cal E}}^\alpha e^{-\beta(1-\alpha)\cal E} } {\sum_{\cal E} e^{(1-\alpha) S_\alpha ( \hat\rho_{{\cal E}-\rm diag} ) } p_{{\cal E}}^\alpha e^{-\beta(1-\alpha)\cal E} } \right], \end{aligned}$$ where $\hat\rho_{{\cal E}-\rm diag} = \Pi( {\left\vert\psi_{\cal E}\right\rangle} {\left\langle\psi_{\cal E}\right\vert})$ is a fully dephased state in the energy eigenspace ${\cal E }$. Then we notice that $S_\alpha ( \hat\rho_{{\cal E}-\rm diag} ) ) > 0$ for any $\alpha \in [0, \infty)$, unless $\hat\rho_{{\cal E}-\rm diag}$ is incoherent (i.e. no internal coherence for $\cal E$). This leads to $ F_\alpha({\cal D}(\hat\rho) ) - F_\alpha(\Pi(\hat\rho)) >0 $ for any finite value of $\alpha$. In the limit $\alpha \rightarrow \infty$, $F_\infty ({\cal D}(\hat\rho) ) - F_\infty (\Pi(\hat\rho)) = \log p_{\cal E^*} e^{\beta \cal E^*} - \max_{{\cal E}, \lambda_{\cal E} } p_{\cal E} \lambda_{\cal E} e^{\beta \cal E} > 0$, unless $\hat\rho_{{\cal E^*}-\rm diag}$ is incoherent. Here, $\lambda_{\cal E}$ are eigenvalues of $\hat\rho_{{\cal E}-\rm diag}$. Thus if a pure state does not contain internal coherence for ${\cal E}^*$, $W_{\rm coh} \leq F_\infty ({\cal D}(\hat\rho) ) - F_\infty (\Pi(\hat\rho)) =0$. Conversely, if the state contains internal coherence for ${\cal E}^*$, $F_\alpha ({\cal D}(\hat\rho) ) - F_\alpha (\Pi(\hat\rho)) >0$ for all $\alpha \in [0,\infty)$, thus positive work can be extracted. Work extraction condition for a bipartite two-level system ---------------------------------------------------------- We consider work extraction from coherence in a two-qubit system with local energy difference $\omega_0$ in each qubit, starting from a pure state of the form $${\left\vert\psi\right\rangle} = \sqrt{p_0} {\left\vert00\right\rangle} + \sqrt{p_1} \left( \frac{ {\left\vert01\right\rangle} + {\left\vert10\right\rangle}}{ \sqrt{2} } \right)+ \sqrt{p_2} {\left\vert11\right\rangle} .$$ Observation 1 says that $p_1$ should be large enough to extract work under a single-shot thermal operation. In this case, the condition from Observation 1 can be written as $$\begin{cases} p_1 e^{\beta \omega_0} > p_0 = 1 - p_1 - p_2 \\ p_1 e^{\beta \omega_0} > p_2 e^{2\beta\omega_0} . \end{cases}$$ This leads to a necessary condition for extracting a positive amount of work $W_{\rm ext} > 0$ from coherence: $$p_1 > \frac{1}{1+e^{\beta\omega_0} + e^{-\beta\omega_0}}$$ Thus if $0< p_1 < (1+e^{\beta\omega_0} + e^{-\beta\omega_0})^{-1}$, we cannot extract work from coherenc,e even though the state definitely contains internal coherence of the form ${\left\vert01\right\rangle} + {\left\vert10\right\rangle}$. On the other hand, a sufficient condition for $W_{\rm ext} >0$ can be obtained: $$p_1 > \frac{e^{\beta\omega_0}}{1+e^{\beta\omega_0}}.$$ An example of the quantum Fisher (skew) information imposing independent constraints from $F_{\alpha}$ or $A_\alpha$ ==================================================================================================================== We present an example showing that our asymmetry quantifiers give constraints on quantum thermodynamics independent from those due to the free energies $F_\alpha$ or the coherence measures $A_\alpha$. Let us consider the transformation by a thermal process of the initial state $$\hat\rho = \left( \begin{array}{cccc} 0.5 & 0 & 0.1 & 0.1 \\ 0 & 0.2 & 0 & 0 \\ 0.1 & 0 & 0.25 & 0.1 \\ 0.1 & 0 & 0.1 & 0.05 \\ \end{array} \right)$$ to the final state $$\hat\sigma = \left( \begin{array}{cccc} 0.5 & 0.099 & 0.099 & 0.099 \\ 0.099 & 0.25 & 0 & 0 \\ 0.099 & 0 & 0.2 & 0 \\ 0.099 & 0 & 0 & 0.05 \\ \end{array} \right)$$ with the Hamiltonian $\hat{H} = \sum_{n=0}^3 n \omega {\left\vertn\right\rangle}{\left\langlen\right\vert}$. It can be checked that the free energies $F_\alpha$ and coherence measures $A_\alpha$ of the initial state $\hat\rho$ are larger than those of the final state $\hat\sigma$. Furthermore, each mode of coherence is decreased from $0.1$ to $0.099$. However, the skew information values for $\alpha=1/2$ are given by $I_{1/2}(\hat\rho,\hat{H}) = 0.153$ and $I_{1/2}(\hat\sigma,\hat{H}) = 0.163$; the quantum Fisher information values are $I_F(\hat\rho,\hat{H}) = 0.843$ and $I_F(\hat\sigma,\hat{H}) = 0.959$ (all in units of $\omega^2$). Thus a thermal process cannot transform $\hat\rho$ into $\hat\sigma$, but such a transformation is not disallowed by the restrictions given by $F_\alpha$ or $A_\alpha$. This is due to that the coherence monotones given by the quantum Fisher information and skew information capture not only the degree of coherence between different energy eigenstates, but also take account of how much energy level spacing exists in each coherence term. Clock/work trade-off relation: Two-level local Hamiltonian systems ================================================================== We first prove the following proposition. For a given energy distribution $p_{\cal E}$, the extractable work from coherence is upper bounded as follows: $$W_{\rm coh} \leq k_B T \sum_{\cal E} p_{\cal E} \log g_{\cal E},$$ where $g_{\cal E}$ is the dimension of the eigenspace of energy level ${\cal E}$. \[Prop1\] Note that $$\begin{aligned} W_{\rm coh} &= \inf_\alpha \left[ F_\alpha({\cal D}(\hat\rho)) - F_\alpha( \Pi(\hat\rho)) \right] \\ &\leq F({\cal D}(\hat\rho)) - F( \Pi(\hat\rho)) \\ &= k_B T \left[ S( \Pi(\hat\rho)) - S({\cal D}(\hat\rho)) \right]. \end{aligned}$$ Since both $\Pi(\hat\rho)$ and ${\cal D}(\hat\rho)$ are energy-block diagonal, we can express $\Pi(\hat\rho) = \sum_{{\cal E}, \lambda} p^\Pi_{{\cal E}, \lambda} {\left\vert{\cal E}, \lambda\right\rangle}{\left\langle{\cal E}, \lambda\right\vert}$ and ${\cal D}(\hat\rho)= \sum_{{\cal E}, \lambda} p^{\cal D}_{{\cal E}, \lambda} {\left\vert{\cal E}, \lambda\right\rangle}{\left\langle{\cal E}, \lambda\right\vert}$ for $\lambda = 1, 2, \cdots, g_{\cal E}$ with $\sum_{\lambda=1}^{g_{\cal E}} p^\Pi_{{\cal E}, \lambda} = \sum_\lambda p^{\cal D}_{{\cal E}, \lambda} =p_{\cal E}$. Then we have $$\begin{aligned} S(\Pi(\hat\rho)) - S({\cal D}(\hat\rho)) &= \sum_{\cal E} p_{\cal E} \sum_{\lambda=1}^{g_{\cal E}} \left[ \frac{p^{\cal D}_{{\cal E}, \lambda}}{p_{\cal E}} \log \frac{p^{\cal D}_{{\cal E}, \lambda}}{p_{\cal E}} - \frac{p^{\Pi}_{{\cal E}, \lambda}}{p_{\cal E}} \log \frac{p^{\Pi}_{{\cal E}, \lambda}}{p_{\cal E}} \right] \\ &\leq \sum_{\cal E} p_{\cal E} \log g_{\cal E}, \end{aligned}$$ since $S(\hat\rho) - S(\hat\sigma) \leq \log d$ for $d$-dimensional states $\hat\rho$ and $\hat\sigma$. Proof of Theorem 1 ------------------ Here we provide a complete proof of Theorem 1 that extractable work from coherence is upper bounded by the quantum Fisher information: $$W_{\rm coh} \leq k_B T N (\log 2) H_b\left( \frac{1}{2} \left[1-\sqrt{\frac{I_F(\hat\rho, \hat{H})}{N^2 \omega_0^2}} \right] \right).$$ In an $N$-particle two-level system with energy difference $\omega_0$, the degeneracy of the energy level ${\cal E}$ is given by $g_{\cal E} = \binom{N}{n}$, where ${\cal E} = \omega_0 n$. By using the fact that $$\binom{N}{n} \leq 2^{ N H_b \left(\frac{n}{N} \right)}$$ for every $N$ and $n$, we obtain $$W_{\rm coh} \leq k_B T \sum_{\cal E} p_{\cal E} \log \binom{N}{n} \leq N k_B T ( \log 2 ) \sum_{\cal E} p_{\cal E} H_b(n/N), \label{ExtBound1}$$ where Proposition 1 has been applied to obtain the first inequality. Furthermore, we can express the binary entropy as $$H_b(x) = 1 - \frac{1}{2 \log 2} \sum_{j=1}^\infty \frac{(1-2x)^{2j}}{j(2j-1)}.$$ For a given probability distribution $p_x$ and $j \geq 1$ we have $$\sum_x p_x (1-2x)^{2j} \geq \left[\sum_x p_x (1- 2x)^2 \right]^j = (1- 2y)^{2j},$$ where $y = \frac{1}{2} \left[ 1 \pm \sqrt{(1-2\bar{x})^2 + 4 {\rm Var}_x} \right]$ with $\bar{x} = \sum_x p_x x$ and ${\rm Var}_x = \sum_x p_x (x - \bar{x})^2$. Then we have $$\begin{aligned} \sum_x p_x H_b(x) &= 1 - \frac{1}{2 \log 2} \sum_{j=1}^\infty \sum_x p_x \frac{(1-2x)^{2j}}{j(2j-1)} \\ & \leq 1 - \frac{1}{2 \log 2} \sum_{j=1}^\infty \frac{(1-2y)^{2j}}{j(2j-1)} \\ & = H_b(y). \end{aligned}$$ By substituting this result into Eq. (\[ExtBound1\]), we obtain $$\label{BoundN} \begin{aligned} W_{\rm coh} & \leq N k_B T (\log 2) H_b \left( \frac{1}{2} \left[ 1 \pm \sqrt{ \left(1- \frac{2 \bar{E}}{N\omega_0}\right)^2 + \frac{ {4 \rm Var}_{\hat{H}}}{N^2 \omega_0^2} } \right]\right), \end{aligned}$$ where $\bar{E} = \langle \hat{H} \rangle_{\hat\rho}$ and ${\rm Var}_{\hat{H}} = \langle (\hat{H} - \bar{E})^2 \rangle_{\hat\rho} $. Note that $H_b$ is symmetric about $x=1/2$ and monotonically increasing for $x \leq 1/2$. We also note that $4 {\rm Var}_{\hat{H}} \geq I_F(\hat\rho, \hat{H})$ for every quantum state $\hat{\rho}$. These observations lead to $H_b \left( \frac{1}{2} \left[ 1 \pm \sqrt{ \left(1- \frac{2 \bar{E}}{N\omega_0}\right)^2 + \frac{ {4 \rm Var}_{\hat{H}}}{N^2 \omega_0^2} } \right] \right) \leq H_b \left( \frac{1}{2} \left[ 1 - \sqrt{ \frac{ I_F(\hat\rho, \hat{H})}{N^2 \omega_0^2} } \right] \right)$, which completes the proof. Tighter bound of the trade-off relation --------------------------------------- The bound from Theorem 1 can be tightened. We have observed that $$\label{BinomBound} \binom{N}{rN} \leq \binom{N}{N/2}^{H_b(r)}$$ or equivalently $$\log \binom{N}{rN} \leq {H_b(r)} \log \binom{N}{N/2}$$ for every $0 \leq r \leq 1$ and $N$ up to $N=100$. The binomial coefficient for an odd number of $N$ is defined as $\binom{N}{N/2} := \frac{\Gamma(N+1)}{\Gamma (N/2+1)^2}$ by using the Gamma function $\Gamma(z) = \int_0^{\infty} x^{z-1} e^{-x} dx$. Figure \[BinomialFig\] shows that the inequality (\[BinomBound\]) well holds for $N \leq100$ and seemingly holds for every number of $N$, yet the proof for a general case has not been found. ![Numerical verification of the bound $\log \binom{N}{n} \leq \log \binom{N}{N/2} H_b(n/N)$ for $N \leq 100$. The value of $N$ increases from the upper most curve ($N=2$) to the lowest curve ($N=100$). []{data-label="BinomialFig"}](BinomialFig.eps){width="0.43\linewidth"} From the inequality above, we obtain a tighter bound of the trade-off relation \[BoundN\] by taking $n=rN$ and replacing $N \log 2$ with $\log \binom{N}{N/2}$: $$\begin{aligned} W_{\rm coh} & \leq k_B T \log \binom{N}{N/2} H_b \left( \frac{1}{2} \left[ 1 - \sqrt{ \frac{ I_F(\hat\rho, \hat{H})}{N^2 \omega_0^2} } \right]\right), \end{aligned}$$ When $N \gg 1$, we note that $N \log 2 \approx \log \binom{N}{N/2}$ then the bound approaches to the bound of Theorem 1. Clock/work trade-off for $N=2$ ------------------------------ We show the tighter trade-off relation between clock/work resources for $N=2$ case: $$W_{\rm coh} + ( k_B T \log 2) \left( \frac{ I_F(\hat\rho, \hat{H})}{4\omega_0^2} \right) \leq k_B T \log 2 .$$ Suppose the state has probability $p_0$, $p_1$, and $p_2$ for each energy level $0$, $\omega_0$ and $2\omega_0$. By using Eq. (\[ExtBound1\]) for $N=2$, we have $W_{\rm coh} \leq k_B T (\log 2) p_1$, since the state has a degeneracy in the energy-eigenspace only for ${\cal E} = \omega_0$. In this case, energy variance ${\rm Var}_{\hat{H}}$ is given by $${\rm Var}_{\hat{H}} = \omega_0^2 (-p_1^2 + p_1 - 4p_2^2 + 4p_2 - 4 p_1 p_2),$$ which leads to the maximum value of $p_1$, $$p_1^{\rm max} = 1 - {\rm Var}_{\hat{H}} / \omega_0^2,$$ for a given value of ${\rm Var}_{\hat{H}}$. Again, we can use $4 {\rm Var}_{\hat{H}} \geq I_F(\hat\rho,\hat{H})$ to get $$W_{\rm coh} \leq k_B T (\log 2) p_1^{\rm max} = k_B T (\log 2) \left( 1 - \frac{I_F(\hat\rho, \hat{H})}{4 \omega_0^2} \right).$$ which is the desired inequality. Clock/work trade-off relation: General case =========================================== Proof of Theorem 2 ------------------ We prove the statement of Theorem 2: $$W_{\rm coh} + k_B T \left( \frac{I_F(\hat\rho,\hat{H})}{2 \Delta_E^2} \right) \leq k_B T \sum_{i=1}^N \log d^{(i)},$$ where $\Delta_E^2 = \sum_{i=1}^N (\Delta_E^{(i)})^2 $ with $\Delta_E^{(i)}$ is the maximum energy difference of the $i$th subsystem. In this case, the degeneracy $g_{\cal E}$ of the energy ${\cal E}$ is given by $$g_{\cal E} = \prod_{i=1}^N d^{(i)} f_{\cal E},$$ where $f_{\cal E} = \sum_{{\cal E}_{\boldsymbol E} = {\cal E}} P(\boldsymbol E)$ is a probability (or frequency) to have the energy $\cal E$ in the $N$-particle system, since $\prod_{i=1}^N d^{(i)}$ is total possible numbers of $\boldsymbol E$. Then $f_{\cal E}$ can be considered as a probability distribution of a variable $X_N = \sum_{i=1}^N E_i$ from the distribution of independent random variables of $E_i$ for $i$th party. In our case, $E_i$ is strictly bounded by $E^{(i)}_1 \leq E_n \leq E^{(i)}_{d^{(i)}}$ and it has the same probability $P(E_i = E^{(i)}_j) = 1/d^{(i)}$ for every $j=1,2,\cdots, d^{(i)}$ and zero for all other cases. Hoeffding’s inequality [@Heffding63], then shows that $$P(X_N- \mu_E \geq t ) \leq \exp\left[{-\frac{2 t^2} {\Delta_E^2}}\right],$$ where $\displaystyle \mu_E := \mathbb{E} (X_N) = \mathbb{E} \left( \sum_{i=1}^N E_i \right) = \sum_{i=1}^N \sum_{j=1}^{d^{(i)}} \left( \frac{E^{(i)}_j}{d^{(i)}} \right)$ and $\Delta_E^2 = \sum_{i=1}^N (\Delta_E^{(i)})^2 $. Using this, the upper bound of $f_{\cal E}$ is given by $$f_{\cal E} = P( X_N = {\cal E}) \leq P( X_N \geq {\cal E}) \leq \exp \left[ -\frac{2( {\cal E} - \mu_E )^2 }{\Delta_E^2} \right].$$ Again using Proposition 1, we have $$\begin{aligned} W_{\rm coh} & \leq k_B T \sum_{\cal E} p_{\cal E} \log g_{\cal E} \\ & = k_B T \sum_{\cal E} p_{\cal E} \log \left( \prod_{i=1}^N d^{(i)} f_{\cal E} \right) \\ &= k_B T \sum_{i=1}^N \log d^{(i)} + k_B T \sum_{\cal E} p_{\cal E} \log f_{\cal E} \\ &\leq k_B T \sum_{i=1}^N \log d^{(i)} - \frac{2 k_B T}{\Delta_E^2} \sum_{\cal E} p_{\cal E} ({\cal E} - \mu_E)^2 \\ &= k_B T \sum_{i=1}^N \log d^{(i)} - \frac{2 k_B T}{\Delta_E^2} {\rm Var}_{\hat{H}} - \frac{2 k_B T}{\Delta_E^2} (\bar{E} - \mu_E)^2 \\ &\leq k_B T \sum_{i=1}^N \log d^{(i)} - k_B T \left( \frac{I_F(\hat\rho, \hat{H})}{ 2 \Delta_E^2} \right), \end{aligned}$$ where the last inequality is from the fact $4 {\rm Var}_{\hat{H}} \geq I_F(\hat\rho, \hat{H})$. Trade-off relation by allowing a small energy resolution window --------------------------------------------------------------- In a real experimental situation, it is hard to access an exact energy level with prefect prevision, so we may permit a finite energy gap $\epsilon$ in energy levels. Under this assumption, we view states with an $\epsilon$-energy gap to be “essentially the same” energy and so can carry internal coherences for approximate work extraction. In order to introduce the energy gap $\epsilon$, we divide the energy spectrum into intervals $${\cal E}_m = \begin{cases} \left[\mu_E + \left(m-\frac{1}{2} \right) \epsilon, \mu_E + \left(m+\frac{1}{2} \right) \epsilon \right) &{\rm for~}m>0\\ \left(\mu_E + \left(m-\frac{1}{2} \right) \epsilon, \mu_E + \left(m+\frac{1}{2} \right) \epsilon \right] &{\rm for~}m<0\\ \left(\mu_E + \left(m-\frac{1}{2} \right) \epsilon, \mu_E + \left(m+\frac{1}{2} \right) \epsilon \right) &{\rm for~}m=0 \end{cases},$$ where each interval has an energy width $\epsilon$. Consequently, we define the energy distribution for $m$th interval $p_m^\epsilon = \sum_{{\cal E} \in {\cal E}_m} p_{\cal E}$ and its degeneracy $g_m^\epsilon = \sum_{{\cal E} \in {\cal E}_m} g_{\cal E}$, respectively. If we allow the $\epsilon$ energy gap for internal coherences, the amount of extractable work is upper bounded by $$\begin{aligned} W_{\rm coh}^\epsilon &\leq k_B T \sum_m p^\epsilon_m \log g^\epsilon_m \\ &= k_B T \sum_{i=1}^N \log d^{(i)} + k_B T\sum_m p^\epsilon_m \log f^\epsilon_m, \end{aligned}$$ where $f_m^\epsilon$ is the frequency to be in the $m$th energy interval. The upper bound of $f_m^\epsilon$ is then given by $$f_m^\epsilon = P ( X_N \in {\cal E}_m) \leq P\left(|X_N - (\mu_E + \epsilon m)| \geq \frac{1}{2}\epsilon \right) \leq \begin{cases} \exp\left[{-\frac{2(|m|-1/2)^2 \epsilon^2}{\Delta^2_E}}\right] & (m \neq 0)\\ 1 & (m=0) \end{cases}.$$ By following the same argument with Theorem 2, we have $$\begin{aligned} \label{EpsilonI} W_{\rm coh}^\epsilon &\leq k_B T \sum_{i=1}^N \log d^{(i)} - k_B T \sum_{m\neq0} p^\epsilon_m \left[ \frac{2(|m|-\frac{1}{2})^2\epsilon^2}{\Delta_E^2} \right] \\ &= k_B T \sum_{i=1}^N \log d^{(i)} - \frac{2k_B T}{\Delta_E^2} \left[ \sum_m p^\epsilon_m |m|^2 \epsilon^2 - \epsilon \sum_{m} p^\epsilon |m| +\frac{1}{4} \epsilon^2 \sum_{m \neq 0}p_m^\epsilon \right]. \end{aligned}$$ When ${\cal E} \in {\cal E}_m$, we use the fact that $\mu_E + (m - 1/2) \epsilon \leq {\cal E} \leq \mu_E + (m + 1/2) \epsilon$ to show $$\begin{aligned} \sum_{\cal E} p_{\cal E} ({\cal E} - \mu_E)^2 &= \sum_m \sum_{{\cal E} \in {\cal E}_m} p_{\cal E} ({\cal E} - \mu_E)^2 \\ &\leq \sum_m \left( \sum_{{\cal E} \in {\cal E}_m} p_m^\epsilon \right) \left( m^2\epsilon^2 + |m| \epsilon + \frac{1}{4} \epsilon^2 \right) \\ &= \sum_m p_m^\epsilon m^2 \epsilon^2 + \epsilon \sum_m p^\epsilon_m |m| + \frac{1}{4} \epsilon^2. \end{aligned}$$ By substituting this inequality into (\[EpsilonI\]), then we finally get the following trade-off relation by allowing a small energy gap $\epsilon$, $$W_{\rm coh}^\epsilon + k_B T \left( \frac{I_F(\hat\rho,\hat{H})}{2 \Delta_E^2} \right) \leq k_B T \sum_{i=1}^N \log d^{(i)} + R(\epsilon),$$ where the correction term is given by $$R(\epsilon) = \frac{k_B T}{\Delta_E^2} \left[ {2 \epsilon} \sum_m p^\epsilon_m |m| - \epsilon^2 p_0^\epsilon \right].$$ We additionally analyse how the quantum Fisher information is perturbed under the same energy resolution window. In this case, the Hamiltonian corresponds to the energy levels $\{ {\cal E}_m\}$ can be expressed as $\hat{H} + (\epsilon/2) \hat{H}_I$, where $(\epsilon/2)\hat{H}_I$ with $|| \hat{H}_I ||_\infty \leq 1$ fills the $\epsilon$-energy gaps of $\hat{H}$. Then the quantum Fisher information with respect to this Hamiltonian is given by $$\label{QFIperturb} \begin{aligned} I_F^\epsilon (\hat\rho, \hat{H}) &= I_F\left(\hat\rho, \hat{H} + \frac{\epsilon}{2} \hat{H}_I\right)\\ &=2 \sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k} |{\left\langle\psi_j\right\vert} \hat{H} + \frac{\epsilon}{2} \hat{H}_I {\left\vert\psi_k\right\rangle}|^2\\ &= 2 \sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k} \left[ |{\left\langle\psi_j\right\vert} \hat{H}{\left\vert\psi_k\right\rangle}|^2 + \epsilon {\left\langle\psi_j\right\vert} \hat{H}{\left\vert\psi_k\right\rangle}{\left\langle\psi_k\right\vert} \hat{H}_I{\left\vert\psi_j\right\rangle}+ \frac{\epsilon^2}{4} |{\left\langle\psi_j\right\vert} \hat{H}_I {\left\vert\psi_k\right\rangle}|^2 \right]\\ &= I_F(\hat\rho, \hat{H}) + 2\epsilon \sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k}{\left\langle\psi_j\right\vert} \hat{H}{\left\vert\psi_k\right\rangle}{\left\langle\psi_k\right\vert} \hat{H}_I{\left\vert\psi_j\right\rangle} + \frac{\epsilon^2}{4} I_F(\hat\rho, \hat{H}_I), \end{aligned}$$ where $\lambda_j$ and ${\left\vert\psi_j\right\rangle}$ are eigenvalue and eigenstate of $\hat\rho$, respectively. By using the fact $\left|\sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k}{\left\langle\psi_j\right\vert} \hat{H}{\left\vert\psi_k\right\rangle}{\left\langle\psi_k\right\vert} \hat{H}_I{\left\vert\psi_j\right\rangle} \right| \leq \left(\sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k} \right) ||\hat{H}||_\infty ||\hat{H}_I||_\infty \leq 2 ||\hat{H}||_\infty $, we observe that the $\epsilon$-energy resolution window perturbs the quantum Fisher information at most $4 \epsilon ||\hat{H}||_\infty + \epsilon^2$. Note that $I_F(\hat\rho, \hat{H}_I) \leq 4 ||\hat{H}_I||_\infty^2 = 4$. Finally, we obtain the following trade-off relation between clock/work resources with the $\epsilon$-energy resolution $$W_{\rm coh}^\epsilon + k_B T \left( \frac{I_F^\epsilon(\hat\rho,\hat{H})}{2 \Delta_E^2} \right) \leq k_B T \sum_{i=1}^N \log d^{(i)} + \tilde{R}(\epsilon),$$ where $\tilde{R}(\epsilon) = \frac{k_B T}{\Delta_E^2} \left[ 2 \epsilon \left(\sum_m p^\epsilon_m |m| + ||\hat{H}||_\infty \right) + \epsilon^2 (\frac{1}{2}-p_0^\epsilon) \right] = {\cal O}(\epsilon)$. Nonlocal energy-diagonal states problem and energy level degeneracy in the Ising model ====================================================================================== The problem of nonlocal energy-diagonal states can be also circumvented by allowing a small energy gap when interaction is weak. As an example, we analyse this in the Ising model. In the 1D transverse-field Ising model, the Hamiltonian is given by $$\hat{H}_{\rm Ising} = - h \sum_{i=1}^N \hat{\sigma}_z^{(i)} - J \sum_{i=1}^N \hat{\sigma}_x^{(i)} \otimes \hat{\sigma}_x^{(i+1)},$$ where $h$ and $J$ are the strength of the transverse field and the coupling between adjacent spins, respectively. Here $\hat{\sigma}^{(i)}_{x,y,z}$ are the Pauli-$x,y,z$ operators for $i$th spin, and we additionally assume a periodic boundary condition $\hat{\sigma}_{x,y,z}^{(N+1)} = \hat{\sigma}_{x,y,z}^{(1)}$. We use ${\left\vert0\right\rangle}$ and ${\left\vert1\right\rangle}$, which are eigenstates of $\hat\sigma_z$ with eigenvalue $-1$ and $+1$, respectively, as a computational basis of local spins. We first consider the case of $N=2$, with a weak interaction $J \ll h$. When there is no interaction, i.e. $J=0$, an energy eigenvalue $E=0$ has degenerate eigenstates ${\left\vert01\right\rangle}$ and ${\left\vert10\right\rangle}$. For $J \neq 0$, the energy level splits into $E = \pm 2J$, where corresponding eigenstates are given by the entangled states ${\left\vert \psi_\pm \right\rangle}= {\left\vert01\right\rangle} \mp {\left\vert10\right\rangle}$. Assuming a weak interaction $J \ll h$, we can allow a small energy gap $\epsilon > 4J$, then we can consider ${\left\vert \psi_\pm \right\rangle}$ to have effectively the same energy level and transformation between them is possible by a thermal process. In this case, [*internal coherence*]{} is now contained in the coherence between ${\left\vert \psi_\pm \right\rangle}$. A thermal process can transform this type of internal coherences to a fully mixed separable form ${\left\vert01\right\rangle}{\left\langle01\right\vert} + {\left\vert10\right\rangle}{\left\langle10\right\vert}$, by which we can extract extra work as in the noninteracting Hamiltonian. The same technique of (\[QFIperturb\]) can be used to calculate perturbation of the QFI by the weak interaction, $$\begin{aligned} I_F(\hat\rho, \hat{H}_{\rm lsing}) &= 2 \sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k} |{\left\langle\psi_j\right\vert} \hat{H}_{\rm Ising} {\left\vert\psi_k\right\rangle}|^2\\ &= 2 \sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k} \\ &\qquad \times \left[ |{\left\langle\psi_j\right\vert} \hat{H}_0 {\left\vert\psi_k\right\rangle}|^2 + 2hJ {\left\langle\psi_j\right\vert}\sum_{i=1}^N \hat{\sigma}_x^{(i)} {\left\vert\psi_k\right\rangle} {\left\langle\psi_k\right\vert} \sum_{i=1}^N \hat{\sigma}_x^{(i)} \otimes \hat{\sigma}_x^{(i+1)} {\left\vert\psi_j\right\rangle} + J^2 |{\left\langle\psi_j\right\vert} \sum_{i=1}^N \hat{\sigma}_x^{(i)} \otimes \hat{\sigma}_x^{(i+1)} {\left\vert\psi_k\right\rangle}|^2\right]\\ &= I_F(\hat\rho, \hat{H}_0) + 4hJ\sum_{j,k} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k} {\left\langle\psi_j\right\vert}\sum_{i=1}^N \hat{\sigma}_x^{(i)} {\left\vert\psi_k\right\rangle} {\left\langle\psi_k\right\vert} \sum_{i=1}^N \hat{\sigma}_x^{(i)} \otimes \hat{\sigma}_x^{(i+1)} {\left\vert\psi_j\right\rangle} + J^2 I_F\left(\hat\rho, \sum_{i=1}^N \hat{\sigma}_x^{(i)} \otimes \hat{\sigma}_x^{(i+1)}\right) \end{aligned}$$ where $\hat{H}_0 = - h \sum_{i=1}^N \hat{\sigma}_z^{(i)}$ is the unperturbed Hamiltonian. Then with the interaction, we have $$I_F(\hat\rho, \hat{H}_0) - 8hJN^2 \leq I_F(\hat\rho, \hat{H}_{\rm lsing}) \leq I_F(\hat\rho, \hat{H}_0) + 8 h J N^2 + 4 J^2 N^2.$$ by using $||\sum_{i=1}^N \hat{\sigma}_x^{(i)}||_\infty \leq N$ and $||\sum_{i=1}^N \hat{\sigma}_x^{(i)} \otimes \hat{\sigma}_x^{(i+1)}||_\infty \leq N$. For general $N$ and an arbitrary strength of interaction $J$, the Ising Hamiltonian can be rewritten in terms of non-interacting quasi-particles [@Sachdev11]. In this viewpoint, we can redefine internal and external coherences between the non-interacting quasi-particles and apply Theorem 2 accordingly. In the transverse Ising model, the spin operators can be mapped to the excitations of spinless fermions by the Jordan-Wigner transformation $\hat{c} = \left(\bigotimes_{j=1}^{l-1} \hat{\sigma}_z^{(j)}\right) \hat{\sigma}_-^{(l)}$, where $\hat{\sigma}_\pm^{(l)} = (\hat{\sigma}_x^{(l)} \pm i \hat{\sigma}_y^{(l)})/2$. Note that the operator $\hat{c}_k$ and $\hat{c}^\dagger_l$ satisfies the following anti-commutation relation: $\{ \hat{c}_k, \hat{c}^\dagger_l \} = \delta_{k,l}$ and $\{ \hat{c}_k, \hat{c}_l \} = 0$. Then we note that the Ising Hamiltonian can be divided into two block diagonal Hamiltonians $\hat{H}_{\rm Ising} = {\hat H}_e \oplus {\hat H}_o$, depending on the parity of the fermion number $\hat{N} = \sum_j \hat{c}_j^\dagger \hat{c}_j$ [@IsingRef]. Each Hilbert space can be diagonalized by taking Fourier transform of fermion operators, $$\hat{c}_{k_n} = \frac{1}{\sqrt{N}} \sum_{j=1}^N \hat{c}_j e^{ i k_n j}~{\rm (even)} \quad {\rm and} \quad \hat{c}_{p_n} = \frac{1}{\sqrt{N}} \sum_{j=1}^N \hat{c}_j e^{ i p_n j}~{\rm (odd)},$$ where $$k_n = \frac{2\pi( n + 1/2)}{N}~{\rm (even)} \quad {\rm and} \quad p_n = \frac{2 \pi n} {N}~{\rm (odd)}$$ for $n = -\frac{N}{2}, \cdots , \frac{N}{2}-1$. Then the following Bogoliubov transformation for both $k_n$ and $p_n$ $$\begin{aligned} \hat{c}_k &= \cos (\theta_k /2) \hat{b}_k + i \sin(\theta_k /2) \hat{b}_{-k}^\dagger \\ \hat{c}_k^\dagger &= i \sin (\theta_k /2) \hat{b}_k + \cos (\theta_k /2) \hat{b}_{-k}^\dagger, \end{aligned}$$ with the Bogoliubov angle $e^{i \theta_k} = (h-J e^{ik})/\sqrt{J^2 + h^2 -2 h J \cos k}$ leads to $$\hat{H}_{e} = \sum_{n} {\cal E}_{k_n} \left[ \hat{b}_{k_n} ^\dagger \hat{b}_{k_n} - \frac{1}{2} \right]~{\rm (even)} \quad {\rm and} \quad \hat{H}_{o} = \sum_{n\neq0} {\cal E}_{p_n} \left[ \hat{b}_{p_n} ^\dagger \hat{b}_{p_n} - \frac{1}{2} \right] - 2(J-h) \left[ \hat{b}^\dagger_{0}\hat{b}_{0} - \frac{1}{2} \right]~{\rm (odd)},$$ with the dispersion relation [@Sachdev11; @IsingRef] $${\cal E}_k = 2 \sqrt{ h^2 + J^2 - 2 h J \cos (k)}.$$ Thus the energy spectrum of the Hamiltonian $\hat{H}_{\rm Ising}$ can be evaluated as ${\cal E}(k_1, \cdots, k_{2m})$ for even numbers of the fermion excitation ${\left\vertk_1, \cdots, k_{2m}\right\rangle} = \prod_{i=1}^{2m} \hat{b}^\dagger_{k_i} {\left\vert0\right\rangle}_{NS} \in {\cal H}_e$ known as Neveu-Schwarz (NS) sector and ${\cal E}(p_1, \cdots, p_{2m})$ for odd numbers of the fermion excitation ${\left\vertp_1, \cdots, p_{2m+1}\right\rangle} = \prod_{i=1}^{2m+1} \hat{b}^\dagger_{p_i} {\left\vert0\right\rangle}_{R} \in {\cal H}_o$ known as Ramond (R) sector. Figure \[IsingFig\] shows the number of degeneracies in the energy eigenspaces with various configurations of $J$ and $h$ for $N=16$. In this case of large $N$, we note that $k_n \approx p_n$ and the diagonalized form of the Ising Hamiltonian can be considered as the $N$-particle non-interacting two-level Hamiltonian model with energy levels $\left\{ -\frac{{\cal E}_{k_n}}{2}, \frac{{\cal E}_{k_n}}{2} \right\}$ in the $n$th site. Then we can apply Theorem 2 to obtain the clock/work trade-off relation by taking $$\Delta_E^2 = \sum_{n=1}^N (\Delta_E^{(n)})^2 \approx \sum_{n=1}^N {\cal E}_{k_n}^2 = \sum_{n=1}^N 4 ( h^2 + J^2 - 2 h J \cos (k_n)) = 4N ( h^2 + J^2).$$ and $d^{(n)} = 2$ for every $n$. Finally, we have the trade-off relation for the 1D-transverse Ising model: $$W_{\rm coh} + k_B T \left( \frac{I_F(\hat\rho,\hat{H})}{8N (h^2 + J^2)} \right) \leq N k_B T \log 2.$$ For general nonintegrable many-body systems, however, a general trade-off relation between clock/work resources is a highly non-trivial question, and deserves further study. ![Degeneracy in energy eigenspaces for the transverse Ising model. Parameters are chosen from $h=1, J=0$ to $h=0, J=1$ with $N=16$ by allowing the energy gap $\epsilon = 0.5$.[]{data-label="IsingFig"}](Ising16.eps){width="0.78\linewidth"}

--- abstract: 'Several results on singularities and convergence of the Yang-Mills flow in dimension four are given. We show that a singularity of pure $+$ or $-$ charge cannot form within finite time, in contrast to the analogous situation of harmonic maps between Riemann surfaces. We deduce long-time existence given low initial self-dual energy, and in this case we study convergence of the flow at infinite time. If a global weak Uhlenbeck limit is anti-self-dual and has vanishing self-dual second cohomology, then the limit exists smoothly and exponential convergence holds. We also recover the classical grafting theorem, and derive asymptotic stability of this class of instantons in the appropriate sense.' author: - Alex Waldron title: 'Instantons and singularities in the Yang-Mills flow' --- The Yang-Mills flow $$\frac{\partial A}{\partial t} = - D_A^* F_A$$ evolves a connection $A$ on a vector or principal bundle by the $L^2$ gradient of the Yang-Mills functional $$\text{YM}(A) = \frac{1}{2} \int |F_A|^2 dV.$$ Over compact base manifolds of dimension two or three, it was shown by G. Daskalopoulos [@dask] and Rade [@rade] that the Yang-Mills flow exists for all time and converges. Finite-time blowup is known to occur in dimension five or higher [@naito], and explicit examples of Type-I shrinking solitons were produced on $\mathbb{R}^n, 5\leq n \leq 9,$ by Weinkove [@weinkove]. Hong and Tian [@hongtian] showed that the singular set has codimension at least four, and gave a complex-analytic description in the compact Kahler case (where an application of the maximum principle shows that singularities can only form at infinite time, see [@siu], Ch. 1). In complex dimension two, Donaldson’s early results [@donsurface] for the flow on stable holomorphic bundles have recently been generalized by Daskalopoulos and Wentworth ([@daskwent04], [@daskwent07]). The behavior of the Yang-Mills flow on Riemannian manifolds of dimension four, however, has not been understood well. The foundational work of Struwe [@struwe] gives a global weak solution with finitely many point singularities, by analogy with harmonic map flow in dimension two [@struwehm]. To date, outside of the Kahler setting, long-time existence and convergence have only been fully established in specific cases, by appealing to energy restrictions on blowup limits [@schlatterglobal] or by imposing a symmetric Ansatz [@sstz]. Moreover, finite-time singularities have long been known as a characteristic feature of critical harmonic map flow [@cdy]. This paper provides several theorems concerning long-time existence and smooth convergence of the Yang-Mills flow in dimension four. As with the classical results of Taubes [@taubes] and Donaldson [@don4d], ours will rely on the splitting of two-forms into self-dual and anti-self-dual parts, as well as a number of useful observations in the parabolic setting. Outline and discussion of results {#outline-and-discussion-of-results .unnumbered} --------------------------------- In Section \[prelims\], we briefly review the Yang-Mills formalism and derive the relevant identities, in particular the split Bochner-Weitzenbock formula. In Section \[selfdualsection\] (p. ), we give a simple yet generic criterion for long-time existence, namely, that either of $F^+$ or $F^-$ does not concentrate in $L^2.$ The proof relies on a borderline Moser iteration (Proposition \[moser\]), together with a manipulation of the local energy inequality with a logarithmic cutoff (Theorem \[selfdual\]). We note that this criterion is not sufficient to rule out singularity formation at infinite time. Moreover, the two results hold simultaneously only in dimension four (see Remark \[simul\]). We draw several conclusions: first, that a singularity of pure positive or negative charge, hence modeled on an instanton, cannot occur at finite time. This suggests that finite-time singularities are very unlikely to form on low-rank bundles, and should be unstable if they do. Second, if the global self-dual energy is less than $\delta,$ a computable constant, then the flow exists for all time and blows up at most exponentially. Third, a proof of long-time existence in the $SO(4)$-equivariant case studied by [@sstz] follows from Theorem \[selfdual\] (see Example \[hmcontrast\]). We also note that finite-time blowup of equivariant harmonic map flow $S^2 \to S^2$ occurs even with low holomorphic energy [@cdy], hence lacks this additional level of “energy quantization.” In this sense, Theorem \[selfdual\][^1] draws a geometric contrast between the dynamics of the two flows (see [@groshat] for a comparative study of the respective scalar equations). The coupling between $F^+$ and $F^-$ also invites a comparison with Topping’s repulsion estimates [@toppingalmost]. In Section \[convergencesection\] (p. ), assuming low initial self-dual energy, we give a characterization of infinite-time singularities along classical gauge-theoretic lines. If the self-dual second cohomology $H^{2+}$ of an anti-self-dual Uhlenbeck limit is zero, e. g. if it is irreducible of charge one, then a Poincaré inequality holds on self-dual two-forms. The estimate is inherited by connections along the flow, implying the exponential decay of $||F^+||^2.$ This results in smooth convergence, once one is sufficiently close to the limit modulo gauge on an open set (Theorem \[convergence\]). The set of bubbling points is therefore empty and the limit unique, in this case. We conclude that an anti-self-dual limit must have $H^{2+} \neq 0,$ if bubbling occurs at infinite time. Since this need not be the case either for a general weakly convergent sequence of instantons, or a priori within Taubes’s framework [@taubesframework], Theorem \[convergence\] may yield additional information about the topology of the instanton moduli spaces. In the final section (p. ), we deduce further properties of the flow at low self-dual energy. We recover the grafting theorem for pointlike instantons [@taubes], which requires a brief new gauge-fixing argument at short time. We also obtain the following (Corollary \[chargeone\]). *Assume the bundle $E$ has structure group $SU(2)$ with $c_2(E) = 1,$ and the base manifold $M$ is simply-connected with $H^{2+}(M)=0.$ If $||F^+||_{L^2} < \delta_1$ initially, then the flow exists for all time and has a smooth subsequential limit. If the limit is anti-self-dual and irreducible then it is unique, and the flow converges exponentially.* Note that on certain manifolds with $H^{2+}(M) \neq 0,$ e. g. ${{\mathbb C}}{{\mathbb P}}^2,$ $SU(2)$-instantons of charge one do not exist, and therefore the flow cannot have a smooth limit. This is also the simplest demonstration that Atiyah-Bott’s description of Morse theory [@atiyahbott] does not generalize naively to dimension four. In the case that the ground state of a certain physical system is not locally unique, the natural question is that of “asymptotic” stability under small perturbations. This has been studied chiefly in the hyperbolic setting, but also by Gustafson, Nakanishi, and Tsai [@gustafson] for the Landau-Lifshitz equations, which include harmonic map flow as the purely parabolic case. In the Yang-Mills context we observe Theorem \[asympstab\], which gives an $H^1$ asymptotic stability result in the parabolic sense for the instantons with $H^{2+}=0.$ Note on dependence of constants {#note-on-dependence-of-constants .unnumbered} ------------------------------- Several of our estimates will have constants, e. g. $C_{\ref{moser}},$ with a particular dependence which we state in the corresponding proposition. The letter $C$ itself denotes a numerical constant which can be taken to be increasing throughout the paper, although it will be used similarly within individual proofs. The constant $C_M$ also depends on the geometry of the fixed base manifold $M.$ In Section \[sobpoinc\] we will also define a Poincaré constant $C_A,$ labeled by the corresponding connection. Preliminaries {#prelims} ============= Let $(M,g)$ be a compact Riemannian manifold of dimension four, $\pi : E \to M$ a vector bundle with fiber ${{\mathbb R}}^n,$ fiberwise inner-product ${\langle}\cdot, \cdot {\rangle}$ and smooth reference connection $A_{ref}.$ We recall the relevant definitions (see [@donkron], [@freeduhl], [@lawson]) and make several basic derivations. The proofs of our main results begin in Section \[selfdualsection\] (p. ). Yang-Mills functional and instantons {#ymfunctional} ------------------------------------ Writing $|F_A|^2$ for the pointwise norm of the curvature form in the fixed metric $g,$ the Yang-Mills energy is defined as above. We may compute its gradient using the formula $$\label{curvderiv} \begin{split} F_{A+a} & = F_A + da + A \wedge a + a \wedge A + a \wedge a \\ & = D_A a + a \wedge a \end{split}$$ in order to obtain $$\begin{split} \frac{d}{dt}YM(A + ta) & = \frac{1}{2} \frac{d}{dt} \left( \int \left(|F_A|^2 + 2 t \, {\langle}F_A, D_A a {\rangle}\right) dV + O(t^2) \right) \\ & = \int {\langle}a, D_A^* F_A {\rangle}dV. \end{split}$$ We conclude that a critical point, or Yang-Mills connection, satisfies $$D_A^* F_A = 0.$$ Moreover the Yang-Mills flow is given in local components $$\frac{\partial}{\partial t} A_{j \beta} ^\alpha = \nabla^i F_{i j}{}^\alpha{}_\beta.$$ By definition, we have the energy inequality $$\text{YM}(A(0)) - \text{YM}(A(T)) = \int_0^T ||D_A^* F_A ||^2 dt$$ as long as the connection is sufficiently smooth. Therefore, if the flow exists for all time, we expect a weak limit which, if not an absolute minimum of $\text{YM},$ is at least a Yang-Mills connection. Note that we will often abbreviate $$||\cdot ||_{L^2(M)} = || \cdot ||.$$ We will write $$F^{\pm} = \frac{1}{2}(F \pm \ast F)$$ for the self-dual and anti-self-dual parts of the curvature form, respectively. In normal coordinates, these satisfy the relations $$\label{asd} F^{\pm}_{12} = \pm F^{\pm}_{34} \quad \quad \quad F^{\pm}_{13} = \mp F^{\pm}_{24} \quad \quad \quad F^{\pm}_{14} = \pm F^{\pm}_{23}.$$ From the second Bianchi identity, remark that $$\label{selfdualbianchi} \begin{split} 2 D_A^*F^\pm & = - \ast ( D \ast F \pm D \ast^2 F ) \\ & = D^*_A F. \end{split}$$ Therefore, if a connection is anti-self-dual ($F^+ = 0$) or self-dual ($F^- = 0$), it is a critical point of YM. These special critical points are called instantons. Recall from Chern-Weil theory that the integer $$\begin{split} \kappa(E) = \frac{1}{8\pi^2} \int Tr F_A \wedge F_A \end{split}$$ is a topological invariant which does not depend on the connection $A$ (for complex bundles, this coincides with the second Chern character). From the definition of the Hodge star operator, we compute $$\begin{split} \int Tr F_A \wedge F_A & = - \int \big{\langle}F^+ + F^-, F^+ - F^- \big{\rangle}dV \\ & = ||F^-||^2 - ||F^+||^2 \end{split}$$ but by orthogonality, also $$||F||^2 = ||F^+||^2 + ||F^-||^2.$$ Changing the orientation of $M$ if necessary, we may assume that $\kappa$ is nonnegative. We obtain the formula $$\label{energyformula} ||F||^2 = 8\pi^2 \kappa + 2 ||F^+||^2.$$ Thus a connection is anti-self-dual if and only if it attains the energy $8\pi^2 \kappa,$ which then must be the absolute minimum for connections on $E.$ Evolution of curvature and Weitzenbock formulae {#weitzenbock} ----------------------------------------------- In what follows, we will always assume that the initial connection is smooth (unless otherwise stated). Although the flow is not strictly parabolic, short-time existence of a solution is guaranteed for smooth data by a De Turck-type trick (see [@donkron], Ch. 6). A solution $A(t)$ was constructed by Struwe so as to achieve the following characterization of long-time existence. We say, for a certain $\epsilon_0 > 0,$ that the curvature $F(t) = F_{A(t)}$ concentrates in $L^2$ at $x\in M$ if $$\inf_{R>0} \limsup_{t \to T} \int_{B_R(x)} |F(t)|^2 dV \geq \epsilon_0.$$ \[struwe\] *(Struwe [@struwe], Theorem 2.3)* The maximal smooth existence time $T$ of $A(t)$ is characterized by concentration of the curvature $F(t)$ at some $x\in M$ as $t\to T.$ The primary remaining task is to study concentration of the curvature along the Yang-Mills flow. From (\[curvderiv\]), we compute the evolution $$\frac{\partial}{\partial t} F_A = D_A (- D_A^* F_A).$$ In view of the second Bianchi identity $D_A F_A = 0,$ we may rewrite this as the tensorial heat equation $$\left( \frac{\partial}{\partial t} + \Delta_A \right) F_A = 0$$ where $\Delta_A = D D^* + D^* D$ is the Hodge Laplacian with respect to the evolving connection. We compute, for $\omega \in \Omega^k({{\mathfrak g}}_E)$ $$\begin{split} (D^*D + D D^*) \omega_{i_1 \cdots i_k} & = - \nabla^j \left(\nabla_j \omega_{i_1 \cdots i_k} - \nabla_{i_1} \omega_{j i_2 \cdots i_k} - \cdots - \nabla_{i_k} \omega_{i_1 \cdots i_{k-1} j} \right) \\ & - \nabla_{i_1} \nabla^j \omega_{j i_2 \cdots i_k} + \nabla_{i_2} \nabla^j \omega_{j i_1 i_3 \cdots i_k} + \cdots + \nabla_{i_k} \nabla^j \omega_{j i_2 \cdots i_{k-1} i_1 }. \end{split}$$ Permuting $j$ and $i_1$ in the positive terms of the second line, we may group all but the very first term into commutators. We obtain the Weitzenbock formula $$(D^* D + D D^*) \omega_{i_1 \cdots i_k} = \nabla^* \nabla \omega_{i_1 \cdots i_k} + Rm \# \omega - {\left[}F_{i_1}{}^j , \omega_{j i_2 \cdots i_k} {\right]}- \cdots - {\left[}F_{i_k}{}^j , \omega_{i_1 \cdots i_{k-1} j} {\right]}$$ In particular, for a two-form, we have $$\label{weitz} \begin{split} - \Delta_A \omega_{ij} & = \nabla^k \nabla_k \omega_{ij} + {\left[}F_i{}^k , \omega_{k j} {\right]}- {\left[}F_j{}^k , \omega_{k i} {\right]}\\ & - R_i{}^k{}^\ell{}_k \omega_{\ell j} - R_i{}^k{}^\ell{}_j \omega_{k \ell} + R_j{}^k{}^\ell{}_k \omega_{\ell i} + R_j{}^k{}^\ell{}_i \omega_{k \ell} \end{split}$$ We now make a simple observation about the zeroth-order terms (see [@lawson], appendix). Assume we are in geodesic coordinates at a point, so (anti)-self-duality is defined as in (\[asd\]). For $\omega \in \Omega^{2+}$ and $\eta \in \Omega^{2-},$ we may write $$\begin{split} \omega_{1k} \eta_{k 2} - \omega_{2k} \eta_{k 1} & = \omega_{13} \eta_{3 2} - \omega_{23} \eta_{31} + \omega_{14} \eta_{4 2} - \omega_{24} \eta_{41} \\ & = (-\omega_{24}) (- \eta_{41}) - \omega_{14} \eta_{42} + \omega_{14} \eta_{4 2} - \omega_{24} \eta_{41} \\ & = 0 \end{split}$$ and similarly for any choice of indices. A similar calculation shows that for $\omega, \omega'$ self-dual, $\omega_{1k} \omega'_{k 2} - \omega_{2k} \omega'_{k 1}$ is again self-dual. These facts amount to the splitting of Lie algebras $$so(4) = so(3) \oplus so(3).$$ For the $Rm $ terms, one notes that the first and third are skew in $i,j,$ as are the second and fourth, and that these are each self-dual if the same is true of $\omega$ (as explained in [@freeduhl], appendix). We conclude that the extra terms of the Weitzenbock formula (\[weitz\]) in fact split into self-dual and anti-self-dual parts. Note also that $\Delta_A \ast = \ast \Delta_A,$ and the trace Laplacian clearly preserves the identites (\[asd\]) in an orthonormal frame. We obtain, finally, for $\omega$ self-dual $$\label{sdweitz} -\Delta_A \omega_{ij} = \nabla^k \nabla_k \omega_{ij} + {\left[}F^+_i{}^k , \omega_{kj} {\right]}- {\left[}F^+_j{}^k , \omega_{ki} {\right]}+ Rm \# \omega$$ as well as a similar formula for anti-self-dual forms. Applied to the self-dual curvature $F^+,$ this yields the key evolution equation $$\label{fplusevolution} \frac{\partial}{\partial t} F^+_{ij} = \nabla^k \nabla_k F^+_{ij} + 2 {\left[}F^+_i{}^k , F^+_{kj} {\right]}+ Rm \# F^+.$$ Sobolev spaces -------------- Any connection can now be uniquely written $A_{ref} + A,$ with $A\in \Omega^1({{\mathfrak g}}_E),$ and any norms applied to a connection will be applied to the global one-form $A.$ We define the Sobolev norms $$||\omega||_{H^k} = \left( \sum_{\ell=0}^k ||\nabla_{ref}^\ell \omega ||_{L^2}^2\right)^{\frac{1}{2}}$$ as well as the corresponding spaces of forms and connections over any open set $\Omega \subset M.$ A different reference connection defines uniformly equivalent norms. Our proofs will not deal directly with Sobolev spaces of gauge transformations and connections, as we are able to cite the highly developed regularity theory. For any $\Omega' \subset \subset \Omega,$ there is a local Sobolev inequality $$\label{localsobolev} ||\omega||_{L^4(\Omega')}^2 < C_{\Omega', \Omega} ||\omega||^2_{H^1(\Omega)}$$ for the norms defined with respect to $A_{ref}.$ The difficulty with Yang-Mills in dimension four and above is that due to the zeroth-order terms of the Weitzenbock formula, the Sobolev constant for $D_A \oplus D_A^*$ blows up as the curvature of $A$ concentrates. (Anti)-self-dual singularities {#selfdualsection} ============================== In order to obtain separate control of the self-dual curvature, we apply the inner product with $F^+$ to (\[fplusevolution\]). Letting $u=|F^+|^2,$ we obtain the differential inequality $$\label{fplusev} \left(\frac{\partial}{\partial t} + \Delta \right) u \leq - 2|\nabla F^+|^2 + A u^{3/2} + B u$$ where $B$ is a multiple of $||Rm||_{L^\infty(M)}.$ \[moser\] Let $u(x,t)\geq 0$ be a smooth function satisfying $$\left( \frac{\partial}{\partial t} + \Delta \right)u \leq A u^{3/2} + B u.$$ on $M \times {\left[}0, T \right),$ with $M$ compact of dimension four. There exist $R_0>0$ (depending on the geometry of $M$) and $\delta>0$ (depending on $A,B, R_0$) as follows: Assume $R < R_0$ is such that $\int_{B_R(x_0)} u(x,t) dx < \delta^2$ for all $x_0 \in M, t \in {\left[}0, T \right).$ Then for $t\in {\left[}\tau, T\right),$ we have $$||u(t)||_{L^\infty(M)} < C_{\ref{moser}} \, ||u||_{L^1\left( M\times {\left[}t-\tau,t {\right]}\right)}.$$ The constant depends on $||u(0)||_{L^2}, R,$ and $\tau.$ If $u$ is defined for all time, then $$\limsup_{t\to \infty} ||u(t)||_{L^\infty(M)} < C_M/R^4.$$ Let $\varphi\in C^\infty_0(B_R(x)).$ Multiplying by $\varphi^2 u$ and integrating by parts, we obtain $$\begin{split} \frac{1}{2} \frac{d}{dt} \left(\int \varphi^2 u^2\right) + \int \nabla(\varphi^2 u) \cdot \nabla u & \leq A\int \varphi^2 u^{5/2} + B \int \varphi^2 u^2 \\ \frac{1}{2} \frac{d}{dt} \left( \int \varphi^2 u^2 \right) + \int |\nabla(\varphi u)|^2 & \leq \int |\nabla \varphi |^2 u^2 + A\int \varphi^2 u^{5/2} + B \int \varphi^2 u^2. \end{split}$$ Applying the Sobolev and Hölder inequalities on $B_R,$ $$\frac{1}{2} \frac{d}{dt} \int \varphi^2 u^2 + \left( \frac{1}{C_S} - A\delta \right) \left(\int (\varphi u)^4 \right)^{1/2} \leq ||\nabla \varphi||^2_{L^{\infty}} \int_{B_R} u^2 + B \int \varphi^2 u^2.$$ Assuming $R<R_0,$ depending on the geometry of $M,$ we have $Vol(B_R(x)) \leq c^2 R^4$ for all $x\in X$ as well as a uniform Sobolev constant $C_S.$ We may also choose a cover of $M$ by geodesic balls $B_{R/2}(x_i)$ in such a way that no more than $N$ of the balls $B_i=B_R(x_i)$ intersect a fixed ball, with $N$ universal in dimension four. For each $i,$ let $\tilde{\varphi}_i$ be a standard cutoff for $B_{R/2}(x_i) \subset B_R(x_i)$ with $||\nabla \tilde{\varphi}_i||_{L^\infty} < 4/R.$ Define $\varphi_i = \tilde{\varphi_i}/\sqrt{\sum_j \tilde{\varphi}_j^2},$ so that $\{ \varphi_i^2 \}$ is a partition of unity with $||\nabla \varphi_i||_{L^\infty} < C/R.$ We now apply the above differential inequality to $\varphi_i$ and sum $$\begin{split} \sum_i \left( \frac{1}{2} \frac{d}{dt} \int \varphi_i^2 u^2 \right. + \left. \left(C_S^{-1} - A \delta \right) \left(\int (\varphi_i u)^4 \right)^{1/2} \right) & \leq \sum_i \left(C R^{-2}\int_{B_i } ( \sum_j \varphi_j^2) u^2 + B \int \varphi_i^2 u^2 \right) \\& \leq \left( \frac{C N}{R^2} + B \right) \sum_i \int \varphi_i^2 u^2. \end{split}$$ Note that for $\theta>0,$ we have by Hölder’s and Young’s inequalities $$\begin{split} \int (\varphi_i u)^2 \leq \delta \left(\int_{B_R} (\varphi_i u)^3 \right)^{1/2} & \leq \delta \left(\int_{B_R} \left(\theta^3 + \dfrac{(\varphi_i u)^4}{\theta}\right) \right)^{1/2} \\& \leq \delta \left((CR^4 \theta^3)^{1/2} + \theta^{-1/2} \left(\int (\varphi_i u)^4 \right)^{1/2} \right). \end{split}$$ Taking $\theta = R^{-4},$ we obtain $$\begin{split} \sum_i \left( \frac{1}{2} \frac{d}{dt} \int \varphi_i^2 u^2 + (C_S^{-1} - A\delta ) \left(\int (\varphi_i u)^4 \right)^{1/2} \right) \leq \delta \left(\dfrac{C}{R^2} + B \right) \sum_i \left(\dfrac{C}{R^4} + R^2 \left(\int (\varphi_i u)^4 \right)^{1/2} \right) \end{split}$$ and subtracting the last term $$\sum_i \left(\frac{d}{dt} \int \varphi_i^2 u^2 + \epsilon \left( \int (\varphi_i u)^4 \right)^{1/2} \right) \leq \dfrac{C \delta \left( 1 + BR^2 \right)}{R^6} \left(\# \mbox{ of balls} \right),$$ where we now choose $\delta$ so that $$\epsilon = 2\left( C_S^{-1} - \delta \left( A+ (C + B R_0^2 ) \right) \right) > 0.$$ We may finally apply Hölder’s inequality to the left-hand side and absorb the partition of unity $$\begin{split} \sum_i \left(\frac{d}{dt} \int \varphi_i^2 u^2 + \dfrac{\epsilon }{c R^2} \int \varphi_i^2 u^2 \right) & \leq \dfrac{C \delta \left(1 + BR^2 \right) }{R^6} \left( \dfrac{ Vol(M)}{R^4} \right) \\\frac{d}{dt} \int u^2 + \dfrac{\epsilon }{c R^2} \int u^2 & \leq \dfrac{ C \delta \left(1 + BR^2 \right) Vol(M)}{R^{10}}. \\\frac{d}{dt} \left(e^{\frac{\epsilon}{c R^2} t} \int u(t)^2 \right) & \leq e^{\frac{\epsilon}{c R^2} t} \dfrac{C \delta \left(1 + BR^2 \right) Vol(M) }{R^{10}}. \end{split}$$ Integrating, we obtain the estimate $$\int u(t)^2 \leq e^{-\frac{\epsilon}{c R^2} t} \int u(0)^2 + \dfrac{ C \delta \left(1 + BR^2 \right) Vol(M)}{\epsilon R^{8}}\left(1-e^{-\frac{\epsilon}{c R^2} t} \right).$$ This gives a uniform $L^2$ bound on $u(t)$ for $t>0,$ hence a uniform $L^4$ bound on $Au^{1/2} + B.$ Standard Moser iteration (see [@li] Lemma 19.1) on cylinders of radius $R_0$ and height $\tau$ then implies the stated $L^\infty$ bounds. \[cutoff\] (C. f. [@donkron], 7.2.10) There is a constant $L$ and for any $N>1, R > 0$ a smooth function $\beta = \beta_{N,R}$ on ${{\mathbb R}}^4$ with $0 \leq \beta(x)\leq 1$ and $$\beta(x)= \begin{cases} 1 & |x| \leq R/N \\ 0 & |x| \geq R \end{cases}$$ and $$||\nabla \beta ||_{L^4}, ||\nabla^2 \beta||_{L^2} < \frac{L}{\sqrt{\log N}}.$$ Assuming $R < R_0,$ the same holds for $\beta (x - x_0)$ on any geodesic ball $B_R(x_0) \subset M.$ We take $$\beta(x) = \tilde{\varphi}\left(\dfrac{\log \frac{N}{R} \,x}{\log N} \right)$$ where $$\tilde{\varphi}(s)= \begin{cases} 1 & s \leq 0 \\ 0 & s \geq 1 \end{cases}$$ is a standard cutoff function (with respect to the cylindrical coordinate $s$). The construction of Lemma \[cutoff\] is possible in dimension four and above due to the scaling of the $L^4$ norm on 1-forms ($L^2$ norm on 2-tensors), together with the failure of these norms to control the supremum. Proposition \[moser\] holds only in dimension less than or equal to four. \[selfdual\] Let $A(t)$ satisfy the Yang-Mills flow equation on $M \times {\left[}0, T \right).$ For $R<R_0$ and $N>1,$ we have the local bound $$\label{selfdualbound} ||F(T)||^2_{L^2(B_{R/N})} \leq ||F(0)||^2_{L^2(B_{R})} + \int_0^T \frac{||F^+(t)||_{L^\infty(B_R)} }{\sqrt{\log(N)}} \left(C+||F(t)||^2_{L^2(B_R)}\right) dt$$ on concentric geodesic balls in $M.$ Therefore if $||F^+||_{L^\infty(M)} \in L^1 \left( {\left[}0, T \right) \right),$ or in particular if $F^+$ does not concentrate in $L^2,$ then the flow extends smoothly past time $T.$ Recall the evolution of the curvature tensor $$\frac{\partial}{\partial t} F_A=-D D^* F.$$ Multiplying by $\varphi^2 F$ and integrating by parts, we obtain $$\frac{1}{2} \frac{d}{dt} ||\varphi F||^2 + ||\varphi D^* F||^2 = 2 (\varphi D\varphi \cdot F,D^*F)$$ On the right-hand side we switch $D^* F = 2D^* F^+$ (\[selfdualbianchi\]), and integrate by parts again to obtain $$\frac{1}{2} \frac{d}{dt} ||\varphi F||^2 + ||\varphi D^* F||^2 = 4 \int_M \Big{\langle}\left( \nabla_i \varphi \nabla^k\varphi + \varphi \nabla_i \nabla^k \varphi \right) F_{kj} + \varphi \nabla^k\varphi \nabla_i F_{kj} \, , (F^+)^{ij} \Big{\rangle}\,\, dV$$ In the inner product with the self-dual 2-form $F^+,$ we may replace the term $\varphi \nabla^k \varphi \nabla_i F_{k j}$ via the identity$$\begin{split} \left( \nabla^k \varphi \left(\nabla_i F_{k j} - \nabla_j F_{k i} \right) \right)^+ & = \left( \nabla^k \varphi \left( \left( - \nabla_j F_{i k} - \nabla_k F_{ji} \right) - \nabla_j F_{k i} \right) \right)^+ \\ & = \left( \nabla^k \varphi \nabla_k F_{ij} \right)^+ \\ & = \nabla^k \varphi \nabla_k F^+_{ij}. \end{split}$$ We then write $$\big{\langle}\nabla_k F^+_{ij}, (F^+)^{ij} \big{\rangle}= \frac{1}{2}\nabla_k|F^+|^2$$ and integrate by parts once more to obtain $$\begin{split}{} \frac{1}{2} \frac{d}{dt} ||\varphi F||^2 + ||\varphi D^* F||^2 & = 4 \int_M \left( \nabla_i\varphi \nabla_k \varphi + \varphi \nabla_i \nabla_k \varphi \right) \left( \Big{\langle}F^k{}_j, (F^+)^{ij} \Big{\rangle}+ g^{ik}\frac{|F^+|^2}{2} \right) \,\, dV \\[2mm] & \leq 4 \, ||F^+||_{L^\infty(B_r)} \left( \epsilon^{-1} ||F||^2_{L^2} + \epsilon \left( ||\nabla \varphi||_{L^4}^4 + ||\varphi \nabla^2 \varphi ||^2_{L^2} \right) \right). \end{split}$$ Choose $\epsilon= 8 \sqrt{\log(N)} $ and $\varphi=\beta_{N,r}$ to obtain the desired estimate. By Theorem \[struwe\] (the work of Struwe [@struwe]), to prove the second claim it suffices to show that the full curvature does not concentrate in $L^2$ at time $T.$ Note that $||F(t)||^2$ is decreasing. Therefore if the curvature on $B_r$ is initially less than $\delta/2,$ then for $N$ sufficiently large, the estimate implies that the full curvature on $B_{r/N}$ remains less than $\delta$ until time $T.$ Moreover by Proposition \[moser\], non-concentration of $F^+$ implies a uniform $L^\infty$ bound, and hence the required $L^1(L^\infty)$ bound at finite time. If the maximal existence time is finite, then both $F^+$ and $F^-$ must concentrate.[^2] \[simul\] In view of the Corollary, one can modify the standard rescaling argument [@schlatterlongtime] at a finite-time singularity to obtain a weak limit which has either nonzero $F^+$ or nonzero $F^-$ (it follows from Proposition \[moser\] that this energy cannot be lost in the limit). Thus one cannot have a finite-time singularity for which every weak blowup limit is strictly self-dual, or anti-self-dual. Since any stable Yang-Mills connection on an $SU(2)$ or $SU(3)$-bundle over $S^4$ is either self-dual, anti-self-dual, or reducible, Theorem \[selfdual\] in this case should imply that finite-time singularities are unstable. \[hmcontrast\] It is straightforward, from the maximum principle applied directly to its evolution, to show that $F^-$ is bounded in the $SO(4)$-symmetric setting of [@sstz]. Hence Theorem \[selfdual\] gives a geometric proof of long-time existence in that case. This clarifies the contrast with harmonic map flow [@cdy] (attributed by [@groshat] to a coefficient in the equivariant ansatz corresponding to the “rotation number”). For, finite-time blowup of equivariant maps readily occurs even with low holomorphic energy (see [@toppingalmost] for definitions). \[tube\] For $\delta$ as in Proposition \[moser\], if an initial $H^1$ connection has self-dual curvature $||F_{A(0)}^+||_{L^2(M)} < \delta$ then the Yang-Mills flow exists for all time and blows up at most exponentially, with asymptotic rate bounded uniformly for $M.$ On any $SU(2)$-bundle, there exists a nonempty $H^1$-open set of initial connections for which the Yang-Mills flow exists for all time, and converges exponentially if $H^{2+}(M)=0.$ The connection is smooth after a short time, modulo gauge. Proposition \[moser\] then implies a uniform bound on $F^+$ for all future time, and long-time existence follows from Theorem \[selfdual\]. Following Freed and Uhlenbeck [@freeduhl], for any $\delta_1$ one can construct smooth pointlike $SU(2)$-connections with $||F^+||_{L^2} < \delta_1$ and $||F^+||_{L^\infty} < C$ (p. 124). Provided $H^{2+}(M)=0,$ Theorem \[parataubes\] (below) yields convergence at infinite time, which holds in an $H^1$-open neighborhood of the resulting instantons (Theorem \[asympstab\]). Convergence at infinite time {#convergencesection} ============================ In this section we assume that all connections have globally small self-dual energy $$||F_A^+||_{L^2(M)} < \delta.$$ By (\[energyformula\]), this condition is preserved by the flow, which exists for all time by Corollary \[tube\]. It is also attained for a nonempty set of connections on bundles with $c_2(E)\geq 0$ and structure group $SU(2),$ and in this case should represent the generic end-behavior of the flow. We first recall and adapt several standard pieces of Yang-Mills theory. For an open set $\Omega \subset M,$ we will write $$\Omega_r = \{ x \in \Omega \mid d(x,\Omega^c) >r \} \subset \subset \Omega.$$ \[epsilonreg\] There exists $\epsilon_0 > 0$ as follows. For $R < R_0,$ if the energy $||F||^2_{L^2(B_R)}<\epsilon_0$ for $t\in {\left[}-R^2 , 0 {\right]},$ then for $k \geq 0$ $$||\nabla_A^k F||_{L^\infty \left(B_{R_k} \times {\left[}- R_k^2,0 {\right]}\right)} < \frac{C_k}{R^{2+k}}$$ where $R_k = R/2^k.$ See [@chenshen], [@hongtian] for standard proofs of the $k=0$ estimate via monotonicity formulae. For $k \geq 1,$ this is the result of the Bernstein-Hamilton-type derivative estimates of [@weinkove].[^3] \[antibubble\] Assume $||F^+(t)||_{L^\infty(\Omega)} < K^+$ for $0 \leq t \leq \tau.$ Let $\epsilon_0$ be as above, and assume that for some $r_0 < R_0$ there holds $$||F(\tau)||^2_{L^2(B_{r_0}(x))} < \epsilon_0 / 3$$ for all $x \in \Omega_{r_0},$ with $0 < r_0^2 < \tau.$ If $$||F(0)||^2_{L^2(M)} - ||F(\tau)||^2_{L^2(M)} \leq \epsilon_0 / 3$$ then we have $$||F(\tau)||_{L^\infty(\Omega_{r_0})} < \frac{C_{\ref{antibubble}}}{r_0^2}.$$ The constant depends on $K^+$ and $||F(0)||^2.$ Assume the contrary. Then by $\epsilon_0$-regularity (Lemma \[epsilonreg\]), for any $N,$ at some time $0<t<\tau$ and $x \in \Omega_r$ we have $$||F(t)||^2_{L^2(B_{r_0/N}(x))} \geq \epsilon_0.$$ Letting $\varphi$ be the cutoff of Lemma \[cutoff\] for $B_{r_0/N} \subset B_{r_0},$ we apply the proof of Theorem \[selfdual\] using $\overline{\varphi} = 1-\varphi.$ This gives $$||F(\tau)||^2_{L^2(M \setminus B_{r_0})} - ||F(t)||^2_{L^2(M \setminus B_{r_0/N})} < \epsilon_0 / 3$$ for $N$ large enough based on $||F||^2$ and $K^+.$ But we also have $$||F(t)||_{L^2(B_{r_0/N})} - ||F(\tau)||_{L^2(B_{r_0})} > 2 \epsilon_0 / 3$$ by assumption. Subtracting, we obtain $$||F(t)||^2 - ||F(\tau)||^2 > \epsilon_0/3$$ which is a contradiction. \[uhllim\] For a sequence $t_j\to \infty,$ we say that $(A_\infty, E_\infty)$ is an *Uhlenbeck limit* for the flow if the following holds. There exists a subsequence of times $t_{j_k}$ and smooth bundle isometries $u_k : E \to E_\infty$ defined on an exhaustion of open sets $$U_1 \subset \cdots \subset U_k \subset \cdots \subset M_0 = M \setminus \{ x_1, \ldots x_n \}$$ such that on any open set $\Omega \subset \subset M_0,$ we have $u_k^* (A_{t_k}) \to A_\infty$ smoothly. \[uhllimexist\] Assuming $||F^+||<\delta,$ any sequence $t_j \to \infty$ necessarily contains an Uhlenbeck limit which is a Yang-Mills connection on $E_\infty.$ This is a standard improvement of the detailed arguments found in [@schlatterlongtime], by analogy with the Kahler case (see [@donkron], Ch. 6). The existence of weak $H^1$ limits on a countable family of balls in $M_0$ is the result of compactness theory for connections with bounded $L^2$ curvature ([@sedlacek], [@uhlenbecklp]). By Lemma \[antibubble\], we in fact have $L^\infty$ bounds on the curvature of $A(t_{j_k})$ and all its derivatives on each ball, for $k$ large enough. By [@donkron], Lemma 2.3.11, upon taking further subsequences, the weak limit can be taken to be a smooth limit over each ball. By the argument of [@donkron], §4.4.2, these gauge transformations can be patched together over the open sets $U_i.$[^4] The fact that the limiting connection is Yang-Mills away from the bubbling points, and therefore extends to a smooth Yang-Mills connection on $E_\infty,$ follows from the energy inequality, [@uhlenbeckremov], and the next estimate. \[dstarf\] Assume $||F(t)||_{L^\infty(B_{R}(x_0))} < K$ on ${\left[}0,T \right).$ Then for $R<R_0$ we have the estimate $$||D^*F(t)||_{L^\infty (B_{R/2})}^2\leq C_{\ref{dstarf}} ||D^* F||^2_{L^2( B_R \times {\left[}t-\tau, t {\right]})}$$ for $t\geq \tau > 0.$ The constant depends on $K,R,$ and $\tau.$ One computes the evolution $$\left( \frac{\partial}{\partial t} + \nabla^* \nabla \right) D^*F_i = 2 {\left[}F_{ij}, D^*F_j {\right]}+ Rm \# D^*F_j.$$ The estimate then follows again from Moser iteration (Li [@li], 19.1). Sobolev and Poincaré inequality {#sobpoinc} ------------------------------- Assuming $||F_A^+|| < \delta,$ Hölder’s inequality applied to the Weitzenbock formula (\[sdweitz\]) implies the Sobolev inequality $$\label{sobolev} ||\omega||_{L^4}^2 + ||\nabla_A \omega||_{L^2}^2 < C_M (||D_A \omega||_{L^2}^2 + ||D_A^* \omega ||_{L^2}^2 + ||\omega||_{L^2}^2).$$ for any $\omega \in \Omega^2_+(\mbox{End} E).$ Recall the basic instanton complex $$0\to {{\mathfrak g}}_E \stackrel{D_A}{\longrightarrow} \Omega^1({{\mathfrak g}}_E) \stackrel{\pi_+ D_A}{\longrightarrow} \Omega^{2+}({{\mathfrak g}}_E).$$ Under the assumption $H^{2+}_A = 0,$ there are no $L^2$ self-dual two-forms $\omega$ with $D_A^* \omega = D_A \omega = 0$ in the distributional sense. Therefore, by the standard compactness argument, we have $$||\omega||_{L^2}^2 < C_A \left( ||D_A \omega||_{L^2}^2 + ||D_A^* \omega ||_{L^2}^2 \right).$$ Hence this term can be dropped from the RHS of (\[sobolev\]), and we obtain $$\label{poincare0} ||\omega||_{L^4}^2 + ||\omega||_{L^2}^2 + ||\nabla_A \omega||_{L^2}^2 < C_A (||D_A \omega||_{L^2}^2 + ||D_A^* \omega ||_{L^2}^2 )$$ for $\omega \in \Omega^{2+}(End E).$ We will always take $C_A \geq C_M.$ \[excision\] Let $A_0$ be a connection on a bundle $E_0$ over $M$ which satisfies the Poincaré inequality $$\label{poincare} ||\omega||_{L^4}^2 + ||\omega||_{L^2}^2 < C_{A_0} (||D_A \omega||_{L^2}^2 + ||D_A^* \omega ||_{L^2}^2 ).$$ Assume $A$ is a connection on $E$ for which there exists a smooth bundle isometry $u : E_0 \to E$ defined over $M_r = M - \overline{B}_r(x_1)\cup \cdots \cup \overline{B}_r(x_n)$ with $||u^*(A)-A_0||_{L^4} \leq \epsilon.$ Then if $r, \epsilon$ are sufficiently small, $A$ satisfies (\[poincare\]) with constant $4C_{A_0}.$ Assume first that $\mbox{Supp}(\omega) \subset M_r.$ Write $\tilde{A} = u^*(A), \tilde{\omega} = u^*(\omega),$ $a = A_0 - \tilde{A}.$ We then have $$\begin{split} ||D_A \omega||_{L^2}^2 + ||D_{A}^* \omega ||_{L^2}^2 & = ||D_{\tilde{A}} \tilde{\omega}||_{L^2}^2 + ||D_{\tilde{A}}^* \tilde{\omega} ||_{L^2}^2 \\ & =||D_{A_0} \tilde{\omega} + a \# \tilde{\omega} ||_{L^2}^2 + ||D_{A_0}^* \tilde{\omega} + a \# \tilde{\omega} ||_{L^2}^2 \\ & \geq \frac{1}{2} \left( ||D_{A_0} \tilde{\omega} ||_{L^2}^2 + ||D_{A_0}^* \tilde{\omega} ||_{L^2}^2 - 2 || a ||_{L^4}^2 ||\omega||_{L^4}^2 \right). \end{split}$$ On the other hand, if $\mbox{Supp}(\omega) \subset B_r(x_1) \cup \cdots \cup B_r(x_n),$ then $||\omega ||^2_{L^2} \leq c n r^2 ||\omega ||^2_{L^4}.$ Now let $\varphi = \sum \beta_{N,r}(x-x_i)$ be a sum of the logarithmic cutoffs of Lemma \[cutoff\], $\overline{\varphi} = 1-\varphi.$ Choose $\epsilon, r, N$ such that $$4\epsilon^2 + c n r^2 + 2 L^2/\log(N) < (4 C_{A_0})^{-1}.$$ Then $$\begin{split} ||\omega||_{L^4}^2 + ||\omega||_{L^2}^2 & \leq 2 \left( ||\varphi \omega||_{L^4}^2 + ||\varphi \omega||_{L^2}^2 + ||\overline{\varphi} \omega||_{L^4}^2 + ||\overline{\varphi} \omega||_{L^2}^2 \right) \\ & \leq 2 C_M \left( ||D_A (\varphi \omega)||^2 + ||D_A^* (\varphi \omega ) ||^2 + ||\varphi \omega||^2 \right) \\ & \quad \quad \quad \quad \quad \quad + 2 C_{A_0} \left( || D_{A_0} (\overline{\varphi} \tilde{\omega}) ||^2 + || D^*_{A_0} (\overline{\varphi} \tilde{\omega}) ||^2 \right) \\ & \leq 2 C_{A_0} \left( ||\varphi D_A \omega ||^2 + ||\varphi D_A^* \omega||^2 + ||\overline{\varphi} D_A \omega ||^2 + ||\overline{\varphi} D_A^* \omega||^2 \right. \\ & \quad \quad \quad \quad \quad \quad \left. + 2||D\varphi \# \omega||^2 + 4 ||a||_{L^4}^2 ||\omega||_{L^4}^2 + cnr^2 ||\omega||_{L^4}^2 \right) \\ & \leq 2 C_{A_0} \left( ||D_A \omega||^2 + ||D_A^* \omega||^2 + \left(2||D\varphi||_{L^4}^2 + 4 \epsilon^2 + cnr^2\right) ||\omega||_{L^4}^2 \right), \end{split}$$ which upon rearranging yields the claim (we replace $r/N$ by $r$ in the statement). Convergence ----------- We now proceed to the proofs of our main convergence results. \[omega\] Assume $||F(0)||^2 - ||F(T)||^2 < 1$ and $||F(t)||_{L^\infty (\Omega)} < K$ for $t \in {\left[}0, T \right).$ Then we have the $L^\infty$ bound $$||A(T) - A(\tau)||^2_{L^\infty(\Omega_r)} < C^0_{\ref{omega}} \left( ||F(0)||^2 - ||F(T)||^2 \right) \left( T-\tau \right).$$ We also have the Sobolev bounds $$||A(T) - A(\tau)||^2_{H^{k}(\Omega_r)} < C^k_{\ref{omega}} \left( ||F(0)||^2 - ||F(T)||^2 \right) \left( T-\tau \right) \left( 1+||A||^{2k}_{L^\infty} \right).$$ The constants depend on $K, r, \tau,$ and $\Omega \subset M.$ Let $\epsilon = ||F(0)||^2 - ||F(T)||^2.$ In this proof the constant $C$ depends on $K,\tau,r$ and $\Omega,$ but not $\epsilon$ or $T.$ From the energy inequality we have $||D^*F||^2_{L^2(M \times {\left[}0, T {\right]}) } = \epsilon,$ and $$||D^*F(t)||^2_{L^\infty(\Omega_r)} < C_{3.5} \, \epsilon \mbox{ for $t>\tau$}.$$ Now let $\varphi$ be a cutoff for $\Omega_{2r} \subset \subset \Omega_r.$ Multiplying the evolution of the full curvature by $\varphi$ and squaring, we obtain $$\begin{split} || \varphi \frac{\partial}{\partial t} F ||^2 + || \varphi D D^* F||^2 & = -2(D D^* F,\varphi^2 \frac{\partial}{\partial t}F) \\ & = -2\left(D^*F, -2 \varphi D\varphi \cdot \frac{\partial}{\partial t}F + \varphi^2 D^* (\frac{\partial}{\partial t}F) \right) \\ & \leq \frac{1}{2} || \varphi \frac{\partial}{\partial t}F||^2 + 8 ||D^*F||^2_{L^4(\Omega_r)}||D\varphi||^2_{L^4}\\ & \quad \quad + 2\left( \varphi^2 D^* F, D^*F \# F - \frac{\partial}{\partial t} \left( D^*F \right) \right) \\ & \leq \frac{1}{2} || \varphi \frac{\partial}{\partial t}F||^2 - \frac{d}{d t}|| \varphi D^*F||^2 + C \left(K + ||D^*F||^2_{L^\infty(\Omega_r)} \right)||D^*F||^2. \end{split}$$ Rearranging and integrating in time yields $$\begin{split} ||\frac{\partial}{\partial t}F||^2_{L^2(\Omega_{2r} \times {\left[}\tau , T \right) )} + 2|| D D^* F||_{L^2(\Omega_{2r} \times {\left[}\tau , T \right) )}^2 & + 2 ||D^*F(T)||^2_{L^2(\Omega_{2r})} \\ & \leq 2 ||D^*F(\tau)||^2_{L^2(\Omega_r)} + C ||D^*F||^2_{L^2(M \times {\left[}\tau , T \right) )} \\ & < C \epsilon. \end{split}$$ Also remark that from Lemma \[dstarf\] applied on cylinders $B_r \times {\left[}t - \tau, t {\right]},$ we have $$\int_\tau^T ||D^*F(t)||^2_{L^\infty(\Omega_r)} dt < C \epsilon.$$ Therefore $$||A(T) - A(\tau)||_{L^\infty(\Omega_{2r})} \leq \int_\tau^T ||D^*F||_{L^\infty} dt \leq C\epsilon^{1/2} (T-t)^{1/2}$$ as claimed. For the $H^1$ norm, we write $F_A = F_{A_{ref}} + D_{ref} A + A\wedge A,$ and $$\frac{\partial}{\partial t} D_{ref} A = \frac{\partial}{\partial t} F_A + D^*F \# A$$ which implies $$\begin{split} ||D_{ref} A(T) - D_{ref} A(\tau)||_{L^2} & \leq \int_\tau^T \left( ||\frac{\partial F}{ \partial t}||_{L^2(\Omega_{2r})} + 2 ||A ||_{L^\infty} ||D^*F||_{L^2} \right) \\ & \leq C \epsilon^{1/2} (T-\tau)^{\frac{1}{2}} (1 + ||A||_{L^\infty} ) \end{split}.$$ Finally, notice that $$\frac{\partial}{\partial t} \left( D_{ref}^* A \right) = D_{ref}^* D^*F = D^* D^* F + A\# D^*F = A\# D^*F$$ which implies a similar bound. This suffices to control the $H^1$ distance, and higher Sobolev norms are controlled similarly. \[convergence\] Assume $||F^+(0)||<\delta,$ and there exists an Uhlenbeck limit $A_\infty$ on $(M,E_\infty)$ which is an instanton with $H^{2+}_{A_\infty}=0.$ Then $E=E_\infty,$ and the flow converges smoothly to a connection which is gauge-equivalent to $A_\infty.$ More precisely, if $A_\infty$ is a connection satisfying (\[poincare\]), then for any $\tau_1\geq \tau_0>0$ there exist $\delta_1, \epsilon_1,$ and $r_1> 0$ as follows. If for some $\tau \geq \tau_1,$ $|| F^+(\tau - \tau_0) || < \delta_1$ and $A(\tau)$ is within $\epsilon_1$ of $A_\infty$ in $H^1(M_{r_1})$ modulo gauge, then for $t \geq \tau$ the flow converges exponentially (in the sense below). The constants $\delta_1,\epsilon_1$ also depend on $||F^+(0)||_{L^4},$ but are independent of it for $\tau_1$ sufficiently large. Let $M_0 = M \setminus \{x_1, \ldots , x_n\}$ be as in Definition \[uhllim\]. Let $r_1=r/3$ (where $r$ is as in Proposition \[excision\]), and choose $r_0 < \min (r_1, R_0, \sqrt{\tau_0})$ such that for every $x \in M_{2 r_1},$ we have $$||F_{ A_\infty }||^2_{L^2(B_{r_0}(x))} < \epsilon_0 / 3.$$ Now, let $\tau \geq \tau_1 $ be such that $$||F^+(\tau - \tau_0)||^2 < \delta_1^2$$ and there exists a smooth isometry $u$ such that $$\label{closeness} ||u^*(A(\tau)) - A_\infty||_{H^1(M_{r_1})} < \epsilon_1.$$ By the local Sobolev inequality,[^5] we have $$||u^*(A(\tau)) - A_\infty||_{L^4(M_{2r_1})} \leq C\epsilon_1.$$ Choosing $\epsilon_1$ such that $C \epsilon_1 < \epsilon / 2$ (where $\epsilon$ is as in Proposition \[excision\]), the Poincaré inequality holds for $A(t)$ with constant $C_\infty = C_{A_\infty}$ on some maximal interval ${\left[}\tau, T \right).$ We will argue that if $\delta_1 > 0$ is small enough, then $T = \infty$ and the flow converges. Note that $|D F^+| = |-*D*F^+| = |D^*F^+|.$ Applied to the global energy inequality for $F^+,$ the Poincaré inequality $$||F^+||^2 \leq C_\infty ||D^* F^+ ||^2$$ yields $$\partial_t ||F^+||^2 + C_\infty^{-1} ||F^+||^2 \leq \partial_t ||F^+||^2 + ||D^*F^+||^2 = 0.$$ This implies the exponential decay for $t \geq \tau$ $$\label{decay} ||D^*F||^2_{L^2( M \times {\left[}t, T {\right]})} \leq ||F^+(t)||^2 \leq \delta_1^2 e^{-(t-\tau)/C_\infty}.$$ By Proposition \[moser\], we have the global $L^\infty$ bound $$||F^+(t)||^2_{L^\infty(M)} \leq K^+(t)^2 := C_{\ref{moser}} \delta_1^2 e^{-(t-\tau)/C_\infty}$$ for $t\geq \tau.$ Therefore, if $\delta_1$ is sufficiently small we have $$\label{kappa} \left( C + ||F(t)||^2 \right) \int_\tau^T K^+(t) dt < \epsilon_0/3.$$ By Theorem \[selfdual\], the full curvature cannot concentrate on $M_{2r_1}$ before time $T,$ and we have a uniform bound $$\label{curvbound} ||F(t)||_{L^\infty(M_{2r_1})} < K$$ for $\tau+r_0^2 < t < T.$ In order to apply Proposition \[omega\], we need this curvature bound on $M_{2r_1}$ also from time $\tau - r_0^2/2.$ Note that $$\delta_1^2 > ||F^+(\tau - r_0^2)||^2 \geq \frac{1}{2}\left(||F(\tau-r_0^2)||^2 - ||F(T)||^2 \right).$$ By Lemma \[antibubble\], provided $\delta_1^2 < \epsilon_0 / 6,$ we in fact have a larger uniform bound (\[curvbound\]) on $M_{2r_1}$ for $\tau-r_0^2/2 < t \leq \tau + r_0^2.$ With this curvature bound, we may now apply Proposition \[omega\] and (\[decay\]) at each time $\tau + i,$ to conclude $$||A(\tau + i + 1) - A(\tau + i)||^2_{L^4(M_{3r_1})} \leq C^0_{\ref{omega}} \left( K^+(\tau + i) \right)^2.$$ By the triangle inequality and geometric series, we have $$||A(T) - A(\tau)||_{L^4(M_{3r_1})} \leq C \sum_i K^+(\tau + i) \leq C K^+(\tau) = C \delta_1.$$ If $\delta_1$ is small enough that $C \delta_1 < \epsilon/2,$ we conclude $$\begin{split} ||u^*(A(T)) - A_\infty ||_{L^4(M_{3r_1})} & \leq ||u^*(A(T)) - u^*(A(\tau))||_{L^4(M_{3r_1})} + ||u^*(A(\tau)) - A_\infty||_{L^4(M_{2r_1})} \\ & \leq C\delta_1 + C\epsilon_1 < \epsilon. \end{split}$$ Therefore $T=\infty,$ and the above estimates continue as $t\to \infty.$ Note that Theorem \[selfdual\] and (\[kappa\]) imply that the curvature does not concentrate anywhere on $M$ as $t\to \infty.$ Therefore the flow converges globally and strongly in $H^1$ (and by Proposition \[omega\] and (\[decay\]) applied on $M,$ at least exponentially). This proves the second statement. In the case that $F^+_{A_\infty}=0,$ by taking $r_1$ and $\epsilon_1$ smaller in the second statement, we can clearly satisfy the assumption $||F^+(\tau - \tau_0)|| < \delta_1.$ Hence the second statement implies the first. Further results {#asdsection} =============== \[parataubes\] *(Taubes’s grafting theorem, parabolic version.)* Let $(E_0,A_0)$ be a flat bundle on $M$ with $H_{A_0}^{2+}=0.$ For any $K^+,$ and points $x_1, \ldots, x_n \in M,$ there exist $\delta_1, \epsilon_1, r_1 > 0$ such that if $A$ is a connection on $E$ with $||F_A^+|| < \delta_1,$ $||F_A^+||_{L^{\infty}(M)} < K^+,$ and $$\label{taubescloseness} ||A - A_0||_{H^1(M_{r_1})} < \epsilon_1$$ then the flow with initial data $A(0) = A$ converges and remains $L^4$-close to $A_0$ modulo gauge on $M_{r_1} = M \setminus \{x_1, \ldots x_n\}.$ By assumption, a Poincaré estimate (\[poincare\]) holds, and we choose $\epsilon_1 \leq \epsilon/2,$ $r_1 = r/2$ according to Lemma \[excision\]. By (\[taubescloseness\]), we have $||F(0)||_{L^2} < C \epsilon_1.$ Applying the maximum principle to the evolution of $|F^+|^2,$ equation (\[fplusev\]), we have $||F^+(t)||_{L^\infty(M)} < 2 K^+$ for $0 \leq t < \tau < 1.$ Therefore, taking $\delta_1$ sufficiently small, Proposition \[moser\] and Theorem \[selfdual\] imply $$||F(t)||_{L^2(M_{2r_1})} < 2 C \epsilon_1$$ for $0\leq t \leq \tau.$ Assume first that $M$ is simply-connected, so we may take $A_0 = 0.$ According to [@donkron], Prop. 4.4.10, there exists a gauge transformation $u$ on $M_{2r_1}$ (also simply-connected) with $$||u^* A(\tau)||_{L^4(M_{2r_1})} < C \epsilon_1$$ for $\delta_1$ sufficiently small.[^6] The claim now follows from the precise statement of Theorem \[convergence\]. If $M$ is not simply-connected, we argue as follows. Let $\pi : \tilde{M} \to M$ be the universal cover, and choose a simply-connected domain $\Omega \subset \tilde{M}$ covering $M_{2r_1},$ which is a finite union of preimages of $B_i \subset M_{r_1},$ with $B_i \cap B_j$ connected.[^7] Now choose a gauge $u$ on $\Omega$ such that $\tilde{A} = \pi^*A(\tau)$ has $$||u^*\tilde{A}||_{L^4(\Omega)} < C \epsilon_1.$$ If this is done using Coulomb gauges on the $ B_i ,$ then $u^{-1}du$ is well-defined on $M.$ Note that we also have $$\begin{split} ||A(\tau) - A_0||_{L^2(M_{r_1})} & \leq ||A(\tau) - A(0)||_{L^2(M_{r_1})} + C ||A(0) - A_0||_{L^4(M_{r_1})} \\ & \leq \tau^{1/2} \left( \int ||D^*F||^2_{L^2(M_{r_1})} dt \right)^{1/2} + C\epsilon_1 \\ & \leq \delta_1 + C \epsilon_1. \end{split}$$ Hence over $\Omega,$ we have $$||du|| = ||u^{-1} du|| \leq ||u^*\tilde{A}|| + ||u \tilde{A} u^{-1}|| < C \epsilon_1.$$ By the Poincaré inequality, in each ball $$||u - \bar{u}||_{L^2(B_i)} < C \epsilon_1.$$ We may therefore choose points $p_i\in B_i$ such that $d(p_i , p_j) \geq c > 0$ and $$|u(\tilde{p}_i) - u(\tilde{p}_j)| < C \epsilon_1$$ for each $\tilde{p}_i, \tilde{p}_j \in \Omega$ such that $\pi(\tilde{p}_i) = p_i$ and $\pi(\tilde{p}_j) = p_j.$ It is clearly possible to construct a frame $v$ over $\Omega$ such that $v(\tilde{p}_i) = u(\tilde{p}_i) \, \forall \, i,$ $||dv||_{L^\infty} < C \epsilon_1,$ (depending on $\Omega$) and $v^{-1} dv$ is well-defined on $M_{r_1}.$ The frame $w = v^{-1} u$ then satisfies $w(\tilde{p}_i) = 1$ for all $\tilde{p}_i,$ and descends to a frame on $E$ over $M_{r_1}.$ Note that $$||w^* \tilde{A}||_{L^4(\Omega)} \leq ||v^{-1} dv + v^{-1}(u^*\tilde{A}) v||_{L^4(\Omega)} \leq 2 C \epsilon_1$$ and so downstairs $$||w^*A(\tau) - A_0||_{L^4(M_{r_1})} \leq C\epsilon_1.$$ Convergence follows for $\epsilon_1$ and $\delta_1$ sufficiently small as before. A similar argument can be used to recover the gluing theorem for connected sums with long necks of small volume, i. e. [@donkron], Theorem 7.2.24. \[chargeone\] Assume $\pi_1(M)$ is abelian or has no nontrivial representations in $SU(2),$ and $H^{2+}(M)=0.$ For any initial connection on the bundle $E$ with structure group $SU(2)$ and $c_2(E)=1,$ assuming $||F^+(0)||<\delta_1,$ no bubbling occurs and the flow has a smooth subsequential limit as $t\to \infty.$ If this limit is an irreducible instanton, then it is unique and the flow converges exponentially. Assume by way of contradiction that bubbling occurs as $t\to \infty.$ The blowup limits of [@schlatterlongtime] at a presumed singularity, as well as the Uhlenbeck limit, preserve the structure group. Due to the $L^\infty$ bound on $F^+,$ the blowup limit at a bubble must be anti-self-dual, and therefore contains all but $2 \delta_1$ of the energy. If the Uhlenbeck limit obtained from Theorem \[uhllimexist\] on the same sequence of times is also anti-self-dual, it must be flat by integrality of $\kappa.$ By the assumptions on $\pi_1(M),$ it acts trivially on the adjoint bundle. But then its cohomology is exactly $H^{2+}(M)=0,$ and by the Theorem the flow converges, which is a contradiction. If the Uhlenbeck limit is not anti-self-dual, it nonetheless must be $L^4$-close to a flat connection by the argument of the previous Theorem, which is still a contradiction. Therefore a smooth Uhlenbeck limit exists. If it is irreducible then $H^{2+}=0,$ and again by Theorem \[convergence\] we have exponential convergence. \[asympstab\] The instantons with $H^{2+} = 0$ are asymptotically stable in the $H^1$ topology. In other words, given an $H^1$ neighborhood $U$ of $A,$ there exists a neighborhood $U' \subset U$ of initial connections for which the limit under the flow will again be an instanton with $H^{2+} = 0,$ lying in $U$ modulo smooth gauge transformations. Moreover, there exists an $H^1$-open neighborhood $N$ for which the flow gives a deformation retraction from $N \cap H^k,$ $k>>1,$ onto the moduli space of instantons with $H^{2+}=0.$ By Struwe’s construction [@struwe], §4.2-4.3, choosing the instanton $A$ itself as reference connection, the gauge-equivalent flow remains in $U$ for a time $\tau,$ long enough for $\epsilon$-regularity to take effect. This gives a uniform bound on the curvature at time $\tau,$ including on $||F^+||_{L^\infty}.$ Choosing $U'$ small enough, we also obtain $||F^+|| < \delta_1.$ We are then in the situation of Theorem \[convergence\], which can be applied with $\{x_i\} = \emptyset.$ The latter refinement follows from standard parabolic theory. 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Repulsion and quantization in almost-harmonic maps. Ann. of Math. (2) 159 (2004), no. 2, 465-534. K. Uhlenbeck. Connections with $L^p$ bounds on curvature, Comm. Math. Phys. 83 (1982), 31-42. K. Uhlenbeck. Removable singularities in Yang-Mills fields, Comm. Math. Phys. 83 (1982), 11-30. B. Weinkove. Singularity formation in the Yang-Mills flow. Calc. Var. 19 (2004), 221-220. [^1]: in particular the scaling of certain Sobolev norms applied to the cutoff [^2]: Since the singularities are isolated, $F^{\pm}$ clearly must concentrate at the same point. This is easily shown by adding a boundary term to Proposion \[moser\]. [^3]: In fact the $k=0$ bound also follows from the derivative estimate, for smooth solutions in dimension four. [^4]: Note that Theorem 1.3(ii) of Schlatter does not include any patching, because this may not be possible with $H^{2,2}$ gauge transformations. [^5]: applied with respect to a smooth reference connection for $E_\infty$ [^6]: Here the $\delta_1$-dependent bounds of Proposition \[omega\] take the place of the anti-self-dual condition used to obtain the curvature-dependent bounds of [@donkron] 2.3.11. [^7]: This can be done for instance by lifting the geodesic balls $B_i$ to $\tilde{M}$ using a set of based paths which form a spanning tree for their incidence graph.

--- abstract: | The accurate springback prediction of dual phase (DP) steels has been reported as a major challenge. It was demonstrated that this was due to the lack of understanding of their nonlinear unloading behavior and especially the dependency of their unloading moduli on the plastic prestrain. An improved compartmentalized finite element model was developed. In this model, each element was assigned a unique linear elastic J2 plastic behavior without hardening. The model’s novelty lied in the fact that: i) a statistical distribution was discretized in a deterministic way and used to assign yield stresses to structures called compartments, ii) those compartments were randomly associated with the elements through a random compartment element mapping (CEM). Multiple CEM were simulated in parallel to investigate the intrinsic randomness of the model. The model was confronted with experimental data extracted from the literature and it was demonstrated that the model was able to reproduce the dependence of the apparent moduli on the tensile prestrain. It was also observed that the evolution of the apparent moduli was predicted even if it was not an explicit input of the experimental dataset used to identify the input parameters of the model. It was then deduced that the shape of the hardening and the dependency of moduli on the prestrain were two manifestations of a single cause: the heterogeneous yield stress in DP steels. address: - 'Univ. Savoie Mont-Blanc, EA 4144, SYMME, F-74940, Annecy-le-Vieux, France' - 'Univ. of Djibouti, Djibouti' author: - Ludovic Charleux - Laurent Tabourot - Emile Roux - Moustapha Issack Farah - Laurent Bizet title: 'Dependency of the Young’s modulus to plastic strain in DP steels: a consequence of heterogeneity ?' --- Dual phase steels, Apparent modulus, Young’s modulus evaluation, Heterogeneity, Springback. Introduction ============ Dual phase (DP) steels exhibit an outstanding combination of strength and ductility. They are widely used in the automotive industry where they contribute to the vehicle mass reduction and thus to greater fuel efficiency [@tasan_overview_2015]. This being said, the complexity of their mechanical behavior is especially high and many scientific challenges have to be addressed before they can be used with full background knowledge. Among those, the accurate prediction of springback is one of the greatest [@wagoner_advanced_2013]. It is known that the apparent Young’s modulus of most metallic alloys is influenced by an applied prestrain [@Yoshida2002; @pham_mechanical_2015] and that this is especially true in the case of DP steels. It has also been demonstrated that this phenomenon affects the ability to predict springback [@eggertsen_constitutive_2010; @Yu2009; @ul2016springback]. While the dependency of the apparent modulus to the prestrain is observed by many researchers, the mechanisms at stake are not yet fully determined. Therefore, several ways have been followed to take this evolution of the apparent modulus into account. The most widely used and arguably the most successful is the phenomenological modeling developed in numerous papers [@Yoshida2003; @Kim2013; @Sun2011; @Lee2013; @Ghaei2015a; @Xue2016; @Zajkani2017; @torkabadi2018nonlinear]. These studies do not focus on the underlying causes of the phenomenon but only aim to reproduce it, in most cases by using the model proposed by @Yoshida2002. The drawback of phenomenological methods is that they rely on a fine-tuning stage of an ever-increasing number of parameters. In parallel, other methods have been aiming at establishing a relation between the material’s physical properties and its macroscopic behavior. As pointed out by @Paul2013, the intrinsic dual phase heterogeneity of DP steels triggers strain incompatibility between the soft ferrite phase and the harder martensite phase. Several two-phase models have been developed using local phenomenological models as well as crystal plasticity to take into account the bimodal mechanical behaviors [@kadkhodapour_micro_2011; @Ramazani2012; @Moeini2017]. Random microstructures have also been used with relative success [@Furushima2009; @Furushima2013; @Furushima2013b; @Ayatollahi2016; @khan2018microstructure]. Still, this global simulation process has been unable to reproduce correctly the non-linear unloading behavior of DP steels and so it has been failing to determine the quantitative changes in the apparent modulus unless finally using the phenomenological Yoshida-Uemori model. As stated by @tasan_overview_2015, the causes of heterogeneity in DP steels are multiple: on the one hand, heterogeneous dislocation microstructure, grain size distribution, presence of impurities and on the other hand, strong differentiation between the mechanical behaviors of its constituent phases which are themselves randomly distributed in the material. Then, a third way has been explored to deal with these observations. It relies on introducing heterogeneity in the model in a more generalized way using a statistical spatial distribution of yield stresses [@Tabourot2012; @Tabourot2013]. This compartmentalized model has shown that it can reproduce experimental observations with fewer adjustable parameters than phenomenological models and that their predictions are more realistic [@Bizet2017]. In this paper, an improved version of the compartmentalized model was used to simulate and analyze the apparent modulus of DP steels. Section 2 is dedicated to the description of the compartmentalized model. Then, the model is applied to retrieve experimental data extracted from the literature. In section 3, the predictions of the model are discussed and confronted with the other existing modeling paradigms. The compartmentalized model ============================ A compartmentalized model is a random heterogeneous finite element model in which the material properties of every element arise from those of a substructure called a compartment. Each compartment has unique material properties. In the presented model each element is a compartment. Compartments are not designed to represent a specific physical structure or scale such as grains. Their only purpose is to introduce a controlled amount of heterogeneity in the model to produce specific effects on the macroscopic mechanical behavior. The implementation of the compartmentalized model described in this section is made available by the authors (python libraries : [Compmod2](https://github.com/lcharleux/compmod2) [@compmod2] and [Argiope](https://github.com/lcharleux/argiope) [@argiope], full code example : [Compdmod2 Documentation](https://compmod2.readthedocs.io/en/latest/notebooks_rst/DP_moduli/DP_steel.html) [@compmod2-doc]) Mesh and boundary conditions ---------------------------- In this paper, the finite element simulations were carried out using the commercial implicit solver Abaqus/Standard (2018 version). Fig. \[fig:model\_3D\] represents the cubic Representative Volume Element (RVE) test sample. The initial dimensions of the RVE were $l\times l \times l$ along $(\hat e_1, \hat e_2, \hat e_3)$ where $l=1$. Only intensive properties such as stress and strains were extracted from the model. Consequently, the problem was dimensionless and the value of $l$ had no impact on the results. A structured $10 \times 10 \times 10 $ hexahedric mesh was used. Periodic boundary conditions were applied in a similar way to [@wu2014applying]. The sample was loaded in tension and the true tensile stress $\sigma$ as well and true tensile strain $\varepsilon$ were calculated. Compartmentalized material definition ------------------------------------- In a compartmentalized model, the material properties are distributed randomly among the elements of the mesh. This procedure has been greatly improved compared to the intuitive one used by the authors in previous articles [@Tabourot2012; @Tabourot2013; @Laurent2014; @Bizet2017]. This legacy procedure is described in Fig. \[fig:algorithm\_old\] while the new one is represented in Fig. \[fig:algorithm\_new\]. Each compartment is associated with a unique but very basic, isotropic, linear elastic and J2 perfectly plastic material. Consequently, no hardening is implemented in any of those materials. Moreover, the elastic behavior of all compartments is homogeneous. Their Young modulus has a fixed value $E = \unit{213}\giga\pascal$ and their Poisson’s ratio is $\nu=0.3$. These values are in agreement with [@Chen2016b]. Yield stress distribution ------------------------- The yield stresses $\sigma_y$ are distributed among the compartments following a statistical distribution noted DIST. It is defined by its probability density function (PDF) noted $f(\sigma_y)$ and its cumulative density function (CDF) noted $F(\sigma_y)$. In this case, the distribution is the weighted sum of two Weibull sub-distributions with PDFs $f_1(\sigma_y)$ and $f_2(\sigma_y)$ that verify: $$f(\sigma_y) = w_1 f_1(\sigma_y) + (1-w_1) f_2(\sigma_y)$$ And: $$f_i(\sigma_y) = \dfrac{k_i}{\lambda_i} \left( \dfrac{\sigma_y}{\lambda_i} \right)^{k_i-1} \exp\left( -\left(\dfrac{\sigma_y}{ \lambda_i}\right)^{k_i} \right)$$ Where: - $k_i$ and $\lambda_i$ are respectively the shape parameters and the scale parameters of the Weibull sub-distributions, - $w_1$ is the weighting factor between each sub-distribution and it verifies $w_1 \in \; ]0, 1[$. The five input parameters $P = \left\lbrace k_1, k_2, \lambda_1, \lambda_2, w_1 \right \rbrace$ fully define the plastic behavior of the RVE. The values of the yield stresses associated with each compartment could then be calculated using a Random Number Generator (RNG) associated with the PDF defined above. However, this solution would mean that for a given value of $P$, multiple different sets of yield stress values could be possible because of the random nature of the RNG. As a consequence, the model would not be deterministic and most optimization scheme to identify the parameters would be compromised. To overcome this issue, a procedure has been developed to discretize the distribution in a deterministic way as described in Fig. \[fig:distribution\_discretization\]. 1. The model contains $N$ compartments, each occupying an equal part of the whole model volume. As a consequence, each compartment $C_i$ is assumed to represent a cumulative probability of $1/N$. The CDF is equally split along the vertical axis in $N$ parts separated by $N+1$ thresholds values noted $F_t$, where $F_{t0} = 0$ and $F_{tN}=1$. 2. The CDF is inverted using the Brent zero finding algorithm [@brent1973] implemented in the Python library Scipy [@scipy], $N+1$ threshold yield stress values $\sigma_t$ are determined. This process is described in Fig. \[fig:distribution\_discretization\]-a. 3. Each individual compartment then represents a unique portion of the distribution. Since only a single value of $\sigma_{y,i}$ has to be associated with each compartment $C_i$, the mean value of distribution on the interval $[\sigma_{t,i}, \sigma_{t,i+1}]$ is chosen: $$\sigma_{y,i} = \dfrac{\int_{\sigma_{t,i}}^{\sigma_{t,i+1}} \sigma f(\sigma)d\sigma}{\int_{\sigma_{t,i}}^{\sigma_{t,i+1}} f(\sigma)d\sigma}$$ The Compartment Element Mapping ------------------------------- Each element $E_j$ is associated with a compartment $C_i$ by a discrete bijective mapping function referred to as the Compartment Element Mapping (CEM) as presented in Fig. \[fig:CEM\]. The CEM is initialized by shuffling the $\lbrace 1 , \ldots , N \rbrace$ list, creating the required association between the compartments and the elements. The CEM is the only source of the randomness of the model and thus it is the way to control it. As a consequence, as long as the CEM is not reinitialized, the model response to a given set of input parameters $P$ is deterministic. In this paper, a set of 10 random CEMs was generated by random shuffling and was used all along. Input parameter identification on a single loading-unloading-reloading cycle ---------------------------------------------------------------------------- The true tensile stress *vs.* true tensile strain curve of a DP980 steel was extracted from Fig. 2 in @Ghaei2015 and is referred to as the experimental curve. This curve was separated into multiple loading, unloading and reloading cycles (MUR). The resulting data-set was split into two subsets represented in Fig. \[fig:exp\_data\]. The first subset noted **EXP-A** contains a monotonic loading up to 8% of strain and the last unloading-reloading cycle (LUR). It is essential to note that the evolution of the elastic moduli with the accumulated plastic strain cannot be observed on this subset because it contains only one unloading-reloading cycle. The second subset noted **EXP-B** contains the remaining experimental data. For any given set of input parameters $P$ and for each CEM $c$, a true tensile stress response can be simulated following the strain path of the test **EXP-A**. As experimental and simulated data are not interpolated on the same grid, it is necessary to interpolate them. The simulated stresses are therefore linearly interpolated on the strain value of the experimental curve **EXP-A**. The interpolated simulated stress is noted $\sigma_{ci}\left( P \right)$, where $i$ denote each point of the **EXP-A** curve. The stress residuals vector $\delta \sigma_{ci}\left( P \right)$ between the calculated set of stresses and the experimental stress $\sigma_{ei}$ is defined as follows: $$\delta \sigma_{ci}\left( P \right) = \sigma_{ci} \left( P \right)- \sigma_{ei}$$ The residual vector $\delta\sigma_{ci}\left( P \right)$ gathers the point to point stress residual for all CEMs at each measurement points. The optimal input parameters $P_{opt}$ were then calculated using the Levenberg-Marquardt least-square optimization algorithm [@leve44] by minimizing the residual vector $\delta\sigma_{ci}\left( P \right)$. With an educated guess of the starting point of the Levenberg-Marquardt algorithm, the convergence was achieved after 70 evaluations of the cost function and thus after 700 individual simulations. The optimal numerical values of $P_{opt}$ are given in the Table \[tab:params\_opti\]. This way to proceed allowed us to determine not only the mean stress value but also the dispersion of the solutions associated with the different CEMs at each strain value. The mean stress value as well as min/max values are represented on Fig. \[fig:model\_optimization\]. The associated yield stress distribution is represented in Fig. \[fig:model\_optimized\_dist\]. The compartments yield stresses statistics are detailed in the Table \[tab:compartment\_stats\]. Moduli calculation on multiple loading-unloading-reloading cycles ----------------------------------------------------------------- A new set of 10 simulations were run using the optimized input parameter set $P_{opt}$. These new simulations included multiple unloading reloading cycles. It is important to note that these multiple unloading reloading cycles were not used in the parameters identification stage. These multiple cycle simulations aimed to calculate the moduli as a function of the prestrain $\varepsilon_u$ to evaluate the capacity of the compartmentalized model. The obtained mean stress value as well as min/max values are represented on Fig. \[fig:model\_multi\_cycle\], and are noted **SIM-MUR** . The moduli $E_1$, $E_2$, $E_3$, $E_4$ and $E_c$ were then calculated following the definitions proposed by [@Chen2016a] as represented on Fig. \[fig:wagoner\_moduli\_definition\] using the stress vs. strain curve **SIM-MUR**. Results and discussion ====================== Overall performances of the compartmentalized model --------------------------------------------------- The Fig. \[fig:model\_optimization\] demonstrates that the compartmentalized model can reproduce efficiently the experimental loading/unloading behavior of a DP steel (DP980 here). This is also true for other existing models such as the QPE model [@Sun2011]. However, the compartmentalized model is more efficient as it only relies on a homogeneous linear elastic coupled to a J2 plastic criterion without hardening behavior and 5 additional parameters (Tab. \[tab:params\_opti\]) to describe the material. This specificity makes the compartmentalized model’s parameters easier to identify than those of its phenomenological counterparts. In this paper, only 70 optimization iterations (700 simulations) were needed to identify the parameters required to fit the **EXP-A** experimental sub-data-set. Compartment statistics {#subsec:stats} ---------------------- The identified parameters $P_{opt}$ solely describe the level of heterogeneity of the local yield stresses. Fig. \[fig:model\_optimized\_dist\] gives an accurate description of the optimal yield stress distribution. Compared to other alloys previously modeled using the compartmentalized model [@Bizet2017], it appears that distinguishing feature of the DP980 steel behavior is due to its exceptional heterogeneity that leads to a wide range of distributed yield stresses. In that respect, 3 compartment sets are defined in the Tab. \[tab:compartment\_stats\], each of them producing a given effect or property. The Soft to Hard compartments (SHC) : constitute the main lobe of the distribution and represent a fraction of $80.5\% $. If this population has to be modeled alone, a single Rayleigh distribution could be used. The upper bound of the yield stress of SHC fixed at  because all the elements having higher yield stress never exhibited plastic strain in any simulation. This also means that the local von Mises stress field can sometimes reach very high values close to $\unit{3}\giga\pascal$. The Elastic compartments (ELC) : represent $12.3\%$ of the compartments. They are key compartments to model the specificity of materials exhibiting hardening at large strains such as DP980. If all compartments would become plastic during loading, the hardening would saturate. Hence, to exhibit hardening even at $\varepsilon \approx 10\%$, a significant proportion of the compartments must always remain elastic. The Ultra-Soft compartments (USC) : represent $7.2\%$ of the compartments. They exhibit very low yield stresses. They don’t play a key role during loading but their existence is strongly connected with the Bauschinger effect observed during unloading and the hysteretic behavior associated with unloading-reloading cycles. Indeed, at the end of the loading step (Fig. \[fig:model\_optimization\], $\varepsilon \approx 8\%$ and $\sigma \approx \unit{1200}\mega\pascal$), when the unloading starts, most of these compartments rapidly become plastic in the compressive direction during the first $\unit{200}\mega\pascal$ of the unloading. Thus, these compartments dissipate energy and increase the curvature of the unloading stress vs. strain curve at the end of the unloading step. The same phenomenon appears symmetrically during reloading. Moduli evolution predicted by the compartmentalized model --------------------------------------------------------- Fig. \[fig:model\_modulus\_Wagoner\] represents the experimental measurements taken from Fig. 5 in @Chen2016a as well as the values obtained by **SIM-MUR** using the protocol described in Fig. \[fig:wagoner\_moduli\_definition\]. Fig. \[fig:model\_modulus\_Wagoner\] shows that the evolution of $E_1$ and $E_3$ is overestimated by the compartmentalized model. However, in their paper, @Chen2016a indicate that the decrease observed in their $E_1$ and $E_3$ measurements is small in the face of measurement uncertainties and could thus be an artifact related to a lack of resolution. In contrast, the variations of $E_2$, $E_4$ and $E_c$ calculated by the simulation **SIM-MUR** using $P_{opt}$ are in very good agreement with the experimental data. This implies, on the one hand, that the compartmentalized model is capable of reproducing the moduli decrease with a constant intrinsic Young’s modulus $E$. On the other hand, since the optimization of the input parameters $P_{opt}$ is carried out with the **EXP-A** dataset, which contains only one loading/unloading/reloading cycle, it follows that, in the context of the compartmentalized model, the decay of the apparent moduli is related to the shape of this cycle. These results indicate that strain-hardening, as well as the evolution of apparent moduli, are only two manifestations of a single cause: the heterogeneity of the material. It also shows that the compartmentalized model allows a physically realistic representation of the mechanisms involved in the plastic deformation of DP steels. Conclusion ========== In this work, the compartmentalized model has been proved to be a relevant way to model the behavior of DP steels. The model was improved by separating its intrinsic spatial heterogeneity and its randomness into two independent contributions driven respectively by the DDP and the CEM. It has been shown that this improvement allows the model to be deterministic and thus that its input parameters $P$ could be easily optimized as long as the CEM is kept unchanged. Results obtained from model simulations have been confronted with experimental pieces of evidence found in the literature and it has been demonstrated that: 1. it can reproduce the overall shape of the stress vs. strain curve of the DP980 steel without needing the use of the variable intrinsic Young’s modulus. 2. the evolution of the apparent moduli with the level of prestrain is predicted spontaneously when the model is optimized to reproduce the experimental LUR cycle. 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[http://dx.doi.org/10.1016/j.jmapro.2016.12.022 http://linkinghub.elsevier.com/retrieve/pii/S1526612516301931](http://dx.doi.org/10.1016/j.jmapro.2016.12.022 http://linkinghub.elsevier.com/retrieve/pii/S1526612516301931) --------------- ---------------------- **Parameter** **Value** $k_1$ $1.64$ $k_2$ $3.17$ $l_1$ $4.16\times 10^{-3}$ $l_2$ $3.42\times 10^{-2}$ $w_1$ $8.67\times 10^{-1}$ --------------- ---------------------- : The parameters $P_{opt}$ resulting from the optimization for the compartmentalized model.[]{data-label="tab:params_opti"} ----------- ------------------ ------------------ ------------------- ------- -------------- **Label** $\sigma_{y,min}$ $\sigma_{y,max}$ $\bar \sigma_{y}$ $N_c$ **Behavior** USC 72 Ultra Soft SHC 805 Soft to Hard ELC $+\infty$ 123 Elastic ----------- ------------------ ------------------ ------------------- ------- -------------- {width=".8\textwidth"}  {width=".8\textwidth"} {width=".8\textwidth"}

--- abstract: 'Given a set of $n$ points $S$ in the plane, a triangulation $T$ of $S$ is a maximal set of non-crossing segments with endpoints in $S$. We present an algorithm that computes the number of triangulations on a given set of $n$ points in time $n^{(11+ o(1))\sqrt{n} }$, significantly improving the previous best running time of $O(2^n n^2)$ by Alvarez and Seidel \[SoCG 2013\]. Our main tool is identifying separators of size $O(\sqrt{n})$ of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers (“cactus graphs”). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in $n^{O(\sqrt{n})}$ time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, $3$-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, $3$-regular graphs, and more.' bibliography: - 'LibSoCG.bib' --- [Peeling and Nibbling the Cactus:\ Subexponential-Time Algorithms for\ Counting Triangulations and Related Problems.\ ]{}  [Dániel Marx Tillmann Miltzow ]{} Institute for Computer Science and Control,\ Hungarian Academy of Sciences (MTA SZTAKI)\ ` dmarx@cs.bme.hu`, `t.miltzow@gmail.com` #### Acknowledgment {#acknowledgment .unnumbered} Both authors are supported by the ERC grant PARAMTIGHT: Parameterized complexity and the search for tight complexity results", no. 280152.

--- abstract: 'This paper addresses the problem of computing the achievable rates for two (and three) users sharing a same frequency band without coordination and thus interfering with each other. It is thus primarily related to the field of cognitive radio studies as we look for the achievable increase in the spectrum use efficiency. It is also strongly related to the long standing problem of the capacity region of a Gaussian interference channel (GIC) because of the assumption of no user coordination (and the underlying assumption that all signals and interferences are Gaussian). We give a geometrical description of the SINR region for the two-user and three-user channels. This geometric approach provides a closed-form expression of the capacity region of the two-user interference channel and an insightful of known optimal power allocation scheme.' author: - title: 'A Geometrical Description of the SINR Region of the Gaussian Interference Channel: the two and three-user case' --- Introduction ============ Interference is a fundamental issue in wireless communication when multiple uncoordinated links share a common communication medium. This paper addresses the problem of computing the achievable rates for two (or three) users sharing a same frequency band without coordination and interfering with each other. It is thus primarily related to the field of cognitive radio studies as we look for the achievable increase in the spectrum use efficiency. It is also strongly related to the long standing problem of the capacity region of a Gaussian interference channel (GIC) because of the assumption of no user coordination (and the underlying assumption that all signals and interferences are Gaussian). Both topics have received a lot of attention in the technical literature where the interference channel is generally addressed via information theoretic tools, see for instance [@Spectrum_breaking][@kramer][@sason] and references herein. To this respect, [@Spectrum_breaking] proposes a definition of cognitive radio as wireless system that makes use of “any available side information about activity, channel conditions, codebooks or messages used by the other users with which it shares the spectrum”. What is the best performance one can achieve without making any a priori assumption on how the common resource is shared? We shall not assume any cooperation between users; they are not able to decode messages from other users, with the consequence that we shall not use sophisticated techniques such that dirty paper coding, rate splitting [@han-kobayashi] and their associated bounds for the achievable rates of each user. Due to its apparent simplicity, the two-user Gaussian interference channel (GIC) was the first to be addressed by the technical literature. Despite some special cases, such as very strong, strong ICs and the trivial case when there is no interference, in general the characterization of its capacity region is said an open problem. The exact characterization of the capacity region of the IC has been derived in the strong interference regime in [@carleial], [@kramer] where it is shown that each user can decode the information transmitted to the other user. The best known achievable strategy is the Han-Kobayashi scheme [@han-kobayashi], where each user splits the information into private and common parts. The common messages are decoded at both the receivers, thereby reducing the level of interference. With the assumption of non cooperating users with power constrained Gaussian signals, the available rate of each of them is given by the $\log_2(1+SINR)$ classical formula, where SINR is the signal to noise plus interference ratio at the receiver. The difficulty we face is that the various SINRs of all users are not independent; they are interrelated in a way involving the channel coefficients as will be seen in the next section. Nevertheless we can have some insight in the shape (the geometry) of the set of possible SINRs, at least for the two or three-user interference channel. We can make use of this geometry to derive some new results: the capacity region of the two-user interference channel, and the SINR region of the three-user channel. Moreover, the way we derive this last result is very general and it allows deriving the $n$-user SINR region provided we know the one corresponding to $(n-1)$ users. This geometric approach provides a closed-form capacity bounds expression of the two-user Gaussian interference channel when interference is considered as noise, although this strategy is known to be suboptimal. The remainder of this paper is organized as follows: we derive the analytical expressions of the capacity bounds, for the two-user Gaussian interference channel, in the Section II. In Section III, we tackle the problem of finding the maximum of the sum rate and we derive two possible areas in the plan where the maximum sum rate point can be. The three-user Gaussian interference channel is considered in Section IV where we find the analytical expressions characterizing the SINR region. Finally, conclusions are given in Section V. THE TWO-USER GAUSSIAN INTERFERENCE CHANNEL: CAPACITY REGION =========================================================== We consider a Gaussian interference channel with two transmitters and two receivers as depicted in the Fig.\[twoUserGIC\]: $$\begin{aligned} Y_1 & = & h_{11}X_1 + h_{12}X_2 + Z_1\\ Y_2 & = & h_{21}X_1 + h_{22}X_2 + Z_2\nonumber\end{aligned}$$ We shall assume that channel inputs are power-limited real Gaussian processes such that $p_i = E\left[ X_{i}^{2}\right]\leq P_i$ , and that there is no cooperation between users, so that interferences can be seen as Gaussian noise. With this assumption the two capacities of users 1 and 2 to their respective receivers are: $$\begin{aligned} C_1 &=& \frac{1}{2}\log_2\left(1 + \frac{g_{11}p_1}{\sigma^2 + g_{12}p_2} \right),\\ C_2 &=& \frac{1}{2}\log_2\left(1 + \frac{g_{22}p_2}{\sigma^2 + g_{21}p_1} \right),\hspace{0.5cm} g_{ij}=\left|h_{ij}\right|^2 \nonumber\end{aligned}$$ With the change of variables $u_i = g_{ii}p_i/\sigma^2$ the above equations can be rewritten as: $$\begin{aligned} C_1 =\frac{1}{2}\log_2\left(1+S_1\right),& S_1=\frac{u_1}{1+a_{12}u_2}\\ C_2 =\frac{1}{2}\log_2\left(1+S_2\right),& S_2=\frac{u_2}{1+a_{21}u_1}\nonumber\end{aligned}$$ where $a_{12}=g_{12}/g_{22}$ and $a_{21}=g_{21}/g_{11}$. The quantities $u_1$, $u_2$ are the SNR values when there is no interference and $S_1$, $S_2$ are the SINRs values (signal to noise plus interference ratios). The introduction of variables $u_1$, $u_2$ is similar to the introduction of the normalized channel in [@kramer] to which the reader is referred, as well as for an account of more results on the Gaussian interference channel. The relation between the SINR variables $S_1$, $S_2$ and the SNR values $u_1$, $u_2$ can be easily inverted to obtain the two following expressions: $$\begin{aligned} u_1 &=& \frac{S_1(1+S_2a_{12})}{1-a_{12}a_{21}S_1S_2}\\ u_2 &=& \frac{S_2(1+S_1a_{21})}{1-a_{12}a_{21}S_1S_2}\nonumber\end{aligned}$$ Expressing the power constraints $0\leq u_i \leq \bar{P_{i}}=g_{ii}P_i/\sigma^2$ allows us to derive corresponding constraints on the SINR variables, namely: $$\begin{aligned} \label{eq1} S_2 &\leq &\frac{1}{a_{12}a_{21}S_1} \\ \label{eq2} S_1 &\leq & \phi_1(S_2)=\frac{\bar{P_1}}{1+a_{12}S_2(1 + a_{21}\bar{P_1})} \\ \label{eq3}S_2 &\leq & \phi_2(S_1)=\frac{\bar{P_2}}{1+a_{21}S_1(1 + a_{12}\bar{P_2})}\end{aligned}$$ The SINR region is thus delimited by these three curves. All variables being positive, the two functions $\phi_1(S_2)$ and $\phi_2(S_1)$ are respectively upper bounded by $(a_{12}\ a_{21}\ S_2)^{-1}$ and $(a_{12}\ a_{21}\ S_1)^{-1}$ so that the first inequality is redundant and is omitted in the sequel of the paper. The SINR region is then the intersection of the regions obeying each of the constraints defined by $\phi_1$, $\phi_2$ : $$\begin{aligned} \mathcal{D}'&= &\left\{(S_1, S_2)| 0\leq S_1 \leq \phi_1(S_2)\right\}\\ &&\cap \left\{(S_1, S_2)| 0\leq S_2 \leq \phi_2(S_1)\right\}\nonumber\end{aligned}$$ We can also notice that $\phi_2(S_1)$ is simply obtained from $\phi_1(S_2)$ by the permutation $\{1,2\}\rightarrow \{2,1\}$, this result will be used later when considering the three-user case. The second inequality (\[eq2\]) above can be written in the equivalent form $S_2 \leq \left(\bar{P_1} -S_1 \right)/\left( a_{12}S_1\left(1+a_{21}\bar{P_1} \right) \right)$ so as to write the following analytic expression for the SINR region as a function of the sole $S_1$ : $$0\leq S_2\leq \min\left(\frac{\bar{P_2}}{1 + a_{21}S_1\left(1+a_{12}\bar{P_2} \right)}, \frac{\bar{P_1}-S_1}{a_{12}S_1\left(1+a_{21}\bar{P_1}\right)} \right)$$ We shall use this expression to derive analytical bound to the capacity region of the interference channel.\ The transformation $(u_1, u_2)\xrightarrow{\phi} (S_1,S_2)$ is a one to one correspondence of the region $\mathcal{D} =\{0\leq u_1 \leq \bar{P_1}, 0\leq u_2 \leq \bar{P_2} \}$ into the transformed region $\mathcal{D}'$, it leaves invariant the two points $(\bar{P_1},0)$ and $(0,\bar{P_2})$. We have $\mathcal{D}'\subset \mathcal{D}$, for $S_i \leq u_i$ . We can already notice that the more $\bar{P_1}$ and $\bar{P_2}$ increase the more the region $\mathcal{D}'$ will be constrained by the red curve in the Fig.\[SINRRegion\] and its shape different from a rectangle. The last transform, $S_i\rightarrow \log_2(1+S_i)$, allows us to give an analytical expression of the capacity region boundary as a parametric curve rather than a simple function giving $R_2$ as a function of $R_1$ : $$\begin{aligned} 0&\leq& t \leq \bar{P_1}\\ C_1 &=&\frac{1}{2}\log_2(1+t)\nonumber\\ C_2 &=&\frac{1}{2}\log_2(1+f(t))\nonumber\\ f(t) &=&\min\left(\frac{\bar{P_2}}{1+a_{21}t\left( 1 + a_{12}\bar{P_2}\right)}, \frac{\bar{P_1}-t}{a_{12}t\left( 1 + a_{21}\bar{P_1}\right)}\right)\nonumber\end{aligned}$$ It is easy to check that $f(0)=\bar{P_2}$ and $f(\bar{P_1})=0$, that are the two cases where all capacity is allocated to only one user. As a result, the same parameterization provides an expression for the sum rate: $$\mathbf{C_{SUM}}=\frac{1}{2}\log_2(1+t)+ \frac{1}{2}\log_2(1+f(t))$$ Depending on the values of $\bar{P_1}$ and $\bar{P_1}$ and the coefficients of the normalized channel $a_{12}$, $a_{21}$, the capacity region and the sum capacity will exhibit different behaviors as depicted below for a symmetric case $a_{12}=a_{21}$ (cf. Fig.\[caparegion1\] and Fig.\[caparegion2\]). The image of the point $(\bar{P_1},\bar{P_2})$ by the SNR to SINR transform is represented by a star; and the dashed curve is the constant sum rate line corresponding to the maximum $\mathbf{C_{SUM}}$ . The following section is devoted to a more thorough analysis of these channel behaviors. SUM RATE MAXIMIZATION ===================== In this part, we consider the maximization problem of the two-user sum rate expressed as a function of the two variables $u_1$, $u_2$ subject to the power constraints $u_{i}=\left(g_{ii}\ p_i/\sigma^2\right)\leq \bar{P_i} $: $$\begin{aligned} \mathbf{C_{SUM}} &=& C_1 + C_2\nonumber\\ &=&\frac{1}{2}\log_{2}\left(1 + \frac{u_1}{1+a_{12}u_2} \right)\nonumber\\ && + \frac{1}{2}\log_{2}\left(1 + \frac{u_2}{1+a_{21}u_1} \right)\end{aligned}$$ It is found in [@gesbert] that the optimal power allocation $(u^{*}_{1},u^{*}_{2})$ to this problem is one of the possible following vectors: $(0,\bar{P_{2}})$, $(\bar{P_{1}},0)$ or $(\bar{P_{1}},\bar{P_{2}})$. The same result is found in [@charafeddine] using the geometric programming method. Following our rate region analysis in section II, we derive two different regions $\mathcal{A}$ and $\mathcal{B}$ (cf. Fig. 5) such that: 1. if the corner point $M\in \mathcal{A}$, then the optimal power allocation is $(\bar{P_1}, \bar{P_2})$; 2. if $M\in \mathcal{B}$, then the optimal power allocation is $(\bar{P_1}, 0)$ or $(0, \bar{P_2})$. Denoting $R^* =\max\left( R_{1}^{max}, R_{2}^{max}\right)$, where $R_{i}^{max}=\frac{1}{2}\log_2\left(1+\bar{P_i} \right)$, the regions $\mathcal{A}$ and $\mathcal{B}$ are separated by the straight line with equation $R_1 + R_2 =R^*$. $\mathcal{A}$ is the region above the separator straight line and $\mathcal{B}$ is the region below.\ Typically, since the point $M$ is reached for the rate vector $(R_{1}^{*}, R_{2}^{*})$, then[^1] the maximum sum rate $R_{sum}^{max}$ verifies $$R_{sum}^{max}\left\{\begin{array}{rlr} =& R^*, & \mbox{if} (R_{1}^{*}+ R_{2}^{*})\leq R^*\\ =& R_{1}^{*} + R_{2}^{*}, & \mbox{if} (R_{1}^{*}+ R_{2}^{*})> R^* \end{array} \right.$$ The Fig.\[sumcapa\] illustrates a case where the corner point $M \in \mathcal{A}$ and $(R_{1}^{*}+ R_{2}^{*})> R^*$, therefore the maximum sum rate $R_{sum}^{max}$ is reached for the power allocation $(\bar{P_1},\bar{P_2})$. THE THREE-USER CASE =================== When considering the three-user case, it is more convenient to write the relations between the SINR variables, $S_1$, $S_2$, $S_3$ and the SNR variables $u_1$, $u_2$, $u_3$ under the following form: $$\begin{aligned} u_1 &=& S_1\left(1+a_{12}u_2 + a_{13}u_3\right)\\ u_2 &=& S_2\left(1+a_{21}u_1 + a_{23}u_3\right)\\ u_3 &=& S_3\left(1+a_{31}u_1 + a_{32}u_2\right)\end{aligned}$$ which we rewrite as a linear system of unknowns $(u_1, u_2, u_3)$: $$\left(\begin{array}{ccc} 1 & -S_1a_{12} & -S_1a_{13}\\ -S_2a_{21} & 1 & -S_2a_{23}\\ -S_3a_{31} & -S_3a_{32} & 1 \end{array} \right)\times \left(\begin{array}{c} u_1\\ u_2\\ u_3 \end{array} \right)=\left(\begin{array}{c} S_1\\ S_2\\ S_3 \end{array} \right)$$ We can make use of the structure of the above $3\times 3$ matrix in order to make apparent the matrix $\mathbf{A}_2$ associated to the two-user problem: $$\left\{ \begin{array}{l} \mathbf{A}_3 =\left(\begin{array}{cc} \mathbf{A}_2 & -\mathbf{a} \\ -S_3\mathbf{b}^t & 1 \end{array}\right)\\ \mathbf{A}_2 =\left(\begin{array}{cc} 1 & -S_1a_{12} \\ -S_2a_{21} & 1 \end{array}\right)\\ \mathbf{a}=\left(\begin{array}{c} S_1a_{13}\\ S_2a_{23} \end{array}\right),\hspace{0.2cm} \mathbf{b}=\left(\begin{array}{c} a_{31}\\ a_{32} \end{array}\right) \end{array} \right.$$ The linear system of unknowns $u_1$, $u_2$, $u_3$ can now be written as: $$\left\{ \begin{array}{rll} \mathbf{A}_2\left(\begin{array}{c} u_1\\ u_2 \end{array} \right)-\mathbf{a}u_3 &=& \left(\begin{array}{c} S_1\\ S_2 \end{array} \right)\\ -S_3\mathbf{b}^t\left(\begin{array}{c} u_1\\ u_2 \end{array} \right)+u_3 &=& S_3 \end{array} \right.$$ After some manipulations, and assuming that $\mathbf{A}_2$ is invertible we can express $u_3$ as: $$u_3=S_3\times \frac{1+\mathbf{b}^t\mathbf{A}_{2}^{-1}\left(\begin{array}{c} S_1\\ S_2 \end{array} \right)}{1-S_3\mathbf{b}^t\mathbf{A}_{2}^{-1}\mathbf{a}}$$ From the constraint $u_3\leq \bar{P}_3$, we have, after some manipulations, a constraint on $S_3$ as a function of $S_1$ and $S_2$: $$S_3 \leq \phi_3(S_1,S_2)=\frac{\bar{P_3}}{1 + (a_{31},a_{32})\mathbf{A}_{2}^{-1}\left(\begin{array}{c} S_1(1+a_{13}\bar{P_3})\\ S_2(1+a_{23}\bar{P_3}) \end{array} \right)}$$ This is the equation of a surface in the three-dimensional space and it is worth noticing that when $S_1=0$ or $S_2=0$ the above upper bound becomes respectively equal to: $$\begin{aligned} S_3&\leq& \frac{\bar{P_3}}{1+a_{32}S_2(1+a_{23}\bar{P_3})}\\ S_3&\leq& \frac{\bar{P_3}}{1+a_{31}S_1(1+a_{13}\bar{P_3})}\end{aligned}$$ In which we recognize the bounds already obtained for the two-user case when the two users are respectively $(2, 3)$ and $(1,3)$. A geometric representation of the constraints on $S_3$, when respectively $S_1=0$ and $S_2=0$, is shown in the Fig.\[georepconst\]. As we also want to derive analogous relations for $S_1$ and $S_2$ we can make use of the invariance of the structure of the linear system under any permutation of the indexes $\{1, 2, 3\}$ to obtain the expressions (\[eqn\_dbl\_x\]). We shall denote these inequalities by $S_i\leq \phi_i(S_j, S_k)$ where $\{i, j, k\}$ is a permutation of the set $\{1, 2, 3\}$; with this notation the SINR region is the intersection of the three regions verifying respectively the three constraints: $$\begin{aligned} \mathcal{D}'&=&\mathcal{D}_1'\cap\mathcal{D}_2'\cap\mathcal{D}_3'\\ \mathcal{D}_1'&=&\left\{(S_1, S_2, S_3)| 0\leq S_1 \leq \phi_1(S_2, S_3) \right\}\nonumber\\ \mathcal{D}_2'&=&\left\{(S_1, S_2, S_3)| 0\leq S_2 \leq \phi_2(S_1, S_3) \right\}\nonumber\\ \mathcal{D}_3'&=&\left\{(S_1, S_2, S_3)| 0\leq S_3 \leq \phi_3(S_1, S_2) \right\}\nonumber\end{aligned}$$ $$\begin{aligned} \label{eqn_dbl_x} S_3 &\leq & \bar{P_3}\frac{1-a_{12}a_{21}S_1S_2}{1-a_{12}a_{21}S_1S_2 + S_1(1 + a_{13}\bar{P_3})(a_{31} + S_{2}a_{32}a_{21}) + S_2(1+ a_{23}\bar{P_3})(a_{32} + S_1a_{31}a_{12})}\\ S_2 &\leq & \bar{P_2}\frac{1-a_{13}a_{31}S_1S_3}{1-a_{13}a_{31}S_1S_3 + S_1(1 + a_{12}\bar{P_2})(a_{21} + S_{3}a_{23}a_{31}) + S_3(1+ a_{32}\bar{P_2})(a_{23} + S_1a_{21}a_{13})}\nonumber\\ S_1 &\leq & \bar{P_1}\frac{1-a_{32}a_{23}S_3S_2}{1-a_{32}a_{23}S_3S_2 + S_3(1 + a_{31}\bar{P_1})(a_{13} + S_{2}a_{12}a_{23}) + S_2(1+ a_{21}\bar{P_1})(a_{12} + S_3a_{13}a_{32})}\nonumber\end{aligned}$$ In the Fig.\[illustsinr\] we give a sketch of $\mathcal{D}'$ with the three sets of intersections on the faces of the positive quadrant. CONCLUSIONS =========== In this paper, we derived the analytical expressions of the SINR bounds for the two and three-user Gaussian interference channel, treating the interference as noise. The way we derive the three-user SINR region is very general as it allows deriving the $n$-user case provided we know the result of the $(n-1)$ user case. Some examples show that an increase in the efficiency of the channel use is possible, depending upon the channel gains: the sum capacity of two-user is greater than the max capacity of a user alone, at the expense of a slight decrease of each user capacity. We have compared this solution to a modified Han-Kobayashi inner bound [@chong]; the comparison is given in the Fig.\[caparegionscomparison\]. We see that, apart a dubious point due possibly to the limited accuracy of picking points in the original figure of [@chong] our capacity region contains the inner bound. Remains a question: our derivation of the capacity region does not involve any cooperation between the two users of the channel, we can expect that any techniques assuming partial knowledge of each user’s message will improve the capacity region, that means it will contain our capacity region. [1]{} A. Goldsmith, S. Jafar, I. Maric and S. Srinivasa. Breaking spectrum gridlock with cognitive radios: an information theoretic perspective. *Proceedings of the IEEE*, to appear 2008. A.B. Carleial. A case where interference does not reduce capacity.*IEEE Trans. Information Theory*, vol. IT-21, pp. 569-570, Sept. 1975. G. Kramer. Review of rate regions for interference channels.*Int. Zurich seminar on communications*,pp. 162-165, Feb. 2006. I.  Sason. On achievable rate regions for the Gaussian interference channel.*IEEE Trans. on Information Theory*, vol. 50, no. 6, pp. 1345 - 1356, June 2004. T. Han and K. Kobayashi. A new achievable rate region for the interference channel.*IEEE Trans. Information Theory*, vol. 27, no. 1, pp. 49-60, Jan. 1981. A. Gjendemsjo, D. Gesbert, G. E. Oien and S. G. Kiani.Optimal power allocation and scheduling for two-cell capacity maximization.*IEEE WiOpt 2006*. H. Mahdavidoost, M. Ebrahimi and A.K. Khandani. Characterization of Rate Region in Interference Channels with Constrained Power.*Proc. IEEE International Symposium on Information Theory (ISIT’07)*,pp. 2441-2445, Nice, France, June 24-29, 2007. A. S. Motahari and A. K. Khandani.Capacity Bounds for the Gaussian Interference Channel.*International Symposium on Information Theory (ISIT’08)*, Toronto, Canada. S. Shamai and A. D. Wyner.Information-theoretic considerations for symmetric, cellular, multiple-access fading channels.*IEEE Trans. Information Theory*, vol. 43, N°6, Part I: pp. 1877-1894, Part II: pp. 1895-1911, Nov. 1997. L. A. Imhof and R. Mathar.The geometry of the capacity region for CDMA systems with general power constraints.*IEEE Trans. Wireless Communications*, vol.4, N° 5,pp. 2040-2044, Sept. 2005. M. Charafeddine and A. Paulray.Sequential Geometric Programming for 2 X 2 Interference Channel Power Control.*IEEE Information Science and Systems, CISS’07*,41st Annual Conference,pp. 185-189, March 2007. H. F. Chong, H. K. Garg and H. El Gamal. On the Han-Kobayashi Region for the Interference Channel.*IEEE Trans. On Information Theory*, vol. 54, no 7, pp. 3188-3195, July 2008. H. SATO.The capacity of the Gaussian interference channel under strong interference.*IEEE Trans. Information Theory*, vol. IT-27, pp. 786-788, Nov. 1981. [^1]: The rate vector $(R_{1}^{*}, R_{2}^{*})$ is reached when each user transmits with his maximum permitted power.

--- abstract: 'We study the sensitivity of top pair production with six-fermion decay at the LHC to the presence and nature of an underlying $Z''$ boson, accounting for full tree-level Standard Model $t\bar{t}$ interference, with all intermediate particles allowed off-shell. We concentrate on the lepton-plus-jets final state and simulate experimental conditions, including kinematic requirements and top quark pair reconstruction in the presence of missing transverse energy and combinatorial ambiguity in jet-top assignment. We focus on the differential mass spectra of the cross section and asymmetry observables, especially demonstrating the use of the latter in probing the coupling structure of a new neutral resonance, in addition to cases in which the asymmetry forms a complementary discovery observable.' author: - 'Lucio Cerrito[^1]' - 'Declan Millar[^2]' - 'Stefano Moretti[^3]' - 'Francesco Spanò[^4]' bibliography: - 'references.bib' nocite: - '[@theatlascollaboration2014c]' - '[@hagiwara1992; @stelzer1994; @lepage1978]' title: 'Using asymmetry observables to discover and distinguish Z’ signals in top pair production with the lepton-plus-jets final state at the LHC' --- Introduction {#sec:introduction} ============ New fundamental, massive, neutral, spin-1 gauge bosons ($Z'$) appear ubiquitously in theories Beyond the Standard Model (BSM). The strongest limits for such a state generally exist for the $e^+e^-$ and $\mu^+\mu^-$ signatures, known collectively as Drell-Yan (DY). However, in addition to their importance in extracting the couplings to top quarks, resonance searches in the $t\bar{t}$ channel can offer additional handles on the properties of a $Z'$ due to uniquely available asymmetry observables, owing to the fact that (anti)tops decay prior to hadronisation and spin information is effectively transmitted to decay products. Their definition in $t\bar{t}$, however, requires the reconstruction of the top quark pair. In these proceedings we summarise our study of the sensitivity to the presence of a single $Z'$ boson at the Large Hadron Collider (LHC) arising from a number of generationally universal benchmark models (section \[sec:models\]), as presented in our recently submitted paper [@cerrito2016]. We simulate top pair production and six-fermion decay mediated by a $Z'$ with full tree-level SM interference and all intermediate particles allowed off-shell, with analysis focused on the lepton-plus-jets final state, and imitating some experimental conditions at the parton level (section \[sec:method\]). We assess the prospect for an LHC analysis to profile a $Z'$ boson mediating $t\bar{t}$ production, using the cross section in combination with asymmetry observables, with results and conclusions in section \[sec:results\] and \[sec:conclusions\], respectively. Models {#sec:models} ====== There are several candidates for a Grand Unified Theory (GUT), a hypothetical enlarged gauge symmetry, motivated by gauge coupling unification at approximately the $10^{16}$ GeV energy scale. $Z'$ often arise due to the residual U$(1)$ gauge symmetries after their spontaneous symmetry breaking to the familiar SM gauge structure. We study a number of benchmark examples of such models. These may be classified into three types: $E_6$ inspired models, generalised Left-Right (GLR) symmetric models and General Sequential Models (GSMs) [@accomando2011]. One may propose that the gauge symmetry group at the GUT scale is E$_6$. When recovering the SM, two residual symmetries U$(1)_\psi$ and U$(1)_\chi$ emerge, which may survive down to the TeV scale. LR symmetric models introduce a new isospin group, SU$(2)_R$, perfectly analogous to the SU$(2)_L$ group of the SM, but which acts on right-handed fields. This symmetry may arise naturally when breaking an SO$(10)$ gauge symmetry. We are particularly interested in the residual U$(1)_R$ and U$(1)_{B-L}$ symmetries, where the former is related to $T^3_R$, and $B$ and $L$ refer to Baryon and Lepton number, respectively. An SSM $Z'$ has fermionic couplings identical to those of the SM $Z$ boson, but is generically heavier. In the SM the $Z$ couplings to fermions are uniquely determined by well defined eigenvalues of the $T^3_L$ and $Q$ generators, the third isospin component and the Electro-Magnetic (EM) charge. For each class we may take a general linear combination of the appropriate operators and fix $g'$, varying the angular parameter dictating the relative strengths of the component generators, until we recover interesting limits. These models are all universal, with the same coupling strength to each generation of fermion. Therefore, as with an SSM $Z'$, the strongest experimental limits come from the DY channel. The limits for these models have been extracted based on DY results, at $\sqrt{s}=7$ and $8$ TeV with an integrated luminosity of $L=20$ fb$^{-1}$, from the CMS collaboration [@thecmscollaboration2015] by Accomando et al. [@accomando2016], with general consensus that such a state is excluded below $3$ TeV. Method {#sec:method} ====== Measuring $\theta$ as the angle between the top and the incoming quark direction, in the parton centre of mass frame, we define the forward-backward asymmetry: $$A_{FB}=\frac{N_{t}(\cos\theta>0)-N_t(\cos\theta<0)}{N_t(\cos\theta>0)+N_t(\cos\theta<0)}, \quad \cos\theta^* =\frac{y_{tt}}{|y_{tt}|}\cos\theta$$ With hadrons in the initial state, the quark direction is indeterminate. However, the $q$ is likely to carry a larger partonic momentum fraction $x$ than the $\bar{q}$ in $\bar{x}$. Therefore, to define $A^{*}_{FB}$ we choose the $z^*$ axis to lie along the boost direction. The top polarisation asymmetry ($A_{L}$), measures the net polarisation of the (anti)top quark by subtracting events with positive and negative helicities: $$A_{L}=\frac{N(+,+)+N(+,-)-N(-,-)-N(-,+)}{N(+,+)+N(+,-)+N(-,-)+N(-,+)}, \quad \frac{1}{\Gamma_l}\frac{d\Gamma_l}{dcos\theta_l}=\frac{1}{2}(1 + A_L \cos\theta_l),$$ where $\lambda_{t}$($\lambda_{\bar{t}}$) denote the eigenvalues under the helicity operator of $t$($\bar{t}$). Information about the top spin is preserved in the distribution of $\cos\theta_l$. We construct two dimensional histograms in $m_{tt}$ and $(\cos\theta_{l})$, and equate the gradient of a fitted straight line to $A_{L}$. In each of the models, the residual U$(1)'$ gauge symmetry is broken around the TeV scale, resulting in a massive $Z'$ boson. This leads to an additional term in the low-energy Lagrangian, from which we may calculate the unique $Z'$ coupling structure for each observable: $$\begin{aligned} \mathcal{L} &\supset g^\prime Z^\prime_\mu \bar{\psi}_f\gamma^\mu(f_V - f_A\gamma_5)\psi_f, \label{eq:zprime_lagrangian}\\ \hat{\sigma} &\propto \left(q_V^2 + q_A^2\right)\left((4 - \beta^2)t_V^2 + t_A^2\right),\\ A_{FB} &\propto q_V q_A t_V t_A,\\ A_{L} &\propto \left(q_V^2 + q_A^2\right)t_V t_A,\end{aligned}$$ where $f_V$ and $f_A$ are the vector and axial-vector couplings of a specific fermion ($f$). While a parton-level analysis, we incorporate restraints encountered with reconstructed data, to assess, in a preliminary way, whether these observables survive. The collider signature for our process is a single $e$ or $\mu$ produced with at least four jets, in addition to missing transverse energy ($E^{\rm miss}_{T}$). Experimentally, the $b$-tagged jet charge is indeterminate and there is ambiguity in $b$-jet (anti)top assignment. We solely identify $E^{\rm miss}_{T}$ with the transverse neutrino momentum. Assuming an on-shell $W^\pm$ we may find approximate solutions for the longitudinal component of the neutrino momentum as the roots of a quadratic equation. In order to reconstruct the event, we account for bottom-top assignment and $p_z^\nu$ solution selection simultaneously, using a chi-square-like test, by minimising the variable $\chi^2$: $$\chi^2 = \left(\frac{m_{bl\nu}-m_{t}}{\Gamma_t}\right)^2 + \left(\frac{m_{bqq}-m_{t}}{\Gamma_t}\right)^2, \label{eq:chi2}$$ where $m_{bl\nu}$ and $m_{bqq}$ are the invariant mass of the leptonic and hadronic (anti)top, respectively. In order to characterise the sensitivity to each of these $Z'$ models, we test the null hypothesis, which includes only the known $t\bar{t}$ processes of the SM, assuming the alternative hypothesis ($H$), which includes the SM processes with the addition of a single $Z'$, using the profile Likelihood ratio as a test statistic, approximated using the large sample limit, as described in [@cowan2011]. This method is fully general for any $n$D histogram, and we test both $1$D histograms in $m_{tt}$, and $2$D in $m_{tt}$ and the defining variable of each asymmetry to assess their combined significance. Results {#sec:results} ======= [0.494]{} ![Expected distributions for each of our observables of interest, with an integrated luminosity of $100$ fb$^{-1}$, at $\sqrt{s}=13$ TeV. The shaded bands indicate the projected statistical uncertainty.[]{data-label="fig:distinguishing"}]({mtt-r-gsm-ggqq-gazx-tt-bbllvv-2-4-5x10m-a-l100}.pdf "fig:"){width="\textwidth"} [0.494]{} ![Expected distributions for each of our observables of interest, with an integrated luminosity of $100$ fb$^{-1}$, at $\sqrt{s}=13$ TeV. The shaded bands indicate the projected statistical uncertainty.[]{data-label="fig:distinguishing"}]({mtt-r-glr-ggqq-gazx-tt-bbllvv-2-4-5x10m-a-l100}.pdf "fig:"){width="\textwidth"} [0.494]{} ![Expected distributions for each of our observables of interest, with an integrated luminosity of $100$ fb$^{-1}$, at $\sqrt{s}=13$ TeV. The shaded bands indicate the projected statistical uncertainty.[]{data-label="fig:distinguishing"}]({afb-r-gsm-ggqq-gazx-tt-bbllvv-2-4-5x10m-a-y0.5-l100}.pdf "fig:"){width="\textwidth"} [0.494]{} ![Expected distributions for each of our observables of interest, with an integrated luminosity of $100$ fb$^{-1}$, at $\sqrt{s}=13$ TeV. The shaded bands indicate the projected statistical uncertainty.[]{data-label="fig:distinguishing"}]({afb-r-glr-ggqq-gazx-tt-bbllvv-2-4-5x10m-a-y0.5-l100}.pdf "fig:"){width="\textwidth"} [0.494]{} ![Expected distributions for each of our observables of interest, with an integrated luminosity of $100$ fb$^{-1}$, at $\sqrt{s}=13$ TeV. The shaded bands indicate the projected statistical uncertainty.[]{data-label="fig:distinguishing"}]({al-r-gsm-ggqq-gazx-tt-bbllvv-2-4-5x10m-a-l100}.pdf "fig:"){width="\textwidth"} [0.494]{} ![Expected distributions for each of our observables of interest, with an integrated luminosity of $100$ fb$^{-1}$, at $\sqrt{s}=13$ TeV. The shaded bands indicate the projected statistical uncertainty.[]{data-label="fig:distinguishing"}]({al-r-glr-ggqq-gazx-tt-bbllvv-2-4-5x10m-a-l100}.pdf "fig:"){width="\textwidth"} Figure \[fig:distinguishing\] shows plots for the differential cross section, $A_{FB}^{*}$ and $A_{L}$. The statistical error is quantified for this luminosity assuming Poisson errors. The absent models, including all of the $E_6$ class, only produce an asymmetry via the interference term, which generally gives an undetectable enhancement with respect to the SM yield. The absence of a corresponding peak in either asymmetry offers an additional handle on diagnosing a discovered $Z'$. The cross section, profiled in $m_{tt}$, shows a very visible peak for all models. The GSM models feature a greater peak, and width, consistent with their stronger couplings, but the impact on the cross section is otherwise similar for both classes. Mirroring the cross section, the $A_{FB}^{*}$ distribution clearly distinguishes between the models and SM, with the difference in width even more readily apparent. The best distinguishing power over all the models investigated comes from the $A_{L}$ distribution, which features an oppositely signed peak for the GLR and GSM classes. To evaluate the significance of each asymmetry as a combined discovery observable we bin in both $m_{tt}$ and its defining variable. For $A_{FB}^*$, the asymmetry is calculated directly. Therefore, we divide the domain of $\cos\theta^*$ into just two equal regions. $A_L$ is extracted from the gradient of the fit to $\cos\theta_l$ for each mass slice, and we calculate the significance directly from this histogram. The final results of the likelihood-based test, as applied to each model, and tested against the SM, are presented in table \[tab:significance\]. The models with non-trivial asymmetries consistently show an increased significance for the 2D histograms compared with using $m_{tt}$ alone, illustrating their potential application in gathering evidence to herald the discovery of new physics. ------- ---------------- ---------- ----------------------------- --------------------------- Class U$(1)'$ $m_{tt}$ $m_{tt}$ & $\cos\theta^{*}$ $m_{tt}$ & $\cos\theta_l$ U$(1)_\chi$ $ 3.7$ - - U$(1)_\psi$ $ 5.0$ - - U$(1)_\eta$ $ 6.1$ - - U$(1)_S$ $ 3.4$ - - U$(1)_I$ $ 3.4$ - - U$(1)_N$ $ 3.5$ - - U$(1)_{R }$ $ 7.7$ $ 8.5$ $ 8.6$ U$(1)_{B-L}$ $ 3.6$ - - U$(1)_{LR}$ $ 5.1$ $ 5.6$ $ 5.8$ U$(1)_{Y }$ $ 6.3$ $ 6.8$ $ 7.0$ U$(1)_{T^3_L}$ $12.1$ $13.0$ $14.0$ U$(1)_{SM}$ $ 7.1$ $ 7.3$ $ 7.6$ U$(1)_{Q}$ $24.8$ - - ------- ---------------- ---------- ----------------------------- --------------------------- : Expected significance, expressed as the Gaussian equivalent of the $p$-value.[]{data-label="tab:significance"} Conclusions {#sec:conclusions} =========== We have investigated the scope of the LHC in accessing semileptonic final states produced by $t\bar t$ pairs emerging from the decay of a heavy $Z'$ state. We tested a variety of BSM scenarios embedding one such a state, and show that asymmetry observables can be used to not only aid the diagnostic capabilities provided by the cross section, in identifying the nature of a possible $Z'$ signal, but also to increase the combined significance for first discovery. While the analysis was performed at the parton level, we have implemented a reconstruction procedure of the (anti)tops that closely mimics experimental conditions. We have, therefore, set the stage for a fully-fledged analysis eventually also to include parton-shower, hadronisation, and detector reconstruction, which will constitute the subject of a forthcoming publication. In short, we believe that our results represent a significant phenomenological advancement in proving that charge and spin asymmetry observables can have a strong impact in accessing and profiling $Z'\to t\bar t$ signals during Run 2 of the LHC. This is all the more important in view of the fact that several BSM scenarios, chiefly those assigning a composite nature to the recently discovered Higgs boson, embed one or more $Z'$ state which are strongly coupled to top (anti)quarks [@barducci2012]. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge the support of ERC-CoG Horizon 2020, NPTEV-TQP2020 grant no. 648723, European Union. DM is supported by the NExT Institute and an ATLAS PhD Grant, awarded in February 2015. SM is supported in part by the NExT Institute and the STFC Consolidated Grant ST/L000296/1. FS is supported by the STFC Consolidated Grant ST/K001264/1. We would like to thank Ken Mimasu for all his prior work on $Z'$ phenomenology in $t\bar{t}$, as well as his input when creating the generation tools used for this analysis. Thanks also go to Juri Fiaschi for helping us to validate our tools in the case of Drell-Yan $Z'$ production. Additionally, we are very grateful to Glen Cowan for discussions on the statistical procedure, and Lorenzo Moneta for aiding with the implementation. [^1]: E-mail: [lucio.cerrito@cern.ch]{} [^2]: E-mail: [declan.millar@cern.ch ]{} [^3]: E-mail: [S.Moretti@soton.ac.uk]{} [^4]: E-mail: [francesco.spano@cern.ch]{}

--- abstract: 'We study the relation between stellar rotation and magnetic activity for a sample of $134$ bright, nearby M dwarfs observed in the Kepler Two-Wheel (K2) mission during campaigns C0 to C4. The K2 lightcurves yield photometrically derived rotation periods for $97$ stars ($79$ of which without previous period measurement), as well as various measures for activity related to cool spots and flares. We find a clear difference between fast and slow rotators with a dividing line at a period of $\sim 10$d at which the activity level changes abruptly. All photometric diagnostics of activity (spot cycle amplitude, flare peak amplitude and residual variability after subtraction of spot and flare variations) display the same dichotomy, pointing to a quick transition between a high-activity mode for fast rotators and a low-activity mode for slow rotators. This unexplained behavior is reminiscent of a dynamo mode-change seen in numerical simulations that separates a dipolar from a multipolar regime. A substantial number of the fast rotators are visual binaries. A tentative explanation is accelerated disk evolution in binaries leading to higher initial rotation rates on the main-sequence and associated longer spin-down and activity lifetimes. We combine the K2 rotation periods with archival X-ray and UV data. X-ray, FUV and NUV detections are found for $26$, $41$, and $11$ stars from our sample, respectively. Separating the fast from the slow rotators, we determine for the first time the X-ray saturation level separately for early- and for mid-M stars.' author: - | B. Stelzer$^{1}$[^1], M. Damasso$^2$, A. Scholz$^3$, S.P. Matt$^4$\ $^{1}$ INAF - Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy\ $^{2}$ INAF - Osservatorio Astrofisico di Torino, via Osservatorio 20, 10025 Pino Torinese, Italy\ $^{3}$ SUPA School of Physics & Astronomy, University of St. Andrews, North Haugh, St.Andrews KY169SS, United Kingdom\ $^{4}$ Department of Physics and Astronomy, University of Exeter, Physics Building, Stocker Road, Exeter, EX4 4QL, United Kingdom bibliography: - 'K2rotactV3.bib' date: 'Accepted 2016 August 2. Received 2016 August 1; in original form 2016 March 22' title: 'A path towards understanding the rotation-activity relation of M dwarfs with K2 mission, X-ray and UV data' --- \[firstpage\] stars: rotation – stars: activity – stars: flare – stars: late-type – ultraviolet: stars – X-rays: stars Introduction {#sect:intro} ============ Together with convection, rotation is the main driver of stellar dynamos and ensuing magnetic activity phenomena (e.g. Kosovichev et al. 2013). In a feedback mechanism, magnetic fields are responsible for the spin-evolution of stars: during part of the pre-main sequence phase the magnetic field couples the star to its accretion disk dictating angular momentum transfer [@Bouvier14.0] and during the main-sequence phase magnetized winds remove angular momentum leading to spin-down [@Kawaler88.0; @Matt15.0]. Rotation and magnetic fields are, therefore, intimately linked and play a fundamental role in stellar evolution. Magnetic field strength and topology can be measured through Zeeman broadening and polarization, respectively. Collecting the required optical high-resolution (polarimetric) spectroscopic observations is time-consuming, and each of these techniques can be applied only to stars with a limited range of rotation rates [e.g. @Donati09.0; @Vidotto14.1]. However, how the stellar dynamo and the spin-evolution are linked can be addressed by measuring both magnetic activity and rotation rate across evolutionary timescales. While the activity-age relation is a proxy for the evolution of the stellar dynamo, the rotation-age relation discriminates between models of angular momentum evolution. In a seminal work by [@Skumanich72.1] the age decay of both activity and rotation of solar-type stars was established by extrapolating between the age of the oldest known open cluster (600Myr) and the Sun (4.5Gyr). Unfortunately, stellar ages are notoriously difficult to assess. Therefore, the direct relation between rotation and activity -– observed first some decades ago [e.g. @Pallavicini81.1; @Vilhu84.1] –- has widely substituted studies which involve age-estimates. The early works cited above have used spectroscopic measurements as measure for stellar rotation ($v \sin{i}$), and carry intrinsic ambiguities related to the unknown inclination angle of the stars. Stellar rotation rates are best derived from the periodic brightness variations induced by cool star spots moving across the line-of-sight, which can be directly associated with the rotation period. In more recent studies of the rotation-activity connection, photometrically measured rotation periods have proven more useful than $v \sin{i}$ [e.g. @Pizzolato03.1; @Wright11.0]. Especially, for M dwarfs ‘saturation’ sets in at relatively small values of $v \sin{i}$ due to their small radii and it is hard to probe the slow-rotator regime with spectroscopic rotation measurements. Theory predicts a qualitative change of the dynamo mechanism at the transition into the fully convective regime [spectral type $\sim$M3; @Stassun11.0]. Fully convective stars lack the tachocline in which solar-like $\alpha\Omega$-dynamos originate. Alternative field generation mechanisms must be at work: a turbulent dynamo was proposed by [@Durney93.1] but it is expected to generate only small-scale fields, in contrast to recent results from Zeeman Doppler Imaging (ZDI) which have shown evidence for large-scale dipolar fields in some fully convective stars [@Morin08.0; @Morin10.0]. Current studies of field generation in the fully convetive regime are, therefore, concentrating on $\alpha^2$-dynamos [@Chabrier06.1]. While rotation has no influence on turbulent dynamo action, it is considered an important ingredient of mean-field $\alpha^2$-dynamos. This attributes studies of the rotation dependence of magnetic activity across the M spectral type range a crucial meaning for understanding fully convective dynamos. Moreover, while improved spin-down models based on stellar wind simulations have been developed for solar-type stars [@Gallet13.0], angular momentum evolution models of M stars are still controversial [@Reiners12.3]. Therefore, for the most abundant type of stars in our Galaxy, both the characteristics of the stellar dynamo and the angular momentum evolution are still widely elusive. Rotation-activity studies have been presented with different diagnostics for activity, the most frequently used ones being H$\alpha$ and X-ray emission. While H$\alpha$ measurements are available for larger samples, especially thanks to surveys such as the [*Sloan Digital Sky Survey*]{} [e.g. @West04.1], X-ray emission was shown to be more sensitive to low activity levels in M dwarfs [@Stelzer13.0]. The samples for the most comprehensive rotation-activity studies involving X-ray data have been assembled from a literature compilation, providing a large number of stars, at the expense of homogeneity. [@Wright11.0] discuss a sample of more than $800$ late-type stars (spectral type FGKM). However, the rotation-activity relation is not studied separately for M stars, possibly due to a strong bias towards X-ray luminous stars which affects especially the M stars as seen from their Fig. 5. Overall, the lack of unbiased overlapping samples with known rotation period and X-ray activity level has left the X-ray - rotation relation of M stars nearly unconstrained [see bottom right panels of Fig.5 and 6 in @Pizzolato03.1]. Studies with optical emission lines (H$\alpha$, Ca[ii]{}H&K) as activity indicator have for convenience mostly been coupled with $v \sin{i}$ as rotation measure because both parameters can be obtained from the same set of spectra [@Browning10.0; @Reiners12.1]. Only lately has it become possible to combine H$\alpha$ data with photometrically measured M star rotation periods, since a larger sample of periods have become available from ground-based planet transit search programs [@West15.0]. M dwarfs have not yet reached a common rotational sequence even at Gyr-ages, suggesting weaker winds and longer spin-down timescales as compared to solar-like stars [@Irwin11.0]. The old and slowly rotating M dwarfs generally have low variability amplitudes resulting from reduced spot coverage and long rotation periods (up to months). From the ground, significant numbers of field M dwarf rotation periods have recently been measured [@Newton16.0]. However, the sample of their study comprises only very low-mass stars ($R_* \leq 0.33\,R_\odot$) and seems to be incomplete in terms of the period detection efficiency [@Irwin11.0]. The Kepler mission [@Borucki10.0] with its ability to provide high-precision, long and uninterrupted photometric lightcurves has led to the detection of rotation periods in $> 2000$ field M dwarfs [@Nielsen13.0; @McQuillan13.0; @McQuillan14.0], a multiple of the number known before. Interesting findings of this Kepler-study are (i) the evidence for a bimodal distribution of rotation periods for M dwarfs with $P_{\rm rot} = 0.4$...$70$d and (ii) the fact that the envelope for the slowest observed rotation periods shifts towards progressively larger periods for stars with mass below $\sim 0.5\,M_\odot$. How these features in the rotational distribution are connected to stellar activity has not yet been examined. Most of the Kepler stars are too distant for detailed characterization in terms of magnetic activity diagnostics. However, the Kepler Two-Wheel (K2) mission is ideally suited to study both rotation and activity for nearby M stars. Since March 2014, with its two remaining reaction wheels, the Kepler spacecraft is restricted to observations in the ecliptic plane changing the pointing direction every $\sim 80$d [@Howell14.0]. With special data processing correcting for the spacecraft’s pointing drift, the photometric precision of K2 is similar to that achieved by the preceding fully functional Kepler mission [@Vanderburg14.0]. A great number of field M dwarfs have been selected as K2 targets with the goal of detecting planet transits. Several lists of planet candidates have already been published [e.g. @ForemanMackey15.0; @Montet15.0; @Vanderburg16.0], and some interesting planet systems have already been validated, including objects from the target list of this study (see Sect \[subsect:results\_planethosts\]). In our program to study the M star rotation-activity connection we limit the sample to nearby, bright M stars which provide the largest signal-to-noise in the K2 lightcurves and are most likely to be detectable at the high energies that are the best probes of magnetic activity. We derive from the K2 mission data both rotation periods and various diagnostics of magnetic activity, and we combine this with X-ray and UV activity from past and present space missions ([*ROSAT*]{}, [*XMM-Newton*]{}, [*GALEX*]{}). As mentioned above, X-ray wavelengths have proven to be more sensitive to low activity levels in M dwarfs than optical emission lines. Moreover, both X-rays and UV photons are known to have a strong impact on close-in planets, providing another motivation for characterizing the high-energy emission of these stars. Given the high occurrence rate of terrestrial planets in the habitable zone of M dwarfs [$\sim 50$% according to @Kopparapu13.0], a substantial number of the stars we survey may soon be found to host potentially habitable worlds. The importance of stellar magnetic activity for exoplanet studies is twofold. First, star spots and chromospheric structures introduce noise in measured radial velocity curves, so-called RV ‘jitter’, which depends strongly on the properties of the spots [@Andersen15.0]. The spectra collected to perform radial velocity measurements can also be used to model starspots [see e.g. @Donati15.0]. However, since it is an impossible task to measure the spot distribution for every potentially interesting star, relations between star spot characteristics and other activity diagnostics such as UV or X-ray emission – if applied to statistical samples – can provide useful estimates of the expected RV noise. Secondly, as mentioned above, the stellar X-ray and UV emission is crucial for the evolution and the photochemistry of planet atmospheres. While the magnetic activity of the star may erode the atmospheres of planets formed in close orbits [e.g. @Penz08.0], it may by the same effect remove the gaseous envelopes of planets migrated inward from beyond the snow line and render them habitable [@Luger15.0]. Until recently, photochemical models for planets around M dwarfs relied exclusively on the observed UV properties of a single strongly active star, ADLeo [@Segura05.0]. Lately, [@Rugheimer15.0] have modeled the effect of an M star radiation field on exoplanet atmospheres based on the Hubble Space Telescope (HST) UV spectra of six exoplanet host stars [@France13.0]. These stars are apparently only weakly active, as none of them displays H$\alpha$ emission. Yet their HST spectra show hot emission lines proving the presence of a chromosphere and transition region. The lower limit of the chromospheric UV flux and its dependence on stellar parameters has not been constrained so far. Similarly, on the high end of the activity range, with exceptions such as ADLeo [e.g. @SanzForcada02.1; @Crespo-Chacon06.0], the frequencies and luminosities of flares on M dwarfs are still largely unknown. There has been significant recent progress in studies of M dwarf flares based on data from the main Kepler mission [@Ramsay13.0; @Hawley14.0; @Davenport14.0; @Lurie15.0]. The time resolution of $1$min obtained in the Kepler short-cadence data proves essential for catching small events, adding to the completeness of the observed flare distributions and enabling the examination of flare morphology. The drawback is that these results are limited to individual objects or a very small group of bright stars. The K2 mission gives access to much larger samples of bright M dwarfs, for which we can examine the relation between flaring and rotation in a statistical way, albeit at lower cadence. In this work we establish, to our knowledge for the first time, a direct connection between white-light flaring and stellar rotation rate. As described above, the sample selection is the key to success in constraining the rotation-activity relation of M dwarfs. We present our sample in Sect. \[sect:sample\]. In Sect. \[sect:stepar\] we derive the stellar parameters. This is necessary in order to investigate the dependence of rotation and activity on effective temperature ($T_{\rm eff}$) and mass ($M_*$), and to compute commonly used activity indices which consist of normalizing the magnetically-induced emission (X-ray, UV, etc.) to the bolometric luminosity. We then describe the analysis of K2 data involving the detection of flares and rotation periods (Sect. \[sect:k2\_analysis\]), archival X-ray (Sect. \[sect:xray\_analysis\]) and UV (Sect. \[sect:uv\_analysis\]) data. We present our results in Sect. \[sect:results\]. The implications are discussed in Sect. \[sect:discussion\], and we provide a summary in Sect. \[sect:summary\]. Sample {#sect:sample} ====== This work is based on all bright and nearby M dwarfs from the Superblink proper motion catalog by @Lepine11.0 [henceforth LG11] observed within the K2 mission’s campaigns C0....C4. The Superblink catalog comprises an All-Sky list of 8889 M dwarfs (spectral type K7 to M7) brighter than $J=10$, within a few tens of parsec. Many other programs focusing on M stars are carried out within the K2 mission, and rotation periods have been measured for more than a thousand M stars during the main Kepler mission [@McQuillan13.0]. However, a careful sample selection comprising stars with already known or easily accessible magnetic activity characteristics is mandatory to nail down the rotation-activity relation. The majority of the Kepler stars are too distant ($> 200$pc) and, therefore, too faint for the [*ROSAT*]{} All-Sky Survey, the main source for X-ray studies of widely dispersed samples. The proper-motion-selected M stars of the LG11 catalog are much closer and consequently brighter, facilitating the detection of both rotation periods and X-ray and UV emission. A total of $134$ Superblink M dwarfs have been observed in K2 campaigns C0...C4. Henceforth we will refer to these objects as the “K2 Superblink M star sample". The target list is given in Table \[tab:targettable\]. We list the identifier from the EPIC catalog, the campaign in which the object was observed, the designation from the [*Third Catalog of Nearby Stars*]{} [CNS3; @Gliese91.1], magnitudes in the Kepler band and further parameters, the calculation of which is described in the next section. Fundamental stellar parameters {#sect:stepar} ============================== We derive physical parameters of the K2 Superblink M stars (effective temperature, mass, radius, and bolometric luminosity) by adopting empirical and semi-empirical calibrations from [@Mann15.0], which are based on the color indices *V$-$J* and *J$-$H*, and on the absolute magnitude in the 2MASS $K$ band, *M*$\rm_{K_{S}}$. The calibrations of [@Mann15.0] are valid for dwarf stars, and can be expected to hold for the K2 Superblink M star sample which has been cleaned by LG11 from contaminating giants. Due to a press error some wrong values appeared in the tables of [@Mann15.0]. We use here the correct values reported in the erratum[^2]. Stellar magnitudes and their uncertainties are obtained from the UCAC4 catalog [@Zacharias13.0] which provides 2MASS near-IR photometry and $V$ band magnitudes from [*The AAVSO Photometric All Sky Survey (APASS)*]{}; [@Henden14.0]. These latter ones are more accurate and have significantly better precision than the $V$ band magnitudes given in LG11. For the $6$ stars with no $V$ magnitude in UCAC4, we found measurements in Data Release9 of the APASS catalog[^3]. We derive an empirical linear calibration to calculate *M*$\rm_{K_{S}}$ for our sample, using a list of $1,078$ M dwarfs with apparent *K*$\rm_{S}$ magnitude from UCAC4 and trigonometric parallax in LG11. This allows us to obtain estimates for $M_{\rm K_s}$ independent on the trigonometric parallax which is available for only $27$ stars in our sample. The best linear least-squares fit to the data is obtained through a Monte-Carlo analysis. This approach provides more realistic errors than simple least-squares fitting because the uncertainties are derived from posterior distributions of the parameters and take into account all the errors affecting the measurements. Specifically, we generate $10,000$ synthetic samples (each composed of $1,078$ stars) drawing $V-J$ and $M_{\rm K_S}$ randomly from 2D normal distributions with mean equal to the observed values and standard deviation (henceforth STD) equal to the uncertainties. We then fit to each of the $10,000$ representations a straight line with the IDL[^4] `FITEXY` routine, assuming for each simulated point the original errors in both variables. The best-fit relation is then defined by the median values and standard deviations of the [*a posteriori*]{} Monte-Carlo distribution for the coefficients in the linear fit, given by $$M_{\rm K_S} = 0.49(\pm 0.02) + 1.539(\pm 0.006) \cdot (V-J)$$ The residuals of this solution, which is applicable in the range $1.54 < V-J < 6.93$, show a rms of $0.56$mag. In Fig. \[fig:pxphoto\_calibration\] we show this relation overplotted on the observed data. All other stellar parameters and their uncertainties are calculated in the same manner through a Monte-Carlo analysis. In particular, the stellar effective temperatures ($T_{\rm eff}$) are obtained from the calibration relation which uses $V-J$ and $J-H$ (Eq. $7$ in @Mann15.0), while radii ($R_*$) and masses ($M_*$) are calculated from relations with *M*$\rm_{K_{S}}$ (Eqs. $4$ and $10$ in @Mann15.0, respectively), and the bolometric correction $BC_{\rm K}$ is derived through a third-degree polynomial with $V-J$ as independent variable [presented in Table 3 of @Mann15.0]. Thus, we first generate for each star a sample of $10,000$ synthetic $V-J$, $J-H$, and $M_{\rm K_S}$ datasets drawn from normal distributions with mean and sigma equal to the observed value and its error. Then we apply to each star the above-mentioned calibrations from [@Mann15.0]. The best estimate of each parameter ($T_{\rm eff}$, $R_*$, $M_*$ and $BC_{\rm K}$) is then obtained as the median value of the corresponding [*a posteriori*]{} distribution, with its standard deviation assumed as the uncertainty. To provide conservative estimates of the stellar parameters, the uncertainties representing the scatter of the relations of @Mann15.0 [see Tab. 1, 2, and 3 therein] are propagated into the Monte-Carlo process. Specifically, for $T_{\rm eff}$ we consider the scatter in the difference between the predicted and the spectroscopically observed temperature ($48$K), and the typical uncertainty on the spectroscopic value of $T_{\rm eff}$ ($60$K) adding both in quadrature, while for $BC_{\rm K}$ we consider the uncertainty of $0.036$mag. These additional uncertainties are taken into account in the Monte-Carlo analysis when drawing randomly the samples. For radius and mass, [@Mann15.0] provide relative uncertainties of $2.89$% and $1.8$%, respectively. These values are calculated from the median values of our posterior distributions for $R_*$ and $M_*$ and are then added in quadrature to their standard deviations. [@Mann15.0] argue that some of the above-mentioned relations for the stellar parameters can be improved by including an additional term involving metallicity (\[Fe/H\]). We found \[Fe/H\] measurements in the literature [@Newton14.0] for only $6$ stars from the K2 Superblink M star sample, and we verified for these objects that the radii and temperatures derived by taking account of \[Fe/H\] (Eqs. 5 and 6 in @Mann15.0) are compatible with our estimates described above. From $BC_{\rm K}$ and *M*$\rm_{K_{S}}$ we calculate the absolute bolometric magnitudes of our sample, which are then converted into luminosities assuming the absolute bolometric magnitude of the Sun is $M_{\rm bol,\odot}=4.7554$. We note, that the distances we infer from our $M_{\rm K_s}$ values and the observed $K_s$ magnitudes are systematically larger, on average by about $\sim 25\,\%$, than the photometric distances presented by LG11 for the same stars. For the $27$ stars with trigonometric parallax in the literature [LG11, @Dittmann14.0] our newly derived photometric distances are in excellent agreement with the trigonometric distances. In the near future, [*Gaia*]{} measurements will provide the ultimate and accurate distances for all K2 Superblink M stars. In the meantime, as corroborated by the comparison to trigonometric parallaxes, our estimates, which are based on the most accurate photometry available to date, can be considered as a fairly reliable guess on the distances. All stars in the K2 Superblink M star sample have a photometric estimate of the spectral type in LG11, based on an empirical relation of spectral type with $V-J$ color which was calibrated with SDSS spectra. Since we use here the higher-precision UCAC4 $V$ band magnitudes, for consistency with our calculation of the other stellar parameters, we derive an analogous relation between $V-J$ and spectral type. To this end, we make use of $1,173$ stars classified as K7 or M-type dwarfs by [@Lepine13.0] based on spectroscopy. We group the stars in bins of $0.5$ spectral subclasses, with K7 corresponding to $-1$, M0 to $0$, and so on until M$4.5$, which is the last sub-type for which we have enough stars in the calibration sample for a useful fit. We calculate the mean and standard deviation of $V-J$ for each spectral type bin, and notice that the data can be fitted with a combination of two straight lines for the ranges \[K7,M2\] and \[M2,M4.5\] (see Fig. \[fig:sptypephoto\_calibration\]). Our fit, performed through a Monte-Carlo procedure as described above, results in the relations $$\begin{aligned} V - J = 2.822( \pm 0.067) + 0.285( \pm 0.061) \cdot SpT \label{eq:vminj_spt_early} \\ V - J = 2.53( \pm 0.29) + 0.432( \pm 0.093) \cdot SpT \label{eq:vminj_spt_late} \end{aligned}$$ which are valid for $2.5 \leqslant V-J \leqslant 3.4$ and $3.4 \leqslant V-J \leqslant 4.5$, for the hotter (Eq. \[eq:vminj\_spt\_early\]) and cooler (Eq. \[eq:vminj\_spt\_late\]) spectral types respectively. We use this calibration to classify the K2 Superblink M star sample, by rounding the results of the linear relations to the closest spectral sub-type. Nine K2 Superblink M stars have $V-J$ colors slightly beyond the boundaries for which we calibrated Eqs. \[eq:vminj\_spt\_early\] and \[eq:vminj\_spt\_late\] and we extrapolate the relations at the ends to spectral types K5 and M5, respectively. No star deviates by more than $0.5$ spectral subclasses from Eqs. \[eq:vminj\_spt\_early\] and \[eq:vminj\_spt\_late\]. The spectroscopically determined spectral types from the literature, which are available for roughly three dozens of the K2 Superblink M stars, are in excellent agreement with our values [see @Reid04.0; @Reiners12.1; @Lepine13.0]. In Table \[tab:targettable\] we provide the photometry (Kepler magnitude $K_{\rm p}$, $V$, $J$, and $K_{\rm s}$), the distances obtained from the absolute $K$ band magnitude, the fundamental parameters ($M_*$, $R_*$, $\log{L_{\rm bol}}$ and $T_{\rm eff}$) and the spectral type (SpT) derived as described above. The few stars with $M_{\rm K_s}$ slightly more than $3\,\sigma$ smaller than the lower boundary of the calibrated range ($4.6 < M_{\rm K_s} < 9.8$) are flagged with an asterisk in Table \[tab:targettable\]. Stars for which the K2 photometry – and in some cases also the optical/IR photometry used by us to calculate the stellar parameters – comprises a potential contribution from a close binary companion are discussed in detail in the Appendix \[sect:appendix\_bin\]. These stars are also highlighted in Table \[tab:targettable\] and flagged in all figures where relevant. The Gl852AB binary is represented in our target list by two objects (EPIC206262223 and EPIC206262336) but they are not resolved in the K2 aperture[^5], i.e. only the combined lightcurve of both stars is at our disposition. We compute the stellar parameters for both components in the binary using the individual $V$ magnitudes from [@Reid04.0]; then we assign the rotational parameters and the X-ray/UV emission to the brighter, more massive star (EPIC206262336) and we do not consider the secondary (EPIC206262223) any further. The distributions of spectral type and mass for the K2 Superblink M star sample are shown in Fig. \[fig:histo\_spt\_mass\]. Covering spectral type K5 to M5 (masses between about $0.2$ and $0.9\,M_\odot$), this is an excellent database for investigating the connection between rotation and activity across the fully convective boundary (SpT $\sim$M3/M4). K2 data analysis {#sect:k2_analysis} ================ We base our analysis of K2 time-series mostly on the lightcurves made publicly available by A.Vanderburg [see @Vanderburg14.0 and Sect. \[subsect:k2\_analysis\_prep\]]. We use the “corrected" fluxes in which the features and trends resulting from the satellite pointing instability have been eliminated. All stars of the K2 Superblink M star sample have been observed in long-cadence (LC) mode with time-resolution of $\Delta t_{\rm LC} = 29.4$min. Nine stars have in addition short-cadence (SC) data available ($\Delta t_{\rm SC} = 1$min). In the following, where not explicitly stated, we refer to the LC data. Our analysis comprises both the measurement of rotation periods and an assessment of photometric activity indicators. In particular, the identification of flares is of prime value both for activity studies and for obtaining a “cleaned" lightcurve allowing to perform more accurate diagnostics on the rotation cycle, e.g. its amplitude. The main limitation of the LC data is the difficulty in identifying short-duration flares, as a result of poor temporal resolution combined with the presence of some residual artefacts from instrumental effects in the lightcurves that have not been removed in the K2 data reduction pipeline. However, in this work we aim at elaborating trends between activity and rotation, and for this purpose completeness of the flare sample is less important than having statistically meaningful numbers of stars. Rotation and activity diagnostics are determined with an iterative process in which we identify “outliers" in the K2 lightcurves. This involves removing any slowly varying signal by subtracting a smoothed lightcurve from the original data. The appropriate width of the boxcar in the smoothing process depends on the time-scale of the variation to be approximated, i.e. on the length of the rotational cycle. Therefore, we start the analysis with a first-guess period search on the original, corrected lightcurve. We use three methods to determine rotation periods which are laid out in Sect. \[subsect:k2\_analysis\_period\]. Before presenting the details of our period search we describe how we prepare the lightcurves and how we extract the flares and “clean" the corrected lightcurves further, thus removing both astrophysical flare events and residual noise from the data reduction. Data preparation {#subsect:k2_analysis_prep} ---------------- We download the lightcurves reduced and made publicly available by A. Vanderburg[^6]. The data reduction steps are described by [@Vanderburg14.0]. In short, the authors extracted raw photometry from K2 images by aperture photometry. The variability in the resulting lightcurves is dominated by a zigzag-like pattern introduced by the instability of the satellite pointing and its correction with help of spacecraft thruster fires taking place approximately every $6$h. This artificial variability can be removed by a “self-flat fielding" process described in detail by [@Vanderburg14.0]. We base our analysis on these “corrected" or “detrended" lightcurves to which we apply some additional corrections described below. Upon visual inspection of each individual corrected lightcurve we notice some flux jumps. As explained by A.Vanderburg in his data release notes[^7] such offsets can arise due to the fact that he divides the lightcurves in pieces and performs the data reduction separately on each individual section. In stars with long-term variations these offsets are clearly seen to be an artefact of the data reduction, and we remove them by applying a vertical shift to the lightcurve rightwards of the feature. Note that, since the absolute fluxes are irrelevant for our analysis it does not matter which side is used as the baseline for the normalization. While such flux jumps are evident in lightcurves with slow variations, for stars with short periods it is much more difficult to identify such systematic offsets and even if they are identified it is impossible to perform the normalization without [*a priori*]{} knowledge of the (periodic) variation pattern. However, since such short-period lightcurves comprise many rotational cycles, the period search is much less sensitive to such residual artifacts than it is for long-period lightcurves. In a second step, we remove all cadences in which the satellite thrusters were on (and the telescope was moving). The thruster fires are shorter than the cadence of observations in LC mode such that each of the corresponding gaps regards a single data point. Several lightcurves have spikes and decrements produced by incomplete background removal or individual null values among the fluxes. We identify such obvious artefacts by visual inspection of each individual lightcurve and remove the respective data points. We then fill all gaps in the K2 lightcurves, i.e. all data points separated by multiples of $\Delta t_{\rm LC}$, by interpolation on the neighboring data points. Evenly spaced data is required for the auto-correlation function, one of the methods we use for the period search (see Sect. \[subsubsect:k2\_analysis\_period\_detrended\]). We add Gaussian noise to the interpolated data points. To avoid that the width of the distribution from which the errors are drawn is dominated by the rotational variation we use only the nearest data points to the right and to the left of the gap to define mean and sigma of the Gaussian. Identification of flares {#subsect:k2_analysis_flares} ------------------------ Then we start the iterative flare search and cleaning process. Our approach is similar to the methods presented in previous systematic Kepler flare studies [@Hawley14.0 and subsequent papers of that series]. Specifically, our procedure consists in (i) boxcar smoothing of the lightcurve, (ii) subtraction of the smoothed from the original lightcurve (i.e. removal of the rotational signal), and (iii) flagging and removal of all data points which deviate by more than a chosen threshold from the subtracted curve. We repeat this procedure three times with successively smaller width of the boxcar. Subsequently, the removed cadences are regenerated by interpolation and addition of white noise as described above. This provides a lightcurve that is free from flares (henceforth referred to as the “cleaned" lightcurve). When subtracted from the original corrected data, the result is a flat lightcurve (henceforth referred to as the “flattened" lightcurve) in which the rotational variation has been removed and the dominating variations are flares, eclipses and artefacts. A significant fraction of the data points that have been removed in the above $\sigma$-clipping process are isolated cadences. Such events are found both as up- and downward excursions in the flattened lightcurves. The number of upward outliers is for most lightcurves much larger than the number of downward outliers, suggesting that many of these events are genuine flares. However, we assume here a conservative approach aimed at avoiding counting spurious events as flare. Therefore, we select all groups of at least two consecutive upwards deviating data points as flare candidates. In practice, this means that the minimum duration of the recognized flares is $\sim 1$hr (two times the cadence of $29.4$min). Note, that as a result of the sigma-clipping, all flare peaks ($F_{\rm peak}$) have a minimum significance of $3\,\sigma$, as measured with respect to the mean and standard deviation of the flattened lightcurve from which outliers have been removed, which is defined and further discussed in Sect. \[subsect:k2\_analysis\_noise\]. Finally, we require that $F_{\rm peak}$ must be at least twice the flux of the last of the data points defining the flare ($F_{\rm last}$). As shown below, this last criterion removes “flat-topped" events from our list of bona-fide flares which we trust less than “fast-decay" events given the possibility of residual artifacts from the data acquisition and reduction. A zoom into two examples of LC lightcurves with flares is shown in Fig. \[fig:flares\] and illustrates our flare search algorithm. The lower panel shows the original, detrended lightcurve and overlaid (in red) the smoothed lightcurve. The upper panel shows the result from the subtraction of these two curves, i. e. the flattened lightcurve. We highlight data points identified as outliers (open circles), and data points that belong to bona-fide flares (filled circles). The example on the right demonstrates the inability of recognizing short flares with our detection procedure. Short-cadence data from the main Kepler mission have shown that many flares on active M stars are, in fact, significantly shorter than one hour [see e.g. @Hawley14.0]. SC lightcurves are available for $9$ stars from the K2 Superblink M star sample. The analysis of SC lightcurves will be described elsewhere. In this work we use the SC data only as a cross-check on the quality of our flare search criteria applied to the LC data (see below and Fig. \[fig:lc\_sc\_lcs\_flares\]). We recall that we aim at a conservative approach, avoiding at best possible spurious events in the flare sample, because our aim is to study trends with rotation. To summarize, the parameters of our flare search algorithm are (i) the width of the boxcar \[adapted individually according to the first-guess period\], (ii) the threshold for outliers identified in the $\sigma$-clipping process \[adopted to be $3\,\sigma$\], (iii) the minimum number of consecutive data points defining a flare \[$2$\], and (iv) the flux ratio between the flare peak bin and the last flare bin \[$F_{\rm peak}/F_{\rm last} \geq 2$\]. The values for these parameters have been chosen by testing various combinations of criteria (i) - (iv) with different parameter values and comparing the results to a by-eye inspection of the “flattened" lightcurves. In particular, criterion (iv) is introduced after a comparison of LC and SC lightcurves which shows that, generally, the LC flare candidates correspond to analogous features in the SC data but in some cases the features are very different from the canonical flare shape (characterized by fast rise and exponential decay). Fig. \[fig:lc\_sc\_lcs\_flares\] demonstrates that with criterion (iv) we de-select such broad events from the list of bona-fide flares: Two flare candidates according to criterion (i) - (iii) are shown; the event on the left panel is a bona-fide flare according to criterion (iv) while the event on the right does not fullfill $F_{\rm peak}/F_{\rm last} \geq 2$. Residual variability and photometric noise {#subsect:k2_analysis_noise} ------------------------------------------ In Fig. \[fig:kp\_stddev\] we show the standard deviations of the flattened lightcurves, $S_{\rm flat}$, for two cases: including and excluding the data points identified as outliers. The ‘outliers’ comprise flares, transits or eclipses, and artefacts from the data reduction. Therefore, for the case without outliers (red circles) the standard deviation is calculated on the residual lightcurve from which the known astrophysical sources of variability have been removed, and it can be expected to represent the noise level in our data. In Fig. \[fig:kp\_stddev\] we compare our standard deviations $S_{\rm flat}$ to the estimated precision of K2 lightcurves provided for campaigns C0 and C1 in the data release notes of A.Vanderburg (see footnote to Sect. \[subsect:k2\_analysis\_prep\]). That estimate represents the $6$hr-precision based on a sample of cool dwarfs that is not clearly specified. Our $S_{\rm flat}$ measurements suggest a somewhat lower precision for the K2 Superblink M star sample. This might be due to differences in the definitions. Vanderburgs’s $6$hr-precisions are medians for their sample and the scatter among their stars is much larger than the factor two difference with our $S_{\rm flat}$ values. Also, we measure the standard deviation on the full lightcurve while Vanderburg’s precisions are based on a running $6$hr mean. An alternative explanation for the apparently different photometric precisions could lie in different activity levels of the two samples, implying residual fluctuations of astrophysical origin in our “noise". In fact, in Sect. \[subsect:results\_noise\] we present evidence that $S_{\rm flat}$ comprises an astrophysical signal. Overall, our analysis presented in Fig. \[fig:kp\_stddev\] confirms the high precision achieved in K2 lightcurves with the detrending method applied by A.Vanderburg. ![Standard deviation for the lightcurves of the Superblink stars after “flattening" by removal of the rotational signal as described in Sect. \[subsect:k2\_analysis\_flares\]. $K_{\rm p}$ is the magnitude in the Kepler band. $S_{\rm flat}$ is calculated for two data sets: the full lightcurve (black circles) and the lightcurve without all data points that were identified as outliers during the clipping process (red circles). Horizontal lines represent the $6$hr-precisions for C0 and C1 calculated by A.Vanderburg (see footnote to Sect.\[subsect:k2\_analysis\_prep\]) for a sample of cool dwarfs drawn from different K2 Guest Observer programs. Stars with a possible contribution in the K2 photometry from an unresolved binary companion are high-lighted with large annuli.[]{data-label="fig:kp_stddev"}](./f6.pdf){width="8.5cm"} Period search {#subsect:k2_analysis_period} ------------- We explore multiple approaches to measure rotation periods on the K2 data. ### Period search on detrended lightcurves {#subsubsect:k2_analysis_period_detrended} We apply standard time-series analysis techniques, the Lomb Scargle (LS) periodogram and the auto-correlation function (ACF), to the detrended K2 lightcurves made publicly available [see @Vanderburg14.0]. As mentioned in Sect. \[sect:k2\_analysis\], as a first step we perform the period search directly on the corrected version of the downloaded lightcurves with the purpose of adapting the boxcar width in the course of the search for flares. We then repeat the period search on the “cleaned" lightcurves obtained after the $\sigma$-clipping process and the regeneration of the missing data points through interpolation, i.e. after removal of the flares and other outliers. The analysis is carried out in the IDL environment using the [scargle]{} and [a\_correlate]{} routines. Periodograms and ACFs have already been used successfully to determine rotation periods in Kepler data [e.g. @McQuillan13.0; @Nielsen13.0; @Rappaport14.0; @McQuillan14.0]. As a cross-check on our procedure, we have downloaded Kepler lightcurves from the [*Mikulski Archive for Space Telescopes*]{} (MAST)[^8] for some M stars from the [@McQuillan13.0] sample and we have verified that we correctly reproduce the published periods. Following [@McQuillan13.0], in our use of the ACF method we generally identify the rotation period as the time lag, $k \cdot \Delta t_{\rm LC}$ with integer number $k$, corresponding to the first peak in the ACF. Subsequent peaks are located at multiples of that period, resulting in the typical oscillatory behavior of the ACF. Exceptions are double-peaked lightcurves where the ACF presents two sequences of equidistant peaks (see e.g. Fig. \[fig:prot\_examples\]). Such lightcurves point to the presence of two dominant spots, and we choose the first peak of the sequence with higher ACF signal as representing the rotation period. [@McQuillan13.0] have performed simulations that demonstrate the typical pattern of the ACF for different effects in the lightcurve, such as changing phase and amplitude, double peaks, and linear trends. All these features are also present in the K2 data, although less pronounced than in the much longer main Kepler mission time-series examined by [@McQuillan13.0]. The classical periodogram is based on a Fourier decomposition of the lightcurve. In the form presented by [@Scargle82.1], it can be applied to unevenly sampled data and is essentially equivalent to least-squares fitting of sine-waves. Realistic time-series deviate from a sine-curve, and are subject to the effects described above. This introduces features in the power spectrum. Since the dominating periodicity in the K2 Superblink stars is reasonably given by the stellar rotation cycle, the highest peak of the periodogram can be interpreted as representing the rotation period. The LS-periodograms are computed here for a false-alarm probability of $0.01$ using the fast-algorithm of [@Press89.1]. In Fig. \[fig:prot\_examples\] we show an example for a detrended K2 lightcurve, its LS-periodogram and ACF, and the lightcurve folded with the derived period. An atlas with the phase-folded lightcurves for all periodic stars is provided in Appendix \[sect:appendix\_folded\]. ### Period search on un-detrended lightcurves {#subsubsect:k2_analysis_period_undetrended} As an independent check we derive the stellar rotation periods with the Systematics-Insensitive Periodogram (SIP) algorithm developed by [@Angus16.0], that produces periodograms calculated from the analysis of the raw K2 photometric time series. For each observing campaign, these are modelled with a linear combination of a set of $150$ ‘eigen light curves’ (ELC), or basis functions, that describe the systematic trends present in K2 data, plus a sum of sine and cosine functions over a range of frequencies[^9]. For each test frequency, the system of linear equations is solved through a least-square fit to the data. The periodogram power is determined as described in [@Angus16.0], by calculating the squared signal-to-noise ratio *(S/N)*$^{2}$ for each frequency. *(S/N)*$^{2}$ is a function of the sine and cosine coefficients (i.e. the amplitudes), where the frequencies corresponding to amplitudes not well constrained by the fit are penalized. The stellar rotation period is finally calculated as the inverse of the frequency having the highest power. ### Sine-fitting of stars with long periods {#subsubsect:k2_analysis_period_sinefits} The techniques described in Sects. \[subsubsect:k2\_analysis\_period\_detrended\] and \[subsubsect:k2\_analysis\_period\_undetrended\] are limited to periods shorter than the duration of the K2 campaigns ($33$d for C0 and $70...80$d for the other campaigns). However, by visual inspection of the lightcurves we identify $11$ stars with clearly sine-like variations that exceed the K2 monitoring time baseline. For these objects a least-squares fit allows us to constrain the rotation periods. The fitting was done with the routine [curve\_fit]{} in the Python package SciPy [@Jones01.0] As initial guesses for the parameters we used four times the standard deviation as amplitude, a period of $30$d, and a phase of $0.0$, but the results do not depend on this choice. For all $11$ lightcurves the routine converges on a unique solution independent of the choice of the initial guesses for the parameters. In three cases the sinecurve provides only a crude approximation because the lightcurve is not symmetric around maximum or minimum and shows signs of spot evolution; in these cases the results are treated with caution. ### Comparison of results from different period search algorithms {#subsubsect:k2_analysis_rot_compare} The results of the different period search methods are compared in Fig. \[fig:period\_comparison\]. Generally, the LS and the ACF periods are in excellent agreement. ![Comparison of the periods derived with the different methods described in Sect. \[subsect:k2\_analysis\_period\]. Reliable periods (green; flag ‘y’ in Table \[tab:rot\]), questionable periods (red; flag ‘?’). Periods equal to or longer than the data set according to the LS and ACF analysis have been determined through sine-fitting and are not shown here.[]{data-label="fig:period_comparison"}](./f8.pdf){width="8.5cm"} For further use, based on the agreement between the periods obtained with the two techniques and considering the appearance of the phase folded lightcurve, we adopt either the ACF or the LS period as rotation period. This selection is made independently by two members of the team (BS and AS), and the results deviate for only few stars. For those dubious cases we make use of the SIP results as cross-check, and we adopt the period (either LS or ACF) which is in better agreement with the SIP “S/N" period. In addition, for the $11$ stars that have their highest peak in the ACF and LS at a period corresponding to the length of the data set ($T_{\rm tot}$) but for which visual inspection reveals a clear (sine-like) pattern indicating a spot-modulation with $P_{\rm rot} > T_{\rm tot}$ we use the periods from the sine-fitting. From a comparison of the values obtained with the ACF and with the LS periodogram we estimate the typical error on our periods to be $\lesssim 3$%. The final, adopted periods are given in Table \[tab:rot\] together with a quality flag and reference to the method with which it was derived. Flag ‘Y’ stands for reliable periods, ‘?’ for questionable period detections, and ‘N’ for no period. These periods are obtained from the ‘cleaned’ lightcurves, but due to the robustness of the detection techniques they are in agreement (within $< 5$%) with the periods found on the original, detrended lightcurves. X-ray emission {#sect:xray_analysis} ============== We perform a systematic archive search for X-ray observations of the K2 Superblink M stars. Specifically, we consult the [*XMM-Newton*]{} Serendipitous Source Catalogue [3XMM-DR5; @Rosen16.0], the [*XMM-Newton*]{} Slew Survey Source Catalogue [XMMSL1\_Delta6; @Saxton08.0], the Second [*ROSAT*]{} Source Catalog of Pointed Observations (2RXP) and the [*ROSAT*]{} Bright and Faint Source catalogs (BSC and FSC). Our procedure for cross-matching the K2 targets with these catalogs and for deriving X-ray fluxes and luminosities follow those described by [@Stelzer13.0]. That work presented the X-ray and UV emission of the M dwarfs within $10$pc of the Sun. Although that sample was drawn from the same catalog (LG11), there are only two stars in common with our K2 study because most of the stars that fall in the K2 fields have distances in the range $20...60$pc. We briefly summarize the individual analysis steps here and we refer to [@Stelzer13.0] for details. First, in order to ensure that no matches are missed due to the high proper motion of the most nearby stars (stars at $< 10$pc have proper motions of $\sim 1^{\prime\prime}$/yr), the cross-correlation between the K2 target list and the X-ray catalogs is done after correcting the object coordinates from the K2 catalog[^10] to the date of the X-ray observation using the proper motions given by LG11. We then use the following match radii between the X-ray catalog positions and the K2 coordinates: $40^{\prime\prime}$ for RASS [@Neuhaeuser95.1], $30^{\prime\prime}$ for XMMSL [@Saxton08.0], $25^{\prime\prime}$ for 2RXP [@Pfeffermann03.0] and $10^{\prime\prime}$ for 3XMM-DR5. With one exception all counterparts have much smaller separations than the respective cross-correlation radius (see Table \[tab:x\]). The only doubtful X-ray counterpart is the XMMSL source associated with EPIC201917390. It has a separation close to the edge of our match circle which – as stated by [@Saxton08.0] – is a generous interpretation of the astrometric uncertainty of the XMMSL. Since the same star is also clearly identified with a RASS source at an X-ray luminosity within a factor of two of the XMMSL source, we decide to keep the XMMSL counterpart. For all but one of the X-ray detected stars the rotation period could be determined. The exception is EPIC210500368 for which hints for pseudo-periodic variations in the K2 lightcurve can be seen by-eye but the ACF and LS periodograms show no dominant peak. After the identification of the X-ray counterparts we compute their $0.2-2$keV flux assuming a $0.3$keV thermal emission subject to an absorbing column of $N_{\rm H} = 10^{19}\,{\rm cm^{-2}}$. The [*ROSAT*]{} count-to-flux conversion factor is determined with PIMMS[^11] to be $CF_{\rm ROSAT} = 2.03 \cdot 10^{11}\,{\rm cts\,erg^{-1}\,cm^{-2}}$ [see also @Stelzer13.0] and we apply it to the count rates given in the 2RXP catalog, the BSC and the FSC. For 3XMM-DR5 sources we use the tabulated EPIC/pn count rates in bands $1-3$ which represent energies of $0.2-0.5$, $0.5-1.0$, and $1.0-2.0$keV, respectively. We sum the count rates in these bands, and perform the flux conversion for the combined $0.2-2.0$keV band. All 3XMM-DR5 counterparts to K2 Superblink M stars were observed with the EPIC/pn medium filter, and for the $N_{\rm H}$ and $kT$ given above we find in PIMMS a count-to-flux conversion factor of $CF_{\rm 3XMM-DR5} = 9.22 \cdot 10^{11}\,{\rm cts\,erg^{-1}\,cm^{-2}}$. The XMMSL1 catalog has the three energy bands already combined in columns ‘B5’. A total of $26$ K2 Superblink stars have an X-ray counterpart in the archival databases that we have consulted. The X-ray fluxes obtained as described above are converted to luminosities using the updated photometric distances of the stars (see Sect. \[sect:stepar\]). The X-ray luminosities are given in Table \[tab:x\] together with the separation between X-ray and optical position and the respective X-ray catalog. For stars with more than one epoch of X-ray detection the luminosities are in agreement within a factor of two and we provide the mean of the two values. The errors of $L_{\rm x}$ comprise the uncertainties of the count rates and an assumed $20$% error of the distances which yield roughly comparable contributions to the error budget. Our assumption on the distance error is motivated by the distance spread between photometric and trigonometric distances for the subsample with both measurements (described in Sect. \[sect:stepar\]). Ultraviolet emission {#sect:uv_analysis} ==================== To assess the UV activity of the K2 Superblink stars we cross-match our target list with the [*GALEX-DR5 sources from AIS and MIS*]{} [@Bianchi12.0]. GALEX performed imaging in two UV bands, far-UV (henceforth FUV; $\lambda_{\rm eff} = 1528$Å, $\Delta \lambda = 1344-1786$Å) and near-UV (henceforth NUV; $\lambda_{\rm eff} =2271$Å, $\Delta \lambda = 1771-2831$Å). The All-Sky Survey (AIS) covered $\sim 85$% of the high Galactic latitude ($\|b\|> 20^\circ$) sky to m$_{AB} \sim 21$mag, and the Medium Imaging Survey (MIS) reached m$_{AB} \sim 23$mag on 1000 deg$^2$ [e.g. @Bianchi09.0]. Analogous to our analysis of the X-ray data, we correct the coordinates from the K2 catalog to the date of the respective UV observation. We use a match radius of $10^{\prime\prime}$, but none of the UV counterparts we identify is further than $3^{\prime\prime}$ from the proper motion corrected K2 position. The GALEX-DR5 catalog provides NUV and FUV magnitudes which we convert to flux densities using the zero points given by [@Morrissey05.0]. We isolate the chromospheric contribution to the UV emission from the photospheric part with help of synthetic [dusty]{} spectra of [@Allard01.1], following the procedure described by [@Stelzer13.0]. We adopt the model spectra with solar metallicity and $\log{g} = 4.5$, and we choose for each star that model from the grid which has $T_{\rm eff}$ closest to the observationally determined value derived in Sect. \[sect:stepar\]. We then obtain the predicted photospheric UV flux density \[$(f_{\rm UV_i,ph})_\lambda$\] in the two GALEX bands ($i=NUV, FUV$) from the UV and $J$ band flux densities of the [dusty]{} model (i.e. the synthetic $UV_i - J$ color) and the observed $J$ band flux density. The model flux densities in the FUV, NUV and $J$ bands are determined by convolving the synthetic spectrum with the respective normalized filter transmission curve. Finally, the FUV and NUV fluxes are obtained by multiplying $(f_{\rm UV_i,ph})_\lambda$ with the effective band width of the respective GALEX filter ($\delta \lambda_{\rm FUV} = 268$Å; $\delta \lambda_{\rm NUV} = 732$Å); [@Morrissey07.0]. The expected photospheric fluxes ($f_{\rm UV_i,ph}$) are then subtracted from the observed ones to yield the chromospheric fluxes. We refer to these values as ‘UV excess’, $f_{\rm UV_i,exc}$. Finally, we define the UV activity index as $R^\prime_{UV_i} = \frac{f_{\rm UV_i,exc}}{f_{\rm bol}}$ where $f_{\rm bol}$ is the bolometric flux. The superscript ($^\prime$) indicates, in the same manner as for the well-known Ca[ii]{}H&K index, that the flux ratio has been corrected for the photospheric contribution. We find NUV detections for $41$ stars from the K2 Superblink M star sample, i.e. roughly $30$%, while only $11$ stars ($\sim 8$%) are identified as FUV sources. [@Stelzer13.0] have shown that the photospheric contribution to the FUV emission of M stars is negligible while the fraction of the NUV emission emitted by the photosphere can be significant. We confirm here for the K2 Superblink M stars with FUV detections that this emission is entirely emitted from the chromosphere, i.e. $f_{\rm FUV,ph}$ is orders of magnitude smaller than the observed FUV flux. The NUV emission of the K2 Superblink M stars is also only weakly affected by photospheric contributions with $f_{\rm NUV,ph}$ less than $\sim 10$% of the observed flux for all stars. In Table \[tab:uv\] we provide the observed FUV and NUV magnitudes and the calculated chromospheric excess, $L_{\rm UV_i,exc}$, of all detected objects. The uncertainties for the UV luminosities comprise the magnitude errors and an assumed $20$% error on the distances (as in Sect. \[sect:xray\_analysis\]). Results {#sect:results} ======= Period statistics and comparison with the literature {#subsect:results_stat_and_lit} ---------------------------------------------------- We could determine reliable periods for $75$ stars (flag ‘Y’ in Table \[tab:rot\]), and periods with lower confidence are found for $22$ stars (flag ‘?’). Twelve stars of our sample have a previously reported period based on the same K2 data in [@Armstrong15.0]. In all but two cases those periods agree within $1-2$% with our values. The exceptions are EPIC-202059204 for which the lightcurves used by us (and produced by A.Vanderburg) show no evidence for the $5.04$d period provided by [@Armstrong15.0], and EPIC-201237257 for which our adopted period is twice the value of $16.2$d presented by [@Armstrong15.0] based on the maximum peak in both our ACF and LS periodogram. Periods for a small number of K2 Superblink stars have been presented previously also in the following studies: Survey in the southern hemisphere using the All-Sky Automated Survey [ASAS; @Kiraga12.0 $6$stars], HATnet survey in the Pleiades [@Hartman10.0 $2$stars], SuperWASP survey in the Hyades and Pleiades [@Delorme11.0 $1$star], and from the compilation of [@Pizzolato03.1 $1$star]. They are all in excellent agreement with our values derived from the K2 lightcurves. For the two stars we have in common with the HATNet survey of field stars presented by [@Hartman11.0], however, we find strongly discrepant values for the periods: $16.1$d vs $39.0$d in [@Hartman11.0] for EPIC-211107998 and $12.9$d vs $0.86$d in [@Hartman11.0] for EPIC-211111803. We see no evidence in the K2 data for the period values determined by [@Hartman11.0]. All in all, a $73$% of the K2 Superblink sample shows periodic variability on timescales up to $\sim 100$d. Our period distribution is shown in Fig. \[fig:periods\_histo\]. Studies of rotation of M stars in the main Kepler mission have come up with $63$% [@McQuillan13.0] and $81$% [@McQuillan14.0] of stars with detected periods. These differences may reflect the different data sets (each K2 campaign provides a lightcurve corresponding to the length of about one quarter of Kepler data) and detection methods (we use sine-fitting in addition to ACF and periodograms). In particular, we establish here in a relatively unbiased sample of M dwarfs periods of $\sim 100$d and longer, in agreement with results from ground-based studies [@Irwin11.0; @Newton16.0]. The period distribution of the Kepler sample from [@McQuillan13.0] shows a cut at $\sim 65$d and [@McQuillan14.0] explicitly limit their sample to periods $< 70$d. Note, that [@McQuillan13.0] have performed the period search on individual Kepler Quarters which are of similar duration as the K2 campaigns. In fact, we are able to detect such long periods only thanks to the least-squares sine-fitting. We find that $\sim 10$% of the periods are longer than $70$d. These would not have been detected by the methods of [@McQuillan13.0; @McQuillan14.0]. An additional possible explanation for the absence of long-period variables in [@McQuillan13.0] – related to photometric sensitivity – is presented in Sect. \[subsect:results\_amplitude\]. ![Distribution of the $97$ rotation periods determined for the K2 Superblink M star sample. The black histogram represents the full sample of periods, and the overlaid green histogram the subsample of reliable periods (flag ‘Y’).[]{data-label="fig:periods_histo"}](./f9_new.pdf){width="8.5cm"} Rotation period and stellar mass {#subsect:results_prot_mass} -------------------------------- We present the newly derived rotation periods for the K2 Superblink M star sample in Fig. \[fig:prot\_mass\] as a function of stellar mass together with results for studies from the main Kepler mission. The sample of [@McQuillan13.0] (black open circles) covered stars in the mass range of $0.3 ... 0.55\,M_\odot$ selected based on the $T_{\rm eff}$ and $log{g}$ values from the Kepler input catalog [@Brown11.0]. Subsequently, [@McQuillan14.0] (black dots) extended this study with similar selection criteria to all stars with $T_{\rm eff} < 6500$K. Among the most notable findings of these Kepler studies was a bimodal period distribution for the lowest masses, and an increasing upper envelope of the period distribution for decreasing mass. While we have too few objects to identify the bimodality, we confirm the upwards trend in the longest periods detected towards stars with lower mass. We are able to measure longer periods than [@McQuillan13.0] and [@McQuillan14.0] because we add sine-fitting to the ACF and periodgram period search methods; see Sect. \[subsect:results\_amplitude\] for a more detailed comparison of the period detection techniques and their implications. The fact that we measure periods in excess of $\sim 100$d only in stars with very low mass ($M \leq 0.45\,M_\odot$) is interesting. If it is a real feature in the rotational distribution, it suggests a change of the spin-down efficiency at the low-mass end of the stellar sequence. Note, however, that the stellar masses at which the upturn is seen to set in does not correspond to the fully convective transition ($\sim 0.35\,M_\odot$) where one might expect some kind of “mode change" in the dynamo. Also, we can not exclude that there are detection biases, e.g. the size and distribution of star spots and their lifetimes could be mass-dependent such that smaller and more quickly changing amplitudes are induced in higher-mass stars which would prevent us from detecting very long periods in them. A more detailed investigation of these features must be deferred to studies on a larger sample. ![Period versus mass for the K2 Superblink M star sample (green and red symbols for periods flagged ‘Y’ and ‘?’, respectively). Binaries are marked with annuli (see Appendix \[sect:appendix\_bin\]). Data from Kepler studies are plotted as open circles [@McQuillan13.0] and black dots [@McQuillan14.0].[]{data-label="fig:prot_mass"}](./f10.pdf){width="8.5cm"} Activity diagnostics from K2 rotation cycles {#subsect:results_amplitude} -------------------------------------------- We examine now various other diagnostics for rotation and activity derived from the K2 data. These are listed together with the rotation periods in Table \[tab:rot\]. The Rossby number (in col.5) is defined as $R_{\rm 0} = P_{\rm rot}/\tau_{\rm conv}$, where $\tau_{\rm conv}$ is the convective turnover time obtained from $T_{\rm eff}$ using Eq. 36 of [@Cranmer11.0] and its extrapolation to $T_{\rm eff} < 3300$K. There is no consensus on the appropriate convective turnover times for M dwarfs beyond the fully convective boundary. As pointed out by [@Cranmer11.0], the extrapolated values for late-M dwarfs ($\tau_{\rm conv} \sim 60...70$d) are in reasonable agreement with semi-empirical values derived by [@Reiners09.3] but significantly lower than the predictions of [@Barnes10.0]. The Rossby number is a crucial indicator of dynamo efficiency and is used in Sect. \[subsect:results\_rotact\] for the description of the rotation - activity relation. The parameters $R_{\rm per}$ (col.6) and $S_{\rm ph}$ (col.7) are measures for the variability in the K2 lightcurve and are examined in this section. ![Period versus amplitude for the K2 Superblink M dwarf sample (large, colored circles) compared to the Kepler field M dwarfs from [@McQuillan13.0] (small, black dots). For our K2 sample we distinguish reliable periods (green) and questionable periods (red). Banner on the right: histogram of $R_{\rm per}$ for the full K2 Superblink star sample (solid line) and for the subsample classified as non-periodic (flag ‘N’). As in the other figures, binaries in our K2 Superblink sample are marked with large annuli.[]{data-label="fig:ampl_vs_period"}](./f11.pdf){width="8cm"} The amplitude of photometric variability associated with star spots is determined by the temperature contrast between spotted and unspotted photosphere and by the spot coverage, and may therefore, to first approximation, be considered a measure for magnetic activity. Various photometric activity indices characterizing the amplitude of Kepler lightcurves have been used in the literature. [@Basri13.0] have introduced the range of variability between the 5th and 95th percentile of the observed flux values, $R_{\rm var}$. This definition is meant to remove the influence of flares which are occasional events involving only a small fraction of a given rotational cycle. To further reduce the influence of outliers, we follow the modified definition of [@McQuillan13.0]: $R_{\rm per}$ is the mean of the $R_{\rm var}$ values measured individually on all observed rotation cycles, expressed in percent. Since in the course of our flare analysis we produce lightcurves where flares and other outliers have been eliminated we could use those “cleaned" lightcurves for the analysis of the rotational variability. This way we could avoid cutting the top and bottom $5$% of the data points. We compute the difference between the full amplitudes measured on the “cleaned” lightcurves and the $R_{\rm per}$ values measured on the original lightcurves and find them to differ by $\sim 0.05 \pm 0.05$dex in logarithmic space. This is negligible to the observed range of amplitudes. In order to enable a direct comparison to results from the literature, we prefer, therefore, to stick to the $R_{\rm per}$ values derived from the original lightcurves. In Fig. \[fig:ampl\_vs\_period\] we show the relation between $P_{\rm rot}$ and $R_{\rm per}$ for the K2 Superblink M stars compared to the much larger Kepler M dwarf sample of [@McQuillan13.0]. The distribution of the two samples is in good agreement. In particular, there is a clear trend for stars with shorter periods to have larger spot amplitudes. We examine this finding in more detail below. Fig. \[fig:ampl\_vs\_period\] also illustrates the difference between our results and those of [@McQuillan13.0] for the longest periods. \[Note that all K2 Superblink stars with periods inferred from sine-fitting have only a lower limit to the variability amplitude $R_{\rm per}$.\] One reason for the absence of long-period stars in [@McQuillan13.0] could be the larger distance of the Kepler stars which results in lower sensitivity for small amplitudes, suggesting that Kepler can find periods only in the more active stars likely to be rotating faster. However, remarkably, the long-period stars in the K2 Superblink M star sample seem to have larger spot amplitudes than stars with lower periods (from $\sim 15 ...50$d). We recall again that we are able to detect such long periods only on stars with clear sine-like variation indicating the presence of a single dominating spot. Therefore, we can only speculate that stars with periods $\gtrsim 100$d and low spot amplitude may exist but their more diffuse spot patterns or changes on time-scales shorter than the rotation period yield a complex lightcurve. If so, one can expect these stars among the ones classified as non-periodic (flag ‘N’) by us. The bar on the right of Fig. \[fig:ampl\_vs\_period\] shows the distribution of $R_{\rm per}$ for all K2 Superblink M stars and for the subsample to which we could not assign a period. There is no clear preference of these latter ones towards small amplitudes, and the above consideration does not allow us to conclude on their periods. Constraining the range of spot amplitude of the slowest rotators should be a prime goal of future studies on larger samples. As described in Sect. \[subsect:results\_prot\_mass\] we find the longest periods exclusively in very low-mass stars. Therefore, the change in the distribution of the $R_{\rm per}$ values for the slowest rotators – if truly existing – might be a mass-dependent effect rather than related to rotation. [@Mathur14.1] defined the standard deviation of the full lightcurve, $S_{\rm ph}$, and $\langle S_{\rm ph,k} \rangle$, the mean of the standard deviations computed for time intervals $k \cdot P_{\rm rot}$. They found that for increasing $k$ the index $\langle S_{\rm ph,k} \rangle$ approaches $S_{\rm ph}$. This way they were able to show that roughly after five rotation cycles ($k = 5$) the full range of flux variation is reached, and they recommend $\langle S_{\rm ph,k=5} \rangle$ as measure of the global evolution of the variability. We compute $S_{\rm ph}$ and $\langle S_{\rm ph,k=5} \rangle$ for the K2 Superblink M stars and show the results in Fig. \[fig:sph\_vs\_period\] versus the rotation periods; filled circles represent $\langle S_{\rm ph,k=5} \rangle$ and open circles mark $S_{\rm ph}$. The sample studied by [@Mathur14.1] is also displayed (black dots for their $\langle S_{\rm ph,k=5} \rangle$ values). That sample consists of $34$ Kepler M stars with $15$ Quarters of continuous observations and $P_{\rm rot} < 15$d from the Kepler study of [@McQuillan13.0]. Our sample improves the period coverage especially for $P \lesssim 12$d. The fact that there are no very fast rotators in the sample studied by [@Mathur14.1] is probably a bias related to their sample selection. We find that stars with short periods have systematically larger $\langle S_{\rm ph,k=5} \rangle$ index than stars with $P \gtrsim 10$d. The upper boundary of $15$d for the periods in the [@Mathur14.1] sample is imposed by their requirement of covering at least $5$ cycles. However, as explained above there is no dramatic difference between $\langle S_{\rm ph,k=5} \rangle$ and $S_{\rm ph}$ for a given star. We verify this on the K2 Superblink M star sample by showing as open circles their values $S_{\rm ph}$. The advantage of $S_{\rm ph}$ is that we can include in Fig. \[fig:sph\_vs\_period\] the stars with $P > 1/5 \cdot \Delta t$. We can see that the pattern over the whole period range is very similar to that of Fig. \[fig:ampl\_vs\_period\], i.e. both spot amplitude and standard deviation of the lightcurve show a dependence on rotation rate which seems to divide the stars in two groups above and below $P \sim 10...12$d. ![Magnetic activity indices defined by [@Mathur14.1] vs period for the K2 Superblink star sample, compared to the subsample of $34$ Kepler field M dwarfs studied by [@Mathur14.1] (black dots). Green and red symbols represent periods flagged ‘Y’ and ‘?’, respectively. Filled circles denote $\langle S_{\rm ph,k=5} \rangle$ values and open circles the $S_{\rm ph}$ values. See text in Sect. \[subsect:results\_amplitude\] for details on the definition of these indices. Large annuli mark binaries.[]{data-label="fig:sph_vs_period"}](./f12.pdf){width="8.5cm"} Activity diagnostics related to flares in K2 lightcurves {#subsect:results_flares} -------------------------------------------------------- Our separate analysis of flares and rotation in the K2 lightcurves enables us to relate flare activity to star spot activity. Fig. \[fig:flares\_vs\_period\] shows the peak amplitudes of all flares defined with respect to the flattened K2 lightcurve (top panel) and the flare frequency of all stars (bottom panel) as function of the rotation period. A clear transition takes place near $P_{\rm rot} \sim 10$d, analogous to the case of the spot activity measures discussed in Sect. \[subsect:results\_amplitude\]. While the absence of small flares in fast rotators is determined by the noise level in the flattened lightcurve (see Fig. \[fig:kp\_stddev\]), there is no bias against the detection of large flares in slowly rotating stars. Note that our algorithm has lower flare detection sensitivity for events on fast-rotating stars because the presence of flares itself impacts on the quality of the smoothing process used to identify the flares. Therefore, especially for the stars with short periods, the number of flares observed per day ($N_{\rm flares}/{\rm day}$) may represent a lower limit to the actual flare frequency. Considering the limitations of the K2 long-cadence data for flare statistics (see discussion in Sect. \[subsect:k2\_analysis\_flares\]) we do not put much weight on the absolute numbers we derive for the flare rates. However, our results are in very good agreement with a dedicated M dwarf flare study based on short-cadence ($1$min) Kepler lightcurves. In particular, for the fast rotators the range we show in Fig. \[fig:flares\_vs\_period\] for the peak flare amplitudes ($\sim 0.01...0.5$) and for the number of flares per day ($\sim 0.05...0.2$), are similar to the numbers obtained by [@Hawley14.0] for the active M star GJ1243 if only flares with duration of more than one hour are considered from that work. ![[*top*]{} - Flare amplitude vs rotation period: All flares are shown and the range of flare amplitudes for a given star is made evident by marking the largest and smallest flare on each star with different colors, red and blue respectively. [*bottom*]{} - Flare frequency vs rotation period: Each star is represented once. Binaries are highlighted in both panels with annuli.[]{data-label="fig:flares_vs_period"}](./f13a.pdf "fig:"){width="8.5cm"} ![[*top*]{} - Flare amplitude vs rotation period: All flares are shown and the range of flare amplitudes for a given star is made evident by marking the largest and smallest flare on each star with different colors, red and blue respectively. [*bottom*]{} - Flare frequency vs rotation period: Each star is represented once. Binaries are highlighted in both panels with annuli.[]{data-label="fig:flares_vs_period"}](./f13b.pdf "fig:"){width="8.5cm"} Residual activity in K2 lightcurves {#subsect:results_noise} ----------------------------------- Above we have shown that both the spot cycle amplitude and the flares display a distinct behavior with rotation period. Here we examine the standard deviation of the “flattened" lightcurves, $S_{\rm flat}$. As described in Sect. \[subsect:k2\_analysis\_noise\], when measured without considering the outliers, this parameter represents a measure for the noise after removal of the rotation cycle and of the flares. We notice a marked trend of $S_{\rm flat}$ with the rotation period (Fig. \[fig:period\_stddev\]). A dependence of the noise level on the brightness of the star is expected and demonstrated in Fig. \[fig:kp\_stddev\], where the lower envelope of the distribution increases towards fainter $K_{\rm p}$ magnitude. However, the difference between the $S_{\rm flat}$ values seen for slow and fast rotators in Fig. \[fig:period\_stddev\] is clearly unrelated to this effect as there is no clustering of stars with large $S_{\rm flat}$ (and fast rotation) at bright magnitudes in Fig. \[fig:kp\_stddev\]. The evidently bimodal distribution with rapid rotators showing larger values of $S_{\rm flat}$, therefore suggests that there is a contribution to the ‘noise’ in the K2 photometry that is astrophysical in origin. ![Standard deviation of the flattened lightcurve excluding the outliers, as in Fig. \[fig:kp\_stddev\], shown here vs rotation period. Green symbols (period flag ‘Y’), red symbols (period flag ‘?’). The clear transition between fast and slowly rotating stars indicates that for fast rotators the origin of this ‘noise’ has an astrophysical component. Binaries are highlighted with annuli.[]{data-label="fig:period_stddev"}](./f14.pdf){width="8.5cm"} The similarity of the period dependence seen in $S_{\rm flat}$ (Fig. \[fig:period\_stddev\]), the spot cycle (Fig. \[fig:ampl\_vs\_period\] and  \[fig:sph\_vs\_period\]) and the flares (Fig. \[fig:flares\_vs\_period\]) may indicate that the ‘noise’ in the fastest rotators could be caused by unresolved spot or flare activity. Many small flares, so-called nano-flares, as well as many small and/or rapidly evolving spots can produce a seemingly stochastic signal. This astrophysical noise sources seem to be limited to fast rotators, while for slow rotators the spot contrast drops below a constant minimum level of the variability which might be identified as the photometric precision (see Sect. \[subsect:k2\_analysis\_noise\]). Photometric activity and binarity {#subsect:results_binaries} --------------------------------- In the relations between various activity indicators and rotation period presented in the previous sections, binary stars that have a possible contribution to the rotational signal from the unresolved companion star are highlighted. Strikingly, the binaries are mostly associated with rotation periods below the transition between fast and slow regimes that we have identified. In Fig. \[fig:period\_stddev\] this could be taken as evidence that the presence of a companion increases the noise in the K2 lightcurve. On the other hand, we have argued above that the coincidence of the bimodality in $S_{\rm flat}$, spot and flare signatures with $P_{\rm rot}$ points at a fundamental transition taking place in these stars. We may speculate that binarity is responsible for the observed dichotomy, e.g. by spinning up the star through tidal interaction or by reducing angular momentum loss. The binary fraction ($BF$) for the fast rotators ($P_{\rm rot} < 10$d) is $8 / 19$, i.e. $8$ stars out of $19$ are known binaries. For slow rotators ($P_{\rm rot} > 10$d) the binary fraction is $2 / 78$. We calculate the $95$% confidence levels for a binomial distribution and find the two samples to be significantly different: $BF_{\rm fast} = 0.42^{+0.67}_{-0.20}$ and $BF_{\rm slow} = 0.03^{+0.09}_{-0.00}$. That said, we caution that no systematic and homogeneous search for multiplicity was done for these stars and our literature compilation (Sect. \[sect:appendix\_bin\]) may be incomplete. The X-ray and UV activity – rotation relation {#subsect:results_rotact} --------------------------------------------- The activity – rotation relation is traditionally expressed using X-rays, Ca[ii]{}H&K and H$\alpha$ emission as activity indicators. Measurements of these diagnostics have historically been easiest to achieve [@Pallavicini81.1; @Noyes84.0]. Yet, as described in Sect. \[sect:intro\], the dependence between magnetic activity and rotation has remained poorly constrained for M stars. In Fig. \[fig:rotact\] we present an updated view using the X-ray data extracted from the archives and the newly derived rotation periods from K2. We also add here, to our knowledge for the first time for field M stars, UV emission as diagnostic of chromospheric activity in conjunction with photometric rotation periods. All but two of the $26$ K2 Superblink stars with X-ray detection have reliable rotation period measurement (flag ‘Y’). The first exception is EPIC-206019392 for which we find through sine-fitting a period of $\sim 75$d. While there are no doubts on a periodic spot-modulation, the slight deviations of its lightcurve from a sinusoidal make the value for the period uncertain (therefore flagged ‘?’ in Table \[tab:rot\]). For the other case, EPIC-210500368, we can not identify a dominating period, yet the lightcurve shows a long-term trend superposed on a variability with a time-scale of $\sim 10$d. Among the NUV detections we could establish the rotation period for $78$% ($32 / 41$), and $46$% ($19 / 41$) of them have a ‘reliable’ period. Nine of $11$ FUV detected stars have a period measurement, of which $7$ are flagged ‘reliable’. The parameters which best describe the connection between activity and rotation are still a matter of debate [@Reiners14.0]. We provide here plots for luminosity versus rotation period (left panels of Fig. \[fig:rotact\]) and for activity index $L_{\rm i}/L_{\rm bol}$ with $i = NUV, FUV, X$ versus Rossby number (right panels). First, it is clear that there is a decrease of the activity levels in all three diagnostics (NUV, FUV, X-rays) for the slowest rotators. While the sample of M stars with FUV detection and rotation period measurement is still very small, a division in a saturated and a correlated regime, historically termed the “linear" regime, can be seen in the relations involving NUV and X-ray emission. The X-ray – rotation relation is still poorly populated for slow rotators, and the turn-over point and the slope of the decaying part of the relation can not be well constrained with the current sample. Interestingly, for the NUV emission the situation is reversed, in a sense that more stars with NUV detection are found among slow rotators. In terms of luminosity NUV saturation seems to hold up to periods of $\sim 40$d, way beyond the critical period of $\sim 10$d identified to represent a transition in the behavior of optical activity indicators extracted from the K2 lightcurves (see Sect. \[subsect:results\_amplitude\] and \[subsect:results\_flares\]). On the other hand, the $L_{\rm NUV}^\prime/L_{\rm bol}$ values are slightly decreased with respect to the levels of the fastest rotators, and the active stars around a $\sim 30...40$d period are all late-K to early-M stars. We also caution that a large fraction of the slowly rotating NUV detected stars have periods that we flagged as less reliable (red symbols in Fig. \[fig:rotact\]). In order to highlight eventual differences emerging at the fully convective transition, we divide the stars in Fig. \[fig:rotact\] into three spectral type groups represented by different plotting symbols. As far as the X-ray emission is concerned, the two order of magnitude scatter in the saturated part of the $L_{\rm x}$ vs $P_{\rm rot}$ relation is clearly determined by the spectral type distribution, with cooler stars having lower X-ray luminosities for given period. This is a consequence of the mass dependence of X-ray luminosity, and was already seen by [@Pizzolato03.1] for coarser bins of stellar mass representing a spectral type range from G to M. We have overplotted in the bottom panels of Fig. \[fig:rotact\] the relation derived by [@Pizzolato03.1] for their lowest mass bin, $M = 0.22 ... 0.60\,M_\odot$ (corresponding to spectral type earlier than M2). It must be noted that in [@Pizzolato03.1] the linear regime was populated by only two stars of their sample and the saturated regime was dominated by upper limits to $P_{\rm rot}$ which were estimated from $v \sin{i}$ measurements. Therefore, even our still limited K2 sample constitutes a significant step forward in constraining the X-ray – rotation relation of M dwarfs. We determine the saturation level for all X-ray detections with $P_{\rm rot} < 10$d in the three spectral type bins K7...M2, M3...M4, and M5...M6 and for the whole sample with spectral types from K7 to M6. The results are summarized in Table \[tab:xraysat\]. If we select the K2 Superblink M star subsample in the same mass range studied by [@Pizzolato03.1] ($M = 0.22 ... 0.60\,M_\odot$) we derive saturation levels of $\log{L_{\rm x,sat}} \,{\rm [erg/s]} = 28.5 \pm 0.5$ and $\log{(L_{\rm x,sat}/L_{\rm bol})} = -3.3 \pm 0.4$, within the uncertainties compatible with their results. We confirm results of previous studies that the saturation level for a sample with mixed spectral types converges to a much narrower distribution if $\log{(L_{\rm x}/L_{\rm bol})}$ is used as activity diagnostic (see Fig. \[fig:rotact\] and last line in Table \[tab:xraysat\]). There is marginal evidence for the very low mass stars (SpT M5...M6) to be underluminous with respect to this level. However, this assertion is not yet statistically sound according to the spread of the data (see standard deviations in Table \[tab:xraysat\]) and two-sample tests carried out with ASURV [@Feigelson85.1] indicate that the $\log{(L_{\rm x}/L_{\rm bol})}$ values of the three spectral type subgroups may be drawn from the same parent distributions (p-values $> 10$%). It has been widely acknowledged that the activity levels show a drop for late-M dwarfs [e.g. @West08.1; @Reiners12.1], but an investigation of whether and how this is related to $P_{\rm rot}$ has come into reach only now with the large number of periods that can be obtained from planet transit search projects. Using rotation periods from the MEarth program, [@West15.0] showed that the average $L_{\rm H\alpha}/L_{\rm bol}$ ratio for fast rotators ($P_{\rm rot} < 10...20$d) decreases by a factor two for late-M dwarfs (SpT M5...M8) compared to early-M dwarfs (SpT M1-M4). Whether a distinct regime exists in which H$\alpha$ activity correlates with $P_{\rm rot}$ could not be established in that study. The X-ray and UV detections we present in this paper also do not adequately sample the regime of long periods. We refrain here from fitting that part of the rotation-activity relation because our upcoming [*Chandra*]{} observations together with the larger sample of periods that will be available for Superblink M stars at the end of the K2 mission will put us in a much better position to address this issue. --------- ------------- ----------------------- ------------------------------------- SpT $N_{\rm *}$ $\log{L_{\rm x,sat}}$ $\log{(L_{\rm x,sat}/L_{\rm bol})}$ \[erg/s\] K7...M2 $5$ $29.2 \pm 0.4$ $-3.0 \pm 0.4$ M3...M4 $7$ $28.6 \pm 0.3$ $-3.1 \pm 0.2$ M5...M6 $4$ $27.9 \pm 0.5$ $-3.5 \pm 0.4$ K7...M6 $16$ $28.7 \pm 0.6$ $-3.2 \pm 0.4$ --------- ------------- ----------------------- ------------------------------------- : X-ray saturation level for M dwarfs determined for X-ray detected stars with $P_{\rm rot} < 10$d.[]{data-label="tab:xraysat"} Activity and rotation of planet host stars {#subsect:results_planethosts} ------------------------------------------ Being bright and nearby, the K2 Superblink stars have special importance for planet search studies. In fact, at the time of writing of this paper two of our targets already have confirmed planets discovered by the K2 mission. K2-3 is a system comprising three super-Earths confirmed through radial velocity monitoring, with the outer planet orbiting close to the inner edge of the habitable zone (EPIC-201367065 observed in campaign C1); see [@Crossfield15.0; @Almenara15.0]. K2-18 (EPIC-201912552, also observed in C1) has a $\sim 2\,R_\oplus$ planet which was estimated to receive $94 \pm 21$% of the Earth’s insolation [@Montet15.0]. Both host stars are prime targets for characterization studies of the planetary atmospheres through transit spectroscopy. Thus, the analysis of their stellar activity is a necessary step toward a global physical description of these systems. Another two stars from our K2 Superblink sample have planet candidates presented by [@Vanderburg16.0]. These objects are not yet verified by radial velocity measurements. Our analysis shows that for both systems the stellar rotation is not synchronized with the planet orbital period ($P_{\rm orb}=1.8$d and $P_{\rm rot} = 17.9$d for EPIC-203099398 and $P_{\rm orb} = 14.6$d and $P_{\rm rot} = 22.8$d for EPIC-205489894, respectively). Discussion {#sect:discussion} ========== We present here the first full flare and rotation period analysis for a statistical sample of K2 lightcurves. Our target list of bright and nearby M dwarfs represents a benchmark sample for exoplanet studies and will be thoroughly characterized by Gaia in the near future. Knowledge of the magnetic activity of these stars is of paramount importance given the potential impact it has on exoplanets. At the moment a planet is detected, the high-energy emission of any given K2 target and its variability becomes a prime interest [see e.g. @Schlieder16.0 for a recent example]. With the study presented here and future analogous work on the remaining K2 campaigns we anticipate such concerns. Our primary aim here is to understand the stellar dynamo and angular momentum evolution at the low-mass end of the stellar sequence through a study of relations between magnetic activity and rotation. We characterize activity with a multi-wavelength approach involving archival X-ray and UV observations as well as parameters extracted directly from the K2 lightcurves which describe spot amplitudes and flares. This way we provide a stratified picture of magnetic activity from the corona over the chromosphere down to the photosphere. To our knowledge this is the first time that the link with photometrically determined rotation periods is made for a well-defined sample of M stars over such a broad range of activity diagnostics. Yet, as of today, only about $25$% of the M dwarfs with K2 rotation periods have a meaningful X-ray measurement, and this percentage is even lower for the NUV and FUV bands. Dedicated X-ray and UV follow-up of these objects can provide the ultimate constraints on the M dwarf rotation-activity relation. Visual inspection of the K2 lightcurves shows that there is not a single non-variable star in this sample of $134$ M dwarfs. We can constrain rotation periods in $73$% of them. The distribution of rotation periods we find for our sample is in general agreement with studies from the main Kepler mission with much larger but less well-characterized M star samples [@McQuillan13.0; @McQuillan14.0]. Contrary to these studies we find long periods up to $\sim 100$d, thanks to our complementary use of direct sine-fitting as period detection method next to ACF and LS periodograms. We detect such long periods only in the lowest mass stars ($M \leq 0.4\,M_\odot$). In this respect, our results resemble those obtained by [@Irwin11.0; @Newton16.0] based on the MEarth program where sine-fitting yielded many long periods. However, unlike that project our sample includes also early-M type stars and, therefore, it has allowed us to establish that there is a dearth of long period detections in early-M dwarfs. Until corroborated by a larger sample, we can only speculate whether this is due to the evolution of spin-down history across the stellar mass sequence or whether it results from a change in spot pattern and related changes in the detection capabilities for the associated periods. The low cadence of the K2 lightcurves allows us to detect only the flares with duration $\geq 1$h, and due to this sparse sampling we refrain from a detailed analysis of flare statistics. Yet, we find an unprecedented link between flares and stellar rotation. The distribution of flare amplitudes and flare frequencies shows a clear transition at $P_{\rm rot} \sim 10$d. The large flares seen in stars rotating faster than this boundary are absent in slow rotators although there is no detection bias against them. The smaller flares on slow rotators have no counterparts in the fast rotators but such events [– if present – would likely be]{} undetectable. We find the same bimodality between fast and slow rotators in the noise level ($S_{\rm flat}$) of the residual lightcurves after the rotational signal, flares and other ‘outliers’ are subtracted off: The residual variability seen in the fast rotators is significantly and systematically larger than in the slow rotators with a dividing line at $P_{\rm rot} \sim 10$d. These new findings can now be added to the rotation-dependence of the spot cycle amplitude ($R_{\rm per}$) already known from the above-mentioned Kepler studies: A cut exists at the same period of $\sim 10$d with faster rotating stars showing larger amplitudes of the rotation cycle. These similarities lead us to speculate that the ‘noise’ (i.e. the high values of $S_{\rm flat}$) seen in the fast rotators is produced by smaller or fast-changing spots or by micro-flares that cause seemingly random variations. The observed dichotomy in photometric activity levels between fast and slow rotators points to a rotation-dependent rapid transition in the magnetic properties of the photospheres in M dwarfs. In fact, an analogous sharp transition is observed in some numerical dynamo models at $R_{\rm 0,l} \sim 0.1$, where $R_{\rm 0,l}$ is the ‘local Rossby number’ [e.g. @Schrinner12.0]. Assuming this theoretical Rossby number corresponds with its empirical definition (see Sect. \[subsect:results\_amplitude\]), this corresponds roughly to our observed critical period of $\sim 10$d. In the simulations, for $R_{\rm 0,l} > 0.1$ (slow rotators) the dipolar component of the dynamo collapses giving way to a multipolar dynamo regime. [@Gastine13.0] have compared these predictions to the magnetic field structure inferred from ZDI of M dwarfs. Such observations are time-consuming and they require substantial modelling effort, and the samples tend to be biased towards fast rotators. When interpreted in terms of the above-mentioned models, our results suggest photometric rotation and activity measures as a new window for observational studies of dynamo flavors in M dwarfs. However, it must be questioned whether these diagnostics, which represent activity on the stellar surface, are sensitive to the large-scale component of the magnetic field. The transition seen in star spots and white-light flares also corresponds approximately to the period where previous studies of the X-ray - rotation relation have placed the transition from the ‘saturated’ to the ‘linear’ regime [e.g. @Pizzolato03.1]. Different explanations have been put forth for this finding involving the filling factor for active regions, the size of coronal loops or the dynamo mechanism. Observationally, those studies have so far shown clear rotation-activity trends only for higher-mass stars. We extend the X-ray – rotation relation here to well-studied M stars. With our data set we can, for the first time, refine the study of X-ray emission from field M dwarfs in the saturated regime (fast rotation) in bins spanning spectral subclasses and we find a continuous decrease of the saturation level $L_{\rm x}$ towards later spectral type which can be understood in terms of the mass dependence of X-ray luminosity. The tentative evidence that the saturated stars in the coolest mass bin (spectral types M5...M6) have lower $L_{\rm x}/L_{\rm bol}$ than the K7...M4 type stars is not statistically solid yet. If confirmed on a larger sample this might represent a change at the fully convective transition, whether due to magnetic field strength or structure, or its coupling to rotation (i.e. the stellar dynamo). It is by now well established that there is a sharp drop of X-ray and H$\alpha$ activity at late-M spectral types [$\sim$ M7...M8; e.g. @Cook14.0; @West08.1] but for mid-M spectral types, so far, X-ray studies have not been resolved in both $P_{\rm rot}$ and spectral type space together. If, e.g., late-M stars remain saturated up to longer periods, the decrease of the saturation level may go unnoticed in samples mixing the whole rotational distribution. We add in this study the first assessment of a link between rotation and chromospheric UV emission in M stars. Similar to the archival X-ray data, the UV data (from the GALEX mission) covers only a fraction of the K2 sample. A curious wealth of stars with high UV emission levels and long periods is seen that seems to be in contrast with the findings regarding all other activity indicators discussed in this work. Finally, our archive search for evidence of multiplicity in our targets raises an interesting point about the possible influence of multiplicity on rotation and activity levels. We find a high incidence of binarity in the group of fast rotators below the critical period at which magnetic activity apparently transitions to a lower level. The difference between the binary fraction of fast and slow rotators is statistically significant. Given the rather large binary separations (of tens to hundreds of AU) this is puzzling because no tidal interaction is expected for such wide systems. Nevertheless, we can speculate about a possible causal connection between binarity and rotation level. It is well established that wide companions accelerate the evolution of pre-main sequence disks [e.g. @Kraus12.0]. Shorter disk lifetimes translate into a shorter period of star-disk interaction and, hence, one may expect higher initial rotation rates on the main sequence for binary stars [@Herbst05.0]. As a result, it may take binaries longer to spin down. Alternatively, we could be seeing the mass-dependence of magnetic braking. With our low-number statistics we can not draw any firm conclusions. Note, however, that a relation between fast rotation and binarity, independent of stellar mass, was also found in a recent K2 study of the Hyades [@Douglas16.0]. Summary and outlook {#sect:summary} =================== From a joint rotation and multi-wavelength activity and variability study of nearby M dwarfs observed in K2 campaigns C0 to C4 we infer a critical period of $\sim 10$d at which photometric star spot and flare activity undergoes a dramatic change. This transition is coincident with the break separating saturated from ‘linear’ regime seen in traditional studies of the rotation-activity relation probing higher atmospheric layers (e.g. the corona through X-rays or the chromosphere through H$\alpha$ emission). We present here an updated view of the X-ray - rotation relation for M dwarfs. The sample analysed in this work has strongly increased the known number of long-period M dwarfs in the X-ray – rotation relation. Nevertheless, at present there is not enough sensitive data in the X-ray archives to constrain the X-ray – rotation relation for periods beyond $\sim 10$d. A key questions is now whether the coronal emission of M dwarfs displays a break-point analogous to the optical photometric activity tracers or whether there is a continuous decrease of activity as seen in FGK stars. This problem will be addressed in the near future with upcoming [*Chandra*]{} observations in which we sample the whole observed K2 rotation period distribution. We will also further examine the UV – rotation relation in the larger M dwarf sample that will be available at the end of the K2 mission. Moreover, in that larger sample we intend to search for a possible mass dependence of the rotation-activity relation within the M spectral sequence. A systematic assessment of multiplicity for these nearby M stars with Gaia will also be useful for examining the influence of a companion star on rotation and activity levels. The observed dichotomy between fast and slow rotators in terms of their magnetic activity level might have interesting consequences for habitability of planets near M stars being fried by flares and high-energy radiation until they have spun down to around $10$d. The time-scale for this process is as yet poorly constrained but certainly on the order of Gyrs, and it becomes longer the lower the mass of the star [@West08.1]. [@Segura10.0] found in models based on ADLeo that UV flares do not strongly affect planet chemistry but the accumulated effect of the exposure to strong flaring over most of the planet’s lifetime has not been studied so far. Phase-folded lightcurves {#sect:appendix_folded} ======================== In the online materials we present the phase-folded lightcurves for all periodic stars in two figures, one for periods flagged ‘Y’ (Fig. A1) and another one for periods flagged ‘?’ (Fig. A2). For each star the lightcurve was folded with the ‘adopted’ period, i.e. either the Lomb-Scargle period (LS), the auto-correlation period (ACF) or the period from the sine-fit (SINE); see Sect. \[subsect:k2\_analysis\_period\] for details. Search for binarity {#sect:appendix_bin} =================== We search all K2 Superblink stars for archival evidence of binarity. We proceed in several steps. First, we perform a visual inspection of POSS1$\_$RED and POSS2$\_$RED photographic plates by using the online Digitized Sky Survey (DSS) and the interactive tools of Aladin. Epochs of each pair of plates are separated by up to $\sim$40 years, with the most recent plates obtained in the 1990s. Comparison of the two epochs can help in identifying possible blends in the K2 photometry. Specifically, we examine if the targets significantly approached other stars due to their proper motion. Then, we search for photometric and astrometric information of each possible contaminant by matching the UCAC4 and 2MASS catalogs in Vizier. Taking into account that the K2 pixel scale is $\sim 4^{\prime\prime}/{\rm pixel}$, for those cases with possible blends we check the K2 imagettes and the photometric mask produced and used by A.Vanderburg in the reduction of the K2 data [^12] to estimate visually the occurrence of blending and its significance. For each target, the inspected imagette represents the sum of all the single imagettes recorded by K2 during a campaign. From our experience such merged imagettes are usually affected by the shift on the sensor of the photometric centroid due to pointing drift of the telescope. The photometric masks are wide enough to take into account the drift of the centroids, making any quantitative analysis of blending with other astrophysical objects rather difficult and beyond the scope of this work. We also search the Washington Double Star catalog (WDS) for information about binarity including sub-arcsec separations, which can not be detected simply by visual inspection of the photographic plates or matching with other catalogues. For binaries in the WDS we adopt the visual magnitude difference between the components indicated in the catalog, when available and other photometric measurements were missing. With this approach, we find evidence for a companion for $25$ stars. However, many of the secondaries have a $J$ magnitude which is more than $4$mag fainter than our target. These secondaries contribute at most a few percent to the flux of the system. They are unlikely to be responsible for the observed rotational signal, and we do not consider them any further. We list the remaining potential companions in Table \[tab:multiplicitytable\]. These objects have either a $J$ magnitude difference of $< 4$mag with respect to the corresponding K2 target, or a small separation according to the WDS catalog without known photometry, or both. Next to an identifier for the putative companion (col.3) we provide the binary separation (col.4), the epoch to which it refers (col.5), the $J$ magnitude or a magnitude difference between the two components according to the WDS (col.6-7), and flags indicating how we identfy it (through visual inspection of photographic plates, as entry in the WDS, or in the K2 imagette; cols.8-10). In a final ‘Notes’ column and in footnotes we add further explanations where needed. The most important fact to note concerns the binary Gl852AB. Both stars are in our target list (EPIC-206262223 and EPIC-206262336) but they are clearly unresolved in A.Vanderburg’s K2 pipeline. In fact, the lightcurves of both stars are identical because the aperture comprises an elongated object, clearly representing the two stars of the $8^{\prime\prime}$ binary. We also add a special note here on EPIC-204927969. Our inspection of the K2 imagette shows that the aperture used by Vanderburg includes other objects but our reconstruction of the lightcurve without the contaminated pixels proved that the rotational modulation is due to the target. Other possible contaminations to be taken serious regard the companions that have $J < 10$mag. There are two such objects listed in Table \[tab:multiplicitytable\]. Another three K2 Superblink stars have companions with $J < 12.5$mag which might contribute somewhat to the variability in the lightcurve. Further three multiples are presented in the literature, one spectroscopic binary and two close visual binaries. For the remaining objects in Table \[tab:multiplicitytable\] we find no photometric measurements, and they are likely faint and may not influence the K2 lightcurves. All stars in Table \[tab:multiplicitytable\] are flagged on the figures involving periods and activity measures from K2 data. Acknowledgments {#acknowledgments .unnumbered} =============== We would especially like to thank A. Vanderburg for his public release of the analysed K2 lightcurves, upon which much of the present work is based. We thank the anonymous referee for a very careful reading of our manuscript. This research has made use of the VizieR catalogue access tool, and the “Aladin sky atlas", both developed at CDS, Strasbourg Observatory, France. \[lastpage\] [^1]: E-mail:stelzer@astropa.inaf.it [^2]: http://iopscience.iop.org/article/10.3847/0004-637X/819/1/87/meta [^3]: https://www.aavso.org/apass [^4]: IDL is a product of the Exelis Visual Information Solutions, Inc. [^5]: Our analysis relies on the data reduction performed by A.Vanderburg; see Sect. \[sect:k2\_analysis\]. [^6]: The reduced K2 lightcurves were downloaded from https://www.cfa.harvard.edu/$\sim$avanderb/k2.html [^7]: The technical reports on the detrending process carried out by A.Vanderburg are accessible at https://www.cfa.harvard.edu/$\sim$avanderb/k2.html [^8]: We downloaded the Kepler lightcurves from the Target Search page at https://archive.stsci.edu/kepler/kepler\_fov/search.php [^9]: K2 raw and ‘eigen’ light curves were downloaded from http://bbq.dfm.io/ketu/lightcurves/ and http://bbq.dfm.io/ketu/elcs/ [^10]: Note, that the coordinates provided in the target lists of the individual K2 campaigns at http://keplerscience.arc.nasa.gov/k2-approved-programs.html refer to epoch 2000, except for campaign C0 where the coordinates seem to refer to the date of observation (Mar - May 2014). [^11]: The Portable Interactive Multi-Mission Simulator is accessible at http://cxc.harvard.edu/toolkit/pimms.jsp [^12]: https://www.cfa.harvard.edu/$\sim$avanderb/k2.html

--- abstract: 'We characterize the nuclearity of the Beurling-Björck spaces $\mathcal{S}^{(\omega)}_{(\eta)}(\mathbb{R}^d)$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}(\mathbb{R}^d)$ in terms of the defining weight functions $\omega$ and $\eta$.' address: | Department of Mathematics: Analysis, Logic and Discrete Mathematics\ Ghent University\ Krijgslaan 281\ 9000 Gent\ Belgium author: - Andreas Debrouwere - Lenny Neyt - Jasson Vindas title: 'Characterization of nuclearity for Beurling-Björck spaces' --- [^1] [^2] [^3] Introduction ============ In recent works Boiti et al. [@B-J-O2019; @B-J-O; @B-J-O-S] have investigated the nuclearity of the Beurling-Björck space $\mathcal{S}_{(\omega)}^{(\omega)}(\mathbb{R}^{d})$ (in our notation below). Their most general result [@B-J-O-S Theorem 3.3] establishes the nuclearity of this Fréchet space when $\omega$ is a Braun-Meise-Taylor type weight function [@braun-meise-taylor] (where non-quasianalyticity is replaced by $\omega(t)=o(t)$ and the condition $\log(t)=o(\omega(t))$ from [@braun-meise-taylor] is relaxed to $\log t=O(\omega(t))$). The aim of this note is to improve and generalize [@B-J-O-S Theorem 3.3] by considerably weakening the set of hypotheses on the weight functions, providing a complete characterization of the nuclearity of these spaces (for radially increasing weight functions), and considering anisotropic spaces and the Roumieu case as well. Particularly, we shall show that the conditions $(\beta)$ and $(\delta)$ from [@B-J-O-S Definition 2.1] play no role in deducing nuclearity. Let us introduce some concepts in order to state our main result. A weight function on $\mathbb{R}^{d}$ is simply a non-negative, measurable, and locally bounded function. We consider the following standard conditions [@bjorck66; @braun-meise-taylor]: - There are $L,C>0$ such that $\omega(x + y) \leq L(\omega(x) + \omega(y)) +\log C,$ for all $x,y \in \mathbb{R}^d.$ - There are $A,B>0$ such that $A\log (1+|x|)\leq \omega(x)+ \log B,$ for all $x\in\mathbb{R}^{d}$. - $\displaystyle \lim_{|x|\to\infty}\frac{\omega(x)}{\log |x|}=\infty. $ A weight function $\omega$ is called radially increasing if $\omega(x) \leq \omega(y)$ whenever $|x| \leq |y|$. Given a weight function $\omega$ and a parameter $\lambda>0$, we introduce the family of norms $$\|\varphi\|_{\omega,\lambda} = \sup_{x\in\mathbb{R}^{d}} |\varphi(x)|e^{\lambda \omega(x)}.$$ If $\eta$ is another weight function, we consider the Banach space $\mathcal{S}^{\lambda}_{\eta,\omega}(\mathbb{R}^{d})$ consisting of all $\varphi \in \mathcal{S}'(\mathbb{R}^{d})$ such that $\|\varphi\|_{\mathcal{S}^{\lambda}_{\eta,\omega}}:= \|\varphi\|_{\eta,\lambda}+\|\widehat{\varphi}\|_{\omega,\lambda}<\infty,$ where $\widehat{\varphi}$ stands for the Fourier transform of $\varphi$. Finally, we define the Beurling-Björck spaces (of Beurling and Roumieu type) as $$\mathcal{S}_{(\eta)}^{(\omega)}(\mathbb{R}^{d})=\varprojlim_{\lambda \to\infty} \mathcal{S}^{\lambda}_{\eta,\omega}(\mathbb{R}^{d}) \quad \mbox{ and }\quad \mathcal{S}_{\{\eta\}}^{\{\omega\}}(\mathbb{R}^{d})=\varinjlim_{\lambda\to0^{+}} \mathcal{S}^{\lambda}_{\eta,\omega}(\mathbb{R}^{d}).$$ \[nuclearity theorem Beurling-Bjorck\] Let $\omega$ and $\eta$ be weight functions satisfying $(\alpha)$. - If $\omega$ and $\eta$ satisfy $(\gamma)$, then $\mathcal{S}_{(\eta)}^{(\omega)}(\mathbb{R}^{d})$ is nuclear. Conversely, if in addition $\omega$ and $\eta$ are radially increasing, then the nuclearity of $\mathcal{S}_{(\eta)}^{(\omega)}(\mathbb{R}^{d})$ implies that $\omega$ and $\eta$ satisfy $(\gamma)$ (provided that $\mathcal{S}_{(\eta)}^{(\omega)}(\mathbb{R}^{d}) \neq \{0\}$). - If $\omega$ and $\eta$ satisfy $(\gamma_0)$, then $\mathcal{S}_{\{\eta\}}^{\{\omega\}}(\mathbb{R}^{d})$ is nuclear. Conversely, if in addition $\omega$ and $\eta$ are radially increasing, then the nuclearity of $\mathcal{S}_{\{\eta\}}^{\{\omega\}}(\mathbb{R}^{d})$ implies that $\omega$ and $\eta$ satisfy $(\gamma_0)$ (provided that $\mathcal{S}_{\{\eta\}}^{\{\omega\}}(\mathbb{R}^{d}) \neq \{0\}$). Furthermore, we discuss the equivalence of the various definitions of Beurling-Björck type spaces given in the literature [@C-C-K; @grochenig-zimmermann; @B-J-O-S] but considered here under milder assumptions. In particular, we show that, if $\omega$ satisfies $(\alpha)$ and $(\gamma)$, our definition of $\mathcal{S}_{(\omega)}^{(\omega)}(\mathbb{R}^{d})$ coincides with the one employed in [@B-J-O-S]. The conditions $(\gamma)$ and $(\gamma_0)$ ========================================== In this preliminary section, we study the connection between the conditions $(\gamma)$ and $(\gamma_0)$ and the equivalence of the various definitions of Beurling-Björck type spaces. Let $\omega$ and $\eta$ be two weight functions. Given parameters $k,l \in \mathbb{N}$ and $\lambda > 0$, we introduce the family of norms $$\|\varphi\|_{\omega,k,l,\lambda} = \max_{|\alpha|\leq k} \max_{|\beta| \leq l} \sup_{x \in \mathbb{R}^d} |x^{\beta}\varphi^{(\alpha)}(x)e^{\lambda\omega(x)}|.$$ We define $\widetilde{\mathcal{S}}^{\lambda}_{\eta,\omega}(\mathbb{R}^{d})$ as the Fréchet space consisting of all $\varphi \in \mathcal{S}(\mathbb{R}^d)$ such that $$\|\varphi\|_{\widetilde{\mathcal{S}}^{k,\lambda}_{\eta,\omega}}:= \|\varphi\|_{\eta,k,k,\lambda}+\|\widehat{\varphi}\|_{\omega,k,k,\lambda}<\infty, \qquad \forall k \in \mathbb{N}.$$ We set $$\widetilde{\mathcal{S}}_{(\eta)}^{(\omega)}(\mathbb{R}^{d})=\varprojlim_{\lambda \to \infty} \widetilde{\mathcal{S}}^{\lambda}_{\eta,\omega}(\mathbb{R}^{d}) \quad \mbox{ and }\quad \widetilde{\mathcal{S}}_{\{\eta\}}^{\{\omega\}}(\mathbb{R}^{d})=\varinjlim_{\lambda\to0^{+}} \widetilde{\mathcal{S}}^{\lambda}_{\eta,\omega}(\mathbb{R}^{d}).$$ The following result is a generalization of [@C-C-K Theorem 3.3] and [@grochenig-zimmermann Corollary 2.9] (see also [@B-J-O-S Theorem 2.3]). We use $\mathcal{S}_{\eta}^{\omega}(\mathbb{R}^{d})$ as a common notation for $\mathcal{S}_{(\eta)}^{(\omega)}(\mathbb{R}^{d})$ and $\mathcal{S}_{\{\eta\}}^{\{\omega\}}(\mathbb{R}^{d})$; a similar convention will be used for other spaces. \[equivalent\] Let $\omega$ and $\eta$ be two weight functions satisfying $(\alpha)$. Suppose that $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) \neq \{ 0 \}$. The following statements are equivalent: 1. $\omega$ and $\eta$ satisfy $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). 2. $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) = \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^d)$ as locally convex spaces. 3. $\displaystyle \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) = \{ \varphi \in \mathcal{S}'(\mathbb{R}^d) \, | \, \forall \lambda > 0 \,(\exists \lambda > 0) \, \forall \alpha \in \mathbb{N}^d \, : \, \newline \phantom{\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) = \{} \sup_{x \in \mathbb{R}^d} |x^\alpha\varphi(x)|e^{\lambda \eta(x)} < \infty \quad \mbox{and} \quad \sup_{\xi \in \mathbb{R}^d} |\xi^{\alpha}\widehat\varphi(\xi)|e^{\lambda \omega(\xi)} < \infty\}$. 4. $\displaystyle \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) = \{ \varphi \in \mathcal{S}'(\mathbb{R}^d) \, | \, \forall \lambda > 0 \, (\exists \lambda > 0) \, \forall \alpha \in \mathbb{N}^d \, : \, \newline \phantom{\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) = \{} \int_{x \in \mathbb{R}^d} |\varphi^{(\alpha)}(x)|e^{\lambda \eta(x)} dx< \infty \quad \mbox{and} \quad \int_{\xi \in \mathbb{R}^d} |\widehat\varphi^{(\alpha)}(\xi)|e^{\lambda \omega(\xi)} d\xi< \infty\}$. 5. $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) \subseteq \mathcal{S}(\mathbb{R}^d)$. Following [@grochenig-zimmermann], our proof of Theorem \[equivalent\] is based on the mapping properties of the short-time Fourier transform (STFT). We fix the constants in the Fourier transform as $$\mathcal{F}(\varphi)(\xi) = \widehat{\varphi}(\xi)=\int_{\mathbb{R}^{d}} \varphi(x) e^{-2\pi i \xi \cdot x}dx.$$ The STFT of $f \in L^{2}(\mathbb{R}^{d})$ with respect to the window $\psi \in L^{2}(\mathbb{R}^{d})$ is given by $$V_{\psi} f(x, \xi) = \int_{\mathbb{R}^{d}} f(t) \overline{\psi(t - x)} e^{- 2 \pi i \xi \cdot t} dt , \qquad (x, \xi) \in \mathbb{R}^{2d} .$$ The adjoint of $V_{\psi}: L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}^{2d})$ is given by the (weak) integral $$V_{\psi}^{*} F(t) = \iint _{\mathbb{R}^{2d}} F(x, \xi) e^{2 \pi i \xi \cdot t}\psi(t-x) dx d\xi .$$ A straight forward calculation shows that, whenever $(\gamma, \psi)_{L^{2}(\mathbb{R}^{d})} \neq 0$, then $$\label{eq:reconstructSTFT} \frac{1}{(\gamma, \psi)_{L^{2}}} V_{\gamma}^{*} \circ V_{\psi} = {\operatorname*{id}}_{L^{2}(\mathbb{R}^{d})} .$$ Next, we introduce two additional function spaces. Given a parameter $\lambda >0$, we define $\mathcal{S}^{\lambda}_{\omega}(\mathbb{R}^{d})$ as the Fréchet space consisting of all $\varphi \in C^\infty(\mathbb{R}^d)$ such that $\|\varphi\|_{\omega,k,\lambda} := \|\varphi\|_{\omega,k,0,\lambda} < \infty$ for all $k \in \mathbb{N}$ and set $$\mathcal{S}_{(\omega)}(\mathbb{R}^{d}) = \varprojlim_{\lambda \to\infty} \mathcal{S}^{\lambda}_{\omega}(\mathbb{R}^{d}) \quad \mbox{and}\quad \mathcal{S}_{\{\omega\}}(\mathbb{R}^{d}) =\varinjlim_{\lambda \to 0^{+}} \mathcal{S}^{\lambda}_{\omega}(\mathbb{R}^{d}).$$ We recall that $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ stands for the common notation to simultaneously treat $\mathcal{S}_{(\omega)}(\mathbb{R}^{d})$ and $\mathcal{S}^{\lambda}_{\omega}(\mathbb{R}^{d})$. Given a parameter $\lambda > 0$, we define $C^{\lambda}_{\omega}(\mathbb{R}^{d})$ as the Banach space consisting of all $\varphi \in C(\mathbb{R}^d)$ such that $\|\varphi\|_{\omega,\lambda} < \infty$ and set $$C_{(\omega)}(\mathbb{R}^{d}) = \varprojlim_{\lambda \to\infty} C^{\lambda}_{\omega}(\mathbb{R}^{d}) \quad \mbox{and}\quad C_{\{\omega\}}(\mathbb{R}^{d}) =\varinjlim_{\lambda \to 0^{+}} C^{\lambda}_{\omega}(\mathbb{R}^{d}).$$ We need the following extension of [@grochenig-zimmermann Theorem 2.7]. We write $\check{f}(t)=f(-t)$. \[STFT Beurling-Bjorck\] Let $\omega$ and $\eta$ be weight functions satisfying $(\alpha)$ and $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). Define the weight $\eta\oplus\omega (x,\xi):=\eta(x)+\omega(\xi)$ for $(x,\xi)\in\mathbb{R}^{2d}$. Fix a window $\psi \in \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d})$. - The linear mappings $$V_{\check{\psi}}: \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d}) \to C_{\eta\oplus\omega}(\mathbb{R}^{2d}) \quad \mbox{ and } \quad V^{\ast}_{\psi}:C_{\eta\oplus\omega}(\mathbb{R}^{2d}) \to \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d})$$ are continuous. - The linear mappings $$V_{\check{\psi}}: \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d}) \to \mathcal{S}_{\eta\oplus\omega}(\mathbb{R}^{2d}) \quad \mbox{ and } \quad V^{\ast}_{\psi}: \mathcal{S}_{\eta\oplus\omega}(\mathbb{R}^{2d}) \to \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d})$$ are continuous. It suffices to show that $V_{\check{\psi}}: \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d}) \to \mathcal{S}_{\eta\oplus\omega}(\mathbb{R}^{2d})$ and $V^{\ast}_{\psi}:C_{\eta\oplus\omega}(\mathbb{R}^{2d}) \to \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d})$ are continuous. Suppose that $\psi\in \widetilde{\mathcal{S}}^{\lambda_0}_{\eta,\omega}(\mathbb{R}^{d})$, so that $\lambda_0 > 0$ is fixed in the Roumieu case but can be taken as large as needed in the Beurling case. Let $A$ and $B=B_A$ be the constants occurring in $(\gamma)$ (in the Roumieu case, $A$ can be taken as large as needed due to $(\gamma_0)$). Furthermore, we assume that all constants occurring in $(\alpha)$ and $(\gamma)$ ($(\gamma_0)$ in the Roumieu case) are the same for both $\omega$ and $\eta$. We first consider $V_{\check\psi}$. Let $\lambda < \lambda_0/L$ be arbitrary. For all $k \in \mathbb{N}$ and $\varphi \in \mathcal{S}^{\lambda L + \frac{k}{A} }_{\eta,\omega}(\mathbb{R}^{d})$ it holds that $$\max_{|\alpha + \beta| \leq k} \sup_{(x,\xi) \in \mathbb{R}^{2d}} |\partial^{\beta}_{\xi}\partial^{\alpha}_{x}V_{\check{\psi}}\varphi(x,\xi)|e^{\lambda \eta(x)} \leq (2\pi)^{k} BC \|\psi\|_{\eta,k,\lambda_0} \|\varphi\|_{\eta, \lambda L + \frac{k}{A}} \int_{\mathbb{R}^d} e^{-(\lambda_0 - \lambda L)\eta(t)} dt$$ and $$\begin{aligned} \max_{|\alpha + \beta| \leq k} \sup_{(x,\xi) \in \mathbb{R}^{2d}} |\partial^{\beta}_{\xi}\partial^{\alpha}_{x}V_{\check{\psi}}\varphi(x,\xi)|e^{\lambda \omega(\xi)} & = \max_{|\alpha +\beta| \leq k} \sup_{(x,\xi) \in \mathbb{R}^{2d}} |\partial^{\beta}_{\xi}\partial^{\alpha}_{x}V_{\check{\mathcal{F}(\psi)}}\widehat{\varphi}(\xi,-x)|e^{\lambda \omega(\xi)} \\ & \leq (2\pi)^{k} BC \|\widehat{\psi}\|_{\omega,k,\lambda_0} \|\widehat{\varphi}\|_{\omega, \lambda L + \frac{k}{A}} \int_{\mathbb{R}^d} e^{-(\lambda_0 - \lambda L)\omega(t)} dt.\end{aligned}$$ These inequalities imply the continuity of $V_{\check{\psi}}$. Next, we treat $V^*_\psi$. Let $\lambda <\lambda_0/L$ be arbitrary. For all $k \in \mathbb{N}$ and $\Phi \in C^{\lambda L + \frac{k}{A}}_{\eta \oplus \omega}(\mathbb{R}^{2d})$ and it holds that $$\|V_{\psi}^{\ast}\Phi\|_{\eta,k,\lambda} \leq (4\pi)^{k}BC\|\psi\|_{\eta,k,\lambda_0}\|\Phi\|_{\eta\oplus\omega,\lambda L + \frac{k}{A}} \int\int_{\mathbb{R}^{2d}} e^{-\lambda L\omega(\xi)- (\lambda_0-\lambda L)\eta(x)} dx d\xi$$ and $$\|\mathcal{F}(V_{\psi}^{\ast}\Phi)\|_{\omega,k,\lambda} \leq (4\pi)^{k}BC\|\widehat{\psi}\|_{\omega,k,\lambda_0}\|\Phi\|_{\eta\oplus\omega,\lambda L+ \frac{k}{A}} \int\int_{\mathbb{R}^{2d}} e^{-\lambda L\eta(x)- (\lambda_0-\lambda L)\omega(\xi)} dx d\xi.$$ Since $\|\, \cdot \, \|_{\eta,k,k,\lambda} \leq B \|\, \cdot \, \|_{\eta,k,\lambda + \frac{k}{A}}$ and $\|\, \cdot \, \|_{\omega,k,k,\lambda} \leq B \|\, \cdot \, \|_{\omega,k,\lambda + \frac{k}{A}}$ for all $\lambda > 0$ and $k \in \mathbb{N}$, the above inequalities show the continuity of $V^*_\psi$. In order to be able to apply Proposition \[STFT Beurling-Bjorck\], we show the ensuing simple lemma. \[non-trivial\] Let $\omega$ and $\eta$ be weight functions satisfying $(\alpha)$. If $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d}) \neq \{0\}$, then also $\widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d}) \neq \{0\}$. Let $\varphi \in \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d}) \backslash \{0\}$. Pick $\psi, \chi \in \mathcal{D}(\mathbb{R}^d)$ such that $\int_{\mathbb{R}^d} \varphi(x)\psi(-x) dx = 1$ and $\int_{\mathbb{R}^d} \chi(x) dx = 1$. Then, $\varphi_0 = (\varphi \ast \chi)\mathcal{F}^{-1}(\psi) \in \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d})$ and $\varphi_0 \not \equiv 0$ (as $\varphi_0(0) = 1$). $(i) \Rightarrow (ii)$ In view of Lemma \[non-trivial\], this follows from Proposition \[STFT Beurling-Bjorck\] and the reconstruction formula . $(ii) \Rightarrow (iii)$ Trivial. $(iii) \Rightarrow (v)$ *and* $(iv) \Rightarrow (v)$ These implications follow from the fact that $\mathcal{S}(\mathbb{R}^d)$ consists precisely of all those $\varphi \in \mathcal{S}'(\mathbb{R^d})$ such that $$\sup_{x \in \mathbb{R}^d} |x^\alpha\varphi(x)| < \infty \quad \mbox{and} \quad \sup_{\xi \in \mathbb{R}^d} |\xi^{\alpha}\widehat\varphi(\xi)| < \infty$$ for all $\alpha \in \mathbb{N}^d$ (see e.g. [@C-C-K Corollary 2.2]). $(v) \Rightarrow (i)$ Since the Fourier transform is an isomorphism from $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d)$ onto $\mathcal{S}_{\omega}^{\eta}(\mathbb{R}^d)$ and from $\mathcal{S}(\mathbb{R}^d)$ onto itself, it is enough to show that $\eta$ satisfies $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). We start by constructing $\varphi_0 \in \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d)$ such that $\varphi(j) = \delta_{j,0}$ for all $j \in \mathbb{Z}^d$. Choose $\varphi \in \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d)$ such that $\varphi(0) = 1$. Set $$\chi(x) = \int_{[-\frac{1}{2}, \frac{1}{2}]^d} e^{-2\pi ix \cdot t} dt, \qquad x \in \mathbb{R}^d.$$ Then, $\chi(j) = \delta_{j,0}$ for all $j \in \mathbb{Z}^d$. Hence, $\varphi_0 = \varphi \chi$ satisfies all requirements. Let $(\lambda_j)_{j \in \mathbb{Z}^d}$ be an arbitrary multi-indexed sequence of positive numbers such that $\lambda_j \to \infty$ as $|j| \to \infty$ ($(\lambda_j)_{j \in \mathbb{Z}^d} = (\lambda)_{j \in \mathbb{Z}^d}$ for $\lambda > 0$ in the Roumieu case). Consider $$\varphi = \sum_{j \in \mathbb{Z}^d} \frac{e^{-\lambda_j\eta(j)}}{(1+|j|)^{d+1}} \varphi_0( \, \cdot - j) \in \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d).$$ Since $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^d) \subseteq \mathcal{S}(\mathbb{R}^d)$, there is $C > 0$ such that $$\frac{e^{-\lambda_j\eta(j)}}{(1+|j|)^{d+1}} = |\varphi(j)| \leq \frac{C}{(1+|j|)^{d+2}}$$ for all $j \in \mathbb{Z}^d$. Hence, $$\log(1+|j|) \leq \lambda_j \eta(j) + \log C$$ for all $j \in \mathbb{Z}^d$. As $\eta$ satisfies $(\alpha)$ and $(\lambda_j)_{j \in \mathbb{Z}^d}$ is arbitrary, the latter inequality is equivalent to $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). $(i) \Rightarrow (iv)$ Let us denote the space in the right-hand side of $(iv)$ by $\mathcal{S}_{\eta,1}^{\omega,1} (\mathbb{R}^d)$. Since we already showed that $(i) \Rightarrow (ii)$ and we have that $\widetilde{\mathcal{S}}^\omega_\eta (\mathbb{R}^d) \subseteq \mathcal{S}_{\eta,1}^{\omega,1} (\mathbb{R}^d)$, it suffices to show that $\mathcal{S}_{\eta,1}^{\omega,1} (\mathbb{R}^d) \subseteq \widetilde{\mathcal{S}}^\omega_\eta (\mathbb{R}^d)$. By Proposition \[STFT Beurling-Bjorck\]$(a)$, Lemma \[non-trivial\] and the reconstruction formula , it suffices to show that $V_{\check{\psi}}(\varphi) \in C_{\eta \oplus \omega}(\mathbb{R}^{2d})$ for all $\varphi \in \mathcal{S}_{\eta,1}^{\omega,1} (\mathbb{R}^d)$, where $\psi \in \widetilde{\mathcal{S}}^\omega_\eta (\mathbb{R}^d)$ is a fixed window. But the latter can be shown by using the same method employed in the first part of the proof of Proposition \[STFT Beurling-Bjorck\]. Proof of Theorem \[nuclearity theorem Beurling-Bjorck\] ======================================================= Our proof of Theorem \[nuclearity theorem Beurling-Bjorck\] is based on Proposition \[STFT Beurling-Bjorck\]$(b)$ and the next two auxiliary results. \[proposition nuclear\] Let $\eta$ be a weight function satisfying $(\alpha)$ and $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). Then, $\mathcal{S}_{\eta}(\mathbb{R}^{d})$ is nuclear. We present two different proofs: $(i)$ The first one is based on a classical result of Gelfand and Shilov [@Gelfand-Shilov3 p. 181]. The nuclearity of $\mathcal{S}_{(\eta)}(\mathbb{R}^{d})$ is a particular case of this result, as the increasing sequence of weight functions $(e^{n\eta})_{n\in \mathbb{N}}$ satisfies the so-called $(P)$ and $(N)$ conditions because of $(\gamma)$. For the Roumieu case, note that $$\mathcal{S}_{\eta}(\mathbb{R}^{d}) = \varinjlim_{n \in \mathbb{Z}_+}\varprojlim_{k \geq n}\mathcal{S}^{\frac{1}{n}-\frac{1}{k}}_{\eta}(\mathbb{R}^{d})$$ as locally convex spaces. The above mentioned result implies that, for each $n \in \mathbb{Z}_+$, the Fréchet space $\varprojlim_{k \geq n}\mathcal{S}^{\frac{1}{n}-\frac{1}{k}}_{\eta}(\mathbb{R}^{d})$ is nuclear, as the increasing sequence of weight functions $(e^{\left(\frac{1}{n}-\frac{1}{k}\right)\eta})_{k \geq n}$ satisfies the conditions $(P)$ and $(N)$ because of $(\gamma_0)$. The result now follows from the fact that the inductive limit of a countable spectrum of nuclear spaces is again nuclear [@Treves]. $(ii)$ Next, we give a proof that simultaneously applies to the Beurling and Roumieu case and only makes use of the fact that $\mathcal{S}(\mathbb{R}^d)$ is nuclear. Our argument adapts an idea of Hasumi [@hasumi61]. Fix a non-negative function $\chi\in\mathcal{D}(\mathbb{R}^{d})$ such that $\int_{\mathbb{R}^{d}}\chi(y)dy=1$ and for each $\lambda>0$ let $$\Psi_\lambda(x)= \exp\left( \lambda L\int_{\mathbb{R}^{d}} \chi(y)\eta(x+y) dy\right).$$ It is clear from the assumption $(\alpha)$ that $\eta$ should have at most polynomial growth. So, we fix $q>0$ such that $(1+|x|)^{-q}\omega(x)$ is bounded. We obtain that there are positive constants $c_{\lambda}$, $C_\lambda$, $C_{\lambda,\beta}$, and $C_{\lambda_1,\lambda_2, \beta}$ such that $$\label{inequalities regularized weight} c_\lambda \exp\left(\lambda \eta( x)\right)\leq \Psi_{\lambda} (x)\leq C_\lambda \exp(L^{2} \lambda \eta(x)), \quad |\Psi_{\lambda}^{(\beta)}(x)|\leq C_{\lambda, \beta}(1+|x|)^{q|\beta|}\Psi_{\lambda}(x) ,$$ and $$\label{inequalities regularized weight 2} \left|\left(\frac{\Psi_{\lambda_1}}{\Psi_{\lambda_2}} \right)^{(\beta)}(x)\right|\leq C_{\lambda_1,\lambda_2, \beta}(1+|x|)^{q|\beta|} ,$$ for each $\beta\in\mathbb{N}^{d}$, and $\lambda_1\leq \lambda_2$. Let $X_\lambda = \Psi_{\lambda }^{-1}\mathcal{S}(\mathbb{R}^{d})$ and topologize each of these spaces in such a way that the multiplier mappings $M_{\Psi_\lambda}: X_\lambda \to \mathcal{S}(\mathbb{R}^{d}): \ \varphi\mapsto \Psi_\lambda \cdot \varphi$ are isomorphisms. The bounds guarantee that the inclusion mappings $X_{\lambda_2}\to X_{\lambda_1}$ are continuous whenever $\lambda_{1}\leq \lambda_{2}$. If $A$ is a constant such that $(\gamma)$ holds for $\eta$, then the inequalities clearly yield $$\max_{|\beta|\leq k} \sup_{x\in\mathbb{R}^{d}} (1+|x|)^{k} |(\Psi_{\lambda}\varphi)^{(\beta)}(x)|\leq B_{k,\lambda,A} \|\varphi\|_{\eta,k,\lambda L^2+(1+q)k/A}$$ and $$\begin{aligned} \|\varphi\|_{\eta,k,\lambda}&\leq \frac{1}{c_\lambda} \max_{|\beta|\leq k}\|\Psi_{\lambda} \varphi^{(\beta)} \|_{L^{\infty}(\mathbb{R}^{d})} \\ & \leq \frac{1}{c_\lambda} \max_{|\beta|\leq k} \left(\|(\Psi_{\lambda} \varphi)^{(\beta)} \|_{L^{\infty}(\mathbb{R}^{d})}+\sum_{\nu<\beta}\binom{\beta}{\nu}\left\| \Psi_{\lambda}^{(\beta-\nu)} \varphi^{(\nu)}\right\|_{L^{\infty}(\mathbb{R}^{d})} \right) \\ & \ll \max_{|\beta|\leq k} \|(\Psi_{\lambda} \varphi)^{(\beta)} \|_{L^{\infty}(\mathbb{R}^{d})}+ \max_{|\beta|\leq k-1}\|(1+|\cdot |)^{qk}\Psi_{\lambda} \varphi^{(\beta)} \|_{L^{\infty}(\mathbb{R}^{d})} \\ & \leq b_{k,\lambda} \max_{|\beta|\leq k} \|(1+|\cdot |)^{qk(k+1))/2}(\Psi_{\lambda} \varphi)^{(\beta)} \|_{L^{\infty}(\mathbb{R}^{d})},\end{aligned}$$ for some positive constants $B_{k,\lambda, A}$ and $b_{k,\lambda}$. This gives, as locally convex spaces, $$\mathcal{S}_{(\eta)}(\mathbb{R}^{d})=\varprojlim_{n \in \mathbb{Z}_+} X_{n}$$ and the continuity of the inclusion $X_{\lambda}\to \mathcal{S}_{\eta}^{\lambda}(\mathbb{R}^{d})$. If in addition $(\gamma_{0})$ holds, we can choose $A$ arbitrarily large above. Consequently, the inclusion $\mathcal{S}_{\eta}^{L^2\lambda+\varepsilon}(\mathbb{R}^{d})\to X_{\lambda} $ is continuous as well for any arbitrary $\varepsilon>0$, whence we infer the topological equality $$\mathcal{S}_{\{\eta\}}(\mathbb{R}^{d})=\varinjlim_{n \in \mathbb{Z}_+} X_{1/n}.$$ The claimed nuclearity of $\mathcal{S}_{(\eta)}(\mathbb{R}^{d})$ and $\mathcal{S}_{\{\eta\}}(\mathbb{R}^{d})$ therefore follows from that of $\mathcal{S}(\mathbb{R}^{d})$ and the well-known stability of this property under projective and (countable) inductive limits [@Treves]. The next result is essentially due to Petzsche [@Petzsche]. Given a multi-indexed sequence $a = (a_j)_{j \in \mathbb{Z}^d}$ of positive numbers, we define $l^r(a) = l^r(\mathbb{Z}^d;a)$, $r \in \{1,\infty\}$, as the Banach space consisting of all $c = (c_j)_{j \in \mathbb{Z}^d} \in \mathbb{C}^{\mathbb{Z}^d}$ such that $\|c\|_{l^r(a)} := \|(c_ja_j)_{j \in \mathbb{Z}^d}\|_{l^r} < \infty$. [@Petzsche Satz 3.5 and Satz 3.6] \[P-trick\] - Let $A = (a_{n})_{n \in \mathbb{N}} = (a_{n,j})_{n \in \mathbb{N}, j \in \mathbb{Z}^d}$ be a matrix of positive numbers such that $a_{n,j} \leq a_{n+1,j}$ for all $n \in \mathbb{N}, j \in \mathbb{Z}^d$. Consider the Köthe echelon spaces $\lambda^r(A) := \varprojlim_{n \in \mathbb{N}} l^r(a_n)$, $r \in \{1,\infty\}$. Let $E$ be a nuclear locally convex Hausdorff space and suppose that there are continuous linear mappings $T:\lambda^1(A) \rightarrow E$ and $S: E \rightarrow \lambda^\infty(A)$ such that $S \circ T = \iota$, where $\iota: \lambda^1(A) \rightarrow \lambda^\infty(A)$ denotes the natural embedding. Then, $\lambda^1(A)$ is nuclear. - Let $A = (a_{n})_{n \in \mathbb{N}} = (a_{n,j})_{n \in \mathbb{N}, j \in \mathbb{Z}^d}$ be a matrix of positive numbers such that $a_{n+1,j} \leq a_{n,j}$ for all $n \in \mathbb{N}, j \in \mathbb{Z}^d$. Consider the Köthe co-echelon spaces $\lambda^r(A) := \varinjlim_{n \in \mathbb{N}} l^r(a_n)$, $r \in \{1,\infty\}$. Let $E$ be a nuclear $(DF)$-space and suppose that there are continuous linear mappings $T:\lambda^1(A) \rightarrow E$ and $S: E \rightarrow \lambda^\infty(A)$ such that $S \circ T = \iota$, where $\iota: \lambda^1(A) \rightarrow \lambda^\infty(A)$ denotes the natural embedding. Then, $\lambda^1(A)$ is nuclear. $(a)$ This follows from an inspection in the second part of the proof of [@Petzsche Satz 3.5]; the conditions stated there are not necessary for this part of the proof. $(b)$ By transposing, we obtain continuous linear mappings $T^t: E'_b \rightarrow (\lambda^1(A))'_b$ and $S: (\lambda^\infty(A))'_b \rightarrow E'_b$ such that $T^t \circ S^t = \iota^t$. Consider the matrix $A^\circ = (1/a_n)_{n \in \mathbb{N}}$ and the natural continuous embeddings $\iota_1: \lambda^1(A^\circ) \rightarrow (\lambda^\infty(A))'_b$ (the continuity of $\iota_1$ follows from the fact that $\lambda^\infty(A)$ is a regular $(LB)$-space [@Bierstedt p. 81]) and $\iota_2: (\lambda^1(A))'_b \rightarrow \lambda^\infty(A^\circ)$. Then, we have that $(\iota^2 \circ T^t) \circ (S^t \circ \iota_1) = \tau$, where $\tau: \lambda^1(A^\circ) \rightarrow \lambda^\infty(A^\circ)$ denotes the natural embedding. Since $E'_b$ is nuclear (as the strong dual of a nuclear $(DF)$-space [@Treves]), part $(a)$ yields that $\lambda^1(A^\circ)$ is nuclear, which in turn implies the nuclearity of $\lambda^1(A)$ (cf. [@M-V Proposition 28.16]). We now ready to prove Theorem \[nuclearity theorem Beurling-Bjorck\]. We first suppose that $\omega$ and $\eta$ satisfy $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). W.l.o.g. we may assume that $\mathcal{S}^\omega_\eta(\mathbb{R}^d) \neq \{0\}$. In view of Lemma \[non-trivial\], Proposition \[STFT Beurling-Bjorck\]$(b)$ and the reconstruction formula imply that $\mathcal{S}^\omega_\eta(\mathbb{R}^d)$ is isomorphic to a (complemented) subspace of $\mathcal{S}_{\eta\oplus\omega}(\mathbb{R}^{2d})$. The latter space is nuclear by Proposition \[proposition nuclear\]. The result now follows from the fact that nuclearity is inherited to subspaces [@Treves]. Next, we suppose that $\omega$ and $\eta$ are radially increasing and that $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d})$ is nuclear and non-trivial. Since the Fourier transform is a topological isomorphism from $\mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d})$ onto $\mathcal{S}_{\omega}^{\eta}(\mathbb{R}^{d})$, it is enough to show that $\eta$ satisfies $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). Set $A_{(\eta)} = (e^{n\eta(j)})_{n \in \mathbb{N}, j \in \mathbb{Z}^d}$ and $A_{\{\eta\}} = (e^{\frac{1}{n}\eta(j)})_{n \in \mathbb{Z}_+, j \in \mathbb{Z}^d}$. Note that $\lambda^1(A_{\eta})$ is nuclear if and only if (cf. [@M-V Proposition 28.16]) $$\exists \lambda > 0 \, (\forall \lambda > 0) \, : \, \sum_{j \in \mathbb{Z}^d} e^{-\lambda \eta(j)} < \infty.$$ As $\eta$ is radially increasing and satisfies $(\alpha)$, the above condition is equivalent to $(\gamma)$ ($(\gamma_0)$ in the Roumieu case). Hence, it suffices to show that $\lambda^1(A_\eta)$ is nuclear. To this end, we use Proposition \[P-trick\] with $A = A_\eta$ and $E = \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d})$. We start by constructing $\varphi_0 \in \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d})$ such that $$\int_{[0,\frac{1}{2}]^d} \varphi_{0}(j+x) dx = \delta_{j,0}, \qquad j \in \mathbb{Z}^d. \label{delta-sum}$$ By Lemma \[non-trivial\], there is $\varphi \in \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d})$ such that $\varphi(0) = 1$. Set $$\chi(x) = \frac{1}{2^d}\int_{[-1, 1]^d} e^{-2\pi ix \cdot t} dt, \qquad x \in \mathbb{R}^d.$$ Then, $\chi(j/2) = \delta_{j,0}$ for all $j \in \mathbb{Z}^d$. Hence, $\psi = \varphi \chi \in \widetilde{\mathcal{S}}^{\omega}_{\eta}(\mathbb{R}^{d})$ and $\psi(j/2) = \delta_{j,0}$ for all $j \in \mathbb{Z}^d$. Then, $\varphi_0 = (-1)^d \partial^d \cdots \partial^1 \psi$ satisfies all requirements. The linear mappings $$T: \lambda^1(A_\eta) \rightarrow \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d}), \quad T( (c_j)_{j \in \mathbb{Z}^d}) = \sum_{j \in \mathbb{Z}^d} c_j \varphi_0(\, \cdot \, - j)$$ and $$S: \mathcal{S}^{\omega}_{\eta}(\mathbb{R}^{d}) \rightarrow \lambda^\infty(A_\eta), \quad S(\varphi) = \left(\int_{[0,\frac{1}{2}]^d}\varphi(x+j) dx \right)_{j \in \mathbb{Z}^d}$$ are continuous. Moreover, by , we have that $S \circ T = \iota$. [99]{} K. D. Bierstedt, *An introduction to locally convex inductive limits*, pp. 35–133, in: *Functional analysis and its applications*, World Sci. Publishing, 1988. G. Björck, *Linear partial differential operators and generalized distributions,* Ark. Mat. **6** (1966), 351–407. C. Boiti, D. Jornet, A. Oliaro, *The Gabor wave front set in spaces of ultradifferentiable functions,* Monatsh. Math. **188** (2019), 199–246. C. Boiti, D. Jornet, A. Oliaro, *About the nuclearity of $\mathcal{S}_{(M_p)}$ and $\mathcal{S}_{\omega}$*, preprint, arXiv:1902.09187. C. Boiti, D. Jornet, A. Oliaro, G. Schindl, *Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis,* preprint, arXiv:1906.05171. R. W. Braun, R. Meise, B. A. Taylor, *Ultradifferentiable functions and Fourier analysis,* Results Math. **17** (1990), 206–237. S. Y. Chung, D. Kim, S. Lee, *Characterization for Beurling-Björck space and Schwartz space*, Proc. Amer. Math. Soc. **125** (1997), 3229–3234. I. M. Gel’fand and G. E. Shilov, [*Generalized functions. Vol. 3: Theory of differential equations,*]{} Academic Press, New York-London, 1967. K. Gr[ö]{}chenig, G. Zimmermann, *Spaces of test functions via the STFT,* J. Funct. Spaces Appl. **2** (2004), 25–53. M. Hasumi, *Note on the n-dimensional tempered ultra-distributions,* Tohoku Math. J. **13** (1961), 94–104. R. Meise, D. Vogt, *Introduction to functional analysis*, Clarendon Press, 1997. H. J. Petzsche, *Die nuklearität der ultradistributionsräume und der satz vom kern I*, Manuscripta Math. **24** (1978), 133–171. F. Trèves, *Topological vector spaces, distributions and kernels*, Academic Press, New York, 1967. [^1]: A. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral grant 12T0519N [^2]: L. Neyt gratefully acknowledges support by Ghent University through the BOF-grant 01J11615. [^3]: J. Vindas was supported by Ghent University through the BOF-grants 01J11615 and 01J04017.

--- abstract: 'We lay out a general method for computing branching distances between labeled transition systems. We translate the quantitative games used for defining these distances to other, path-building games which are amenable to methods from the theory of quantitative games. We then show for all common types of branching distances how the resulting path-building games can be solved. In the end, we achieve a method which can be used to compute all branching distances in the linear-time–branching-time spectrum.' author: - 'Uli Fahrenberg1[^1]' - Axel Legay23 - Karin Quaas4 title: | Computing Branching Distances\ Using Quantitative Games --- Introduction ============ During the last decade, formal verification has seen a trend towards modeling and analyzing systems which contain quantitative information. This is motivated by applications in real-time systems, hybrid systems, embedded systems and others. Quantitative information can thus be a variety of things: probabilities, time, tank pressure, energy intake, [*etc.*]{} A number of quantitative models have hence been developed: probabilistic automata [@DBLP:conf/concur/SegalaL94], stochastic process algebras [@book/Hillston96], timed automata [@DBLP:journals/tcs/AlurD94], hybrid automata [@DBLP:journals/tcs/AlurCHHHNOSY95], timed variants of Petri nets [@journal/transcom/MerlinF76; @DBLP:conf/apn/Hanisch93], continuous-time Markov chains [@book/Stewart94], [*etc.*]{} Similarly, there is a number of specification formalisms for expressing quantitative properties: timed computation tree logic [@DBLP:journals/iandc/HenzingerNSY94], probabilistic computation tree logic [@DBLP:journals/fac/HanssonJ94], metric temporal logic [@DBLP:journals/rts/Koymans90], stochastic continuous logic [@DBLP:journals/tocl/AzizSSB00], [*etc.*]{} Quantitative verification, [*i.e.*,]{} the checking of quantitative properties for quantitative systems, has also seen rapid development: for probabilistic systems in PRISM [@DBLP:conf/tacas/KwiatkowskaNP02] and PEPA [@DBLP:conf/cpe/GilmoreH94], for real-time systems in Uppaal [@DBLP:journals/sttt/LarsenPY97], RED [@DBLP:conf/fm/WangME93], TAPAAL [@DBLP:conf/atva/BygJS09] and Romeo [@DBLP:conf/cav/GardeyLMR05], and for hybrid systems in HyTech [@DBLP:journals/sttt/HenzingerHW97], SpaceEx [@DBLP:conf/cav/FrehseGDCRLRGDM11] and HySAT [@DBLP:journals/fmsd/FranzleH07], to name but a few. Quantitative verification has, however, a problem of *robustness*. When the answers to model checking problems are Boolean—either a system meets its specification or it does not—then small perturbations in the system’s parameters may invalidate the result. This means that, from a model checking point of view, small, perhaps unimportant, deviations in quantities are indistinguishable from larger ones which may be critical. As an example, Fig. \[fi:tatrain\] shows three simple timed-automaton models of a train crossing, each modeling that once the gates are closed, some time will pass before the train arrives. Now assume that the specification of the system is $$\textit{The gates have to be closed 60 seconds before the train arrives.}$$ Model $A$ does guarantee this property, hence satisfies the specification. Model $B$ only guarantees that the gates are closed 58 seconds before the train arrives, and in model $C$, only one second may pass between the gates closing and the train. Neither of models $B$ and $C$ satisfies the specification, so this is the result which a model checker like for example Uppaal would output. What this does not tell us, however, is that model $C$ is dangerously far away from the specification, whereas model $B$ only violates it slightly (and may be acceptable from a practical point of view given other constraints on the system which we have not modeled here). =\[font=\] =\[inner sep=.5mm,minimum size=3.5mm\] \(0) at (0,0) ; (1) at (0,-1.5) ; (2) at (0,-3) ; (0) edge node \[left,anchor=base east\] [$x:= 0$]{} node \[right,anchor=base west\] [close]{} (1); (1) edge node \[left,anchor=base east\] [$x\ge 60$]{} node \[right,anchor=base west\] [train]{} (2); at (0,-3.8) [$A$]{}; \(0) at (0,0) ; (1) at (0,-1.5) ; (2) at (0,-3) ; (0) edge node \[left,anchor=base east\] [$x:= 0$]{} node \[right,anchor=base west\] [close]{} (1); (1) edge node \[left,anchor=base east\] [$x\ge 58$]{} node \[right,anchor=base west\] [train]{} (2); at (0,-3.8) [$B$]{}; \(0) at (0,0) ; (1) at (0,-1.5) ; (2) at (0,-3) ; (0) edge node \[left,anchor=base east\] [$x:= 0$]{} node \[right,anchor=base west\] [close]{} (1); (1) edge node \[left,anchor=base east\] [$x\ge 1$]{} node \[right,anchor=base west\] [train]{} (2); at (0,-3.8) [$C$]{}; In order to address the robustness problem, one approach is to replace the Boolean yes-no answers of standard verification with distances. That is, the Boolean co-domain of model checking is replaced by the non-negative real numbers. In this setting, the Boolean `true` corresponds to a distance of zero and `false` to the non-zero numbers, so that quantitative model checking can now tell us not only that a specification is violated, but also *how much* it is violated, or *how far* the system is from corresponding to its specification. In the example of Fig. \[fi:tatrain\], and depending on precisely how one wishes to measure distances, the distance from $A$ to our specification would be $0$, whereas the distances from $B$ and $C$ to the specification may be $2$ and $59$, for example. The precise interpretation of distance values will be application-dependent; but in any case, it is clear that $C$ is much farther away from the specification than $B$ is. The distance-based approach to quantitative verification has been developed in [@DBLP:journals/tcs/DesharnaisGJP04; @DBLP:journals/tcs/BreugelW05; @DBLP:journals/tcs/AlfaroFHMS05; @DBLP:conf/formats/HenzingerMP05; @DBLP:journals/tac/GirardP07; @DBLP:journals/tcs/Breugel01; @DBLP:journals/jlp/ThraneFL10] and many other papers. Common to all these approaches is that they introduce distances between systems, or between systems and specifications, and then employ these for approximate or quantitative verification. However, depending on the application context, a plethora of different distances are being used. Consequently, there is a need for a general theory of quantitative verification which depends as little as possible on the concrete distances being used. Different applications foster different types of quantitative verification, but it turns out that most of these essentially measure some type of distances between labeled transition systems. We have in [@DBLP:journals/tcs/FahrenbergL14] laid out a unifying framework which allows one to reason about such distance-based quantitative verification independently of the precise distance. This is essentially a general metric theory of labeled transition systems, with infinite quantitative games as its main theoretical ingredient and general fixed-point equations for linear and branching distances as one of its main results. The work in [@DBLP:journals/tcs/FahrenbergL14] generalizes the linear-time–branching-time spectrum of preorders and equivalences from van Glabbeek’s [@inbook/hpa/Glabbeek01] to a quantitative linear-time–branching-time spectrum of distances, all parameterized on a given distance on traces, or executions; [*cf.*]{} Fig. \[fi:spectrum\]. This is done by generalizing Stirling’s bisimulation game [@DBLP:conf/banff/Stirling95] along two directions, both to cover all other preorders and equivalences in the linear-time–branching-time spectrum and into a game with quantitative (instead of Boolean) objectives. =\[font=,text badly centered\] (traceeq) at (0,.3) [$\infty$-nested trace equivalence]{}; (k+1-r-trace) at (-2,-2.4) [$( k+ 1)$-nested ready inclusion]{}; (k+1-traceeq) at (2,-1.6) [$( k+ 1)$-nested trace equivalence]{}; (k-r-traceeq) at (-2,-4.4) [$k$-nested ready equivalence]{}; (k+1-trace) at (2,-3.6) [$( k+ 1)$-nested trace inclusion]{}; (k-r-trace) at (-2,-6.4) [$k$-nested ready inclusion]{}; (k-traceeq) at (2,-5.6) [$k$-nested trace equivalence]{}; (2-r-trace) at (-2,-9.4) [$2$-nested ready inclusion]{}; (2-traceeq) at (2,-8.6) [$2$-nested trace equivalence\ *possible-futures equivalence*]{}; (1-r-traceeq) at (-2,-11.4) [$1$-nested ready equivalence\ *ready equivalence*]{}; (2-trace) at (2,-10.6) [$2$-nested trace inclusion\ *possible-futures inclusion*]{}; (1-r-trace) at (-2,-13.4) [$1$-nested ready inclusion\ *ready inclusion*]{}; (1-traceeq) at (2,-12.6) [$1$-nested trace equivalence\ *trace equivalence*]{}; (1-trace) at (2,-14.6) [$1$-nested trace inclusion\ *trace inclusion*]{}; (bisim) at (7,.3) [$\infty$-nested simulation equivalence\ *bisimulation*]{}; (k+1-r-sim) at (5,-2.4) [$( k+ 1)$-ready sim. equivalence]{}; (k+1-simeq) at (9,-1.6) [$( k+ 1)$-nested sim. equivalence]{}; (k-r-simeq) at (5,-4.4) [$k$-nested ready sim. equivalence]{}; (k+1-sim) at (9,-3.6) [$( k+ 1)$-nested simulation]{}; (k-r-sim) at (5,-6.4) [$k$-nested ready simulation]{}; (k-simeq) at (9,-5.6) [$k$-nested sim. equivalence]{}; (2-r-sim) at (5,-9.4) [$2$-nested ready simulation]{}; (2-simeq) at (9,-8.6) [$2$-nested sim. equivalence]{}; (1-r-simeq) at (5,-11.4) [$1$-nested ready sim. equivalence\ *ready simulation equivalence*]{}; (2-sim) at (9,-10.6) [$2$-nested simulation]{}; (1-r-sim) at (5,-13.4) [$1$-nested ready simulation\ *ready simulation*]{}; (1-simeq) at (9,-12.6) [$1$-nested sim. equivalence\ *simulation equivalence*]{}; (1-sim) at (9,-14.6) [$1$-nested simulation\ *simulation*]{}; (bisim) edge (traceeq); (k+1-r-sim) edge (k+1-r-trace); (k+1-simeq) edge (k+1-traceeq); (k-r-simeq) edge (k-r-traceeq); (k+1-sim) edge (k+1-trace); (k-r-sim) edge (k-r-trace); (k-simeq) edge (k-traceeq); (2-simeq) edge (2-traceeq); (2-r-sim) edge (2-r-trace); (2-sim) edge (2-trace); (1-r-simeq) edge (1-r-traceeq); (1-simeq) edge (1-traceeq); (1-r-sim) edge (1-r-trace); (1-sim) edge (1-trace); (traceeq) edge (k+1-r-trace); (traceeq) edge (k+1-traceeq); (k+1-r-trace) edge (k-r-traceeq); (k+1-r-trace) edge (k+1-trace); (k+1-traceeq) edge (k-r-traceeq); (k+1-traceeq) edge (k+1-trace); (k-r-traceeq) edge (k-r-trace); (k-r-traceeq) edge (k-traceeq); (k+1-trace) edge (k-r-trace); (k+1-trace) edge (k-traceeq); (k-r-trace) edge (2-r-trace); (k-r-trace) edge (2-traceeq); (k-traceeq) edge (2-r-trace); (k-traceeq) edge (2-traceeq); (2-r-trace) edge (1-r-traceeq); (2-r-trace) edge (2-trace); (2-traceeq) edge (1-r-traceeq); (2-traceeq) edge (2-trace); (1-r-traceeq) edge (1-r-trace); (1-r-traceeq) edge (1-traceeq); (2-trace) edge (1-r-trace); (2-trace) edge (1-traceeq); (1-r-trace) edge (1-trace); (1-traceeq) edge (1-trace); (bisim) edge (k+1-r-sim); (bisim) edge (k+1-simeq); (k+1-r-sim) edge (k-r-simeq); (k+1-r-sim) edge (k+1-sim); (k+1-simeq) edge (k-r-simeq); (k+1-simeq) edge (k+1-sim); (k-r-simeq) edge (k-r-sim); (k-r-simeq) edge (k-simeq); (k+1-sim) edge (k-r-sim); (k+1-sim) edge (k-simeq); (k-r-sim) edge (2-r-sim); (k-r-sim) edge (2-simeq); (k-simeq) edge (2-r-sim); (k-simeq) edge (2-simeq); (2-r-sim) edge (1-r-simeq); (2-r-sim) edge (2-sim); (2-simeq) edge (1-r-simeq); (2-simeq) edge (2-sim); (1-r-simeq) edge (1-r-sim); (1-r-simeq) edge (1-simeq); (2-sim) edge (1-r-sim); (2-sim) edge (1-simeq); (1-r-sim) edge (1-sim); (1-simeq) edge (1-sim); What is missing in [@DBLP:journals/tcs/FahrenbergL14] are actual *algorithms* for computing the different types of distances. (The fixed-point equations mentioned above are generally defined over infinite lattices, hence Tarski’s fixed-point theorem does not help here.) In this paper, we take a different route to compute them. We translate the general quantitative games used in [@DBLP:journals/tcs/FahrenbergL14] to other, path-building games. We show that under mild conditions, this translation can always be effectuated, and that for all common trace distances, the resulting path-building games can be solved using various methods which we develop. We start the paper by reviewing the quantitative games used to define linear and branching distances in [@DBLP:journals/tcs/FahrenbergL14] in Section \[se:lbdist\]. Then we show the reduction to path-building games in Section \[se:reduc\] and apply this to show how to compute all common branching distances in Section \[se:comput\]. We collect our results in the concluding section \[se:conc\]. The contributions of this paper are the following: 1. \[en:contri.reduce\] A general method to reduce quantitative bisimulation-type games to path-building games. The former can be posed as *double* path-building games, where the players alternate to build *two* paths; we show how to transform such games into a form where the players instead build *one* common path. 2. \[en:contri.solve\] A collection of methods for solving different types of path-building games. Standard methods are available for solving discounted games and mean-payoff games; for other types we develop new methods. 3. The application of the methods in  to compute various types of distances between labeled transition systems defined by the games of . Linear and Branching Distances {#se:lbdist} ============================== Let $\Sigma$ be a set of labels. $\Sigma^\omega$ denotes the set of infinite traces over $\Sigma$. We generally count sequences from index $0$, so that $\sigma=( \sigma_0, \sigma_1,\dotsc)$. Let ${{\mathbbm{R}}_*}= {{\mathbbm{R}}_{ \ge 0}}\cup\{ \infty\}$ denote the extended non-negative real numbers. Trace Distances {#se:trace_distances} --------------- A *trace distance* is a hemimetric $D: \Sigma^\omega\times \Sigma^\omega\to {{\mathbbm{R}}_*}$, [*i.e.*,]{} a function which satisfies $D( \sigma, \sigma)= 0$ and $D( \sigma, \tau)+ D( \tau, \upsilon)\ge D( \sigma, \upsilon)$ for all $\sigma, \tau, \upsilon\in \Sigma^\omega$. The following is an exhaustive list of different trace distances which have been used in different applications. We refer to [@DBLP:journals/tcs/FahrenbergL14] for more details and motivation. #### The discrete trace distance: $D_\textup{disc}( \sigma, \tau)= 0$ if $\sigma= \tau$ and $\infty$ otherwise. This is equivalent to the standard Boolean setting: traces are either equal (distance $0$) or not (distance $\infty$). #### The point-wise trace distance: $D_\textup{sup}( \sigma, \tau)= \sup_{ n\ge 0} d( \sigma_n, \tau_n)$, for any given label distance $d: \Sigma\times \Sigma\to {{\mathbbm{R}}_*}$. This measures the greatest individual symbol distance in the traces and has been used for quantitative verification in, among others, [@DBLP:journals/tse/AlfaroFS09; @DBLP:conf/qest/DesharnaisLT08; @FahrenbergLT10; @LarsenFT11-Axioms; @DBLP:journals/jlp/ThraneFL10; @conf/icalp/AlfaroHM03]. #### The discounted trace distance: $D_+( \sigma, \tau)= \sum_{ n= 0}^\infty \lambda^n d( \sigma_n, \tau_n)$, for any given *discounting factor* $\lambda\in[ 0, 1\mathclose[$. Sometimes also called *accumulating* trace distance, this accumulates individual symbol distances along traces, using discounting to adjust the values of distances further off. It has been used in, for example, [@FahrenbergLT10; @LarsenFT11-Axioms; @DBLP:journals/jlp/ThraneFL10; @DBLP:journals/tcs/CernyHR12]. #### The limit-average trace distance: $D_\textup{lavg}( \sigma, \tau)= \liminf_{ n\ge 1} \frac1 n \sum_{ i= 0}^{ n- 1} d( \sigma_i, \tau_i)$. This again accumulates individual symbol distances along traces and has been used in, among others, [@conf/csl/ChatterjeeDH08; @DBLP:journals/tcs/CernyHR12]. Both discounted and limit-average distances are well-known from the theory of discounted and mean-payoff games [@EhrenfeuchtM79; @DBLP:journals/tcs/ZwickP96]. #### The Cantor trace distance: $D_\textup{C}( \sigma, \tau)= \frac1{ 1+ \inf\{ n\mid \sigma_n\ne \tau_n\}}$. This measures the (inverse of the) length of the common prefix of the traces and has been used for verification in [@DBLP:conf/acsd/DoyenHLN10]. #### The maximum-lead trace distance: $D_\pm( \sigma, \tau)= \sup_{ n\ge0}\bigl| \sum_{ i= 0}^n( \sigma_i- \tau_i)\bigr|$. Here it is assumed that $\Sigma$ admits arithmetic operations of $+$ and $-$, for instance $\Sigma\subseteq {\mathbbm{R}}$. As this measures differences of accumulated labels along runs, it is especially useful for real-time systems, [*cf.*]{} [@DBLP:conf/formats/HenzingerMP05; @conf/fit/FahrenbergL12; @DBLP:journals/jlp/ThraneFL10]. Labeled Transition Systems -------------------------- A *labeled transition system* (LTS) over $\Sigma$ is a tuple $( S, i, T)$ consisting of a set of states $S$, with initial state $i\in S$, and a set of transitions $T\subseteq S\times \Sigma\times S$. We often write $s{\xrightarrow{a}} t$ to mean $( s, a, t)\in T$. We say that $(S,i,T)$ is *finite* if $S$ and $T$ are finite. We assume our LTS to be *non-blocking* in the sense that for every state $s\in S$ there is a transition $( s, a, t)\in T$. We have shown in [@DBLP:journals/tcs/FahrenbergL14] how any given trace distance $D$ can be lifted to a quantitative linear-time–branching-time spectrum of distances on LTS. This is done via quantitative games as we shall review below. The point of [@DBLP:journals/tcs/FahrenbergL14] was that if the given trace distance has a recursive formulation, which, as we show in [@DBLP:journals/tcs/FahrenbergL14], every commonly used trace distance has, then the corresponding linear and branching distances can be formulated as fixed points for certain monotone functionals. The fixed-point formulation of [@DBLP:journals/tcs/FahrenbergL14] does not, however, give rise to actual algorithms for computing linear and branching distances, as it happens more often than not that the mentioned monotone functionals are defined over infinite lattices. Concretely, this is the case for all but the point-wise trace distances in Section \[se:trace\_distances\]. Hence other methods are required for computing them; developing these is the purpose of this paper. Quantitative Ehrenfeucht-Fra[ï]{}ss[é]{} Games ---------------------------------------------- We review the quantitative games used in [@DBLP:journals/tcs/FahrenbergL14] to define different types of linear and branching distances for any given trace distance $D$. For conciseness, we only introduce *simulation games* and *bisimulation games* here, but similar definitions may be given for all equivalences and preorders in the linear-time–branching-time spectrum [@inbook/hpa/Glabbeek01]. ### Quantitative Simulation Games {#quantitative-simulation-games .unnumbered} Let ${\mathcal S}=( S, i, T)$ and ${\mathcal S}'=( S', i', T')$ be LTS and $D: \Sigma^\omega\times \Sigma^\omega\to {{\mathbbm{R}}_*}$ a trace distance. The *simulation game* from ${\mathcal S}$ to ${\mathcal S}'$ is played by two players, the maximizer and the minimizer. A play begins with the maximizer choosing a transition $( s_0, a_0, s_1)\in T$ with $s_0= i$. Then the minimizer chooses a transition $( s_0', a_0', s_1')\in T'$ with $s_0'= i'$. Now the maximizer chooses a transition $( s_1, a_1, s_2)\in T$, then the minimizer chooses a transition $( s_1', a_1', s_2')\in T'$, and so on indefinitely. Hence this is what should be called a *double path-building game*: the players each build, independently, an infinite path in their respective LTS. A *play* hence consists of two infinite paths, $\pi$ starting from $i$, and $\pi'$ starting from $i'$. The *utility* of this play is the distance $D( \sigma, \sigma')$ between the traces $\sigma$, $\sigma'$ of the paths $\pi$ and $\pi'$, which the maximizer wants to maximize and the minimizer wants to minimize. The *value* of the game is, then, the utility of the play which results when both maximizer and minimizer are playing optimally. To formalize the above intuition, we define a *configuration* for the maximizer to be a pair $( \pi, \pi')$ of finite paths of equal length, $\pi$ in ${\mathcal S}$ and starting in $i$, $\pi'$ in ${\mathcal S}'$ starting in $i'$. The intuition is that this covers the *history* of a play; the choices both players have made up to a certain point in the game. Hence a configuration for the minimizer is a similar pair $( \pi, \pi')$ of finite paths, but now $\pi$ is one step longer than $\pi'$. A *strategy* for the maximizer is a mapping from maximizer configurations to transitions in ${\mathcal S}$, fixing the maximizer’s choice of a move in the given configuration. Denoting the set of maximizer configurations by ${\textup{\textsf{Conf}}}$, such a strategy is hence a mapping $\theta: {\textup{\textsf{Conf}}}\to T$ such that for all $( \pi, \pi')\in {\textup{\textsf{Conf}}}$ with $\theta( \pi, \pi')=( s, a, t)$, we have ${\textup{\textsf{end}}}( \pi)= s$. Here ${\textup{\textsf{end}}}( \pi)$ denotes the last state of $\pi$. Similarly, and denoting the set of minimizer configurations by ${\textup{\textsf{Conf}}}'$, a strategy for the minimizer is a mapping $\theta': {\textup{\textsf{Conf}}}'\to T'$ such that for all $( \pi, \pi')\in {\textup{\textsf{Conf}}}'$ with $\theta'( \pi, \pi')=( s', a', t')$, ${\textup{\textsf{end}}}( \pi')= s'$. Denoting the sets of these strategies by $\Theta$ and $\Theta'$, respectively, we can now define the *simulation distance* from ${\mathcal S}$ to ${\mathcal S}'$ induced by the trace distance $D$, denoted $D^\textup{sim}( {\mathcal S}, {\mathcal S}')$, by $$D^\textup{sim}( {\mathcal S}, {\mathcal S}')= \adjustlimits \sup_{ \theta\in \Theta} \inf_{ \theta'\in \Theta'} D( \sigma( \theta, \theta'), \sigma'( \theta, \theta'))\,,$$ where $\sigma( \theta, \theta')$ and $\sigma'( \theta, \theta')$ are the traces of the paths $\pi( \theta, \theta')$ and $\pi'( \theta, \theta')$ induced by the pair of strategies $( \theta, \theta')$. \[re:simulation\_game\] If the trace distance $D$ is discrete, [*i.e.*,]{} $D= D_\textup{disc}$ as in Section \[se:trace\_distances\], then the quantitative game described above reduces to the well-known *simulation game* [@DBLP:conf/banff/Stirling95]: The only choice the minimizer has for minimizing the value of the game is to always choose a transition with the same label as the one just chosen by the maximizer; similarly, the maximizer needs to try to force the game into states where she can choose a transition which the minimizer cannot match. Hence the value of the game will be $0$ if the minimizer always can match the maximizer’s labels, that is, iff ${\mathcal S}$ is simulated by ${\mathcal S}'$. ### Quantitative Bisimulation Games {#quantitative-bisimulation-games .unnumbered} There is a similar game for computing the *bisimulation distance* between LTS ${\mathcal S}$ and ${\mathcal S}'$. Here we give the maximizer the choice, at each step, to either choose a transition from $s_k$ as before, or to “switch sides” and choose a transition from $s_k'$ instead; the minimizer then has to answer with a transition on the other side. Hence the players are still building two paths, one in each LTS, but now they are *both* contributing to *both* paths. The utility of such a play is still the distance between these two paths, which the maximizer wants to maximize and the minimizer wants to minimize. The *bisimulation distance* between ${\mathcal S}$ and ${\mathcal S}'$, denoted $D^\textup{bisim}( {\mathcal S}, {\mathcal S}')$, is then defined to be the value of this quantitative bisimulation game. If the trace distance $D= D_\textup{disc}$ is discrete, then using the same arguments as in Remark \[re:simulation\_game\], we see that $D_\textup{disc}^\textup{bisim}( {\mathcal S}, {\mathcal S}')= 0$ iff ${\mathcal S}$ and ${\mathcal S}'$ are *bisimilar*. The game which results being played is precisely the bisimulation game of [@DBLP:conf/banff/Stirling95], which also has been introduced by Fra[ï]{}ss[é]{} [@Fraisse54] and Ehrenfeucht [@Ehrenfeucht61] in other contexts. ### The Quantitative Linear-Time–Branching-Time Spectrum {#the-quantitative-linear-timebranching-time-spectrum .unnumbered} The above-defined quantitative simulation and bisimulation games can be generalized using different methods. One is to introduce a *switch counter* ${\textup{\textsf{sc}}}$ into the game which counts how often the maximizer has switched sides during an ongoing game. Then one can limit the maximizer’s capabilities by imposing limits on ${\textup{\textsf{sc}}}$: if the limit is ${\textup{\textsf{sc}}}= 0$, then the players are playing a simulation game; if there is no limit (${\textup{\textsf{sc}}}\le \infty$), they are playing a bisimulation game. Other limits ${\textup{\textsf{sc}}}\le k$, for $k\in {\mathbbm{N}}$, can be used to define *$k$-nested simulation distances*, generalizing the equivalences and preorders from [@DBLP:journals/iandc/GrooteV92; @DBLP:journals/jacm/HennessyM85]. Another method of generalization is to introduce *ready moves* into the game. These consist of the maximizer challenging her opponent by switching sides, but only requiring that the minimizer match the chosen transition; afterwards the game finishes. This can be employed to introduce the *ready simulation distance* of [@DBLP:conf/popl/LarsenS89] and, combined with the switch counter method above, the *ready $k$-nested simulation distance*. We refer to [@DBLP:journals/tcs/FahrenbergL14] for further details on these and other variants of quantitative (bi)simulation games. For reasons of exposition, we will below introduce our reduction to path-building games only for the quantitative simulation and bisimulation games; but all our work can easily be transferred to the general setting of [@DBLP:journals/tcs/FahrenbergL14]. Reduction {#se:reduc} ========= In order to compute simulation and bisimulation distances, we translate the games of the previous section to path-building games [à]{} la Ehrenfeucht-Mycielski [@EhrenfeuchtM79]. Let $D: \Sigma^\omega\times \Sigma^\omega\to {{\mathbbm{R}}_*}$ be a trace distance, and assume that there are functions ${\textup{\textsf{val}}}_D: {{\mathbbm{R}}_*}^\omega\to {{\mathbbm{R}}_*}$ and $f_D: \Sigma\times \Sigma\to {{\mathbbm{R}}_*}$ for which it holds, for all $\sigma, \tau\in \Sigma^\infty$, that $$\label{eq:DtoGf} D( \sigma, \tau)= {\textup{\textsf{val}}}_D( 0, f_D( \sigma_0, \tau_0), 0, f_D( \sigma_1, \tau_1), 0,\dotsc)\,.$$ We will need these functions in our translation, and we show in Section \[se:examples-red\] below that they exist for all common trace distances. Simulation Distance ------------------- Let ${\mathcal S}=( S, i, T)$ and ${\mathcal S}'=( S', i', T')$ be LTS. We construct a turn-based game ${\mathcal U}= {\mathcal U}( {\mathcal S}, {\mathcal S}')=( U, u_0, {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}})$ as follows, with $U= U_1\cup U_2$: $$\begin{gathered} U_1= S\times S'\qquad U_2= S\times S'\times \Sigma\qquad u_0=( i, i') \\[.5ex] \begin{aligned} {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}&= \{( s, s'){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}}( t, s', a)\mid( s, a, t)\in T\} \\[-.5ex] &\quad\cup \{( t, s', a){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{ f_D( a, a')}}}( t, t')\mid( s', a', t')\in T'\} \end{aligned}\end{gathered}$$ This is a two-player game. We again call the players maximizer and minimizer, with the maximizer controlling the states in $U_1$ and the minimizer the ones in $U_2$. Transitions are labeled with extended real numbers, but as the image of $f_D$ in ${{\mathbbm{R}}_*}$ is finite, the set of transition labels in $U$ is finite. The game on ${\mathcal U}$ is played as follows. A play begins with the maximizer choosing a transition $(u_0,a_0,u_1)\in {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}$ with $u_0= i$. Then the minimizer chooses a transition $(u_1,a_1,u_2)\in {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}$. Then the maximizer chooses a transition $(u_2,a_2,u_3)\in {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}$, and so on indefinitely (note that ${\mathcal U}$ is non-blocking). A play thus induces an infinite path $\pi=(u_0,a_0,u_1), (u_1,a_1,u_2),\dotsc$ in ${\mathcal U}$ with $u_0= i$. The goal of the maximizer is to maximize the value ${\textup{\textsf{val}}}_D( {\mathcal U}):= {\textup{\textsf{val}}}_D(a_0, a_1,\dotsc)$ of the trace of $\pi$; the goal of the minimizer is to minimize this value. This is hence a path-building game, variations of which (for different valuation functions) have been studied widely in both economics and computer science since Ehrenfeucht-Mycielski’s [@EhrenfeuchtM79]. Formally, configurations and strategies are given as follows. A configuration of the maximizer is a path $\pi_1$ in ${\mathcal U}$ with ${\textup{\textsf{end}}}( \pi_1)\in U_1$, and a configuration of the minimizer is a path $\pi_2$ in ${\mathcal U}$ with ${\textup{\textsf{end}}}( \pi_2)\in U_2$. Denote the sets of these configurations by ${\textup{\textsf{Conf}}}_1$ and ${\textup{\textsf{Conf}}}_2$, respectively. A strategy for the maximizer is, then, a mapping $\theta_1: {\textup{\textsf{Conf}}}_1\to {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}$ such that for all $\pi_1\in {\textup{\textsf{Conf}}}_1$ with $\theta_1( \pi_1)=( u, x, v)$, ${\textup{\textsf{end}}}( \pi_1)= u$. A strategy for the minimizer is a mapping $\theta_2: {\textup{\textsf{Conf}}}_2\to {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}$ such that for all $\pi_2\in {\textup{\textsf{Conf}}}_2$ with $\theta_2( \pi_2)=( u, x, v)$, ${\textup{\textsf{end}}}( \pi_2)= u$. Denoting the sets of these strategies by $\Theta_1$ and $\Theta_2$, respectively, we can now define $${\textup{\textsf{val}}}_D( {\mathcal U})= \adjustlimits \sup_{ \theta_1\in \Theta_1} \inf_{ \theta_2\in \Theta_2} {\textup{\textsf{val}}}_D( \sigma( \theta_1, \theta_2))\,,$$ where $\sigma( \theta_1, \theta_2)$ is the trace of the path $\pi( \theta_1, \theta_2)$ induced by the pair of strategies $( \theta_1, \theta_2)$. By the next theorem, the value of ${\mathcal U}$ is precisely the simulation distance from ${\mathcal S}$ to ${\mathcal S}'$. \[th:sim=u\] For all LTS ${\mathcal S}$, ${\mathcal S}'$, $D^\textup{sim}( {\mathcal S}, {\mathcal S}')= {\textup{\textsf{val}}}_D( {\mathcal U}( {\mathcal S}, {\mathcal S}'))$. Write ${\mathcal S}=( S, i, T)$ and ${\mathcal S}'=( S', i', T')$. Informally, the reason for the equality is that any move $( s, a, t)\in T$ of the maximizer in the simulation distance game can be copied to a move $( s, s'){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{ 0}}}( t, s', a)$, regardless of $s'$, in ${\mathcal U}$. Similarly, any move $( s', a', t')$ of the minimizer can be copied to a move $( t, s', a){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{ f_D( a, a')}}}( t, t')$, and all the moves in ${\mathcal U}$ are of this form. To turn this idea into a formal proof, we show that there are bijections between configurations and strategies in the two games, and that under these bijections, the utilities of the two games are equal. For $( \pi, \pi')\in {\textup{\textsf{Conf}}}$ in the simulation distance game, with $\pi=( s_0, a_0, s_1),\dotsc,$ $( s_{ n- 1}, a_{ n- 1}, s_n)$ and $\pi'=( s_0', a_0', s_1'),\dotsc,( s_{ n- 1}', a_{ n- 1}', s_n')$, define $$\begin{gathered} \phi_1( \pi, \pi')=(( s_0, s_0'), 0,( s_1, s_0', a_0)),(( s_1, s_0', a_0), f_D( a_0, a_0'),( s_1, s_1')),\dotsc, \\ (( s_n, s_{ n- 1}', a_{ n- 1}), f_D( a_{ n- 1}, a_{ n- 1}'),( s_n, s_n'))\,. \end{gathered}$$ It is clear that this defines a bijection $\phi_1: {\textup{\textsf{Conf}}}\to {\textup{\textsf{Conf}}}_1$, and that one can similarly define a bijection $\phi_2: {\textup{\textsf{Conf}}}'\to {\textup{\textsf{Conf}}}_2$. Now for every strategy $\theta: {\textup{\textsf{Conf}}}\to T$ in the simulation distance game, define a strategy $\psi_1( \theta)= \theta_1\in \Theta_1$ as follows. For $\pi_1\in {\textup{\textsf{Conf}}}_1$, let $( \pi, \pi')= \phi_1^{ -1}( \pi_1)$ and $s'= {\textup{\textsf{end}}}( \pi')$. Let $\theta( \pi, \pi')=( s, a, t)$ and define $\theta_1( \pi_1)=(( s, s'), 0,( t, s', a))$. Similarly we define a mapping $\psi_2: \Theta'\to \Theta_2$ as follows. For $\theta': {\textup{\textsf{Conf}}}'\to T'$ and $\pi_2\in {\textup{\textsf{Conf}}}_2$, let $( \pi, \pi')= \phi_2^{ -1}( \pi_2)$ with $\pi=( s_0, a_0, s_1),\dotsc,( s_n, a_n, s_{ n+ 1})$. Let $\theta'( \pi, \pi')=( s', a', t')$ and define $\psi_2( \theta')( \pi_2)=((s_{ n+ 1}, s', a_n), f_D( a_n, a'),( s_{ n+ 1}, t'))$. It is clear that $\psi_1$ and $\psi_2$ indeed map strategies in the simulation distance game to strategies in ${\mathcal U}$ and that both are bijections. Also, for each pair $( \theta, \theta')\in \Theta\times \Theta'$, $D( \sigma( \theta, \theta'), \sigma'( \theta, \theta'))= {\textup{\textsf{val}}}_D( \sigma( \psi_1( \theta), \psi_2( \theta')))$ by construction. But then $$\begin{aligned} D^\textup{sim}( {\mathcal S}, {\mathcal S}') &= \adjustlimits \sup_{ \theta\in \Theta} \inf_{ \theta'\in \Theta'} D( \sigma( \theta, \theta'), \sigma'( \theta, \theta')) \\ &= \adjustlimits \sup_{ \theta\in \Theta} \inf_{ \theta'\in \Theta'} {\textup{\textsf{val}}}_D( \sigma( \psi_1( \theta), \psi_2( \theta'))) \\ &= \adjustlimits \sup_{ \theta_1\in \Theta_1} \inf_{ \theta_2\in \Theta_2} {\textup{\textsf{val}}}_D( \sigma( \theta_1, \theta_2))= {\textup{\textsf{val}}}_D( {\mathcal U})\,, \end{aligned}$$ the third equality because $\psi_1$ and $\psi_2$ are bijections. Examples {#se:examples-red} -------- We show that the reduction applies to all trace distances from Section \[se:trace\_distances\]. 1. \[ex:red.discr\] For the discrete trace distance $D= D_\textup{disc}$, we let $${\textup{\textsf{val}}}_D( x)= \sum_{ n= 0}^\infty x_n\,, \qquad f_D( a, b)= \begin{cases} 0 &\text{if } a= b\,, \\ \infty &\text{otherwise}\,, \end{cases}$$ then  holds. In the game on ${\mathcal U}$, the minimizer needs to play $0$-labeled transitions to keep the distance at $0$. 2. \[ex:red.pw\] For the point-wise trace distance $D= D_\textup{sup}$, we can let $${\textup{\textsf{val}}}_D( x)= \sup_{ n\ge 0} x_n\,, \qquad f_D( a, b)= d( a, b)\,.$$ Hence the game on ${\mathcal U}$ computes the sup of a trace. 3. \[ex:red.disc\] For the discounted trace distance $D= D_+$, let $${\textup{\textsf{val}}}_D( x)= \sum_{ n= 0}^\infty \sqrt \lambda^n x_n\,, \qquad f_D( a, b)= \sqrt \lambda\, d( a, b)\,,$$ then  holds. Hence the game on ${\mathcal U}$ is a standard discounted game [@DBLP:journals/tcs/ZwickP96]. 4. \[ex:red.lavg\] For the limit-average trace distance $D= D_\textup{lavg}$, we can let $${\textup{\textsf{val}}}_D( x)= \liminf_{ n\ge 1} \frac 1 n \sum_{ i= 0}^{ n- 1} x_i\,, \qquad f_D( a, b)= 2 d( a, b)\,;$$ we will show below that  holds. Hence the game on ${\mathcal U}$ is a mean-payoff game [@DBLP:journals/tcs/ZwickP96]. 5. \[ex:red.cant\] For the Cantor trace distance $D= D_\textup{C}$, let $${\textup{\textsf{val}}}_D( x)= \frac2{ 1+ \inf\{ n\mid x_n\ne 0\}}\,, \qquad f_D( a, b)= \begin{cases} 0 &\text{if } a= b\,, \\ 1 &\text{otherwise}\,. \end{cases}$$ The objective of the maximizer in this game is to reach a transition with weight $1$ *as soon as possible*. 6. \[ex:red.maxl\] For the maximum-lead trace distance $D= D_\pm$, we can let $${\textup{\textsf{val}}}_D( x)= \sup_{ n\ge 0}\bigl| \sum_{ i= 0}^n x_i\bigr|\,, \qquad f_D( a, b)= a- b\,,$$ then  holds. Bisimulation Distance --------------------- We can construct a similar turn-based game to compute the bisimulation distance. Let ${\mathcal S}=( S, i, T)$ and ${\mathcal S}'=( S', i', T')$ be LTS and define ${\mathcal V}= {\mathcal V}( {\mathcal S}, {\mathcal S}')=( V, v_0, {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}})$ as follows, with $V= V_1\cup V_2$: $$\begin{gathered} V_1= S\times S'\qquad V_2= S\times S'\times \Sigma\times\{ 1, 2\}\qquad v_0=( i, i') \\[.5ex] \begin{aligned} {\mathord{{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{}}}}}&= \{( s, s'){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}}( t, s', a, 1)\mid( s, a, t)\in T\} \\[-.5ex] &\quad\cup \{( s, s'){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}}( s, t', a', 2)\mid( s', a', t')\in T'\} \\[-.5ex] &\quad\cup \{( t, s', a, 1){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{ f_D( a, a')}}}( t, t')\mid( s', a', t')\in T'\} \\[-.5ex] &\quad\cup \{( s, t', a', 2){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{ f_D( a, a')}}}( t, t')\mid( s, a, t)\in T\} \end{aligned}\end{gathered}$$ Here we have used the minimizer’s states to both remember the label choice of the maximizer and which side of the bisimulation game she plays on. By suitable modifications, we can construct similar games for all distances in the spectrum of [@DBLP:journals/tcs/FahrenbergL14]. The next theorem states that the value of ${\mathcal V}$ is precisely the bisimulation distance between ${\mathcal S}$ and ${\mathcal S}'$. For all LTS ${\mathcal S}$, ${\mathcal S}'$, $D^\textup{bisim}( {\mathcal S}, {\mathcal S}')= {\textup{\textsf{val}}}_D( {\mathcal V}( {\mathcal S}, {\mathcal S}'))$. This proof is similar to the one of Theorem \[th:sim=u\], only that now, we have to take into account that the maximizer may “switch sides”. The intuition is that maximizer moves $( s, a, t)$ in the ${\mathcal S}$ component of the bisimulation distance games are emulated by moves $( s, s'){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}}( t, s', a, 1)$, maximizer moves $( s', a', t')$ in the ${\mathcal S}'$ component are emulated by moves $( s, s'){{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}}( s, t', a', 2)$, and similarly for the minimizer. The values $1$ and $2$ in the last component of the $V_2$ states ensure that the minimizer only has moves available which correspond to playing in the correct component in the bisimulation distance game ([*i.e.*,]{} that $\psi_2$ is a bijection). Computing the Values of Path-Building Games {#se:comput} =========================================== We show here how to compute the values of the different path-building games which we saw in the last section. This will give us algorithms to compute all simulation and bisimulation distances associated with the trace distances of Section \[se:trace\_distances\]. We will generally only refer to the games ${\mathcal U}$ for computing simulation distance here, but the bisimulation distance games ${\mathcal V}$ are very similar, and everything we say also applies to them. #### Discrete distance: The game to compute the discrete simulation distances is a reachability game, in that the goal of the maximizer is to force the minimizer into a state from which she can only choose $\infty$-labeled transitions. We can hence solve them using the standard controllable-predecessor operator defined, for any set $S\subseteq U_1$ of maximizer states, by $${\textup{\textsf{cpre}}}( S)=\{ u_1\in U_1\mid \exists u_1{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}} u_2: \forall u_2{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{x}}} u_3: u_3\in S\}\,.$$ Now let $S\subseteq U_1$ be the set of states from which the maximizer can force the game into a state from which the minimizer only has $\infty$-labeled transitions, [*i.e.*,]{}$$S=\{ u_1\in U_1\mid \exists u_1{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}} u_2: \forall u_2{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{x}}} u_3: x= \infty\}\,,$$ and compute $S^*= {\textup{\textsf{cpre}}}^*( S)= \bigcup_{ n\ge 0} {\textup{\textsf{cpre}}}^n( S)$. By monotonicity of ${\textup{\textsf{cpre}}}$ and as the subset lattice of $U_1$ is complete and finite, this computation finishes in at most $| U_1|$ steps. ${\textup{\textsf{val}}}_D( {\mathcal U})= 0$ iff $u_0\notin S^*$. As we are working with the discrete distance, we have either ${\textup{\textsf{val}}}_D( {\mathcal U})= 0$ or ${\textup{\textsf{val}}}_D( {\mathcal U})= \infty$. Now $u_o\in S^*$ iff the maximizer can force, using finitely many steps, the game into a state from which the minimizer only has $\infty$-labeled transitions, which is the same as ${\textup{\textsf{val}}}_D( {\mathcal U})= \infty$. #### Point-wise distance: To compute the value of the point-wise simulation distance game, let $W=\{ w_1,\dotsc, w_m\}$ be the (finite) set of weights of the minimizer’s transitions, ordered such that $w_1<\dotsm< w_m$. For each $i= 1,\dotsc, m$, let $S_i=\{ u_1\in U_1: \exists u_1{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{0}}} u_2: \forall u_2{{ \ext@arrow 0099\xrightarrowtrianglefill@{}{x}}} u_3: x\ge w_i\}$ be the set of maximizer states from which the maximizer can force the minimizer into a transition with weight at least $w_i$; note that $S_m\subseteq S_{ m- 1}\subseteq\dotsm\subseteq S_1= U_1$. For each $i= 1,\dotsc, m$, compute $S_i^*= {\textup{\textsf{cpre}}}^*( S_i)$, then $S_m^*\subseteq S_{ m- 1}^*\subseteq\dotsm\subseteq S_1^*= U_1$. Let $p$ be the greatest index for which $u_0\in S_p^*$, then $p= {\textup{\textsf{val}}}_D( {\mathcal U})$. For any $k$, we have $u_0\in S_k^*$ iff the maximizer can force, using finitely many steps, the game into a state from which the minimizer only has transitions with weight at least $w_k$. Thus $u_0\in S_p^*$ iff (1) the maximizer can force the minimizer into a $w_p$-weighted transition; (2) the maximizer *cannot* force the minimizer into a $w_{ p+ 1}$-weighted transition. #### Discounted distance: The game to compute the discounted simulation distance is a standard discounted game and can be solved by standard methods [@DBLP:journals/tcs/ZwickP96]. #### Limit-average distance: For the limit-average simulation distance game, let $( y_n)_{ n\ge 1}$ be the sequence $( 1, 1, \frac32, 1, \frac54,\dotsc)$ and note that $\lim_{ n\to \infty} y_n= 1$. Then $$\begin{aligned} {\textup{\textsf{val}}}_D( x)= {\textup{\textsf{val}}}_D( x) \lim_{ n\to \infty} y_n &= \liminf_{ n\ge 1} \frac{ y_n}{ n} \sum_{ i= 0}^{ n- 1} x_i \\ &= \liminf_{ 2k\ge 1} \frac 1{ 2k} \sum_{ i= 0}^{ k- 1} f_D( \sigma_i, \tau_i) \\ &= \liminf_{ k\ge 1} \frac 1 k \sum_{ i= 0}^{ k- 1} d( \sigma_i, \tau_i)= D_\textup{lavg}( \sigma, \tau)\,,\end{aligned}$$ so, indeed,  holds. The game is a standard mean-payoff game and can be solved by standard methods, see for example [@DBLP:conf/valuetools/DhingraG06]. #### Cantor distance: To compute the value of the Cantor simulation distance game, let $S_1\subseteq U_1$ be the set of states from which the maximizer can force the game into a state from which the minimizer only has $1$-labeled transitions, [*i.e.*,]{} $S_1=\{ u_1\in U_1\mid \exists u_1{\xrightarrow{0}} u_2: \forall u_2{\xrightarrow{x}} u_3: x= 1\}$. Now recursively compute $S_{ i+ 1}= S_i\cup {\textup{\textsf{cpre}}}( S_i)$, for $i= 1, 2,\dotsc$, until $S_{ i+ 1}= S_i$ (which, as $S_i\subseteq S_{ i+ 1}$ for all $i$ and $U_1$ is finite, will happen eventually). Then $S_i$ is the set of states from which the maximizer can force the game to a $1$-labeled minimizer transition which is at most $2i$ steps away. Hence ${\textup{\textsf{val}}}_D( {\mathcal U})= 0$ if there is no $p$ for which $u_0\in S_p$, and otherwise ${\textup{\textsf{val}}}_D( {\mathcal U})= \frac1 p$, where $p$ is the least index for which $u_0\in S_p$. #### Maximum-lead distance: For the maximum-lead simulation distance game, we note that the maximizer wants to maximize $\sup_{ n\ge 0}\bigl| \sum_{ i= 0}^n x_i\bigr|$, [*i.e.*,]{} wants the accumulated values $\sum_{ i= 0}^n x_i$ or $-\sum_{ i= 0}^n x_i$ to exceed any prescribed bounds. A weighted game in which one player wants to keep accumulated values inside some given bounds, while the opponent wants to exceed these bounds, is called an *interval-bound energy game*. It is shown in [@DBLP:conf/formats/BouyerFLMS08] that solving general interval-bound energy games is EXPTIME-complete. We can reduce the problem of computing maximum-lead simulation distance to an interval-bound energy game by first non-deterministically choosing a bound $k$ and then checking whether player 1 wins the interval-bound energy game on ${\mathcal U}$ for bounds $[ -k, k]$. (There is a slight problem in that in [@DBLP:conf/formats/BouyerFLMS08], energy games are defined only for *integer*-weighted transition systems, whereas we are dealing with real weights here. However, it is easily seen that the results of [@DBLP:conf/formats/BouyerFLMS08] also apply to *rational* weights and bounds; and as our transition systems are finite, one can always find a sound and complete rational approximation.) We can thus compute maximum-lead simulation distance in non-deterministic exponential time; we leave open for now the question whether there is a more efficient algorithm. Conclusion and Future Work {#se:conc} ========================== We sum up our results in the following corollary which gives the complexities of the decision problems associated with the respective distance computations. Note that the first part restates the well-known fact that simulation and bisimulation are decidable in polynomial time. 1. Discrete simulation and bisimulation distances are computable in PTIME. 2. Point-wise simulation and bisimulation distances are computable in PTIME. 3. Discounted simulation and bisimulation distances are computable in $\text{NP}\cap\text{coNP}$. 4. Limit-average simulation and bisimulation distances are computable in $\text{NP}\cap\text{coNP}$. 5. Cantor simulation and bisimulation distances are computable in PTIME. 6. Maximum-lead simulation and bisimulation distances are computable in NEXPTIME. In the future, we intend to expand our work to also cover *quantitative specification theories*. 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[^1]: This author’s work is supported by the *Chaire ISC : Engineering Complex Systems* – cole polytechnique – Thales – FX – DGA – Dassault Aviation – DCNS Research – ENSTA ParisTech – T[é]{}l[é]{}com ParisTech

--- abstract: 'We report the detection of GeV $\gamma$-ray emission from supernova remnant HESS J1731-347 using 9 years of [*Fermi*]{} Large Area Telescope data. We find a slightly extended GeV source in the direction of HESS J1731-347. The spectrum above 1 GeV can be fitted by a power-law with an index of $\Gamma = 1.77\pm0.14$, and the GeV spectrum connects smoothly with the TeV spectrum of HESS J1731-347. Either a hadronic-leptonic or a pure leptonic model can fit the multi-wavelength spectral energy distribution of the source. However, the hard GeV $\gamma$-ray spectrum is more naturally produced in a leptonic (inverse Compton scattering) scenario, under the framework of diffusive shock acceleration. We also searched for the GeV $\gamma$-ray emission from the nearby TeV source HESS J1729-345. No significant GeV $\gamma$-ray emission is found, and upper limits are derived.' author: - 'Xiao-Lei Guo$^{1,2}$, Yu-Liang Xin$^{1,3}$, Neng-Hui Liao$^{1,2}$, Qiang Yuan$^{1,2}$, Wei-Hong Gao$^{4,5}$, Yi-Zhong Fan$^{1,2}$' title: 'Detection of GeV $\gamma$-ray emission in the direction of HESS J1731-347 with Fermi-LAT' --- Introduction ============ It is widely believed that supernova remnants (SNRs) are the main accelerators of Galactic cosmic rays (CRs) with energies up to the knee. This is supported by the non-thermal X-ray emission detected in many SNRs, which indicates the acceleration of electrons to hundreds of TeV energies [e.g., @Koyama1995]. The GeV and/or TeV $\gamma$-rays have also been detected in some SNRs, for example, RCW 86 [@Aharonian2009; @Yuan2014]; Cas A [@Albert2007a; @Abdo2010a]; CTB 37B [@Aharonian2008a; @Xin2016]; Puppis A [@Hewitt2012; @Xin2017]; IC 443 [@Albert2007b; @Acciari2009; @Ackermann2013]; W44 [@Abdo2010b; @Ackermann2013]. Gamma-rays can be produced by the decay of neutral pions due to the inelastic $pp$ collisions (the hadronic process), the Inverse Compton Scattering (ICS) or bremsstrahlung process of relativistic electrons (the leptonic process). For some SNRs interacting with dense molecular clouds, the evidence for acceleration of nuclei has been suggested by GeV/TeV $\gamma$-ray observations [@Li2010; @Li2012; @Ackermann2013]. HESS J1731-347 (G353.6-0.7) was first observed as an unidentified very-high-energy (VHE; $>$100 GeV) $\gamma$-ray source by the High Energy Stereoscopic System (HESS) [@Aharonian2008b]. @Tian2008 discovered the radio and X-ray counterparts of HESS J1731-347 and identified it as a shell-type SNR. @Abramowski2011 carried out an additional $\gamma$-ray observation with HESS and detected its shell-type morphology. Together with RX J1713.7-3946 [@Aharonian2004; @Aharonian2006; @Aharonian2007a], RX J0852.0-4622 [@Aharonian2005; @Aharonian2007b], RCW 86 [@Aharonian2009; @Abramowski2016] and SN 1006 [@Acero2010], HESS J1731-347 becomes one of five firmly identified TeV shell-type SNRs [@Rieger2013]. The distance of HESS J1731-347 is under debated. @Tian2008 argued that HESS J1731-347 locates at $\sim$3.2 kpc if it is associated with the nearby HII region G353.42-0.37. By comparing the absorption column density derived from the X-ray observation and that obtained from $^{12}$CO and HI observations, @Abramowski2011 set 3.2 kpc as a lower limit of its distance, which is reinforced by @Doroshenko2017. In addition, @Fukuda2014 suggested that HESS J1731-347 is correlated with the interstellar proton cavity at a velocity range from $-90$ km s$^{-1}$ to $-75$ km s$^{-1}$, indicating a distance of 5.2$-$6.1 kpc. However, no significant emission from dense molecular gas traced by CS(1-0) line coincides with HESS J1731-347 at that distance [@Maxted2017]. Due to uncertainties of the distance and other parameters, the age of HESS J1731-347 is estimated to be in a wide range of $2-27$ kyr [@Tian2008; @Abramowski2011; @Fukuda2014; @Acero2015b]. HESS J1731-347 and its sub-regions were detected in the X-ray band by [*ROSAT*]{}, [*XMM-Newton*]{} and [*Suzaku*]{}, with an X-ray morphology consistent with the radio shell [@Tian2008; @Tian2010; @Abramowski2011; @Bamba2012; @Doroshenko2017]. The X-ray emission from the complete SNR and its sub-regions are found to be non-thermal. The X-ray spectral index is 2.66 for the entire SNR [@Doroshenko2017], and is somewhat harder ($\Gamma=2.28$) for the north-east region [@Abramowski2011]. A compact object, XMMS J173203-344518, located near the geometrical center of the remnant, was detected by [*XMM-Newton*]{}, which was considered to be the central compact object (CCO) associated with HESS J1731-347 [@Tian2010; @Halpern2010; @Abramowski2011]. @Yang2014 and @Acero2015b searched for the GeV $\gamma$-ray emission from HESS J1731-347 with the [*Fermi*]{} Large Area Telescope [[*Fermi*]{}-LAT; @Atwood2009] data. No significant signal was detected and only the upper limits were given. Furthermore, no candidate source in the third [*Fermi*]{}-LAT source catalog [3FGL; @Acero2015a] is found to be associated with HESS J1731-347. In this paper, we revisit the GeV $\gamma$-ray emission in the direction of HESS J1731-347, with 9 year Pass 8 data recorded by [*Fermi*]{}-LAT. A statistically significant excess which is positionally consistent with HESS J1731-347 is found. In Section 2, we present the data analysis and results, including the spatial and spectral analysis. Based on the multi-wavelength observations of HESS J1731-347, we model the non-thermal radiation of it in Section 3. The conclusion of this work is presented in Section 4. Data Analysis ============= Data Reduction -------------- {width="3.5in"} {width="3.5in"} We select the latest Pass 8 version of the [*Fermi*]{}-LAT data with “Source” event class (evclass=128 & evtype=3), recorded from August 4, 2008 (Mission Elapsed Time 239557418) to August 4, 2017 (Mission Elapsed Time 523497605). The region of interest (ROI) is chosen to be a $14^\circ \times 14^\circ$ box centered at HESS J1731-347. In order to have a good angular resolution, we adopt the events with energies between 1 GeV and 300 GeV in this analysis. In addition, the events whose zenith angles are larger than $90^\circ$ are excluded to reduce the contamination from the Earth Limb. The data are analyzed with the [*Fermi*]{}-LAT Science Tools v10r0p5[^1], and the standard binned likelihood analysis method [gtlike]{}. The diffuse backgrounds used are [gll\_iem\_v06.fits]{} and [iso\_P8R2\_SOURCE\_V6\_v06.txt]{}, which can be found from the Fermi Science Support Center[^2]. All sources listed in the 3FGL and the two diffuse backgrounds are included in the model. During the fitting procedure, the spectral parameters and the normalizations of sources within $5^\circ$ around HESS J1731-347, together with the normalizations of the two diffuse backgrounds, are left free. Results ------- We create a $4^\circ \times 4^\circ$ TS (test statistic, which is essentially the logarithmic likelihood ratio between different models) map centered at HESS J1731-347 by slice the center of the box along each axis, after subtracting the 3FGL sources and the diffuse backgrounds. There are still excesses in this TS map, as marked out by green crosses. At the center of the TS map, a weak excess (labelled as Source T) is found to be spatially coincident with HESS J1731-347. It is noted that Newpts C was also detected in @Yang2014 with a TS value of about 20. We add Source T and the other six new sources, from A to F, in the model as additional point sources with power-law (PL) spectra, and re-do the likelihood fitting. The positions of these new sources are optimized by the [gtfindsrc]{} tool. Best-fitting results of their coordinates and TS values are listed in Table \[table:newpts\]. The TS value of Source T is about 25.9, and its best-fitting position is R.A.$=262.902^\circ$, Dec.$=-34.775^\circ$ with 1$\sigma$ error circle of $0.022^\circ$. The residual TS map after subtracting the additional sources A to F is shown in the right panel Figure \[fig:tsmap\]. Figure \[fig:tsmap-small\] gives a $1.3^\circ \times 1.3^\circ$ zoom-in of Figure \[fig:tsmap\]b in order to better show the spatial distribution of the target source T and its relationship with HESS J1731-347. The GeV $\gamma$-ray emission overlaps with part of the VHE emission region shown by the contours [@Abramowski2011]. However, the GeV TS map does not fully overlap with the TeV image, which is possibly due to the large point spread function (PSF) of Fermi-LAT and/or the fluctuation of the weak signal. Similar cases were also shown for SN 1006 [@Xing2016] and HESS J1534-571 [@Araya2017]. ---------- ----------- ----------- --------- ---------------- Name R.A. Dec. TS $\Delta\theta$ \[deg\] \[deg\] \[deg\] Source T $262.902$ $-34.775$ $25.9$ $0.093$ Newpts A $264.048$ $-34.370$ $130.8$ $0.936$ Newpts B $262.629$ $-33.882$ $29.3$ $0.929$ Newpts C $262.280$ $-35.051$ $59.8$ $0.670$ Newpts D $260.867$ $-33.704$ $59.5$ $2.062$ Newpts E $262.266$ $-36.233$ $34.9$ $1.598$ Newpts F $265.161$ $-34.500$ $26.5$ $1.786$ ---------- ----------- ----------- --------- ---------------- : Coordinates, TS values, and angular separations from the center of HESS J1731-347 of the newly added point sources \[table:newpts\] ![Zoom-in of the right panel of Figure \[fig:tsmap\], for a region of $1.3^\circ \times 1.3^\circ$ centered at HESS J1731-347, overlaid with the contours of the VHE image by HESS [@Abramowski2011]. The green contours to the west show the VHE image of HESS J1729-345. The radius of Gaussian smooth kernel is $\sigma = 0.04^\circ$. The black and white circles mark the radii of $0.1^\circ$ and $0.2^\circ$ uniform disk, respectively.[]{data-label="fig:tsmap-small"}](fig2.eps){width="3.5in"} ### Spatial Extension Considering that HESS J1731-347 has an extended morphology in radio, X-ray, and TeV $\gamma$-ray bands, we carried out an extension test with different spatial models. We used a uniform disk centered at the best-fitting position of Source T with radius of $0.1^\circ$, $0.15^\circ$, $0.2^\circ$, and $0.25^\circ$, as well as the TeV $\gamma$-ray image of HESS J1731-347, as spatial templates of Source T. The TS values for different spatial models are listed in Table \[table:template\]. We found that a $0.15^\circ$ disk template gives the highest TS value, 33.9, which corresponds to a significance of $\sim4.7\sigma$ for five (2 for the coordinates, 1 for the radius, and 2 for the spectrum) degrees of freedom (dof). For the four adopted disk templates, the TS values do not differ much from each other. Compared with the point source hypothesis, the data favors slightly an extended morphology. Using the TeV $\gamma$-ray template, a TS value of 25 is found. These results are quite consistent with that of @Condon2017, which used the data with different energy ranges and observation time-series. In the following analysis, we adopt the $0.15^\circ$ disk template for Source T. We also try to search for $\gamma$-ray emission from the nearby TeV source HESS J1729-345. The TeV image of HESS J1729-345 is used as the spatial template. No significant GeV $\gamma$-ray emission from the direction of HESS J1729-345 is detected. The TS value of HESS J1729-345 is about 4, and its flux upper limits will be derived (see the next sub-section). Spatial Model TS Degrees of Freedom --------------------------- -------- -------------------- -- Point Source $25.9$ $4$ $0.1^\circ$ Uniform Disk $32.1$ $5$ $0.15^\circ$ Uniform Disk $33.9$ $5$ $0.2^\circ$ Uniform Disk $32.4$ $5$ $0.25^\circ$ Uniform Disk $32.4$ $5$ TeV Image $25.0$ $2$ : TS values of Source T with different spatial models \[table:template\] ### Spectral Analysis For Source T, the global fit in the 1$-$300 GeV energy range with a $0.15^\circ$ disk template gives a spectral index of $\Gamma = 1.77\pm0.14$, and an integral photon flux of $(6.92\pm2.06)\times10^{-10}$ photon cm$^{-2}$ s$^{-1}$ with statistical errors only. Assuming a distance of 3.2 kpc [@Tian2008; @Nayana2017], the $\gamma$-ray luminosity between 1 GeV and 300 GeV is $1.26\times 10^{34}\,(d/3.2\ {\rm kpc})^2$ erg s$^{-1}$. The data is further divided into four energy bins with equal width in the logarithmic space to study is spectral energy distribution (SED). For each energy bin, we repeat the likelihood analysis, with only the normalizations of the sources within $5^\circ$ around Source T and the diffuse backgrounds in the model free. The spectral parameters of these sources are fixed to be the best-fitting values obtained in the global likelihood analysis. If the TS value of Source T is smaller than 4 in an energy bin, a 95% confidence level upper limit is given. The results of the SED are shown in Figure \[fig:sed\]. The GeV SED connects smoothly with the TeV spectrum of HESS J1731-347. The spatial coincidence and a smoothly connected $\gamma$-ray spectrum suggest that Source T is the GeV counterpart of HESS J1731-347. ![The Fermi-LAT SED of Source T (black dots). The red solid and dashed lines show the best-fitting power-law spectrum and its 1$\sigma$ statistic error band. Shaded gray regions (right axis) show the TS values of the four energy bins. The blue dots are the HESS observations of HESS J1731-347 in the VHE band [@Abramowski2011].[]{data-label="fig:sed"}](fig3.eps){width="50.00000%" height="0.3\textheight"} The significance of HESS J1729-345 is not high enough, and we derive the flux upper limits in energy bins of $1-6.7$, $6.7-44.8$, and $44.8-300$ GeV, which are shown in Figure \[fig:sed2\]. ![Upper limits of GeV $\gamma$-ray emission from the direction of HESS J1729-345, together with the HESS observations in the VHE band [@Abramowski2011].[]{data-label="fig:sed2"}](fig4.eps){width="50.00000%" height="0.3\textheight"} Discussion ========== The radio counterpart of HESS J131-347 was firstly identified by @Tian2008. The integrated flux density was derived to be $2.2 \pm 0.9$ Jy at 1420 MHz, through extrapolating that of one half of the remnant at low Galactic latitudes to the total SNR. With the Giant Metrewave Radio Telescope (GMRT), @Nayana2017 observed the complete shell of HESS J131-347 at 325 MHz, and obtained an integrated flux density of $1.84 \pm 0.15$ Jy. In the following models, we use the result of @Nayana2017 to constrain the model parameters. The X-ray flux of the full SNR given by @Doroshenko2017 is also used. We assume either a pure leptonic model or a hadronic-leptonic hybrid one to fit the wide-band SED from radio to TeV $\gamma$-rays. The spectrum of electrons or protons is assumed to be an exponential cutoff power-law form $$dN/dE \propto E^{-\alpha_{i}} {\rm exp}[-(E/E_{c,i})^{\delta}]\,, \nonumber$$ where [*i*]{} = [*e*]{} or [*p*]{}, $\alpha_{i}$ is the spectral index, $E_{c,i}$ is the cutoff energy of particles. and $\beta$ describes the sharpness of the cutoff. $\delta$ describes the sharpness of the cutoff, and we adopt the typical values of 0.5, 0.6 and 1.0 to constrain the parameters in the model. The radius of the SNR is nearly $0.25^\circ$ in the radio band [@Tian2008; @Nayana2017], and $0.27^\circ$ in the TeV band [@Abramowski2011]. Such an angular size corresponds to a physical radius of about $14-15$ pc for a distance of 3.2 kpc. The gas density in the vicinity of HESS J1731-347 is quite uncertain, due to the lack of thermal X-ray emissions. We assume a nominal value of $n = 1.0$ cm$^{-3}$. For the leptonic model, the background radiation field considered includes the cosmic microwave background (CMB), and an infrared (IR) radiation field with a temperature of 40 K and an energy density of 1 eV cm$^{-3}$ [@Abramowski2011]. The magnetic field strength is taken as a free parameter, which is determined through fitting to the multi-wavelength data. The derived model parameters are given in Table \[table:model\]. The corresponding multi-wavelength SED of the model calculation is shown in the left panel of Figure \[fig:multi-sed\]. The leptonic models with the three different values of $\delta$ can reproduce the muti-wavelength SED with little differences. Compared with the results of @Yang2014, the spectral index of electrons $\alpha_e$ and cutoff energy $E_{c,e}$ are both slightly smaller in this work. This may due to the updated radio, X-ray and GeV data we used in the model. The magnetic field strength, $B \sim 28~\mu$G, is consistent with that given in @Yang2014. Such a magnetic field strength is slightly larger than that of several other SNRs which show similar GeV-TeV $\gamma$-ray spectra, e.g. RX J1713.7-3946 [@Abdo2011; @Yuan2011; @Zeng2017], RX J0852-4622 , and RCW 86 [@Yuan2014]. These SNRs are believed to be a class of sources with leptonic origin of the $\gamma$-ray emission [@Yuan2012; @Funk2015; @Guo2017]. The cutoff of the spectrum may be due to the (synchrotron) cooling of electrons. The synchrotron cooling time scale of HESS J1731-347 is estimated to be $$t_{\rm syn} \approx 1800 \left(\frac{E_{c,e}}{9\, \mathrm{TeV}}\right)^{-1} \left(\frac{B}{28\,\mu\mathrm{G}}\right)^{-2}\, \mathrm{yr}. \nonumber$$ This time scale is close to the minimum value of the age of HESS J1731-347 inferred with other methods [@Tian2008; @Abramowski2011; @Fukuda2014; @Acero2015b]. @Nayana2017 reported an anti-correlation between the TeV $\gamma$-ray emission and radio brightness profile, and ascribed such an anti-correlation to the synchrotron cooling effect with a non-uniform magnetic field. This result supports the leptonic scenario for the multi-wavelength emission of HESS J1731-347. The right panel of Figure \[fig:multi-sed\] shows the multi-wavelength SED of the hadronic-leptonic hybrid model, in which the radio to X-ray data is acounted for by the synchrotron emission of electrons, and the GeV-TeV $\gamma$-ray emission is produced by the decay of neutral pions from $pp$ collisions. The model parameters are also summarized in Table \[table:model\]. For the hybrid models with different values of $\delta$, a hard spectral index of protons with $\alpha_p \sim 1.7$ even $\alpha_p \sim 1.5$, is needed to explain the hard GeV $\gamma$-ray spectrum. However, such spectrum of protons is difficult to be produced in the conventional diffusive shock acceleration model of strong shocks. The total energy of protons above 1$~$GeV is estimated to be $W_p \sim 1.5 \times 10^{50} (n/1.0\,\mathrm{cm}^{-3})^{-1}\ (d/3.2\,\mathrm{kpc})^{2}\ \mathrm{erg}$, corresponding to $\sim$ 15% particle acceleration efficiency for a typical total energy of $E_{\rm SN}\sim10^{51}$ erg released by a core-collapse supernova. The total energy $W_p$ depends on the distance and ambient gas density of HESS J1731-347. Since there is no thermal X-ray emission observed, the gas density would be very low, (e.g., @Abramowski2011 derived an upper limit of gas density of $\sim 0.01$ cm$^{-3}$ assuming an electron plasma temperature of 1 keV), and hence the corresponding $W_p$ would be much higher. However, if HESS J1731-347 expands in an inhomogeneous environment with dense gas clumps, the hard $\gamma$-ray emission and the high energy budget can be solved [@Inoue2012; @Gabici2014; @Fukui2013]. A spatial correlation between the TeV $\gamma$-ray shell and the interstellar protons at a distance of $\sim$ 5.2 kpc was reported in @Fukuda2014. It was suggested that the hadronic process contributes a large fraction of the $\gamma$-ray emission of HESS J1731-347 [@Fukuda2014]. This is similar to the cases of RX J1713.7-3946 and RX J0852.0-4622 [@Fukui2012; @Fukui2013]. However, no significant emission from dense molecular gas at such a distance was detected by @Maxted2017, which seems to be challenge to the hadronic scenario. {width="3.5in"} {width="3.5in"} ---------- ---------- -------------- -------------- --------------- ----------------- ----------- ----------------- ----------- -------------- -- -- -- -- -- -- Model $\delta$ $\alpha_{p}$ $\alpha_{e}$ $B_{\rm SNR}$ $W_{e}$ $E_{c,e}$ $W_{p}$ $E_{c,p}$ $\chi^2$ ($\mu$G) ($10^{47}$ erg) (TeV) ($10^{50}$ erg) (TeV) Leptonic $0.5$ $-$ $1.6$ $29.0$ $1.7$ $1.2$ $-$ $-$ $529.27/320$ $0.6$ $-$ $1.7$ $27.0$ $2.0$ $2.8$ $-$ $-$ $428.96/320$ $1.0$ $-$ $1.8$ $28.0$ $1.6$ $8.9$ $-$ $-$ $406.27/320$ Hybrid $0.5$ $1.5$ $1.6$ $85.0$ $0.35$ $0.7$ $1.4$ $10.0$ $472.04/317$ $0.6$ $1.5$ $1.7$ $87.0$ $0.35$ $1.5$ $1.5$ $15.0$ $372.33/317$ $1.0$ $1.7$ $1.8$ $80.0$ $0.3$ $5.5$ $1.5$ $38.0$ $377.31/317$ ---------- ---------- -------------- -------------- --------------- ----------------- ----------- ----------------- ----------- -------------- -- -- -- -- -- -- \[table:model\] HESS J1729-345 is an unidentified TeV source near HESS J1731-347 [@Abramowski2011]. Assuming that HESS J1731-347 locates at a distance of $\sim$3.2 kpc, @Cui2016 suggested that the TeV $\gamma$-ray emission of HESS J1729-345 possibly originates from the nearby molecular clouds illuminated by the CRs escaped from HESS J1731-347. @Capasso2016 reported a good spatial coincidence between the TeV $\gamma$-ray image in the bridge region and the dense gas at a distance of 3.2 kpc, which further supports the scenario of [@Cui2016]. @Nayana2017 detected possible radio counterparts of HESS J1729-345 at 843 MHz and 1.4 GHz. However, the multi-wavelength data of HESS J1729-345 is still lack. Future multi-wavelength observations are need to explore its nature. Conclusion ========== In this paper, we report the GeV $\gamma$-ray emission from the direction of HESS J1731-347 at a significance level of $\sim 4.7 \sigma$, with nine years of Pass 8 data recorded by the [*Fermi*]{}-LAT. The spatial morphology of HESS J1731-347 is found to be slightly extended in the GeV band. The GeV spectrum can be described by a hard PL for with an index of $\Gamma = 1.77\pm0.14$. The $\gamma$-ray characteristics of HESS J1731-347 is similar with several shell-type SNRs, including RX J1713.7-3946, RX J0852-4622, RCW 86, and SN 1006. A pure leptonic model can account for the wide-band SED of HESS J1731-347. If the hadronic process is adopted to explain the $\gamma$-ray emission, a very hard ($\sim1.6$) proton spectrum is required. In addition, the energy budget of CR protons may be a problem, given a potentially low gas density environment implied by the lack of thermal X-ray emission. We also search for GeV $\gamma$-ray emission from the nearby source HESS J1729-345. No significant excess is detected in its direction, and the upper limits are given. More multi-wavelength observations are necessary to address its emission mechanism, and test the proposed scenario of the interaction between CRs escaped from HESS J1731-347 and the molecular clouds. 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--- abstract: 'The Kirchhoff-Helmholtz principle of least heat dissipation is applied in order to derive the stationary state of the spin-Hall effect. Spin-accumulation due to spin-orbit interaction, spin-flip relaxation, and electrostatic interaction due to charge accumulation are treated on an equal footing. A nonlinear differential equation is derived, that describes both surface and bulk currents and spin-dependent chemical potentials. It is shown that if the ratio of the spin-flip relaxation length over the Debye-Fermi length is small, the stationary state is defined by a linear spin-accumulation potential and zero pure spin-current.' author: - 'J.-E. Wegrowe' - 'P.-M. Déjardin' title: 'Variational approach to the stationary spin-Hall effect' --- The classical bulk spin-Hall effect (SHE) is an ohmic conduction process occurring in non-ferromagnetic conductors, in which spin-orbit interaction leads to a spin-accumulation process [@Dyakonov; @Dyakonov2; @Hirsch; @Zhang; @Tse; @Maekawa; @Review; @Saslow; @JPhys; @EPL]. In the framework of the two channel model [@Hirsch; @Zhang; @Tse; @Maekawa], the system can be described as two sub-systems equivalent to two usual Hall devices, with an effective magnetic field that is acting in opposite directions (see Fig.1). In the Hall bar geometry [@EPL], charge accumulation is produced inside each spin-channel over the Debye-Fermi length scale $\lambda_D$. Due to the symmetry of the spin-orbit effective field, the total charge accumulation for the two channels cancels out and the total electric potential between the two edges of the device is zero. However, spin accumulation of both channels adds up and the consequences can be exploited in terms of spin-accumulation [@Awschalom; @Jungwirth; @Valenzuela; @Otani; @Gambardella; @Bottegoni]. In the usual descriptions of SHE [@Dyakonov; @Dyakonov2; @Hirsch; @Zhang; @Tse; @Maekawa; @Review; @Saslow; @JPhys], the system is defined with two sets of equations: the Dyakonov-Perel (DP) transport equations and the conservation laws for the spin-dependent electric charges. However, as far as we known, the conservation equations used in order to describe the drift currents in both spin-channels and the spin-flip relaxation from one-channel to the other do not take into account the electrostatic interaction and screening effects that govern the electric potential along the $y$ axis. Indeed, the conservation equation used for SHE is that derived in the framework of spin-injection effects [@Valet-Fert; @PRB2000; @Shibata; @Zhang; @Tse; @Maekawa; @Review; @Saslow; @JPhys] - i.e. without electric charge accumulation - that leads to a spin-accumulation spreading over the typical length scale $l_{sf}$. In order to derive the equations corresponding to the stationary states of the SHE, we apply the second law of thermodynamics through the Kirshhoff-Helmholz principle of least heat production [@Jaynes]. All three fundamental components of the SHE are taken into account on equal footing. Namely: the effect of the effective magnetic field due to spin- orbit scattering, the electric charge accumulation with electrostatic interactions and screening, and the spin-flip relaxation effect described by the chemical potential difference between the two spin-channels. A nonlinear fourth order differential equation is then derived for the chemical potentials, that describes non-trivial spin-currents flowing at the surface (defined over the length $\lambda_D$), and the bulk spin-dependent electric fields. In the case of small charge accumulation, it is shown that the stationary state is reached for linear spin-accumulation potential and zero pure spin-current at the limit $l_{sf} \gg \lambda_D$. This work shows that the variational approach yields a firm basis for the modeling of complex phenomena occurring in spintronic devices, like spin Hall magnetoresistance, Spin-pumping effect, and Spin-Seebeck or spin-Peltier effects.\ \[h!\] -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![ \[fig:Fig2\] : Schematic representation of the spin-Hall effect with the electrostatic charge accumulation $\delta n_{\updownarrow}$ at the boundaries. (a) usual Hall effect with the effective spin-orbit magnetic field $\vec H_{\uparrow}$ and (b) $\vec H_{\downarrow} = - \vec H_{\uparrow}$. (c ) the addition of configurations (a) and (b) leads to an effective magnetic field acting on the two different electric carriers. ](Fig1Seb.pdf "fig:"){height="8cm"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The system under interest is a Hall bar of finite width contacted to an electric generator (see Fig.1), in which the invariance along the $x$ axis is assumed (the role of Corbino geometries [@Benda] or the presence of lateral contacts [@Popovic] are not under consideration here). The density of electric carriers is described along the $y$ direction, inside each spin-channels, as $n_{\updownarrow} = n_0 + \delta n_{\updownarrow}$, where $n_0$ is the density of electric carriers for an electrically neutral system, and $\delta n_{\updownarrow}$ is the accumulation of the electric charge along the $y$ axis. The charge accumulation is governed by the Poisson law, that defines the electric potential along the $y$ axis : $\nabla^2 V = \frac{\partial^2 V}{\partial y^2} = - \frac{q \delta n}{\epsilon} $, where $\delta n = \delta n_{\uparrow} + \delta n_{\downarrow} $, $q$ is the electric charge, and $\epsilon$ is the electric permittivity. We assume that there is no accumulation of electric charges along the $x$ axis, so that the electric field is reduced to the drift force $n_0E_x^0$ in this direction, and $\partial n_{\updownarrow}/\partial x= 0$. The electro-chemical potential $\mu_{\updownarrow}$ is spin-dependent as it takes into account not only the electric potential $V$, but also the diffusion of the electric carriers due to the charge accumulation $\delta n_{\updownarrow}$, and the chemical potential $\mu^{ch}_{\updownarrow}$ that accounts for the spin-flip relaxation of the internal spin degrees of freedom (which is analogous to a chemical reaction [@DeGroot]). We have $\mu_{\updownarrow} = k \tilde T ln \left (\frac{n_{\updownarrow}}{n_0} \right ) + V + \mu^{ch}_{\updownarrow}$ [@Rubi; @Moi2007; @Entropy; @MagDiff] where $k$ is the Boltzmann constant and the temperature $\tilde T$ is the Fermi temperature $\tilde T= T_F$ in the case of a fully degenerated conductors, or the temperature of the heat bath $\tilde T=T$ in the case of a non-degenerated semiconductors. The Ohm’s law applied to the two channels reads $\vec J_{\updownarrow} = - q \hat \eta n_{\updownarrow} \vec \nabla \mu_{\updownarrow}$, where the mobility tensors $\hat \eta$ is a four by four matrix defined by the diagonal coefficients $\eta$ (the mobility of the charge carriers), and by the off-diagonal coefficients $\eta_{so}$ (the effective Hall mobility due to spin-orbit coupling). The off-diagonal coefficients obey the Onsager reciprocity relations $\eta_{xy \uparrow} = - \eta_{yx \uparrow} = \eta_{so}$ for the up spin channel and $\eta_{xy \downarrow} = - \eta_{yx \downarrow} = - \eta_{so}$ for the down spin-channel. The Ohm’s law then reads [@JPhys; @EPL]: $$\vec J_{\updownarrow}= q \eta n_{\updownarrow} \vec E_{\updownarrow} - D \vec \nabla n_{\updownarrow} \pm \vec e_z \times \left ( - q n_{\updownarrow} \eta_{so} \vec E_{\updownarrow} + D_{so} \vec \nabla n_{\updownarrow})\right ) \label{DP}$$ where $\vec E_{\updownarrow} = - \vec \nabla (V + \mu_{\updownarrow}^{ch})$ and $D = \eta kT$, $D_{so} = \eta_{so} kT$ are the diffusion constants [@JPhys]. The DP equations are recovered in the case $\vec \nabla \mu_{\updownarrow}^{ch} = 0$. The heat dissipation is due to the Joule heating for the two channels $- \vec J_{\updownarrow}. \vec \nabla \mu_{\updownarrow}$ and to the contribution due to the spin-flip relaxation. This last contribution reads $\mathcal L \Delta \mu^2 $, where $\Delta \mu = \mu_{\uparrow} - \mu_{\downarrow}$ is the spin-accumulation potential, and $\mathcal L$ is the Onsager transport coefficient related to the spin-flip relaxation process [@DeGroot; @PRB2000; @Moi2007; @Entropy]. Inserting equations (\[DP\]) we have: $$P_J = \int_{\mathcal D} \left \{ q \eta n_{\uparrow} \left( \vec \nabla \mu_{\uparrow} \right)^2 + q \eta n_{\downarrow} \left( \vec \nabla \mu_{\downarrow} \right)^2 + \mathcal L \Delta \mu^2 \right \} {{d}^{3}}\vec{r}. \label{PJ}$$ where $\mathcal D$ is the volume of the device. Note that the expression $P_J$ of the Joule power is the same with and without Hall or Spin-Hall effects (i.e. with or without cross coefficients in the Ohm’s law $\eta_{so}$, $D_{so}$ in Eq. (\[DP\])), since these effects are nondissipative. In order to illustrate the efficiency of the variational approach, we apply the Kirchhoff-Helmholz principle to the Joule power $P_J$ without any constraint : we observe that the functional derivative $\left ( \delta P_J/\delta \mu_{\updownarrow}\right )_{\mu_{\updownarrow}^{st}} = 0$ leads directly to the well known spin-accumulation equation that characterizes the stationary state for spin-injection through an interface between a ferromagnetic and a non-ferromagnetic conductor [@Johnson; @Wyder; @Valet-Fert; @PRB2000; @Shibata]: $$\nabla^2 \Delta \mu^{st} - \frac{\Delta \mu^{st}}{l_{sf}^2}=0 \label{SpinAcc}$$ where $\Delta \mu^{st}$ is the stationary value for the spin-accumulation $\Delta \mu$ and the spin-diffusion length is given by the relation $1/l_{sf}^2 = 1/l_{\uparrow}^2 + 1/l_{\downarrow}^2$ with $ l_{\updownarrow} = \sqrt{q \eta n_{\updownarrow} /(4 \mathcal L)}$. However, the description of the lateral charge accumulation at the edges of the Hall bar imposes the electrostatic interactions (that takes the form of a Poisson’s equation) to be introduced as a constraint with a first Lagrange multiplier $\lambda(y)$. Furthermore, in Hall devices, the electric generator imposes a constant current $J_x^{\circ}$ with the constant field $E_x^0$ along the $x$ axis, while the chemical potentials $\mu_{\updownarrow}(y)$ are left free along the $y$ axis. This constraint is described by a second Lagrange multiplayer $\beta(y)$ related to the projection of equation (\[DP\]) on the unit vector $\vec e_x$. The functional $\mathcal I$ to be optimized is given by: $$\begin{aligned} && \mathcal I [\mu_{\updownarrow}, \vec \nabla \mu_{\updownarrow}, \nabla^2 \mu_{\updownarrow}, n_{\updownarrow}, \nabla^2 n_{\updownarrow}] = \int_{\mathcal D} \bigg \{ q \eta n_{\uparrow} \left( \vec \nabla \mu_{\uparrow} \right)^2 + q \eta n_{\downarrow} \left( \vec \nabla \mu_{\downarrow} \right)^2 + \mathcal L (\mu_{\uparrow} - \mu_{\downarrow})^2 \nonumber \\ && - \lambda(y) \left [ \nabla^2 \left( \mu_{\uparrow}+\mu_{\downarrow} \right) - \frac{k \tilde T}{q} \nabla^2 \left (ln(n_{\uparrow}) + ln(n_{\downarrow}) \right ) +2q \frac{\delta n}{\epsilon} \right] \nonumber \\ && - \beta(y) \left [ q\left ( \eta n_0 E_x^{\circ} \vec e_x - \vec e_z \times \eta_{so} \left ( n_{\uparrow} \vec \nabla \mu_{\uparrow} - n_{\downarrow} \vec \nabla\mu_{\downarrow} \right) \right ). \vec e_x - J_{x}^{\circ} \right] \, \, \bigg \} \, dx dy. \label{I_Min}\end{aligned}$$ Note that this variational problem has been solved in a previous work [@Entropy] in the absence of spin-flip relaxation and with constant conductivity. Here, the minimization of Eq.(\[I\_Min\]) leads to the four Euler-Lagrange equations for the problem under interest. For simplicity, we will omit in what follows the superscript $st$ for the stationary values of the variables ($\mu_{\updownarrow} \equiv \mu_{\updownarrow}^{st} $, $\Delta \mu = \Delta \mu^{st}$, etc). Thus, on one hand, the Euler-Lagrange equations $\frac{\delta \mathcal I}{\delta \mu_{\updownarrow}} = 0$ explictly read : $$\begin{aligned} 2 \mathcal L (\mu_{\uparrow} - \mu_{\downarrow}) - 2 \eta q \frac{\partial }{\partial y}\left (n_{\updownarrow} \frac{\partial \mu_{\updownarrow}}{\partial y} \right ) - \frac{\partial^2 \lambda}{\partial y^2} \mp q \eta_{so} \frac{\partial \left (\beta n_{\updownarrow} \right)}{\partial y} =0. \label{deltaImu}\end{aligned}$$ On the other hand, the Euler-Lagrange equations $\frac{\delta \mathcal I}{\delta n_{\updownarrow}} = 0$ are : $$\begin{aligned} \eta q n_{\updownarrow} \left ( \frac{\partial \mu_{\updownarrow}}{\partial y} \right)^2 + \frac{k \tilde T}{q} \frac{\partial^2 \lambda}{\partial y^2} - \frac{2q n_{\updownarrow}}{\epsilon} \lambda \pm q \eta_{so} n_{\updownarrow} \frac{\partial \mu_{\updownarrow}}{\partial y} \beta =0 \label{deltaIn}\end{aligned}$$ On combining Eqs.(\[deltaIn\]) into Eqs.(\[deltaImu\]) we arrive at the relation between the Lagrange multipliers $\lambda$ and $\beta$: $$\begin{aligned} \lambda = - \frac{\epsilon kT}{2q^2} G_{\updownarrow} + \frac{\epsilon}{2} \left [\eta \left ( \frac{\partial \mu_{\updownarrow}}{\partial y} \right )^2 \pm \beta \eta_{so} \frac{\partial \mu_{\updownarrow}}{\partial y} \right ], \label{lambda}\end{aligned}$$ where $G_{\updownarrow} = \frac{1}{n_{\updownarrow}}\left ( 2 \eta q \frac{\partial }{\partial y} \left ( n_{\updownarrow} \frac{\partial \mu_{\updownarrow}}{\partial y} \right ) \pm q \eta_{so} \frac{\partial }{\partial y}(\beta n_{\updownarrow}) \mp 2 \mathcal L \Delta \mu \right )$. Injecting Eq.(\[lambda\]) into Eq.(\[deltaImu\]) yields: $$\begin{aligned} \frac{\partial^2 G_{\updownarrow}}{\partial y^2} - \frac{2q^2 n_{\updownarrow}}{\epsilon kT} G_{\updownarrow} = \frac{q^2}{kT} \frac{\partial^2}{\partial y^2} \left [ \left (\eta \frac{\partial \mu_{\updownarrow}}{\partial y} \right )^2 \pm \beta \eta_{so} \frac{\partial \mu_{\updownarrow}}{\partial y} \right ]. \label{G}\end{aligned}$$ The stationary state is verified if the Lagrange multipliers $\lambda$ and $\beta$ are related by Eqs.(\[lambda\]) and (\[G\]). The parameter $\beta$ is then free, provided Eq. (\[G\]) is verified. Therefore, we choose $\beta$ such that the right hand side of Eq.(\[G\]) vanishes, namely : $$\beta = \mp \frac{\eta}{\eta_{so}} \frac{\partial \mu_{\updownarrow}}{\partial y} \label{beta}$$ As $\beta$ does not depend on the spin state, it follows immediately that $\frac{\partial \mu_{\uparrow}}{\partial y} = - \frac{\partial \mu_{\downarrow}}{\partial y}$. Hence Eq.(\[G\]) reduces to: $$\begin{aligned} \frac{\partial^2}{\partial y^2} \left ( \frac{n_0}{n_{\updownarrow}} \frac{\partial }{\partial y} \left ( \frac{n_{\updownarrow}}{n_0} \frac{\partial \mu_{\updownarrow}}{\partial y} \right ) \mp \frac{\Delta \mu}{2 l_{sf}^2} \right ) - \frac{1}{\lambda_D^2} \left ( \frac{\partial }{\partial y} \left ( \frac{n_{\updownarrow}}{n_0} \frac{\partial \mu_{\updownarrow}}{\partial y} \right ) \mp \frac{ \Delta \mu}{2 l_{sf}^2} \right ) = 0 \label{Result}\end{aligned}$$ where we have introduced the Debye-Fermi length $\lambda_D = \sqrt{\frac{\epsilon k \tilde T}{2q^2 n_0}}$ and the spin-flip diffusion length $l_{sf} = \sqrt{\frac{q \eta n_0}{4 \mathcal L}}$ . The non-linear equation (\[Result\]) is a fourth order differential equation for the chemical potential $\mu_{\updownarrow}$, that has no simple analytical solution. This equation together with the symmetry of the spin-dependent electric fields $\frac{\partial \mu_{\uparrow}}{\partial y} = - \frac{\partial \mu_{\downarrow}}{\partial y}$, is an [*exact formulation of the stationary problem for the SHE*]{}. This is the main result of this work. Interestingly, Eq.(\[Result\]) does not depend on the Hall or Spin-Hall terms $\eta_{so}$ (as for the power $P_J$ in Eq.(\[PJ\])), so that it can also be apply to the case of the diffusive spin-accumulation in the so-called non-local or lateral geometry. Note also that Eq.(\[Result\]) is non-trivial in the case without spin-flip relaxation $\Delta \mu = 0$, as discussed in reference [@EPL]. The physical significance of this result can be analyzed further by formulating Eq.(\[Result\]) in terms of the current divergence $\vec{\nabla}\cdot\vec J_{\updownarrow}$. Inserting the divergence of Eqs.(\[DP\]) into Eq.(\[Result\]) yields: $$\begin{aligned} \frac{\partial^2}{\partial y^2}\left \{ \frac{n_0}{n_{\updownarrow}} \left ( \vec{\nabla}\cdot\vec J_{\updownarrow} \pm 2 \mathcal L \, \Delta \mu \right ) \right \} - \frac{1}{\lambda_D^2} \left ( \vec{\nabla}\cdot\vec J_{\updownarrow} \pm 2 \mathcal L \, \Delta \mu \right ) = 0 \label{Conservation1}\end{aligned}$$ A trivial solution is found with the usual conservation equations for the two channels $\vec{\nabla}\cdot\vec J_{\updownarrow} \pm 2 \mathcal L \, \Delta \mu = 0$. But the interesting point is that this simple solution corresponds to the situation in which the charge accumulation is ignored, which is the situation treated in the literature so far [@Valet-Fert; @PRB2000; @Shibata; @Zhang; @Tse; @Maekawa; @Review; @Saslow; @JPhys]. In contrast, Eq.(\[Conservation1\]) shows that due to electrostatic interactions surface currents are flowing within the region defined by the characteristic length $\lambda_D$. In order to analyze further the solutions of Eq.(\[Result\]), we assume a small charge accumulation $\delta n_{\updownarrow}/n_0 \ll1$ and use perturbation theory in terms of this small parameter. At zero order of perturbation we have: $$\begin{aligned} \frac{\partial^2}{\partial y} \left ( \frac{\partial^2 \mu_{\updownarrow}}{\partial y^2} \mp \frac{\Delta \mu}{2l_{sf}^2} \right ) - \frac{1}{\lambda_D^2} \left ( \frac{\partial^2 \mu_{\updownarrow}}{\partial y^2} \mp \frac{\Delta \mu}{2l_{sf}^2} \right ) = 0. \label{ResultZero}\end{aligned}$$ Due to the two characteristic length scales $\lambda_D$ and $l_{sf}$ (such that $\lambda_D/l_{sf} \ll 1$), we have to define the dimensionless variable $\tilde y = y/\lambda_D$ (the limit $\lambda_D \rightarrow 0$ directly applied on Eq.(\[ResultZero\]) is not correct due to the other limit $\lambda_D/l_{sf} \ll 1$). Taking the difference between the two channels in Eq.(\[ResultZero\]) yields : $$\begin{aligned} \frac{\partial^4 \Delta \mu}{\partial \tilde y^4} - \left ( 1+ \frac{\lambda_D^2}{l_{sf}^2} \right) \frac{\partial^2 \Delta \mu}{\partial \tilde y^2} + \frac{\lambda_D^2}{l_{sf}^2}\Delta \mu = 0 \label{ChargeSpinDiff0}\end{aligned}$$ The limit $\lambda_D/l_{sf} \ll 1$ leads to $\frac{\partial^4 \Delta \mu}{\partial \tilde y^4} - \frac{\partial^2 \Delta \mu}{\partial \tilde y^2} = 0$, or, in terms of the variable $y$: $$\frac{{{\partial }^{4}}\Delta \mu }{\partial {{y}^{4}}}-\frac{1}{\lambda _{D}^{2}}\frac{{{\partial }^{2}}\Delta \mu }{\partial {{y}^{2}}} \approx 0 \label{SpinDiffSHE}$$ Note that Eq.(\[SpinDiffSHE\]) deviates from the well-known spin-accumulation equation Eq.(\[SpinAcc\]) derived in the case of spin-injection. In particular, far away from the edges, we have: $$\frac{{{\partial }^{2}}\Delta \mu }{\partial {{y}^{2}}}\approx 0, \label{linear}$$ hence the profile of the spin-accumulation $\Delta\mu(y)$ is linear in the bulk (i.e. for $y \gg \lambda_D$) [@Bottegoni]. Inserting the solution $\frac{\partial \Delta \mu }{\partial y} = cst$ in the transport equation, we have $$\vec{J}_{\updownarrow} \cdot \vec{e}_{y}=0 \label{zeroCurrent}$$ This stationary state is defined by zero spin-current, and an effective electric field such that $E_{\uparrow} = - E_{\downarrow}$. Analysis of the first order of perturbation of Eq.(\[Result\]) shows that charges accumulate on the boundaries and therefore that the above discussion is unchanged. Interestingly, the linear solution was also that found in the case without spin-flip scattering (i.e. with $l_{sf} \rightarrow \infty$) [@Benda]. This is due to the fact that, in the framework of the SHE, the spin-flip scattering is related to the free variables $\mu_{\updownarrow}$ or $J_{y \updownarrow}$. In other terms, the spin-flip relaxation process cannot force the Spin-Hall device to dissipate more at stationary state. This is the opposite in the case of the usual spin-injection that leads to the giant magnetoresistance effect, for which the spin-flip relaxation is related to the constrained variables $E_{x}^0$ or $J_{x \updownarrow}$: spin-flip scattering is then forced by the generator along the $x$ direction. At last, the physical meaning of the linear solution just found can be best understood by inserting Eq.(\[linear\]) into the exact equation (\[Result\]). We obtain the well-known screening equation for $\lambda_D/l_{sf} \ll 1$ and at first order in $\delta n / n_0$ : $$\frac{{{\partial }^{2}} \delta n_{\updownarrow}}{\partial {{y}^{2}}} - \frac{\delta n_{\updownarrow}}{\lambda_D^2} = 0, \label{Screening}$$ Accordingly, this linear solution of the exact equation Eq.(\[Result\]) is the stationary state that corresponds to [*equilibrium*]{} along the $y$ axis. In conclusion, we studied the stationary state of the spin-Hall effect with taking into account both the electrostatic interactions and the spin-flip relaxation. We defined the Spin-Hall effect by the corresponding DP transport equations and by the expression of the power. In the framework of the Kirchhoff-Helmoltz variational principle, the stationary state is defined by the minimization of the dissipated power under the constraints specified by the electrostatic interactions and by a uniform charge current injected along the $x$ axis. The minimization leads to a fourth order differential equation, that describes the system, including both surface and bulk currents and fields. This equation shows that the form of the usual conservation laws used in the context of spin-injection and giant magnetoresistance should be modified in order to take into account electrostatic interaction and screening effects. We show that the solution for small charge accumulation and at the limit $\lambda_D/l_{sf} \ll 1$ is the same as that without spin-flip scattering, *whatever the absolute value of* $l_{sf}$. This solution corresponds to the linear behavior of the spin-accumulation $\Delta \mu(y)$ and chemical potentials $\mu_{\updownarrow}(y)$. 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--- abstract: 'In this letter, we propose a new routing strategy with a single free parameter $\alpha$ only based on local information of network topology. In order to maximize the packets handling capacity of underlying structure that can be measured by the critical point of continuous phase transition from free flow to congestion, the optimal value of $\alpha$ is sought out. By investigating the distributions of queue length on each node in free state, we give an explanation why the delivering capacity of the network can be enhanced by choosing the optimal $\alpha$. Furthermore, dynamic properties right after the critical point are also studied. Interestingly, it is found that although the system enters the congestion state, it still possesses partial delivering capability which do not depend on $\alpha$. This phenomenon suggests that the capacity of the network can be enhanced by increasing the forwarding ability of small important nodes which bear severe congestion.' author: - 'Chuan-Yang Yin' - 'Bing-Hong Wang' - 'Wen-Xu Wang' - Tao Zhou - 'Hui-Jie Yang' title: 'Efficient routing on scale-free networks based on local information' --- Since the seminal work on the small-world phenomenon by Watts and Strogatz [@WS] and scale-free networks by Barabási and Albert [@BA], the evolution mechanism of the structure and the dynamics on the networks have recently generated a lot of interests among physics community [@Review1; @Review2]. One of the ultimate goals of the current studies on complex networks is to understand and explain the workings of systems built upon them[@Epidemic1; @Epidemic2; @Epidemic3; @Cascade1; @Cascade2; @Yang], and relatively, how the dynamics affect the network topology[@WWX1; @WWX2; @WWX3; @Zhu]. We focus on the traffic dynamics upon complex networks, which can be applied everywhere, especially the vehicle flow problem on networks of roads and the information flow dynamic on interconnection computer networks. Some previous works have focused on finding the optimal strategies for searching target on the scale-free networks [@BJKim] and others have investigated the dynamics of information flow with respect to the packets handling capacity of the communication networks [@loadKim; @orderpara; @optimal; @Tadic; @Lai; @alleviate; @Yan], however, few of which incorporate these two parts. In this letter, we address a new routing strategy based on the local information in order to both minimize the packets delivering time and maximize the capacity of huge communication networks. In order to obtained the shortest path between any pair of nodes, one has to know the whole network structure completely. However, due to the huge size of the modern communication networks and continuous growth and variance of the networks’ structure, it is usually an impossible task. Even though the network is invariant, for the sake of routing packet along the shortest path each node has to put all the shortest paths between any pair of nodes into its routing table, which is also impractical for huge size because of limited storage capacity. Therefore, In contrast to previous works allowing the data packets forwarding along the shortest path, in our model, we assume each node only has the topology knowledge of it’s neighbors. For simplicity, we treat all nodes as both hosts and routers for generating and delivering packets. The node capacity, that is the number of data packets a node can forward to other nodes each time step, is also assumed to be a constant for simplicity. In this letter, we set $C=10$. \[0.80\][![\[fig:epsart\] (color online). The order parameter $\eta$ versus $R$ for BA network with different free parameter $\alpha$.](phase_tran.eps "fig:")]{} Recent studies indicate that many communication networks such as Internet and WWW are not homogeneous like random and regular networks, but heterogeneous with degree distribution following the power-law distribution $P(k)\thicksim k^{-\gamma}$. Barabási and Albert proposed a simple and famous model (BA for short) called scale-free networks [@BA] of which the degree distribution are in good accordance with real observation of communication networks. Here we use BA model with $m=5$ and network size $N=1000$ fixed for simulation. Our model is described as follows: at each time step, there are $R$ packets generated in the system, with randomly chosen sources and destinations, and all nodes can deliver at most $C$ packets toward their destinations. To navigate packets, each node performs a local search among its neighbors. If the packet’s destination is found within the searched area, it is delivered directly to its target, otherwise, it is forwarded to a neighbor node according to the preferential probability of each node: $$\Pi_{i}=\frac{k_i^\alpha}{\sum_jk_j^\alpha},$$ where the sum runs over the neighbors of node $i$ and $\alpha$ is an adjustable parameter. Once the packet arrives at its destination, it will be removed from the system. We should also note that the queue length of each node is assumed to be unlimited and the FIFO (first in first out) discipline is applied at each queue [@Lai]. Another important rule called path iteration avoidance (PIA) is that a path between a pair of nodes can not be visited more than twice by the same packet. Without this rule the capacity of the network is very low due to many times of unnecessary visiting of the same links by the same packets, which does not exist in the real traffic systems. \[0.80\][![\[fig:epsart\] The critical $R_c$ versus $\alpha$. The maximum of $R_c$ corresponds to $\alpha=-1$ marked by dot line.](Rc_alpha.eps "fig:")]{} \[0.80\][![\[fig:epsart\] The queue length cumulative distribution on each node by choosing different $\alpha$ more than zero. Data are consistent with power-law behavior.](pack_dis1.eps "fig:")]{} One of the most interesting properties of traffic system is the packets handling and delivering capacity of the whole network. As a remark, it is different between the capacity of network and nodes. The capacity of each node is set to be constant, otherwise the capacity of the entire network is measured by the critical generating rate $R_c$ at which a continuous phase transition will occur from free state to congestion. The free state refers to the balance between created packets and removed packets at the same time. While if the system enters the jam state, it means the continuous packets accumulating in the system and finally few packets can reach their destinations. In order to describe the critical point accurately, we use the order parameter[@orderpara]: $$\eta(R)=\lim_{t\rightarrow \infty}\frac{C}{R}\frac{\langle\Delta N_p\rangle}{\Delta t},$$ where $\Delta N_p=N(t+\Delta t)-N(t)$ with $\langle\cdots\rangle$ indicates average over time windows of width $\Delta t$ and $N_p(t)$ represents the number of data packets within the networks at time $t$. For $R<R_c$, $\langle\Delta N\rangle=0$ and $\eta=0$, indicating that the system is in the free state with no traffic congestion. Otherwise for $R>R_c$, $\eta \rightarrow\infty$, the system will collapse ultimately. As shown in Fig. 1, the order parameter versus generating rate $R$ by choosing different value of parameter $\alpha$ is reported. It is easy to find that the capacity of the system is not the same with variance of $\alpha$, thus, a natural question is addressed: what is the optimal value of $\alpha$ for maximizing the network’s capacity? Simulation results demonstrate that the optimal performance of the system corresponds to $\alpha\thickapprox -1$. Compared to previous work by Kim et al. [@BJKim], one of the best strategies is PRF corresponding to our strategy with $\alpha=1$. By adopting this strategy a packet can reach its target node most rapidly without considering the capacity of the network. This result may be very useful for search engine such as google, but for traffic systems the factor of traffic jam can not be neglected. Actually, average time of the packets spending on the network can also be reflected by system capacity. It will indeed reduce the network’s capacity if packets spend much time before arriving at their destinations. Therefore, choosing the optimal value of $\alpha=-1$ can not only maximize the capacity of the system but also minimize the average delivering time of packets in our model. \[0.80\][![\[fig:epsart\] The queue length cumulative distribution on each node by choosing different $\alpha$ less than zero. $P(n_p)$ approximately exhibits a Poisson distribution.](pack_dis2.eps "fig:")]{} \[0.80\][![\[fig:epsart\] The evolution of $N_p$ for $R>R_c$. Here, $\alpha_c$ takes $-1.5$ corresponding to the critical point $R_c=39$.](Np_t.eps "fig:")]{} To better understand why $\alpha=-1$ is the optimal choice, we also investigate the distribution of queue length on each node with different $\alpha$ in the stable state. Fig. 3 shows that when $\alpha\geq 0$, the queue length of the network follows the power-law distribution which indicates the highly heterogenous traffic on each node. Some nodes with large degree bear severe traffic congestion while few packets pass through the others and the heterogenous behavior is more obviously correspondent to the slope reduction with $\alpha$ increase from zero. But due to the same delivering capacity of all nodes, this phenomenon will undoubtedly do harm to the system because of the severe overburden of small numbers of nodes. In contrast to Fig. 3, Fig. 4 shows better condition of the networks with queue length approximately displays the Poisson distribution which represents the homogenous of each node like the degree distribution of random graph. From this aspect, we find that the capacity of the system with $\alpha<0$ is larger than that with $\alpha>0$. But it’s not the whole story, in fact, the system’s capacity is not only determined by the capacity of each node, but also by the actual path length of each packet from its source to destination. Supposing that if all packets bypass the large degree nodes, it will also cause the inefficient routing for ignoring the important effect of hub nodes on scale-free networks. By the competition of these two factors, the nontrivial value $\alpha=-1$ is obtained with the maximal network’s capacity. \[0.80\][![\[fig:epsart\] The ratio between $\Delta N_p$ and time step interval $\Delta t$ versus $R$ (a) and versus $R-R_c$ the rescaling of $R$ (b) for different $\alpha$. In (b) three curves collapse to a single line with the slope $\approx 0.7$ marked by a dashed line.](two_in_one.eps "fig:")]{} The behavior in jam state is also interesting for alleviating traffic congestion. Fig. 5 displays the evolution of $N_p(t)$ i.e. the number of packets within the network with distinct $R$. $\alpha$ is fixed to be $-1.5$ and $R_c$ for $\alpha=-1.5$ is $39$. All the curves in this figure can be approximately separated into two ranges. The starting section shows the superposition of all curves which can be explained by the fact that few packets reach their destinations in a short time so that the increasing velocity of $N_p$ is equal to $R$. Then after transient time, $N_p$ turns to be a linear function of $t$. Contrary to our intuition, the slope of each line is not $R-R_c$. We investigate the increasing speed of $N_p$ with variance of $R$ by choosing different parameter $\alpha$. In Fig. 6(a), in the congestion state $N_p$ increases linearly with the increment of $R$. Surprisingly, after $x$ axis is rescaled to be $R-R_c$, three curves approximately collapse to a single line with the slope $\approx0.7$ in Fig. 6(b). On one hand, this result indicates that in the jam state and $R$ is not so large, the dynamics of the system do not depend on $\alpha$. On the other hand the slope less than $1$ reveals that not all the $R-R_c$ packets are accumulated per step in the network, but about $30$ percent packets do not pass through any congested nodes, thus they can reach their destination without contribution to the network congestion. This point also shows that when $R$ is not too large in the congestion state, the congested nodes in the network only take the minority, while most other nodes can still work. Therefore, the congestion of the system can be alleviated just by enhancing the processing capacity of a small number of heavily congested nodes. Furthermore, we study the variance of critical point $R_c$ affected by the link density of BA network. As shown in Fig. 7, increasing of $m$ obviously enhances the capacity of BA network measured by $R_c$ due to the fact that with high link density, packets can more easily find their target nodes. \[0.80\][![\[fig:epsart\] The variance of $R_c$ with the increasing of $m$. ](Rc_m.eps "fig:")]{} Motivated by the problem of traffic congestion in large communication networks, we have introduced a new routing strategy only based on local information. Influenced by two factors of each node’s capacity and navigation efficiency of packets, the optimal parameter $\alpha=-1$ is obtained with maximizing the whole system’s capacity. Dynamic behavior such as increasing velocity of $N_p$ in the jam state shows the universal properties which do not depend on $\alpha$. In addition, the property that scale-free network with occurrence of congestion still possesses partial delivering ability suggests that only improving processing ability of the minority of heavily congested nodes can obviously enhance the capacity of the system. The variance of critical value $R_c$ with the increasing of $m$ is also discussed. Our study may be useful for designing communication protocols for large scale-free communication networks due to the local information the strategy only based on and the simplicity for application. The results of current work also shed some light on alleviating the congestion of modern technological networks. The authors wish to thank Na-Fang Chu, Gan Yan, Bo Hu and Yan-Bo Xie for their valuable comments and suggestions. This work is funded by NNSFC under Grants No. 10472116, 70271070 and 70471033, and by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP No.20020358009). 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--- abstract: 'We revisit the proposal that the resolution of the Cosmological Constant Problem involves a sub-millimeter breakdown of the point-particle approximation for gravitons. No fundamental description of such a breakdown, which simultaneously preserves the point-particle nature of matter particles, is yet known. However, basic aspects of the self-consistency of the idea, such as preservation of the macroscopic Equivalence Principle while satisfying quantum naturalness of the cosmological constant, are addressed in this paper within a Soft Graviton Effective Theory. It builds on Weinberg’s analysis of soft graviton couplings and standard heavy particle effective theory, and minimally encompasses the experimental regime of soft gravity coupled to hard matter. A qualitatively distinct signature for short-distance tests of gravity is discussed, bounded by naturalness to appear above approximately 20 microns.' author: - | **Raman Sundrum[^1]**\ \ *Department of Physics and Astronomy*\ *Johns Hopkins University*\ *3400 North Charles St*.\ *Baltimore, MD 21218-2686* title: | **Fat Gravitons, the Cosmological Constant\ and Sub-millimeter Tests** --- Introduction ============ Imagine an alien species, sophisticated enough to know the overarching principles of quantum mechanics and relativity, but whose particle physics expertise (or funding) can only engineer or observe momentum transfers below $10^{-3}$ eV. While they have access to heavy, macroscopic sources, the only fundamental fields and particles they know are the metric of General Relativity (GR), soft electromagnetic fields and perhaps some neutrinos. The alien theoreticians have nevertheless synthesized the various tools of quantum field theory from the big principles. Superstring theory is also flourishing. Phenomenologists have put in place a minimal effective field theory cut off by $10^{-3}$ eV, which accomodates the data below this scale while being agnostic about physics above. The aliens have also run into the Cosmological Constant Problem (CCP). (For a review see Ref. [@weinberg].) Actually, since the observed “dark energy” density of the cosmos is $\sim (10^{-3}$ eV)$^4$ [@ccexpt] [@pdg], their minimal effective theory is not presently fine-tuned. However, if new experiments above $10^{-3}$ eV continue to support the minimal effective theory, now with a larger UV cutoff, then the cosmological constant would be fine-tuned. Naturalness therefore predicts new physics just above $10^{-3}$ eV, acting to cut off the quartic divergences in the cosmological constant within the effective theory. The aliens are therefore quite excited about new short distance tests of gravity, $< 0.1$ mm $\sim (10^{-3}$ eV)$^{-1}$, along with other “high-energy” experiments. They imagine that they might discover sub-millimeter strings cutting off all of point-particle effective field theory, or supersymmetry enforcing cancellations in radiative corrections to the cosmological constant. Or perhaps something no one has thought of. We, on the other hand, seem less excited that experiments have the answer. We already know too much. Our particle physicists have probed momentum transfers all the way up to a TeV without finding sub-millimeter supersymmetry or strings. Effective field theory of the Standard Model (SM) coupled to GR with a TeV cutoff beautifully accounts for all the data, but now the cosmological constant is unavoidably fine-tuned. There is “no-go” theorem forbidding new light fields from relaxing the cosmological constant [@weinberg]. The door through which new sub-millimeter gravitational physics might enter into a solution of the CCP seems firmly shut. The purpose of this paper is to pry open this door a little. An important first step is to notice that the TeV scale effective theory which leads to the CCP involves a tremendous extrapolation of standard GR to far shorter distances than gravity has been experimentally probed, in order to accomodate the wealth of SM data.[^2] Naively, this observation has no bearing on the CCP, since SM corrections to the IR effective cosmological constant reside in the gravitational effective action, $\Gamma_{eff}[g_{\mu \nu}]$, evaluated for extremely soft gravitational fields. The SM fields are hard and off-shell in general in such corrections, but then hard SM processes are well understood up to a TeV. Nevertheless, the central point of Ref. [@97] was to argue that (virtual) high energy contributions to an effective action, $\Gamma_{eff}[A]$, of a sector, A, from integrating out a different sector, B, cannot be robustly determined (or even roughly estimated) without knowing the high energy dynamics and degrees of freedom of [*both sectors, A and B*]{}. This conclusion does not follow from standard Feynman diagram calculations, but rather by re-thinking whether certain diagrams are warranted at all. Ref. [@97] illustrated the general claim by studying an analog system built out of QCD, where sector A undergoes a radical, but well hidden, change in its degrees of freedom, from light pions at low energies to quarks and gluons at high energies. The tentative conclusion drawn for the real CCP was that a drastic change in gravitational degrees of freedom above $10^{-3}$ eV, akin to compositeness[^3], could suppress high energy SM contributions to $\Gamma_{eff}[g_{\mu \nu}]$. If this is correct, then even though we know much more than the aliens about the SM, we have every reason to be as excited about sub-millimeter tests of gravity [@gravexpt1] [@gravexpt2] where the new degrees of freedom would be revealed. The present paper will further discuss the case for such a resolution to the CCP by a “fat graviton”, and how it can be tested experimentally in the near future. We do not construct a fundamental theoretical model of fat gravitons and point-like matter here. Instead we pursue a more modest goal. We argue that such a resolution to the CCP, and the relevance therefore of short-distance gravity experiments, is not ruled out by general considerations and principles, despite the fact that these considerations seem at first sight to strongly exclude any such scenario. Many aspects of self-consistency are addressed by an effective field theory formalism we will call Soft Graviton Effective Theory (SGET). It blends together aspects of Weinberg’s analysis of soft graviton couplings [@wsoft] with standard heavy particle effective theory [@hpet]. It is hoped that the development of such an effective field theory description will make the ideas precise enough to pursue more fundamental model-building, say within string theory, or to identify and pursue phenomenological implications. On the other hand, the more precise description of fat gravity may lead to falsification, either by experimental means or by proving “no-go” theorems. At least we will know for sure then that the door is shut. The discussion of the CCP in this paper suffers from some significant limitations. There are issues related to the CCP which involve cosmological time evolution. The discussion here takes a rather static view of the problem, focussing on SM quantum corrections. A key diagnostic tool for any new mechanism for the CCP is to consider its behavior in cases where there are multiple (metastable) vacua. It is certainly very interesting to pursue these considerations in the case of the present proposal, but the result is not yet conclusive and a discussion is deferred for later presentation. There is undeniably a new scale in gravity provided by the observed dark energy density [@ccexpt]. While in this paper it is related to the “size” of the fat graviton, this size is not predicted from other considerations but taken as input. Its constancy over cosmological times is also not determined. The proposition that the small vacuum energy density might translate by naturalness into a sub-millimeter scale for new gravitational physics was made in Ref. [@banks] (although the primary subject of Ref. [@banks] is a quite different approach to the CCP). The idea that this new physics involves a sub-millimeter breakdown in point-like gravity [@97] has been further discussed in the extra-dimensional proposal of Ref. [@gia]. The layout of this paper is as follows. Section 2 re-analyses the robustness of the CCP in standard effective field theory and how it rests on the presumption that the graviton is point-like and able to mediate hard momentum transfers. Section 3 considers the consequences of rejecting this presumption, that is, entertaining the possibility of a “fat graviton”. We see that there is now a loop-hole in the CCP, but that the macroscopic consequences of GR and the Equivalence Principle are necessarily preserved. Section 4 discusses experimental/observational contraints and predictions following from a fat graviton resolution of the CCP. In particular, naturalness predicts a non-zero cosmological constant, now however set only by the graviton “size”. There is rather a sharp prediction for where fat graviton modifications of Newton’s Law should appear, and the qualitative form they take. In Section 5, we begin construction of a Soft Graviton Effective Theory (SGET) which satisfies basic principles, captures the physics of hard SM processes as well as soft graviton exchanges between SM matter, but [*does not extrapolate standard GR to short distances*]{}. It is demonstrated that this effective theory, capable of minimally capturing our present experimental regimes, does not give robust contributions to the cosmological constant from heavy SM physics, thereby clarifying the fat-graviton loophole. Section 6 provides conclusions. Robustness of the CCP ===================== The quantum contributions to the cosmological constant which dominate in standard effective field theory, and appear most robust, arise from Feynman diagrams such as Fig. 1, with SM matter loops and very soft graviton external lines. Diagrams with different numbers of external gravitons correspond to different terms in the expansion of the cosmological term about flat space, $$\label{sqrtg} \sqrt{-g} = 1 + \frac{ h_{\mu}^{\mu}}{2 M_{Pl}} + ..., ~ ~ g_{\mu \nu} \equiv \eta_{\mu \nu} + \frac{ h_{\mu \nu} }{ M_{Pl}}.$$ Of course the tadpole term implies that flat space is destabilized by a cosmological constant, but as long as it is small the flat space diagrams still provide a convenient way to expand the leading effects. Diagrams such as Fig. 1 make ${\cal O}(\Lambda_{UV}^4/16 \pi^2)$ contributions to the effective cosmological constant, when $\Lambda_{UV}$ is taken to be a typical general coordinate invariant (GCI) cutoff. For $\Lambda_{UV} \geq$ TeV, the contribution is many orders of magnitude larger than the observed cosmological constant and the situation is highly unnatural. To search for the most robust contributions to the cosmological constant let us be more optimistic about the sensitivity to the true nature of the physics above a TeV which cuts off the diagrams (perhaps some type of stringiness). A simple way to do this is to calculate using dimensional regularization. However, this still results in finite contributions to the vacuum energy from known SM masses and interactions, $$\label{smrobust} \sim ~ \sum_{SM} \frac{(-1)^{F_{SM}}}{16 \pi^2} m_{SM}^4 \ln m_{SM} + {\cal O}(\frac{\lambda_{SM}}{(16 \pi^2)^2} m_{SM}^4),$$ which are still so large that the CCP is not much diminished. Here, the first term is due to just the free particle zero-point energies, while the second term is sensitive to SM couplings, $\lambda_{SM}$. These contributions of diagrams such as Fig. 1 seem theoretically very robust. Afterall, the couplings of the graviton lines are being evaluated for soft momenta, precisely where we are most confident about their couplings to SM matter given the experimental success of GR. We can therefore use Eq. (\[sqrtg\]) to evaluate the diagrams with any number of graviton legs, the one with no legs being the simplest way to compute the cosmological constant contribution of course. The remainder of the calculation involves the propagation and soft and hard quantum interactions of SM particles. Again, we have tested all this extensively in particle physics experiments up to a TeV. Yet it is the purpose of this paper is to look for loop-holes in the apparent robustness of the contributions to the cosmological constant from known SM physics. Let us digress here from the main thrust of this paper to briefly discuss another well-known approach to the CCP which naively avoids the robustness of the contributions, Eq. (\[smrobust\]). In this approach, GCI is replaced as the guardian symmetry of massless gravitons by Special Coordinate Invariance (SCI) [@sci] (for a review see Ref. [@weinberg]), consisting of only coordinate transformations with unit Jacobian and metrics with $\sqrt{-g} = 1$. In this approach, the cosmological constant appears as an extra integration constant of the (quantum) equations of motion, rather than being dynamically determined. In this way the CCP becomes an issue of [*initial conditions*]{} and does not relate to quantum corrections or, by naturalness, to any testable new gravitational physics (which would be a shame, but of course this is not an argument against the idea). Even with SCI, however there is a formal (and quite possibly physically relevant in a more fundamental description of gravity) objective meaning to Eq. (\[smrobust\]). It follows by thinking of all $m_{SM}$ as having their origins as the VEVs of some source external fields, which may even vary somewhat in different parts of the universe. We can formally write $$m_{SM} = \langle \chi(x) \rangle.$$ Then Eq. (\[smrobust\]) does contribute to dark energy, and if $\chi$ varies in different parts of the universe so do these contributions. In such a setting, the extra integration constant of SCI remains an exact spacetime constant which adds to the theoretically distinct and robust effects of Eq. (\[smrobust\]). From this perspective, there is not much difference between the implications of GCI and SCI, except that the ability in effective field theory to simply add an arbitrary cosmological constant counterterm to any particular model is replaced by the ability to add the effect of an arbitrary integration constant. The need to fine-tune away quantum corrections from Eq. (\[smrobust\]) remains intact, although with SCI this is done using the integration constant. From now on we return to taking GCI as the symmetry protecting the massless graviton. There are also subleading contributions (for $\Lambda_{UV} < M_{Pl}$) from diagrams such as Fig. 2, with graviton lines in the quantum loops. These quantum gravity contributions to the cosmological constant of ${\cal O}(\Lambda_{UV}^6/(16 \pi^2 M_{Pl}^2))$ are still significant from the point of view of naturalness (for $\Lambda_{UV} \geq$ TeV) but one might hope that our imperfect understanding of quantum gravity makes these contributions a less robust problem than diagrams such as Fig. 1. Further, they are certainly Planck-suppressed, and it is consistent for us to first neglect these effects and tackle instead the leading quantum corrections. In this paper, for simplicity we will neglect quantum gravity corrections all together, deferring a treatment of this topic for later presentation. Let us return to consider the dominant contributions coming from purely SM loops. When we allow general graviton momenta, Fig. 1 is a contribution to the gravitational effective action, $\Gamma_{eff}[g_{\mu \nu}]$, not just the cosmological constant. Let us ask, since the diagram is the cause of such concern, why we bother to include its contribution to $\Gamma_{eff}[g_{\mu \nu}]$ at all. A first response is that we are simply following the Feynman rules, but let us inquire more deeply what fundamental principles are at stake if we simply throw out these diagrams, but not the (well-tested) loop diagrams contributing to SM processes. Three principles stand out. \(I) Unitarity: Fig. 1 unitarizes lower-order tree and loop processes of the form gravitons $\rightarrow$ SM ($+$ gravitons). That is, Fig. 1 has imaginary parts for general momenta following from unitarity and these lower order processes. We have not yet seen such processes experimentally. Furthermore, when there are massive SM particles in the loops, the imaginary parts only exist once the external graviton momenta are above the SM thresholds. These are momenta for which quite generally we have not probed gravity (except that we know it is still so weak as to be invisible in experiments). If gravity is radically modified below such SM thresholds then we would have to radically modify diagrams such as Fig. 1. \(II) GCI: There are diagrams without imaginary parts in any momentum regime, but which are required when we include diagrams with imaginary parts so as to maintain the GCI Ward identities, ultimately needed to protect theories of massless spin-2 particles. Note however that throwing out [*all*]{} SM contributions to $\Gamma_{eff}[g_{\mu \nu}]$ is a perfectly GCI thing to do. \(III) Locality: In standard effective field theory one also has non-vacuum SM diagrams with soft gravitons attached, such as Fig. 3. where soft gravitons couple to, and thereby measure, loop corrections to a SM self-energy. We certainly do not want to throw this away since these contributions are absolutely crucial in maintaining the precisely tested equivalence of gravitational and inertial masses of SM particles and their composites. Now, when Figs. 3 and 1 are viewed as position space Feynman diagrams (or better yet as first quantized sums over particle histories) it is clear that they are [*indistinguishable locally in spacetime*]{}, only globally can we make out their topological difference. Locality of couplings of the point particles in the diagrams does not allow us to contemplate throwing away Fig. 1 which we do not want, while retaining Fig. 3 which we need. The gravitons cannot take a global view of which diagram they are entering into before “deciding” whether to couple or not. Thus our earlier argument for the robustness of the CCP hinges on locality. We can dissect diagrams contributing to $\Gamma_{eff}[g_{\mu \nu}]$ into small spacetime windows, and all the ingredient windows are well tested in other physical processes, albeit in globally different ways. We might contemplate dispensing with locality, but it seems to be the only way we explicitly know to have point-particle dynamics in a relativistic quantum setting. However, see Refs. [@moffat] for an approach to the CCP with sub-millimeter non-locality, as well as Refs. [@banks] and [@savas] for extremely non-local approaches to the CCP. Room for a Fat Graviton ======================= Basic Notions ------------- If some particles appearing in the Feynman diagrams are secretly extended states then the constraints of locality, and the consequent robustness of the CCP, are weakened. Since we have tested the point-like nature of SM particles to very short distances, the only candidate for an extended state is the graviton itself. Indeed, direct probes of the “size” of the graviton only bound it to be smaller than $0.2$ mm, following from short-distance tests of Newton’s Law [@gravexpt1]. Let us grant the graviton a size, $$\ell_{grav} \equiv 1/\Lambda_{grav}.$$ Such a “fat graviton” does not have to couple with point-like locality to SM loops, but rather with locality up to $\ell_{grav}$. In particular for $m_{SM} \gg \Lambda_{grav}$ a fat graviton can couple to SM loops [*globally*]{}, thereby evading reason (III) for the robustness of the CCP. To see this, note that locality up to $\ell_{grav}$ corresponds to standard locality of SM loops when only graviton wavelengths $> \ell_{grav}$ are allowed. Of course, for $m_{SM} \gg \Lambda_{grav}$ and graviton momenta $< \Lambda_{grav}$, diagrams such as Fig. 1 can be expanded as a series in the external momenta, that is a set of local vertices in spacetime. A cartoon of all this is given in Fig. 4. Thus a theory with a fat graviton could distinguish between Figs. 3 and 1, possibly suppressing Fig. 1 while retaining Fig. 3 needed for the Equivalence Principle. It is at least conceivable. One naive objection is that among the diagrams contributing to the cosmological constant is the one with no graviton external legs, corresponding to the first term on the right-hand side of Eq. (\[sqrtg\]), that is, pure SM vacuum energy. Since the graviton does not appear in the diagram the size of the graviton appears irrelevant and incapable of suppressing the contribution. However, physically the cosmological term is a self-interaction of the graviton field (defined about flat space say). Once we trust point-like diagrams we can use GCI to relate all of them for soft gravitons to the diagrams with no gravitons and then it becomes a mathematical convenience to compute this latter class of diagrams. They do not have any direct physical significance except as a short-hand for the diagrams with gravitons interacting. If the diagrams with graviton external lines are modified because gravitons are fat, there is no meaning to the diagram with no external lines. Indeed, notice that there is no physical consequence in an effective lagrangian if in addition to a cosmological constant multiplying $\sqrt{-g}$, we add a pure constant term with no gravitational field attached. When we expand about flat space the extra constant modifies the first term in Eq. (\[sqrtg\]). This shows that the cosmological term only has physical importance as a graviton coupling, and the size of the graviton is most certainly relevant to how SM loops can induce it. Let us now return to the issue (I) of the need to unitarize processes, gravitons $\rightarrow$ SM states ($+$ gravitons). We can most easily deal with this by making a precise assumption for what is an intuitive property of fat objects. We assume that hard momentum transfers $\gg \Lambda_{grav}$ via gravitational interactions are essentially forbidden, that is they are extremely highly suppressed even beyond the usual Planck suppressions of gravitational interactions. In particular for $m_{SM} \gg \Lambda_{grav}$, gravitons $\rightarrow$ SM states ($+$ gravitons) is suppressed, and the related unitarizing SM loop contributions, as well as GCI-related diagrams, are not required. Throwing out all massive SM loop contributions to $\Gamma_{eff}[g_{\mu \nu}]$ is entirely consistent with the GCI of the soft graviton effective field theory below $\Lambda_{grav}$. Naturalness of the Equivalence Principle ----------------------------------------- While we have seen that the general considerations (I – III) for the robustness of massive SM loop contributions to the cosmological constant, and indeed the whole of $\Gamma_{eff}[g_{\mu \nu}]$, are evaded by a fat graviton in principle, one can ask whether it requires fine tunings even more terrible than the original CCP in order to understand why the fat graviton manages to couple to self-energies of SM states in the precise way to maintain the equivalence between gravitational and inertial mass. After all, these self-energies, for example the mass of a proton or of hydrogen, are determined by short-distance physics $\ll \ell_{grav}$, which unlike a pointlike graviton, a fat graviton cannot probe. In fact we shall see that there is really no option but that soft fat gravitons couple according to the dictates of the Equivalence Principle. The only miracle is that the fat graviton has a mode which is massless with spin 2 and couples somehow to matter. The only way for soft massless spin-2 particles to consistently couple is under the protection of GCI [@protect], which in turn leads to the Equivalence Principle macroscopically. The energy and momentum of SM states are macroscopic features which the fat graviton could imaginably couple to. They are determined microscopically but are measurable macroscopically, just as one can measure the mass density of a chunk of lead macroscopically, even though this density has its origins in and is sensitive to microscopic physics. In fact in Section 5 we will show how the leading couplings of gravity to SM masses and interaction energies can be recovered as a consequence only of GCI below $\Lambda_{grav}$, forced on us once we accept that our fat object contains an interacting massless spin-2 mode. Soft and Hard SM effects ------------------------- Until now, we have been careless of the complication that the SM contains particles which can be lighter than $\Lambda_{grav}$, such as the photon, as well as particles which are heavier. Even with the fat graviton we cannot throw away loops of these lighter SM fields contributing to $\Gamma_{eff}[g_{\mu \nu}]$ because the soft components of these fields are part of a standard effective field theory, including GR below $\Lambda_{grav}$ (that is, the effective theory known to the aliens). Soft gravitons can therefore scatter into soft electromagnetic radiation, so there are unitarity-required loops in the gravitational effective action. Indeed, we know that soft massless SM loops are not local on the scale $\Lambda_{grav}$, that is such loops are not expandable as local vertices for soft graviton momenta. Therefore, even a fat graviton cannot couple globally to such loops and the arguments for their robustness from locality now apply. A further complication is that the soft SM particles can couple to hard or massive SM particles. Let us consider a concrete example from QED coupled to soft gravitons, Fig. 5. We cannot throw away this whole contribution to the gravity effective action because even for soft graviton external lines there are imaginary parts which unitarize soft gravitons $\rightarrow$ soft photons processes (and their reverse), as well as soft photon-photon scattering due to the electron loop. On the other hand if we keep this diagram, we get a contribution to the cosmological constant set by the electron mass, which is too big. The way too disentangle the hard and soft SM contributions is to simply do effective field theory below $\Lambda_{grav} \ll m_e$, where the soft photons “see” the electron loop as a local vertex, Fig. 6, the leading behaviour being of the rough form, $${\cal L}_{eff} \ni \alpha_{em}^2 \frac{F^4}{m_e^4},$$ $F_{\mu \nu}$ being the electromagnetic field strength. Thus we recover all the same soft graviton and photon imaginary parts of Fig. 5 with the diagram of Fig. 7, but now the contribution to the cosmological constant vanishes when we compute with dimensional regularization, since there is no mass scale in the propagators of the diagram itself, only in the overall coefficient. Thus a more precise statement is that with fat gravitons, massive SM ($m_{SM} \gg \Lambda_{grav}$) and hard light SM pieces of loop contributions to $\Gamma_{eff}[g_{\mu \nu}]$ may consistently be suppressed while soft light SM contributions are not. “Consistent” refers to the principles we have discussed before, unitarity and GCI relations in the regime we trust GR as an effective field theory now, namely $< \Lambda_{grav}$, and locality down to $\ell_{grav}$. There are no robust contributions to the cosmological constant from mass scales above $\Lambda_{grav}$. The Cosmological Constant in Fat Gravity ---------------------------------------- Below $\Lambda_{grav}$, we have a standard effective field theory of GR coupled to SM light states. Here, [*all*]{} these states behave in a point-like manner. At the edge of this effective theory there are $\Lambda_{grav}$-mass vibrational excitations of the fat graviton, since in relativistic theory an extended object cannot be rigid. At least in this standard effective field theory regime the quantum contributions to the cosmological constant should follow by the usual power-counting, that is $\sim {\cal O}(\Lambda_{grav}^4/16\pi^2)$. The details however depend on the details of the fat graviton. Thus in a fat graviton theory naturalness implies that the lower bound on the full cosmological constant is $\sim {\cal O}(\Lambda_{grav}^4/16\pi^2)$. Experimental/Observational Constraints/Predictions ================================================== When we apply the above naturalness bound for a fat graviton to the observed dark energy [@ccexpt] [@pdg], we can derive a bound on the graviton size, $$\ell_{grav} > 20 ~{\rm microns}.$$ Of course since the naturalness bound involved an order of magnitude estimate for the fat graviton quantum corrections to the cosmological constant, the bound on $\ell_{grav}$ is not an exact prediction. However, it is a reasonably sharp prediction because much of the uncertainty is suppressed upon taking the requisite fourth root of the dark energy. If we can experimentally exclude $\ell_{grav}$ being in this regime we can falsify the idea that the fat graviton is (part of) the solution to the CCP. How can we probe $\ell_{grav}$? This is the finite range of the vibrational excitations of the fat graviton. They can therefore mediate deviations from Newton’s Law at or below $\ell_{grav}$. The precise details of the deviations are sensitive to the unknown fat graviton theory, but the qualitative behavior follows from the essential feature of the fat graviton as unable to mediate momentum transfers harder than $\Lambda_{grav}$. In position space this implies Fig. 8, with the gravitational force being suppressed at distances below $\ell_{grav}$. Of course this would be a striking signal to observe in sub-millimeter tests of gravity! It sharply contrasts with the rapid short distance enhancement expected in theories with (only) large extra dimensions for gravity [@add]. Indeed there is no natural way (in the absence of supersymmetry [@scherk]) to have such suppression within standard point-like effective field theory. Present bounds from such tests [@gravexpt1] demonstrate that $\ell_{grav} < 0.2$ mm, so there is a fairly narrow regime to be explored in order to rule in or rule out a fat graviton approach to the CCP. Soft Graviton Effective Theory (SGET) ===================================== Basic Notions ------------- Can the graviton really be fat when SM matter is pointlike? Of course the only way to confidently answer in the affirmative is to build a consistent fundamental theory of this type. It is presently not known how to do this, say within string theory, the only known fundamental theory of any type of quantum gravity, where $\ell_{grav} = \ell_{string}$. One might worry that there is some sort of theorem anwering in the negative, that in a regime of pointlike matter the graviton must be pointlike too. Such a theorem would have to show that our macroscopic tests of GR and microscopic tests of SM quantum field theory imply robust (pointlike) features of microscopic gravity. The best way to argue against such a theorem is to construct a consistent effective field theory description which encapsulates precisely the present asymmetric experimental regimes for gravity and matter, but which does not commit itself to the details of microscopic gravity such as whether the graviton is pointlike. In particular, it would not suffer the usual CCP. We now turn to such a construction, generalizing the methods of heavy particle effective theory [@hpet]. Our discussion is closely related to the analysis of soft graviton couplings in Ref. [@wsoft]. Earlier discussions of heavy particle effective theory applied to GR appear in Refs. [@donohue] and [@97]. SGET is constructed from the fat graviton’s “point of view”. While the fat graviton can itself only mediate soft momentum transfers, it can “witness” and couple to hard momentum transfers taking place within the SM sector. The momentum of a freely propagating SM particle can be expressed in terms of its $4$-velocity, $$p_{\mu} = m v_{\mu}, ~ v^2 = 1, ~ v_0 \geq 1.$$ Interactions with fat gravitons can change this momentum but by less than $\Lambda_{grav}$, $$\label{momsplit} p_{\mu} = m v_{\mu} + k_{\mu}, ~ |k_{\mu}| < \Lambda_{grav}.$$ The basic idea of the effective theory is to integrate out all components of the SM field which are not of this form, that is far off-shell. The result is all that the fat graviton can “see” and couple to. In this regime, many SM loop effects which used to appear non-local in spacetime will appear local to the fat graviton. When coupling to gravity we will consider integrated out all massive vibrational excitations of the fat graviton, leaving only the soft massless graviton mode. We know that GCI is a necessary feature for basic consistency of the couplings of gravitons softer than $\Lambda_{grav}$ [@protect]. The IR use of GCI will be enough to recapture the standard macroscopic tests of GR. Integrating out only SM fields which are far off-shell will not eliminate hard but on-shell SM particles, and therefore when properly matched, the effective theory should reproduce the high energy S-matrix of the SM as well. That is, the effective theory must reproduce what we have seen of Nature, hard matter coupled to soft gravity. A very important distinction should be made. One might consider for example a proton, propagating along, only interacting with gravity. In the effective field theory, we must treat the proton as an elementary particle, not a composite of far off-shell quarks and gluons. All these off-shell particles are integrated out but the elementary proton is integrated “in” to match the usual SM physics. Intuitively, the fat graviton cannot distinguish the substructure of the proton. In this manner, there will be many more “elementary” fields in the effective field theory than in the usual SM quantum field theory. In this paper a simplified problem is considered, where the SM is replaced by a toy model consisting of a single massive real scalar field, $\phi$, with $\lambda \phi^4$ coupling. The scalar has its own “hierarchy” fine-tuning problem, but we will ignore this irrelevant issue here. The central limitation is not the restriction to spin 0 (non-zero spin is tedious but straightforward), but rather the absence of light SM particles. We will rectify this omission elsewhere. We will refer to the renormalizable toy model as the “SM” or “fundamental theory” for the remainder of this section, hopefully without causing confusion. We will also defer here the consideration of soft graviton quantum corrections, where soft graviton lines carry loop momenta. Again this will be presented elsewhere. Single Heavy Particle Effective Theory -------------------------------------- To begin, let us consider a state consisting of a single massive $\phi$-particle. We describe it with an effective field $\phi_v(x)$ which respects the split of momentum in Eq. (\[momsplit\]). The fact that the $4$-velocity $v$ is formally unaffected by gravity means that it is just an unchanging label for the field, while the Fourier components of $\phi_v(x)$ correspond to the “residual” momentum, $k_{\mu} < \Lambda_{grav}$, that alone can fluctuate with gravity interactions. However the split of Eq. (\[momsplit\]) has an inherent redundancy formalized in terms of a symmetry known as reparametrization invariance (RPI) [@rpi]: $$\begin{aligned} v &\rightarrow& v + \delta v \nonumber \\ k &\rightarrow& k - m \delta v, \end{aligned}$$ where $\delta v$ is an infinitesimal change in velocity, $\delta v.v = 0$. Obviously this transformation results in the same physical momentum $p$ and therefore the effective theory must identify the pairs $(v,k)$ and $(v + \delta v, k - m \delta v)$. Let us begin in flat space, without gravity. In terms of the effective field, RPI requires the identification $$\label{rpieq} \phi_v(x) \longleftrightarrow e^{i m \delta v.x} \phi_{v + \delta v}(x).$$ The simplest way to implement this is to treat this RPI as a “gauge” symmetry and ensure that the effective lagrangian is invariant under it. One must then be careful to only choose one element of any “gauge orbit” when extracting physics. It is straightforward to see that a covariant derivative given by $$D_{\mu} = \partial_{\mu} + i m v_{\mu}$$ is required in order to build RPI effective lagrangians. For an isolated SM particle we need only consider quadratic lagrangians. Using the fewest derivatives (corresponding to the fewest powers of $k_{\mu}/m < \Lambda_{grav}/m$) we have $$\begin{aligned} \label{freeeff} {\cal L}_{eff} &=& \frac{1}{2m} |D_{\mu} \phi_v|^2 - \frac{m}{2} \phi^{\dagger}_v \phi_v \nonumber \\ &=& \phi^{\dagger}_v [iv.\partial + \frac{\partial^2}{2m}] \phi_v, \end{aligned}$$ where we integrated by parts to get the last line. In the first line we chose a particular linear combination of two RPI terms. The overall coefficient is just a conventional wavefunction renormalization, but the relative coefficient is chosen to satisfy the physical requirement that the propagator have a pole at $k = 0$, given the interpretation of Eq. (\[momsplit\]). Formally, in the derivative expansion ($k/m$ expansion) the dominant term in ${\cal L}_{eff}$ is $ \phi^{\dagger}_v iv.\partial \phi_v$, while the subleading term, $\phi^{\dagger}_v \frac{\partial^2}{2m} \phi_v$ can be treated perturbatively, that is, as a higher derivative “interaction” vertex. The effective propagator is then given by $$\label{prop} \frac{i}{k.v + i \epsilon}.$$ This treatment will suffice for most of the examples given below. An important property of the effective propagator is that it contains a single pole, rather than the two poles at positive and negative energy of a standard field theory propagator. The reason is simple to understand. Given that (when gravity is finally included) the maximum residual momentum $k$ is $< \Lambda_{grav}$, even though the sign of $k_0$ is not fixed the sign of the total energy $p_0$ is clearly positive. Fat gravitons cannot impart the momentum transfers needed to go near the usual negative energy pole. Without negative total energies the effective field is necessarily complex, as indicated, and $\phi_v$ is purely a creation operator while $\phi_v^{\dagger}$ is purely a destruction operator. Such a split would look non-local in a fundamental quantum field theory, but not in the SGET “seen” by the fat graviton. Sometimes one studies processes where components of $k$ orthogonal to $v$ are larger than $k.v$, such that one cannot treat $\phi^{\dagger}_v \frac{\partial^2}{2m} \phi_v$ perturbatively. It would seem then that if one uses all of Eq. (\[freeeff\]) to determine the propagator one would find two poles again. However, in these circumstances one only needs to resum the $\partial_{\perp v}^2/2m$ part of the $\partial^2/2m$ term in the propagator. The other $(v.\partial)^2/2m$ piece of $\partial^2/2m$ would then be of even higher order and could still be treated as a perturbtation. In this way the resulting propagator, $$\label{nlprop} \frac{i}{k.v + k_{\perp v}^2/2m + i \epsilon},$$ again has a single pole. We will see such an example in what follows. Let us now couple soft gravity to ${\cal L}_{eff}$. Interactions for soft massless spin-2 modes of the fat graviton only make sense if protected by GCI, so our effective theory must be exactly GCI. The only assumption is that there is a sensible theory of a fat object with a massless spin-2 mode coupling to matter. To determine the possible couplings we must first determine the spacetime transformation properties of $\phi_v(x)$. Naively, one would guess that since the particle has spin 0, that in flat space $\phi_v(x)$ is a scalar field of Poincare invariance, and becomes a GCI scalar field once we turn on gravity. However the first presumption is incorrect. To see this consider a flat space Poincare transformation defined by $$x^{\mu} \rightarrow \Lambda^{\mu}_{~ \nu} x^{\nu} + a^{\mu}.$$ Restricting to an [*infinitesimal*]{} transformation of this type we have $$\begin{aligned} \phi_v(x) &\rightarrow& \phi_{\Lambda v}(\Lambda x + a) \nonumber \\ &=& \phi_{v + \delta v}(\Lambda x + a) \nonumber \\ &\equiv& e^{- i m \delta v.(\Lambda x + a)} \phi_{v}(\Lambda x + a),\end{aligned}$$ where in the second line we have used the fact that for an infinitesimal Lorentz transformation we can always write $\Lambda v$ in the form $v + \delta v$ where $v.\delta v = 0$, and in the last line we have used the RPI equivalence relation, Eq. (\[rpieq\]). This is not the transformation property of a Poincare scalar field. But clearly $e^{i m v.x} \phi_v(x)$ [*is*]{} a Poincare scalar. When we couple to gravity, it is this combination that remains a scalar field. A generally covariant derivative of the scalar is easy to form, $$\partial_{\mu} [e^{i m v.x} \phi_v(x)] = e^{i m v.x} (\partial_{\mu} + i m v_{\mu}) \phi_v(x).$$ Thus the combination $D_{\mu} \equiv (\partial_{\mu} + i m v_{\mu})$ here is forced on us by both GCI and RPI. The leading GCI and RPI effective lagrangian is then given by $$\begin{aligned} {\cal L}_{eff} &=& \sqrt{-g} \{ \frac{g^{\mu \nu}}{2m} D_{\mu} \phi_v^{\dagger} D_{\nu} \phi_v - \frac{m}{2} \phi^{\dagger}_v \phi_v \}. $$ To leading order in the expansions in $k/m$ and $m/M_{Pl}$, this yields $${\cal L}_{eff} = \phi^{\dagger}_v \{ i v.\partial - \frac{m}{2 M_{Pl}} v_{\mu} v_{\nu} h^{\mu \nu} \} \phi_v,$$ which reproduces the standard equivalence of gravitational and inertial mass. Note that in our derivation of this equivalence we did not make use of the existence of standard GR at short distances, even though short distance physics may well contribute in complicated ways to the mass of the SM state. Let us now ask what robust loop contribution the effective theory makes to the cosmological constant. Since the effective theory does have GCI we can just calculate the pure SM vacuum energy with no graviton external legs. Apparently this requires us to interpret the expression, $$\int d^4 k ~ \ln (k.v + i \epsilon).$$ Note that there is no physical SM mass scale in this expression so there is no robust contribution from known physics here at all even though the effective theory does reproduce the coupling of massive SM particles to gravity. We can simply set the above expression to zero. This can be thought of as normal ordering. The reason for having no robust contribution to the cosmological constant is because the effective theory of the soft gravitons does not know whether the graviton is fat or not, or indeed whether the heavy matter is highly composite, say the Earth (if the gravitons are very soft), or solitonic like a 0-brane in string theory. All these possibilites lead to and effective theory of the same form. The effective theory cannot commit to (even the rough size of) a cosmological constant contribution without knowing the differentiating physics which lies beyond itself. Effective theory with SM Interactions ------------------------------------- Let us now consider how to generalize the effective theory for processes involving several interacting SM particles coupled to soft gravitons. The SGET general form can be compactly expressed, $$\label{generaleff} {\cal L}_{eff} = \sqrt{-g} ~ \sum_{v, v'} \frac{\kappa_{v v'}(D_{\mu}/m)}{m^{3/2(N + N') - 4} } ~ \phi^{\dagger}_{v'_1} ... \phi^{\dagger}_{v'_{N'}} ~ \phi_{v_1} ... \phi_{v_N} e^{i m (\sum v - \sum v').x},$$ where there are dimensionless coefficients $\kappa_{v v'}$ which in general can contain GCI and RPI derivatives acting on any of the effective fields, contracted with the inverse metric, $g^{\mu \nu}$. Of course the non-trivial parts of such derivatives correspond to residual momenta, $k$, balanced by powers of $1/m$. Therefore they are relevant for subleading effects in $\Lambda_{grav}/m$. The general procedure for specifying the (leading terms) of the SGET is to match the effective theory to the fundamental SM in the absence of gravity, and then to covariantize minimally with respect to (soft) gravity. The matched correlators are those with nearly on-shell external lines, Eq. (\[momsplit\]). This will reproduce the soft graviton amplitudes of the fundamental SM directly coupled to gravity in the standard way, but now without any reference to pointlike graviton couplings. Therefore it is compatible with a having a fat graviton whose massless mode is protected only by an infrared GCI. Note that the hard SM momentum transfers are to be described in the effective theory by $v_i \rightarrow v'_j$, that is by a change of labels. Changes in effective field momenta, that is residual momenta, are necessarily soft. This is how a fat graviton sees SM processes, the hard momentum transfers are a given feature of such processes which the fat graviton cannot influence. See Ref. [@iramike] for an effective field theory formalism exhibiting similar dynamical label-changing in the context of non-relativistic QCD. It is unusual to see explicit $x$-dependence in (effective) lagrangians such as appears in the phase factor. Here it is required by overall momentum conservation, rather than conservation of residual momenta. More formally, it is required by RPI, as well as GCI (recalling that it is $\phi_v e^{i m v.x}$ which is a scalar of GCI). It is possible, and perhaps prettier, to partially gauge fix the RPI, by reparametrizing (as can always be done) such that $\sum v = \sum v'$, so that the phase factor $\rightarrow 1$ without compromising GCI. We will call this the “label-conserving gauge”. However, when loops are considered we will find it convenient (but not necessary) to depart infinitesimally from this gauge fixing and consider an infinitesimal phase. Below, we will work to leading order about the limits $M_{Pl}, m, \Lambda_{grav} \rightarrow \infty, ~ \Lambda_{grav}/m \ll 1,$ with $m/M_{Pl}$ fixed. This formal limit simplifies the effective theory. It is similar to doing standard effective field theory calculations, including matching, without an explicit cutoff, even though physically one imagines new physics cutting off the effective theory at a finite scale. ### Tree-level Matching Let us follow the general procedure outlined above and first shut off gravity and work in flat space. Consider the tree level diagram for $2 \rightarrow 2$ scattering in the fundamental theory, Fig. 9, where the external momenta are nearly on-shell. We can then express these external momenta in the form of Eq. (\[momsplit\]), where we choose label-conserving gauge, $$v_1 + v_2 = v'_1 + v'_2.$$ This amplitude is then straightforwardly matched by including a $2 \rightarrow 2$ effective vertex, $$\label{eff4} {\cal L}_{eff} \ni \frac{\lambda}{4m^2} \phi^{\dagger}_{v_1'} \phi^{\dagger}_{v_2'} ~ \phi_{v_1} \phi_{v_2}.$$ The $1/4m^2$ factor only arises due to the different normalizations of the interpolating fields between the fundamental and effective theories. Next, consider the $3 \rightarrow 3$ process in the fundamental theory, Fig. 10. Again, the external lines are nearly on-shell, so we can express them as $$p_i = m v_i + k_i, ~ p_i' = m v'_i + k'_i, ~ i = 1,2,3,$$ with label conservation. The internal line has momentum, $$p_{int} = m(v_1 + v_2 - v_1') + k_1 + k_2 - k_1'.$$ Generically in such hard SM collisions, in the limit $\Lambda_{grav}/m \ll 1$, the internal lines will be far off-shell and to leading order we can drop the $k$’s, $$p_{int} \approx m(v_1 + v_2 - v_1').$$ We can then match the fundamental diagram with an effective vertex, Fig. 11, given by $$\label{eff6} {\cal L}_{eff} \ni \frac{\lambda^2}{8m^5[(v_1 + v_2 - v_1')^2 - 1]} ~ \phi^{\dagger}_{v_1'} \phi^{\dagger}_{v_2'} \phi^{\dagger}_{v_3'} ~ \phi_{v_1} \phi_{v_2} \phi_{v_3}.$$ Notice that what was a fundamentally non-local exchange requiring an off-shell internal line in the fundamental theory, that is with non-analytic dependence on the external total momenta, is replaced in the effective vertex by a local interaction with a non-analytic dependence only on the labels, $v, v'$. Physically, this is because the process is fundamentally non-local, but is local down to $\Lambda_{grav} \ll m$, that is local “enough” for a fat graviton. If we had worked to higher order in $\Lambda_{grav}/m$, matching would have resulted in higher derivative effective vertices, corresponding to having retained higher powers of $k/m$ in expanding $1/(p_{int}^2 - m^2)$ for small $k < \Lambda_{grav}$. For processes of the form of Fig. 10, there are also exceptional situations which result in $p_{int}$ being nearly on-shell. These arise when one considers experiments (in position space) where the interaction region for wavepackets of particles $1$ and $2$ is greatly displaced from the trajectory of the wavepacket for particle $3$, compared with the size of the wavepackets. Thus the three-particle scattering is dominated by a sequence of two-particle scatterings, $$\begin{aligned} p_1 + p_2 &\rightarrow& p'_1 + p_{int} \nonumber \\ p_3 + p_{int} &\rightarrow& p'_2 + p'_3, \end{aligned}$$ where all momenta are nearly on-shell. In these exceptional cases we can express $p_{int} = m v_{int} + k_{int}$ with label conservation: $$\begin{aligned} v_1 + v_2 &=& v'_1 + v_{int} \nonumber \\ v_3 + v_{int} &=& v'_2 + v'_3. \end{aligned}$$ In the limits we are considering, the fundamental internal propagator then approaches the effective propagator, Eq. (\[prop\]), up to the convention-dependent field normalization, $$\begin{aligned} \frac{1}{ p_{int}^2 - m^2 + i \epsilon} &=& \frac{1}{2m} ~ \frac{1} {k_{int}.v_{int} + k_{int}^2/2m + i \epsilon} \nonumber \\ &\approx& \frac{1}{2m} ~ \frac{1} {k_{int}.v_{int} + i \epsilon}.\end{aligned}$$ Thus the exceptional fundamental diagram of the form of Fig. 10 is matched by an effective diagram of the same form, but where we use the effective $4$-point vertices matched earlier, $${\cal L}_{eff} \ni \frac{\lambda}{4m^2} \phi^{\dagger}_{v_1'} \phi^{\dagger}_{v_{int}} ~ \phi_{v_1} \phi_{v_2} + \frac{\lambda}{4m^2} \phi^{\dagger}_{v_2'} \phi^{\dagger}_{v_3'} ~ \phi_{v_3} \phi_{v_{int}},$$ and the effective propagator, Eq. (\[prop\]), for the internal line. More general tree level amplitudes are generally matched by a combination of the two procedures illustrated above, introducing new effective vertices and connecting effective vertices with effective propagators. Tree-level unitarity in the effective theory arises from the imaginary parts of amplitudes due to the $i \epsilon$-prescription when internal lines go on shell, precisely matching the fundamental theory. There is a situation one can imagine for finite $\Lambda_{grav}$ where we carefully tune the external momenta on Fig. 10 so that the internal line is intermediate between the two (more generic) situations we have considered of being far off-shell or nearly on-shell, that is, the internal line is of order $\Lambda_{grav}$ off-shell. In that case our simple procedure does not always allow the fundamental graph to be matched by a local effective vertex or an effective theory graph. It appears that there is a more complicated scheme for matching even these cases within SGET, and that they pose a technical rather than conceptual challenge. We will study these cases elsewhere. ### Coupling the effective matter theory to soft gravity Coupling the vertices of ${\cal L}_{eff}$ to gravity is very simple. For example, the minimal covariantization of Eq. (\[eff6\]) is given by multiplying by $\sqrt{-g}$. If we had worked to higher order in the derivative expansion we would have to also covariantize these derivatives with respect to GCI and contract them using $g^{\mu \nu}$. We can now use the GCI effective theory to compute hard SM processes coupled to soft gravitons, such as say Fig. 12. The results automatically match with the leading behavior of the analogous diagrams if we coupled soft gravitons directly to the fundamental theory in the usual way. There is no extra tuning of couplings needed to recover standard gravitational results beyond imposing infrared GCI. The dominant couplings of soft gravitons therefore do not distinguish whether (a) the graviton is fat and GCI is only a guiding symmetry of the couplings in the far infrared, or (b) the gravitons are point-like and GCI governs their couplings to the fundamental theory in the standard way usually assumed. ### Matching Loops Here, we will match the simplest fundamental non-trivial loop diagram, Fig. 13, in flat space. It illustrates the essential new complication that loops bring in the presence of a “gauge” symmetry like RPI, namely the need to gauge fix and determine the right integration measure. We will first consider the case where the incoming momenta are far away from the two-particle threshold. We can decompose the momenta as usual, $$p_i = m v_i + k_i, ~ p_i' = m v'_i + k'_i,$$ with label conservation. Denote the loop amplitude by $\Gamma_{fund}(p, p')$. There is a general result for Feynman diagrams [@bj], which is straightforward to explicitly check in this example, that $\Gamma_{fund}$ is locally analytic in the external momenta except when near a threshold, which we assumed above is not the case. That is, $\Gamma_{fund}(mv + k, mv' + k')$ is analytic in $k$ and $k'$ and has a series expansion. Naively, we might try to match the whole diagram in the effective theory by Fourier transforming this series expansion into local operators in the effective theory. However, we cannot do this as it violates hermiticity of the effective lagrangian and ultimately unitarity. To see this focus on the leading term in the expansion and how it would appear as an effective vertex, depicted in Fig. 14, $$\label{naivevx} {\cal L}_{eff} \ni \sum_{v, v'} \frac{\Gamma_{fund}(v, v')}{4m^2} \phi^{\dagger}_{v_1'} \phi^{\dagger}_{v_2'} ~ \phi_{v_1} \phi_{v_2} ~ ?$$ Hermiticity of the effective lagrangian requires that the coupling $\Gamma_{fund}(v, v')$ be real, but by unitarity in the fundamental theory or direct calculation we know that $\Gamma_{fund}(v, v')$ has an imaginary part corresponding to the region of integration where the internal propagators are on-shell. Of course we are free to replace $\Gamma_{fund}(v, v') \rightarrow {\rm Re} ~ \Gamma(v, v')$ in Eq. (\[naivevx\]), but then the imaginary piece must be matched from another source. Obviously, in the effective theory there is also a diagram of the form of Fig. 13, but where the vertices and propagators are replaced by effective vertices, Eq. (\[eff4\]), and propagators, Eq. (\[prop\]). There are two internal momenta now, $$\begin{aligned} p_{int} &=& mv_{int} + k_{int} \nonumber \\ p'_{int} &=& mv'_{int} + k'_{int},\end{aligned}$$ so naively the loop momentum integration measure has the form $$\sum_{v_{int}} \sum_{v'_{int}} \int d^4 k_{int} \int d^4 k'_{int} ~ \delta^4(mv_{int} + k_{int} + mv'_{int} + k'_{int} - p_1 - p_2).$$ This is ill-defined, there are too many sums going on because we are multiple-counting combinations $(v,k)$ that should be indentified by RPI. That is the correct measure has the form $$\frac{\sum_{v_{int}} \sum_{v'_{int}} \int d^4 k_{int} \int d^4 k'_{int}}{RPI} ~ \delta^4(mv_{int} + k_{int} + mv'_{int} + k'_{int} - p_1 - p_2),$$ where the denominator means to identify RPI related combinations. Our job is to do this by gauge fixing RPI, so that we are summing just one representative of each RPI equivalence class. This is the central subtlety in computing with the effective theory at loop level. We will fix the following gauge. The total incoming momentum $p_1 + p_2$ is necessarily timelike. We will define our coordinates in its rest frame for convenience, not because the gauge fixing breaks manifest relativistic invariance (which in any case would not be a disaster if properly treated). In this frame we will gauge fix $$\vec{k}_{int} = \vec{k}'_{int} = 0,$$ that is, only $k^0_{int}, k^{'0}_{int} \neq 0$. It is obvious that any nearly on-shell momenta like those in the internal lines of effective theory diagrams, $p_{int}, p'_{int}$, can be decomposed in this gauge, $$\begin{aligned} \label{gfix} \vec{v} &\equiv& \frac{\vec{p}}{m} \nonumber \\ v_0 &\equiv& \sqrt{1 + \vec{v}^2} \nonumber \\ k_0 &\equiv& p_0 - mv_0.\end{aligned}$$ Thus, an internal line is now specified by four real numbers, $k_0, \vec{v}$ rather than seven, $k_{\mu}, v_{\mu}: v^2 = 1$. This is the right counting. We can get the correct RPI measure of integration for the internal momenta, by noting that obviously $\int d^4 p_{int}$ is a RPI measure, since the $(v,k)$ split has not been made. Using Eq. (\[gfix\]) then leads to the RPI measure $$m^3 \int d^3 \vec{v}_{int} \int d k^0_{int}.$$ Thus the effective theory version of Fig. 13 is given by, $$\begin{aligned} \Gamma_{SGET loop} &=& \frac{i \lambda^2}{(2 \pi)^4} ~ \frac{\sum_{v_{int}} \sum_{v'_{int}} \int d^4 k_{int} \int d^4 k'_{int}}{RPI} ~ \delta^4(mv_{int} + k_{int} + mv'_{int} + k'_{int} - p_1 - p_2) \nonumber \\ &~& ~ \times \frac{1}{k_{int}.v_{int} + i \epsilon} ~ \frac{1}{k'_{int}.v'_{int} + i \epsilon} \nonumber \\ &=& \frac{i \lambda^2 m^6}{(2 \pi)^4} ~ \int d^3 \vec{v}_{int} \int d k^0_{int} \int d^3 \vec{v}'_{int} \int d k'^{0}_{int} ~ \delta^3(m \vec{v}_{int} + m \vec{v}'_{int} - \vec{p_1} - \vec{p_2}) \nonumber \\ &~& ~ \times \delta(m v^0_{int} + m v'^{0}_{int} + k^0_{int} + k'^{0}_{int} - p^0_{1} - p^0_{2}) ~ \frac{1}{k^0_{int}v^0_{int} + i \epsilon} ~ \frac{1}{k'^{0}_{int}.v'^{0}_{int} + i \epsilon}.\end{aligned}$$ Note that the two tree effective vertices here do not satisfy label conservation. This is because with our present gauge fixing it would be inconsistent to also insist on label-conserving gauge. Therefore we relax the latter requirement, which in any case has only a cosmetic value. We will work to leading (zeroth) order in the [*external*]{} residual momenta, so that we can simply take $p_i = m v_i, p'_i = m v'_i$. In our choice of frame we then have $$\begin{aligned} \vec{v}_1 + \vec{v}_2 &=& 0 \nonumber \\ E_{tot}/2 &\equiv& v^0_{1} = v^0_{2}.\end{aligned}$$ Substituting this in and integrating the $\delta$-functions gives, $$\begin{aligned} \Gamma_{SGET loop} &=& \frac{i \lambda^2 m^3}{(2 \pi)^4} \int \frac{d^3 \vec{v}_{int}}{(v^0_{int})^2} \int d k^0_{int} ~ \frac{1}{k^0_{int} + i \epsilon} ~ \frac{1}{E_{tot} - 2 m v^0_{int} - k^0_{int} + i \epsilon}.\end{aligned}$$ The $k^0_{int}$-integral is finite and done by contour integration,, $$\Gamma_{SGET loop} = \frac{\lambda^2 m^2}{2 (2 \pi)^3} \int \frac{d^3 \vec{v}_{int}}{(v^0_{int})^2} \frac{1}{v^0_{int} - E_{tot}/2m + i \epsilon}.$$ Now, the remaining $\vec{v}_{int}$-integral representation of $\Gamma_{SGET loop}$ is logarithmically divergent just as the familiar $\Gamma_{fund}$. However, it is straightforward to see that the imaginary parts, related by unitarity to tree-level two-particle scattering, are finite and agree (up to the usual difference in normalization of states), $${\rm Im} ~ \Gamma_{fund} = {\rm Im} ~ \Gamma_{SGET loop}/4m^2 = \frac{\lambda^2}{16 \pi^2} \int \frac{d^3 \vec{v_{int}}}{(2 v^0_{int})^2} \delta(v^0_{int} - E_{tot}/2m).$$ This of course just corresponds to integrating over the phase space for on-shell $2$-particle intermediate states. Thus the imaginary parts are matched between the effective and fundamental theories. It is the real parts which diverge. In both cases the integrals converge in a $(4 - \delta)$-dimensional spacetime, that is with dimensional regularization. The important point is that not just Re $\Gamma_{fund}$, discussed above, but also Re $\Gamma_{SGET loop}$ are local analytic functions of the external momentum. The latter is easily seen by deformation of the integration contour for $|\vec{v}_{int}|$ to avoid the $E_{tot}/2m$ pole, as long as $E_{tot}/2m > 1$ as we assumed (that is we are above the two-particle threshold). Therefore $\Gamma_{SGET loop}$ is locally analytic in $E_{tot} = \sqrt{(m v_1 + k_1 + m v_2 + k_2)^2}$, that is, analytic in the $k_i$. Thus we can introduce a local [and hermitian ]{} effective vertex given by $4 m^2 {\rm Re} ~ \Gamma_{fund} - {\rm Re} ~\Gamma_{SGET loop}$, of the form of Fig. 14 to match the real parts, thereby completing the matching procedure. Of course we can also reverse our starting assumption and reconsider matching if we chose $p_1 + p_2$ to be nearly at the two-particle threshold. In that case, we can decompose all the external particles of Fig. 13 to have a common 4-velocity label, $$p_{i} = m v + k_i, ~ p'_{i} = m v + k'_i.$$ The problem now is that even the real part of the diagram can be non-analytic in the $k_i$ and therefore cannot be captured by a local effective vertex, but must emerge from an effective diagram of the form of Fig. 13 just as the imaginary part must. But in just this near-threshold case, this proves to be possible. This situation corresponds to a rather standard case in heavy particle effective theory. For example, see Ref. [@nucl]. Nevertheless, we will verify below that things work. By a standard calculation we have $$\begin{aligned} \Gamma_{fund} &\equiv& \frac{i \lambda^2}{(2 \pi)^4} \int d^4 p_{int} ~ \frac{1}{p_{int}^2 - m^2 + i \epsilon} ~ \frac{1}{(p_1 + p_2 - p_{int})^2 - m^2 + i \epsilon} \nonumber \\ &=& \frac{\lambda^2}{16 \pi^2} \sqrt{ 1 - 4m^2/(p_1 + p_2)^2} ~ \ln \{ \frac{\sqrt{ 1 - 4m^2/(p_1 + p_2)^2} + 1}{\sqrt{ 1 - 4m^2/(p_1 + p_2)^2} - 1} \} + {\rm constant},\end{aligned}$$ where the constant term contains the usual divergence. Matching the constant term is trivial so we will focus on the term non-analytic in the external momenta. We will work to leading non-trivial order in $k_i/m$, $$(p_1 + p_2)^2 = 4m^2 + 2mv.(k_1 + k_2) + (k_1 + k_2)^2 \approx 4m^2 + 2mv.(k_1 + k_2).$$ Substituting into $\Gamma_{fund}$ and working to leading order yields $$\Gamma_{fund} \ni \frac{i \lambda^2}{16 \pi} \sqrt{\frac{(k_1 + k_2).v}{m}}.$$ Because we are near threshold this is not analytic in even the residual momenta. This allows it to have the behavior required by unitarity, imaginary above threshold, $(k_1 + k_2).v > 0$, but real below, $(k_1 + k_2).v < 0$. We now compute the analogous loop diagram in SGET of the form of Fig. 13. Again we must gauge fix. The simplest procedure is to write $$p_{int} = m v + k_{int}, ~ p'_{int} = m v + k'_{int},$$ with the [*same*]{} fixed velocity as the external lines. Thus the RPI integration measure is obvious, $\int d^4 p_{int} = \int d^4 k_{int}$. Let us first try to work strictly to leading order in the $1/m$ expansion. After integrating the momentum-conserving $\delta$-functions we have $$\begin{aligned} \Gamma_{SGET loop} &=& \frac{i \lambda^2}{(2 \pi)^4} \int d^4 k_{int} ~ \frac{1}{k_{int}.v + i \epsilon} ~ \frac{1}{(k_1 + k_2 - k_{int}).v + i \epsilon} \nonumber \\ &=& \frac{\lambda^2}{(2 \pi)^3} ~ \frac{1}{(k_1 + k_2).v} ~ \int d^3 k_{int \perp v} ~ 1 \nonumber \\ &=& 0,\end{aligned}$$ where in the second line we have done the finite $\int d (k_{int}.v)$ by contour integration. We see a cubic divergence. Formally, this corresponds to an ${\cal O}( m/(k_1 + k_2).v)$ effect. However, in dimensional regularization $\int d^3 k_{int \perp v} ~ 1 = 0$. Therefore we must work to higher order in $1/m$ by keeping the dominant subleading terms in the propagators, as in Eq. (\[nlprop\]), $$\begin{aligned} \Gamma_{SGET loop} &=& \frac{i \lambda^2}{(2 \pi)^4} \int d^4 k_{int} ~ \frac{1}{k_{int}.v + k_{int \perp v}^2/2m + i \epsilon} \nonumber \\ &~& ~ \times \frac{1}{(k_1 + k_2 - k_{int}).v + (k_1 + k_2 - k_{int})_{\perp v}^2/2m + i \epsilon} \nonumber \\ &\approx& \frac{\lambda^2}{(2 \pi)^3} \int d^3 k_{int \perp v} ~ \frac{1}{(k_1 + k_2 - k_{int}).v + k_{int \perp v}^2/m + i \epsilon} \nonumber \\ &=& \frac{i m^2 \lambda^2}{4 \pi} \sqrt{\frac{(k_1 + k_2).v}{m}},\end{aligned}$$ where in the second line we have again done the finite $\int d (k_{int}.v)$ by contour integration and kept only the leading terms in $(k_1 + k_2)/m$, and in the last line we have evaluated the linearly divergent $\int d^3 k_{int \perp v} $ using dimensional regularization. As can be seen, this precisely matches the non-analytic terms in $\Gamma_{fund}$ (taking into account the different state normalization as usual). The Cosmological Constant in SGET --------------------------------- Let us finally consider what robust contributions to the infrared cosmological constant emerge within SGET. Because of the procedure for obtaining the SGET by [*first*]{} matching to the fundamental flat space theory [*and then*]{} covariantizing with respect to gravity, there is no cosmological constant term in the effective lagrangian, Eq. (\[generaleff\]). Of course we could simply add one, but there is no robust reason to do so except quantum naturalness. Therefore let us consider loop contributions to the infrared cosmological constant within SGET. We can make use of the GCI enjoyed by SGET to simply compute the pure vacuum diagrams in the complete absence of graviton lines, that is, the coefficient of the “$1$” term in the expansion of $\sqrt{-g}$. Let us consider some typical diagrams. We have already discussed the non-interacting diagram of Fig. 15 in the previous section and explained why it must be set to zero. Fig. 16 is an example of a vacuum diagram involving propagators from a vertex to itself, which apparently requires us to make sense of expressions such as $$\frac{\sum_{v} \int d^4 k}{RPI} ~ \frac{1}{k.v + i \epsilon}.$$ However, as was the case in Fig. 15, there is a physical reason why we must take the diagram to vanish. The reason is that Fig. 16 requires a vertex of the form $\phi_{v_1} \phi_{v_2} ~\phi^{\dagger}_{v_1} \phi^{\dagger}_{v_2}$, obtained by matching in situations which there is only a soft momentum transfer in two-particle scattering. As in the discussion of Fig. 15, under these circumstances the effective theory cannot distinguish whether the heavy particle is itself a composite of very light particles or solitonic, in which case it cannot make robust large loop contributions. We can summarize the vanishing of diagrams like Fig. 15 and 16 by saying that in the SGET we must normal order. Fig. 17 is a vacuum diagram one can draw despite normal ordering. We will compute it by first gauge fixing in a similar manner to the non-vacuum loop diagram we examined above. However, unlike that case where there was a natural frame selected by the incoming (or outgoing) momenta, we must simply choose a frame arbitrarily, and fix the gauge $\vec{k}_{int} = 0$ in that frame. One might worry that this will lead to a Lorentz non-invariant answer, but it will not as we will see below. There are originally four $k^0_{int}$ residual energies for the four internal lines, but after integrating the energy-conserving $\delta$-function we are left with three residual energy integrals to be done. Let us focus on any single one of these. It clearly has the form $$\begin{aligned} \int d k^0_{int} ~ \frac{1}{k^0_{int} v^0_{int} + i \epsilon} ~ \frac{1}{k^0_{int} v'^{0}_{int} + ... + i \epsilon} ~ = ~ 0,\end{aligned}$$ where the ellipsis refers to a (real) combination of other energies external to this integral. This integral is finite and evaluates to zero by contour integration. To see this note that $v^0_{int}, v'^{0}_{int} > 0$ and therefore both poles lie on the same side of the real $k^0_{int}$ axis, so the contour can be closed at infinity without enclosing any poles. If we repeated this same type of analysis for Fig. 18, we would get residual energy integrals of the form $$\begin{aligned} \int d k^0_{int} ~ \frac{1}{k^0_{int} v^0_{int} + i \epsilon} ~~ \frac{1}{- k^0_{int} v'^{0}_{int} + ... + i \epsilon},\end{aligned}$$ which does not vanish. However, the diagram does not exist in the SGET because it requires vertices of the form $\phi \phi \phi \phi$ rather than the $\phi^{\dagger} \phi^{\dagger} \phi \phi$ vertices appearing in Fig. 17. Recalling that SGET vertices arise from matching to fundamental correlators with nearly on-shell external lines, we see that no effective vertex such as $\phi \phi \phi \phi$ could have arisen upon matching. Therefore Fig. 18 simply does not exist. Recalling that $\phi^{\dagger}_v$ and $\phi_v$ are creation and destruction operators, one might wonder whether the presence of $\phi^{\dagger} \phi^{\dagger} \phi \phi$ is correlated with that of $\phi \phi \phi \phi$ as it is in fundamental field theories. In fundamental theories this is a consequence of locality, there is no local way of separating positive and negative energy operators. However, as we have seen above, locality only down to $\Lambda_{grav} \ll m$ does not imply such a correlation. It is straightforward to check that the examples of Figs. 15 to 18 exhaust all the possible cases arising in general vacuum diagrams. In SGET there are simply no robust contributions from heavy matter to the cosmological constant! For a fat graviton there is new gravitational physics at $\Lambda_{grav}$, its vibrational excitations. We do not explicitly know this physics but we can estimate its loop contributions to the cosmological constant by standard power-counting, ${\cal O}(\Lambda_{grav}^4/16 \pi^2)$. This sets the minimal natural size of the cosmological constant. Conclusions =========== The soft graviton effective theory demonstrates a clear qualitative distinction between (a) loop effects of heavy SM physics on SM processes, (b) soft graviton exchanges between such ongoing SM processes, and (c) loop effects of heavy SM physics on the low-energy gravitational effective action, and in particular the infrared cosmological constant. The effective theory can match (a) to the fundamental SM theory (which is of course very well tested), and describe (b) constrained only by general coordinate invariance in the infrared, such as even a fat graviton must have. In this manner the effective theory captures the two pillars of our experimental knowledge, soft gravitation and the SM of high-energy physics. However, none of this implies any robust contributions to (c). There is therefore a loop-hole in the cosmological constant problem for a fat graviton which is absent for a point-like graviton, and it makes sense to vigorously hunt for its realization experimentally and within more fundamental theories such as string theory. Quantum naturalness related to the contributions to the cosmological constant of the vibrational excitations of the fat graviton, as well as from soft gravitons, photons and neutrinos, implies that Newton’s Law should yield to a suppression of the gravitational force below distances of [*roughly*]{} 20 microns, as illustrated in Fig. 8. Of course the onset of such modifications of Newton’s Law may be seen at somewhat larger distances. Future work will focus on generalizing the effective theory to include massless or light SM particles as well as quantum gravity corrections, and to looking for potential signals for experiment and observation due to higher order (in the derivative expansion) effects, that are allowed by the effective theory but forbidden by the standard theory where point-like gravity is extrapolated to at least a TeV. The compatibility of fat graviton ideas with multiple matter vacua will also be investigated. Acknowledgements {#acknowledgements .unnumbered} ================ In thinking about fat gravitons, I greatly benefitted from comments, criticisms, leads and help from Nima Arkani-Hamed, Tom Banks, Andy Cohen, Savas Dimopoulos, Michael Douglas, Gia Dvali, Adam Falk, Shamit Kachru, Mike Luke, Markus Luty, Ann Nelson, Dan Pirjol, Joe Polchinski, Martin Schmaltz, Matt Strassler and Andy Strominger. This research was supported in part by NSF Grant P420D3620434350 and in part by DOE Outstanding Junior Investigator Award Grant P442D3620444350. [99]{} S. Weinberg, Rev. Mod. Phys. 61 (1989) 1. A. G. Riess [*et al.*]{}, Astron. J. 116 (1998) 1009, astro-ph/9805201; P. M. Garnavich [*et al.*]{}, Astrophys. J. 509 (1998) 74, astro-ph/9806396; S. Perlmutter [*et al.*]{}, Astrophys. J. 517 (1999) 565, astro-ph/9812133. K. Hagiwara [*et al.*]{}, Phys. Rev. D66 (2002) 010001. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, hep-ph/9803315, Phys. Lett. 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Drell, “Relativistic Quantum Fields”, McGraw-Hill (1964). D. B. Kaplan, M. J. Savage and M. B. Wise, Nucl. Phys. B534 (1998) 329, nucl-th/9802075. [^1]: email:   sundrum@pha.jhu.edu [^2]: Such an extrapolation should certainly not be taken for granted, as dramatically illustrated by noting that present data cannot distinguish a theory with a gravity-only large extra dimension with a size of order $0.1$ mm (similar to the well-known proposal of Ref. [@add] with two extra dimensions) from the usual 4D theory. [^3]: The graviton could not literally be a composite of a Poincare invariant quantum field theory, by the theorem of Ref. [@ww]. However, the physical manifestations of compositeness are compatible with the graviton, as string theory perfectly illustrates.

--- abstract: 'We present a model to understand the redshift evolution of the UV luminosity and stellar mass functions of Lyman Break Galaxies. Our approach is based on the assumption that the luminosity and stellar mass of a galaxy is related to its dark matter halo assembly and gas infall rate. Specifically, galaxies experience a burst of star formation at the halo assembly time, followed by a constant star formation rate, representing a secular star formation activity sustained by steady gas accretion. Star formation from steady gas accretion is the dominant contribution to the galaxy UV luminosity at all redshifts. The model is calibrated by constructing a galaxy luminosity versus halo mass relation at $z=4$ via abundance matching. After this luminosity calibration, the model naturally fits the $z=4$ stellar mass function, and correctly predicts the evolution of both luminosity and stellar mass functions from $z=0$ to $z=8$. While the details of star formation efficiency and feedback are hidden within our calibrated luminosity versus halo mass relation, our study highlights that the primary driver of galaxy evolution across cosmic time is the build-up of dark matter halos, without the need to invoke a redshift dependent efficiency in converting gas into stars.' author: - 'Sandro Tacchella, Michele Trenti $\&$ C. Marcella Carollo' title: 'A physical model for the $0\la z\la8$ redshift evolution of the galaxy UV luminosity and stellar mass functions' --- Introduction {#sec:intro} ============  The galaxy luminosity function (LF) and the stellar mass function (MF), along with their redshift evolution, summarize key information on galaxy properties and on their evolution with cosmic time. The rest-frame UV $1500~\mathrm{\AA}$ LF in particular can be traced with current technology across the whole redshift range from $z\sim0$ (e.g., @blanton03 [@arnouts05; @oesch10]) to $z\sim10$ (e.g., @reddy09 [@bradley12; @oesch12; @ellis13]), the current frontier of detection of Lyman Break Galaxy (LBG) populations. Similarly, the stellar MF can be derived from observations in the rest-frame optical [@arnouts07; @stark13; @gonzalez11]. These data enable a self-consistent comparison of star forming galaxies over the entire span of cosmic history. A powerful approach to link the properties of galaxies to those of their host dark-matter (DM) halos in a $\Lambda$ Cold Dark Matter ($\Lambda$CDM) cosmology is to use halo occupation distribution models, which give the probability that a halo of mass $M_h$ hosts a galaxy [@jing98; @peacock00]; these can be generalized into a conditional luminosity function modeling, giving the probability a halo of mass $M_h$ hosts a galaxy with luminosity $L$ [@vdbosch03; @cooray05a]. This approach provides a $L(M_h)$ relation between galaxy luminosity and DM halo mass, derived at each redshift through “abundance matching” [e.g., @mo96; @vale04], which generally includes a duty cycle parameter so as to populate with UV luminous galaxies only a fraction of DM halos [@cooray05a; @lee09]. It is quite successful in providing a description of the LF, but it does not provide a physical explanation for it. Our approach aims at identifying the key drivers of the evolution of galaxy properties with the least amount of assumptions. We showed in @trenti10 that the LF at $z\gtrsim5$ is successfully modeled by assuming that UV bright galaxies are present, at any cosmic epoch, only in halos assembled within $\Delta t$ ($\Delta t\sim10^8~\mathrm{Myr}$). This results in a duty-cycle which is physically motivated, defined without free parameters, and dependent on redshift and halo mass. While this model well reproduces the observed rest-frame UV LF evolution at very high redshifts, it cannot be extrapolated down to redshifts $z\la4$, since at such late epochs DM halos older than a few $10^8~\mathrm{yr}$ are likely to host UV bright galaxies. In this Letter we expand the @trenti10 model by making the more realistic assumption that, at any epoch, all massive DM halos host a galaxy with a star formation history (SFH) that is related to the time of halo assembly. Note that the duty cycle inserted by @trenti10 is not necessary in our model, which adopts a physical prescription to connect the UV luminosity to a given halo. We anchor our model to the observed LF at $z=4$, and evolve it towards higher ($z\approx8$) and lower ($z\approx0$) redshifts with a simple physical prescription that enables us to explore the origin of the observed UV LF evolution. Our new model features $(i)$ a burst of star formation at halo assembly time, followed by (ii) constant star formation with rate inversely proportional to the halo assembly time (halos at a given mass and different redshift accrete the same gas but over a different timescale). These assumptions, calibrated at $z=4$, are able to reproduce the evolution of the LF, cosmic mass density ($\rho_{M_{\star}}$), and specific star formation rate (sSFR) across 13 billion years of cosmic time. This good match between model and observations is achieved with a dominant contribution to the UV luminosity, at all epochs, of a continuous mode of star formation, fueled by gas accretion. This Letter is organized as follows. Section \[sec:Obs\] summarizes the observational datasets we aim at modeling. Section \[sec:model\] describes the model and its calibration. Section \[sec:Res\] presents our results and discusses model uncertainty. Section \[sec:Con\] highlights some concluding remarks. We adopt WMAP5 cosmology: $\Omega_{\Lambda,0}=0.72$, $\Omega_{m,0}=0.28$, $\Omega_{b,0}=0.0462$, $\sigma_8=0.817$, $n_s=0.96$, and $h=0.7$ [@komatsu09]. Observational Data {#sec:Obs} ================== ![Top panel: Star formation rate density ($\dot{\rho}_{M_{\star}}$) and luminosity density ($\rho_L$) derived by integrating the model UV LFs in comparison with the observations before (gray points) and after dust correction (black points). Both $\rho_L$ and $\dot{\rho}_{M_{\star}}$ are given for $L\geq0.05~L^{\ast}_{z=3}$ ($M_{AB}\leq-17.7$). Middle: Evolution in the stellar mass density ($\rho_{M_{\star}}$), computed by integrating the stellar mass function to a fixed stellar mass limit of $10^8~\mathrm{M_{\sun}}$. Bottom: Evolution in the sSFR as a function of redshift for a galaxy with $M_{\ast}=5\times10^9~\mathrm{M_{\sun}}$. The red lines show our standard model predictions, while the dashed green lines show the model that includes the full probability distribution of the halo assembly time (i.e. with scatter in the $L(M_h,z)$ relation).[]{data-label="fig:SFRD"}](fig2.eps) Figure \[fig:CompareData\] and Table \[tbl:LFEvolution\] summarize the literature compilation of observed UV LFs we use, including the parameters for the best-fit Schechter LF functions ($\phi(L)=\phi^*(L/L^*)^{\alpha}\exp{-(L/L^*)}$). Observed stellar mass densities ($\rho_{M_*}$) integrated above $M_{*,min}=10^8~\mathrm{M_{\sun}}$ and sSFR at $M_{\star}=5\times10^9~\mathrm{M_{\sun}}$ are shown in Figure \[fig:SFRD\]. The figure also includes the cosmic star formation rate density $\dot{\rho}_{M_{\star}}$, obtained by converting, using the @madau98 relation, the luminosity density $\rho_L$ integrated to $L_{min}=0.05~L^{\ast}_{z=3}$ (corresponding to $M_{AB}=-17.7$). Over plotted to observations are our model predictions, obtained with the prescriptions described below. Model Description {#sec:model} ================= Our model links the UV LF to abundance of DM halos at the same epoch, from $z=0$ to $z\sim10$, adopting a physical recipe for star formation with dependence on halo assembly time. Halo Assembly Time {#subsec:AssemTime} ------------------ We adopt the halo assembly time (redshift) as defined by @lacey93 as typical timescale for galaxy formation. The assembly redshift $z_a$ of a halo of mass $M_h$ at redshift $z$ is the redshift at which the mass of the main progenitor is $M_h/2$, which can be calculated within the extended Press-Schechter formalism [@bond91]. For this, we use the ellipsoidal collapse model [@sheth01], which reproduces well numerical simulation results [@giocoli07]. We adopt, as the fiducial assembly time for each halo, the median of the probability distribution associated with each halo (shown in the top-left panel of Figure \[fig:model\] for different redshifts), but we also account for the full probability distribution of $z_a$ to compute the scatter in the $L(M_h,z)$ relation and validate our simpler assumption of a median value for $z_a$. At a given mass, halos are assembled faster at higher $z$, with important consequences on the UV properties of stellar populations. Star Formation Modeling {#subsec:GalLum} ----------------------- We populate halos with stars based on the Simple Stellar Population (SSP) models of @bruzual03, adopting a Salpeter IMF [@salpeter55] between $M_L=0.1~\mathrm{M_{\sun}}$ and $M_U=100~\mathrm{M_{\sun}}$. We use constant stellar metallicity $Z=0.02~\mathrm{Z_{\sun}}$, neglecting redshift evolution as there is little dependence of the UV luminosity on metallicity. We define as $l(t)$ the resulting luminosity at $1500$ Å for a SSP of age $t$ and stellar mass $1~\mathrm{M_{\sun}}$. For a halo at a given redshift, we set the start of the SFH to coincide with the halo assembly time $t_H(z_a)$. Specifically, we parametrize the SFHs through a short-duration burst at the halo assembly time, followed by a constant SFR period. This latter term is normalized by $1/t_{age}$, with $t_{age}=t_H(z)-t_H(z_a)$ and $t_H(z)$ the age of the Universe at redshift $z$. The “burst mode” integrates the total stellar mass produced from the earliest epochs down to $t_{H}(z_a)$. This integrated contribution of early star formation activity makes up for about half of the stellar mass; as $t_{H}(z_a)\ga10^8$ yr, the burst adds up however relatively little, i.e., $\la20\%$, to the UV LF (see Section \[sec:Res\]). During the continuous “accretion mode”, halos accrete of order $M_h/2$ within the $t_{age}$ time scale. Since $t_{age}$ depends strongly on $z$ (see Figure \[fig:model\] top left panel), the accretion rate changes as well: halos of the same mass at higher redshifts have naturally higher accretion rates, as also indicated by other studies [@genel08; @dekel09]. The resulting halo luminosity is: $$\label{eq:SFH}\begin{split} L(M_h,z) & =x\cdot\left[\eta(M_h)M_hl(t_{age})\right] \\ & +(1-x)\cdot\left[\varepsilon(M_h)M_h\frac{1}{t_{age}}\int_0^{t_{age}}l(t)dt\right], \end{split}$$ The first term $\eta(M_h)$ describes the efficiency of the burst episode, while the second term $\varepsilon(M_h)$ describes the efficiency of the accretion mode. Both efficiencies are assumed to be redshift independent (see @behroozi13). The free parameter $x$ controls the relative contribution of the initial burst to the total luminosity of the galaxy at $z=4$. From Equation \[eq:SFH\], the resulting stellar mass, i.e., the time integral of the star formation rate is: $$\label{eq:StellarMass} M_{\star}(M_h)=x\cdot\eta(M_h)M_h+(1-x)\cdot\varepsilon(M_h)M_h,$$ which can be used to obtain stellar densities and specific star formation rates. Dust Extinction {#subsec:Dust} --------------- Dust extinction significantly affects the observed UV flux, especially at $z\lesssim4$ (see Figure \[fig:SFRD\], top panel). Following @smit12, for a spectrum modeled as $f_{\lambda}\sim\lambda^{\beta}$, we assume a linear relation between the UV-continuum slope $\beta$ and luminosity ($<\beta>=\frac{d\beta}{dM_{\mathrm{UV}}}(M_{\mathrm{UV,AB}}+19.5)+\beta_{M_{\mathrm{UV,AB}}}$). Assuming a dependence of UV extinction on $\beta$ as $A_{\mathrm{UV}}=4.43+1.99\beta$ [@meurer99], and a Gaussian distribution for $\beta$ at each $M_{\mathrm{UV}}$ value (with dispersion $\sigma_{\beta}=0.34$), the average $<A_{\mathrm{UV}}>$ is given by $<A_{\mathrm{M_{UV}}}>=4.43+0.79\ln(10)\sigma_{\beta}^2+1.99<\beta>$. We adopted the value of $0$ for any negative $<A_{\mathrm{UV}}>$. Values for $\frac{d\beta}{dM_{\mathrm{UV}}}$ and $\beta_{M_{\mathrm{UV,AB}}}$ are taken from Table 5 of @bouwens12a and are listed in Table \[tbl:LFEvolution\]. We extrapolated $\beta_{M_{\mathrm{UV,AB}}}$ to higher and lower redshifts, while letting $\frac{d\beta}{dM_{\mathrm{UV}}}$ constant at the $z = 4$ value, since uncertainties in this latter parameter are large. [ccccccccccccc]{} $z=0.3$ & & $4.2\pm0.1$ & $-18.9\pm0.1$ & $-1.29\pm0.05$ & & $6.2\pm1.8$ & $-18.4\pm0.3$ & $-1.19\pm0.15$ & (1) & & $-1.45$ & $-0.13$\ $z=1$ & & $1.6^{+0.2}_{-0.1}$ & $-19.9\pm0.1$ & $-1.63^{+0.04}_{-0.02}$ & & $1.1\pm0.8$ & $-20.1\pm0.5$ & $-1.63\pm0.45$ & (1) & & $-1.55$ & $-0.13$\ $z=2$ & & $2.2^{+0.2}_{-0.1}$ & $-20.3^{+0.2}_{-0.1}$ & $-1.60^{+0.04}_{-0.06}$ & & $2.2\pm1.8$ & $-20.2\pm0.5$ & $-1.60\pm0.51$ & (2) & & $-1.70$ & $-0.13$\ $z=3$ & & $1.72\pm0.01$ & $-20.9^{+0.3}_{-0.1}$ & $-1.68^{+0.05}_{-0.07}$ & & $1.7\pm0.5$ & $-21.0\pm0.1$ & $-1.73\pm0.13$ & (3) & & $-1.85$ & $-0.13$\ $z=4$ & & $1.30\pm0.01$ & $-21.0^{+0.2}_{-0.3}$ & $-1.73^{+0.07}_{-0.05}$ & & $1.3\pm0.2$ & $-21.0\pm0.1$ & $-1.73\pm0.05$ & (4) & & $-2.00$ & $-0.13$\ $z=5$ & & $1.4\pm0.1$ & $-20.6^{+0.2}_{-0.3}$ & $-1.77^{+0.11}_{-0.05}$ & & $1.4^{+0.7}_{-0.5}$ & $-20.6\pm0.2$ & $-1.79\pm0.12$ & (5) & & $-2.08$ & $-0.16$\ $z=6$ & & $1.4\pm0.1$ & $-20.4^{+0.4}_{-0.2}$ & $-1.76^{+0.14}_{-0.12}$ & & $1.4^{+1.1}_{-0.6}$ & $-20.4\pm0.3$ & $-1.73\pm0.20$ & (5) & & $-2.20$ & $-0.17$\ $z=7$ & & $0.9\pm0.1$ & $-20.2\pm0.2$ & $-1.84^{+0.12}_{-0.17}$ & & $0.9^{+0.7}_{-0.4}$ & $-20.1\pm0.3$ & $-2.01\pm0.21$ & (6) & & $-2.27$ & $-0.21$\ $z=8$ & & $0.5\pm0.1$ & $-20.2^{+0.4}_{-0.2}$ & $-1.92^{+0.11}_{-0.15}$ & & $0.4^{+0.4}_{-0.2}$ & $-20.3^{+0.3}_{-0.3}$ & $-1.98^{+0.2}_{-0.2}$ & (7) & & $-2.34$ & $-0.25$\ $z=10$ & & $0.2\pm0.1$ & $-19.74^{+0.3}_{-0.5}$ & $-2.18^{+0.25}_{-0.02}$ & & $0.1\pm0.1 $ & $-19.6^b$ & $-1.73^b$ & (8) & & —$^c$ & —$^c$\ Model Calibration {#subsec:Derive_LMh} ----------------- To calibrate $\eta(M_h)$ and $\varepsilon(M_h)$ we perform abundance matching at $z=4$, assuming one galaxy per halo and equating the number of galaxies with luminosity greater than $L$ (after dust correction) to the number of halos with mass greater than $M_h$: $$\label{eq:AbMatch} \int_{M_h}^{+\infty}n(\tilde{M_h},z=4)d\tilde{M_h}=\int_L^{+\infty}\phi(\tilde{L},z=4)d\tilde{L},$$ where $n(M_h, z)$ is the MF of DM halos obtained adopting @sheth99 MF. This gives us a luminosity versus halo mass relation at $z=4$, $L(M_h, z=4)$, shown in the bottom-right panel of Figure \[fig:model\]. From this we can then infer $\eta (M_h)$ and $\varepsilon (M_h)$ by solving Equation \[eq:SFH\] (bottom-left panel of Figure \[fig:model\]). We calibrate these two quantities independently, so that their linear combination also satisfies $L(M_h, z=4)$ by construction. The shaded areas in both panels represent the uncertainty in the model calibration derived by varying the $z=4$ LF parameters within the $1\sigma$ confidence regions in Figure 3 of @bouwens07. From the bottom-left panel of Figure \[fig:model\] it is immediate to see that halos with $M_h\sim10^{11}-10^{12}~\mathrm{M_{\sun}}$ have the highest specific star formation efficiencies. This is not surprising, given the shapes of the LFs and DM MF. Our final calibration step is selecting a value for the only free parameter in the model, $x$, i.e., the contribution of the burst to the total luminosity at $z=4$. For this we compute model predictions over the redshift range $0\la z\la8$ with varying $x$ values, and adopt the value of $x$ which minimizes the residuals relative to the observed LFs. The best match to observations is given by $x=0.1$, which is a $10\%$ of contribution from the initial burst to the total halo luminosity at $z=4$; the model is however not very sensitive to the exact value for as long as $x\ll1$ (see top panels of Figure \[fig:sensitivity\]).  Results and Discussion {#sec:Res} ====================== The curves in the bottom-right panel of Figure \[fig:model\] show our predictions for the observed $L(M_h,z)$ relation at different redshifts. A decreasing contribution from dust is the main cause for the brightening of the relation towards higher redshifts at fixed halo mass. Note that these assume a single assembly time for a given halo (see Section \[subsec:AssemTime\]). Taking into account the whole probability distribution for the halo assembly time leads to scatter in the $L(M_h,z)$ relation (shown in the inset of Figure \[fig:model\], bottom-right panel), but such model has overall similar predictions in terms of the observed LF as shown in Figure \[fig:CompareData\] for $z=2$ and $z=8$. Therefore, we focus primarily on our canonical model without scatter. The predictions for the LFs over the $z\la10$ time span are shown overplotted to the observations in Figure \[fig:CompareData\]. In addition, the model reproduces both low-$z$ and high-$z$ UV LFs remarkably well, suggesting that the evolution of the UV LF across most of cosmic time can indeed result from the redshift evolution of the halo MF, coupled with simple star formation histories beginning at the halo assembly time. The model, calibrated to the observed Schechter LF at $z=4$, produces Schechter functions at all other epochs. Furthermore, the predicted LFs well approximates the Schechter functions with the observed best-fit parameters reported in Table \[tbl:LFEvolution\] (Figure \[fig:CompareData\]). In particular, the model correctly describes the evolution of the faint-end slope $\alpha$, from its shallow low-$z$ value $\alpha(z\approx0)\sim-1.3$ to the steepening observed at $z\gtrsim6$, where $\alpha\la-1.7$. At high-$z$, the model LFs are similar to the LFs predicted by @trenti10; in addition, our new model is also successful in reproducing the observed LFs also at low redshifts, all the way down to $z\simeq0$, resolving the puzzling quick rise of $\alpha$ from $z\simeq0$ to 1 [e.g., @oesch10]. Figure \[fig:SFRD\] shows the model predictions for the redshift evolution of the star formation rate density $\dot{\rho}_{M_{\star}}(z)$ and luminosity density $\rho_L$($z$) (top panel). The model-observation agreement for $\dot{\rho}_{M_{\star}}(z)$ is again very good at all epochs from $z\approx0$ to $z\approx8$. At $z\sim10$ the model appears to over-predict $\dot{\rho}_{M_{\star}}$ as measured by @oesch12 by $\sim0.8$ dex. This might be due to very short assembly times for $z\sim10$ halos ($t_{age}\lesssim10^8~\mathrm{Myr}$), hence to a dominant contribution to the UV light from the very young stellar populations produced in the burst mode. Another possibility is sample variance in the observations. In fact, @zheng12 derive from a gravitationally lensed source in CLASH $\dot{\rho}_{M_{\star}}(z=10)=(1.8^{+4.3}_{-1.1})\times10^{-3}~\mathrm{M_{\sun}Mpc^{-3}yr^{-1}}$, in agreement with our model predictions.  Figure \[fig:sensitivity\] (panels (a)-(b)) shows model-observations comparisons for the $z=2$ and 6 LFs; the model results are plotted for different values of the burst fractional contribution $x$ to the total UV luminosity, and for different dust extinction corrections (as parametrized by $\beta$). Varying $\beta$ within the uncertainty given by @bouwens12a has little impact on the predicted LFs, which are thus robust against uncertainties in the amount and treatment of dust obscuration. In contrast, the predicted LFs do depend on the choice of $x$. As the star formation histories are modeled to approach a single episode of star formation at the halo assembly time, i.e., $x\rightarrow1$, the $L(M_h,z)$ relation shows an increasingly stronger dependence on the assembly time: at low redshift, halos become too faint relative to the observations, and the LFs and $\rho_L$ are under-estimated; at high redshifts, halos are too UV-bright and LFs and $\rho_L$ are overestimated. By construction, the initial burst phase contributes modestly to the UV luminosity of galaxies at all epochs, especially at lower redshifts, i.e., as $t_{age}$ increases. This is shown in panel (c) of Figure \[fig:sensitivity\], which plots the contribution of the burst mode to the total galaxy luminosity, for halos of different masses. The redshift evolution of the $L_{burst}/L_{tot}$ ratio is faster for smaller halos, as for these $t_{age}$ evolves faster (Figure \[fig:model\]). In contrast, the initial star formation burst, occurring at the time when the halo has already assembled half of its total mass, consistently contributes of order a half of the stellar mass budget (Figure \[fig:sensitivity\], panel (c)). Specifically, the contribution of the burst phase to the stellar mass is roughly 50% at $M_{h}\sim10^{14}~\mathrm{M_\sun}$, and increases to about 70% at $M_{h}\sim10^{9}~\mathrm{M_\sun}$. Figure \[fig:sensitivity\], panel (d), shows stellar mass as a function of magnitude. The red points show the relation at $z=4$ of @stark09, and the blue points are re-normalized for accounting for emission lines (see @stark13 [@de-barros12]). The model data are slightly below the $z=4$ observations; in contrast, at $z=7$ the model predicts a $M_{\star}-M_{AB,1500}$ relation which well matches the data of @stark13, once these are corrected for emission line contamination and for a redshift-dependent equivalent width of nebular emission (increasing with increasing redshift). The middle panel of Figure \[fig:SFRD\] shows the comparison between model and observations for $\rho_{M_{\star}}$. Overall, the model (with or without scatter in $L(M_h,z)$) fits the data well at all redshifts. At $z=7$, we under-estimate by 0.25 dex the (not emission-corrected) @gonzalez11 data point. On the other hand, we are broadly consistent with the @stark13 measurements. The sSFR is shown in the bottom panel of Figure \[fig:SFRD\] and shows overall a good agreement with the observations. Conclusion {#sec:Con} ========== We have presented a model for the evolution of the UV LF based on the simple assumption that all massive DM halos host a galaxy with a star formation history that is closely related to the halo assembly time. We specifically adopt for the star formation histories the combination of an initial star formation burst at the halo assembly time, representing the integrated galaxy star formation histories down to this epoch, plus a constant SFR phase, a proxy for secular, low-level star formation activity fueled by steady gas accretion. While the assumption that each DM halo hosts only one galaxy is clearly a simplification, especially towards lower redshifts, the model is remarkably successful in reproducing major features of the evolving star forming galaxy population since $z\simeq8$; this is also due to the fact that massive galaxies which, at later times, will share a common halo, will be mostly quenched of their star formation activity. It is remarkable that this simple parametrization reproduces very well the evolution of the UV LFs over the whole cosmic time since $z\simeq8$ down to $z\simeq0$, as well as the evolution of the cosmic specific star formation rate, luminosity density and stellar mass density. In our model, the cosmic star formation rate density rises and then falls towards lower $z$ naturally as it is observed, which can be explained by the drop in the accretion rate at low redshifts of the individual DM halos - overtaking the increase in abundance of galaxies at $z\la2$. This demonstrates the key role played by DM halo assembly in shaping the properties of the luminous galaxies. 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--- author: - | [^1]\ Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany\ [*INTEGRAL*]{} Science Data Centre, Université de Genève, Chemin d’Ecogia 16, CH–1290 Versoix, Switzerland\ E-mail: - | V. Bosch-Ramon[^2]\ Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland.\ E-mail: - | M. Perucho[^3]\ Departament d’Astronomia i Astrofísica. Universitat de València. C/ Dr. Moliner 50, 46100 Burjassot (València), Spain\ E-mail: title: 'Jet/medium interactions at large scales' --- Introduction ============ The jets of Fanaroff-Riley galaxies (of type FR-I and FR-II, [@fr74]) deliver kinetic energy to the surrounding interstellar and intergalactic medium (ISM and IGM respectively) at a rate between $\sim 10^{42}$ and $\sim 10^{46}\,\rm{erg~s}^{-1}$. The ejections of microquasars ($\mu$Q), on the other hand, can also inject large amounts of energy into the ISM, at a level $\sim 10^{37}-10^{39}$ erg s$^{-1}$. Both of $\mu$Q and FR-I jet/medium interactions could be strong enough to accelerate particles and produce non-thermal radiation [@br09; @br10]. Extended X-ray emission from the jets and/or lobes of the radio galaxies 3C 15 [@ka03], Cen A [@cr09], Fornax A [@fe95] and M87 [@ka05] have been observed with *Chandra*. Furthermore, the [*Fermi*]{} Collaboration recently reported on the extended GeV emission from Cen A [@ab10a]. Efficient particle acceleration is therefore taking place at least in some FR-Is. In the case of $\mu$Qs, evidence of particle acceleration is found in the non-thermal radio and/or X-ray emission observed e.g. in SS 433 [@zealey80], XTE J1550$-$564 [@corbel02] and Cir X-1 [@tudose06]), and perhaps also in GRS 1915+105 [@Kaiser04] and LS I +61 303 [@paredes07]. In this work we explore the leptonic non-thermal emission expected in both $\mu$Q and FR-I interaction scenarios, using the results from numerical simulations coupled to a radiation model for the jet shocked material (reconfinement region and cocoon) and the ambient material shocked by the bow shock (the shell). Our results are used to make predictions for the broadband non-thermal fluxes (from radio to $\gamma$-rays) and we compare them with current and future instrument capabilities. Jet/medium interaction model ============================ Two symmetric jets emerge from the central source, expanding freely until they are recollimated when their ram pressure equals that of the surrounding cocoon. Further on, the jet is decelerated once the accumulated mass of the swept up ISM/ICM gas becomes similar to that carried by the jet. A forward shock is produced propagating into the external medium, whilst the shocked matter from the jet inflates the cocoon. In the FR-I scenario, the jet gets disrupted before reaching the termination regions, and we take the jet recollimation shock as the cocoon particle accelerator. In the $\mu$Q scenario the jet is assumed to remain undisrupted, and a strong reverse shock is formed near the jet head. In our model the mass density and pressure of the shocked regions are taken homogeneous, although they can evolve with time. We assume the presence of a randomly oriented magnetic field $B$ in the downstream regions, derived taking the magnetic energy density to be $\sim$ 10 % of the internal energy density. In each shocked region, the fraction of kinetic power transferred to non-thermal particles is taken to be $\sim$ 1 %. We consider the CMB and the radiation field energy density from the central engine (the companion star in the case of $\mu$Q and the galaxy nucleous in the case of FR-Is), although the latter is only important at the very initial stages of evolution. We adopt a power-law spectral distribution ($N(E)=K\,E^{-2}$, with $p = 2$ and $K$ such that $\int E\,N(E)\,dE = 0.1 \times Q_{\rm jet}$) for the leptons injected at the reconfinement, bow and reverse shock fronts (we remark that the latter is only considered in the undisrupted jets of $\mu$Qs). Maximum energies are calculated equating the energy gain to synchrotron, relativistic Bremsstrahlung, Inverse Compton (IC) and adiabatic losses. Differently evolved populations injected all along the source age are considered at a given $t_{\rm src}$. For the synchrotron losses ($t_{\rm syn}\approx 4\times 10^{12}\,(B/{\rm 10 \mu~G})^{-2}\,(E/{\rm 1 TeV})^{-1}\,{\rm s}$), we use the magnetic field considered above for each interaction region. Relativistic Bremsstrahlung is calculated accounting for the densities $n$ in the downstream regions ($t_{\rm rel.br}\sim 10^{18}\,(n/10^{-3}\,{\rm cm}^{-3})\,{\rm s}$). To compute IC losses ($t_{\rm IC}\approx 1.6\times 10^{13}\,(u_{\rm rad}/10^{-12}{\rm erg~cm}^{-3})^{-1}\,(E/{\rm 1 TeV})^{-1}\,{\rm s}$), we consider the total radiation field energy density $u_{\rm rad}$. Adiabatic losses, $\dot{E} \approx [v/r]\,E$, are computed from the size $r$ and the expansion velocity $v$ of the emitters. Escape losses are also consideredd by taking the gyroradius of the most energetic particles equal to the accelerator size [@Hillas1984]. Hydrodynamical simulations ========================== Hydrodynamical simulations have been performed to further study the interaction of both $\mu$Q and FR-I jets with their surroundings. A two-dimensional finite-difference code based on a high-resolution shock-capturing scheme has been used, which solves the equations of relativistic hydrodynamics written in conservation form. The reader is referred to [@pe+05; @pm07] for further details on the simulation code (see also [@br09]). For the $\mu$Q scenario, the jet is injected at a distance of $10^{18}$ cm from the compact object, with initial radius $ = 10^{17}$ cm. Both the jet and the ambient medium are characterized with an adiabatic exponent $\Gamma=5/3$. The number density in the ambient medium is $n_{\rm ISM}=0.3$ cm$^{-3}$. The velocity of the jet at injection is $0.6\,c$, its number density $n_{\rm j}=1.4\times 10^{-5}\,\rm{cm^{-3}}$, and its temperature $T\sim 10^{11}$ K. These parameter values result in a jet power $Q_{\rm jet}=3\times 10^{36}$ erg s$^{-1}$. For the numerical simulations of FR-I sources, the jet is injected in the numerical grid at $500\,\rm{pc}$ from the active nucleus, with a radius of $60\,\rm{pc}$. The ambient medium is composed by neutral hydrogen. Its profile in pressure, density and temperature includes the contribution from a core region and from the galaxy group, which dominates at large distances. The initial jet velocity is $\sim 0.87\,c$, its temperature $\sim 4\times10^9\,\rm{K}$, and a density and pressure ratio with respect to the ambient are $\sim 10^{-5}$ and $\simeq 8$, respectively, resulting in a jet kinetic luminosity $Q_{\rm j}=10^{44}$ erg s$^{-1}$. Non-thermal emission from $\mu$Q ================================ Figure 1 show the spectral energy distribution (SED) for the shell, the cocoon and the jet reconfinement regions for the different set of parameters explored. The emission of only one jet is accounted for. Synchrotron emission is the channel through which the highest radiation output is obtained, with bolometric luminosities up to $\sim~10^{33}$ erg s$^{-1}$ for powerful ($Q_{\rm jet}= 10^{37}$ erg s$^{-1}$) systems. At high and very high energies (HE and VHE, respectively), IC emission is the dominant process in the cocoon and reconfinement regions, reaching a few $\times 10^{30}$ erg s$^{-1}$, while in the shell relativistic Bremsstrahlung dominates at this energy range, with luminosities up to $\sim~10^{32}$ erg s$^{-1}$. Notable differences are found in the reported SEDs when varying the source age $t_{\rm MQ}$ from $10^{4}$ yr to $10^{5}$ yr. For older sources, the shell and cocoon are located at larger distances from the companion star, $u_{\rm rad}$ decreases and the IC contribution gets slightly lower. Relativistic Bremsstrahlung emission is also larger for older sources. Higher values of $n_{\rm ISM}$ make the jet to be braked at shorter distances from the central engine. The interaction regions have higher $u_{\rm rad}$ from the companion star, and the IC emission is again slightly enhanced. The relativistic Bremsstrahlung emission in the shell zone is also higher for denser mediums, since the luminosity is proportional to the target ion field density, $n_{\rm t} \sim 4\,n_{\rm ISM}$. Finally, we note that in our model all the non-thermal luminosities scale roughly linearly with the jet power $Q_{\rm jet}$. {width="100.00000%"} \[SEDS\_mq\_TMQ4\_TMQ5\] Non-thermal emsission from FR-I sources ======================================= The SEDs for the cocoon and the shell regions at $t_{\rm src}=10^{5}$, $3 \times 10^{6}$ and $10^{8}$ yr, are shown in Fig. \[SED\_FRI\_totes\]. The obtained radio and X-ray synchrotron luminosities in both regions are at the level of $2\times 10^{41}$ erg s$^{-1}$. The synchrotron break frequency, corresponding to the electron energy at which $t_{\rm syn}(E)\approx t_{\rm src}$, and the highest synchrotron frequency, $\nu_{\rm syn~max}\propto B\,E_{\rm max}^2$, are shifted down for older sources. The former effect makes the radio luminosity to increase at the late stages of the evolution of both the cocoon and the shell, whereas the latter decreases the X-ray luminosity in the shell due to the decrease of $\nu_{\rm syn~max}$ with time. The slightly different conditions in the shell yield a higher break frequency, which implies a factor $\sim 2$ lower radio luminosity in this region compared to that of the cocoon. The IC luminosity grows as long as this process becomes more efficient compared to synchrotron and adiabatic cooling. The cocoon and the shell have similar HE luminosities, but the cocoon is few times brighter at VHE than the shell. In both regions the bolometric IC luminosities grow similarly with time, reaching $\sim 10^{42}$ and $10^{41}$ erg s$^{-1}$ at HE and VHE, respectively. ![SEDs of the non-thermal emission produced in the interaction of FR-I jets with their surroundings. Panels a), b) and c) correspond respectively to a source age $t_{\rm src} = 10^{5}, 3 \times 10^{6}$ and $10^{8}$ yr. Blue dashed lines show the contribution from the shell region and green dashed lines that of the cocoon. The overall emission is marked with a thick solid black line.[]{data-label="SED_FRI_totes"}](SED_FRI_totes.eps){width="100.00000%"} Discussion ========== The obtained radio fluxes in a $\mu$Q scenario would imply a flux density of $\sim 150$ mJy at 5 GHz for a source located at $\sim$ 3 kpc. The emitting size would be of a few arcminutes, since the electron cooling timescale is longer than the source lifetime and they can fill the whole cocoon/shell structures. Considering this angular extension and taking a radio telescope beam size of $10''$, radio emission at a level of $\sim 1$ mJy beam$^{-1}$ could be expected. In X-rays, we find a bolometric flux in the range 1–10 keV of $F_{\rm 1-10 keV}$ $\sim 2 \times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$. The electrons emitting at X-rays by synchrotron have very short time-scales, and the emitter size cannot be significantly larger than the accelerator itself. Although the X-rays produced in the shell through relativistic Bremsstrahlung are expected to be quite diluted, the X-rays from the cocoon would come from a relatively small region close to the reverse shock, and could be detectable by [*XMM-Newton*]{} and [*Chandra*]{} at scales of few arcseconds. At hard X-rays, *INTEGRAL* could detect the high energy tail of the particle population, although its moderate angular resolution would make difficult to resolve the sources. At HE, the flux between 100 MeV and 100 GeV is $F_{\rm 100~MeV<E<100~GeV}\sim 10^{-14}$ erg s$^{-1}$ cm$^{-2}$, below the *Fermi* sensitivity, while the integrated flux above 100 GeV is $F_{\rm E>100GeV}\sim 10^{-15}$ erg s$^{-1}$ cm$^{-2}$, also too low to be detectable by current Cherenkov telescopes. Taking into account the rough linearity between $Q_{\rm jet}$, $n_{\rm ISM}$, $t_{\rm src}$ and $d^{-2}$ with the gamma-ray fluxes obtained, sources with higher values of these quantities than the ones used here may render the $\mu$Q jet termination regions detectable by current and next future gamma-ray facilities. For FR-I sources radio fluxes as high as $\sim 10^{-12}\,(d/100~$Mpc$)^{-2}$ erg cm$^{-2}$ s$^{-1}$ or $\sim 10$ Jy at 5 GHz from a region of few times $10'\,(d/100~$Mpc$)^{-1}$ angular size are obtained. The properties of the radio emission from the interaction jet-medium structure are comparable with those observed for instance in 3c 15 [@ka03], in which fluxes of a few $\times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$ ($d\sim$ 300 Mpc) are found. At X-rays, the dominant emission comes from the cocoon, with fluxes $\sim 10^{-13}\,(d/100~$Mpc$)^{-2}$ erg cm$^{-2}$ s$^{-1}$, although limb brightening effects may increase the shell detectability. The lifetime of X-ray synchrotron electrons, $\sim 10^{11}\,{\rm s}$, is much shorter than in radio, and $\ll t_{\rm src}$ as well, so their radiation may come mostly from the inner regions of the cocoon or the bow-shock apex. The total non-thermal X-ray flux is roughly constant for the explored range of $t_{\rm src}$, although the shell contribution decreases significantly with time. At HE and VHE energies, the obtained fluxes are similar for both the cocoon and the shell, although the latter shows a lower maximum photon energy. Gamma-ray emission increases with time mainly due to the increasing efficiency of the CMB IC channel. The obtained fluxes, for a source with $t_{\rm src}\sim 10^8$ yr, are around $\sim 10^{-12}\,(d/100\,{\rm Mpc})$ erg cm$^{-2}$ s$^{-1}$. At HE, such a source may require very long exposures to be detected. Nevertheless, [*Fermi*]{} has recently detected some FR-I galaxies at a few hundred Mpc distances, including the extended radio lobes of Cen A (at a distance $\sim 4$ Mpc), presenting fluxes similar to those predicted here. At VHE, the fluxes would be detectable by the current instruments, although the extension of the source, of tens of arcminutes at 100 Mpc, and the steepness of the spectrum above $\sim 100$ GeV, may render its detection difficult. In the case of Cen A, detected by HESS [@ah09], the emission seems to come only from the core, but this is expected given the large angular size of the lobes of this source, which would dilute its surface brightness, requiring very long observation times to render the source detectable. Finally, we note that the fluxes showed above strongly depend on the non-thermal luminosity fraction going to non-thermal particles, for which we use a quite conservative value $=0.01$. In the case of a source able to accelerate particles at a higher efficiency at the interaction shock fronts, then the expected non-thermal fluxes would be enhanced by a similar factor. [99]{} Fanaroff, B. L. & Riley, J. M., 1974, MNRAS, 167, 31 Bordas P., Bosch-Ramon V., Paredes J. M., Perucho M., 2009, A&A, 497, 325 Bordas, P., Bosch-Ramon, V. & Perucho, M. Kataoka, J., Leahy, J. P., Edwards, et al. 2003, A&A, 410, 833 Croston J. H., Kraft R. P., Hardcastle M. J., et al. 2009, MNRAS, 395, 1999 Feigelson, E. D., Laurent-Muehleisen, S. A., Kataoka, J. & Stawarz, L. 2005, ApJ, 622, 797 Abdo, A. A. et al. 2010, Science, 328, 725 Zealey, W. J., Dopita, M. A., & Malin, D. F. 1980, MNRAS, 192, 731 Corbel, S., & Fender, R. P. 2002, ApJ, 573, L35 Corbel, S., Kaaret, P., Fender, R. P., et al. 2004, ApJ, 617, 1272 Tudose, V., Fender, R. P., & Kaiser, C. R., et al. 2006, MNRAS, 372, 417 Kaiser, C. R., Gunn, K. F., Brocksopp, C., & Sokoloski, J. L. 2004, ApJ, 612, 332 Paredes, J. M., Ribó, M., & Bosch-Ramon, V., et al. 2007a, ApJ, 664, L39 Hillas, A. M. 1984, ARA&A, 22, 425 Perucho, M., Martí, J. M., & Hanasz, M. 2005, A&A, 443, 863 Perucho M., Martí J. M., 2007, MNRAS, 382, 526 (PM07) Aharonian F. A. et al., 2009, ApJ, 695, L40 [^1]: P.B has been supported by grant DLR 50 OG 0601 during this work. P.B acknowledges support by the Spanish DGI of MEC under grant AYA2007-6803407171-C03-01. P.B also acknowledges the excellent work conditions at the *INTEGRAL* Science Data Center. [^2]: V.B-R. acknowledges support of the Spanish MICINN under grant AYA2007-68034-C03-1 and FEDER funds. V.B-R. also acknowledges the support of the European Community under a Marie Curie Intra-European fellowship. [^3]: M.P acknowledges support from the Spanish MEC and the European Fund for Regional Development through grants AYA2010-21322-C03-01, AYA2010-21097-C03-01 and CONSOLIDER2007-00050, and from the “Generalitat Valenciana” grant “PROMETEO-2009-103”. M.P acknowledges support from MICINN through a “Juan de la Cierva” contract.

--- abstract: 'In this article, we study the possibility of sustaining a static and spherically symmetric traversable wormhole geometries admitting conformal motion in Einstein gravity, which presents a more systematic approach to search a relation between matter and geometry. In wormhole physics, the presence of exotic matter is a fundamental ingredient and we show that this exotic source can be dark energy type which support the existence of wormhole spacetimes. In this work we model a wormhole supported by dark energy which admits conformal motion. We also discuss the possibility of detection of wormholes in the outer regions of galactic halos by means of gravitational lensing. The studies of the total gravitational energy for the exotic matter inside a static wormhole configuration are also done.' author: - Piyali Bhar - Farook Rahaman - Tuhina Manna - Ayan Banerjee title: Wormhole supported by dark energy admitting conformal motion --- Introduction ============ In last two decades, there has been a considerable interest in the field of wormhole physics after seminal work by Morris-Thorne [@Morris]. They proposed the possibility of traversable wormholes in the theoretical context of the general relativity as a teaching tool. Topologically, wormholes acts as a tunnels in the geometry of space and time that connect two space-times of same universe or of different universes altogether by a minimal surface called the throat of the wormhole, satisfying flare-out condition [@HV1997], through which a traveler can freely traverse in both directions. Today, most of the efforts are directed to study the necessary conditions to ensure their traversability. The most striking of these properties is a special type of matter that violates the energy conditions, called exotic matter which is necessary to construct traversable wormholes. Recent astronomical observations have confirmed that the universe is undergoing a phase of accelerated expansion which was conformed by the measurements of supernovae of type Ia (SNe Ia) and the cosmic microwave background anisotropy [@Riess]. It has been suggested that dark energy is still an unknown component with a relativistic negative pressure, is a possible candidate for the present cosmic expansion and our Universe is composed of approximately 70 percent of it. The simplest candidate for explaining the dark energy is the cosmological constant $\Lambda$ [@Carmelli], which is usually interpreted physically as a vacuum energy, with $p= -\rho$. Another possible way to explain the dark energy by invoking an equation of state, p= $\omega\rho$ with $\omega< 0$, where p is the spatially homogeneous pressure and $\rho$ the energy density of the dark energy, instead of the constant vacuum energy density. As a particular range of the $-1 <$ $\omega<-1/3$, is a widely accepted results known as quintessence is often considered. The ratio $\omega< -1$ has been denoted phantom energy, corresponding to violation of the null energy condition, thus providing a theoretically supported scenario for the existence of wormholes [@Sushkov]. The presence of phantom energy in the universe leads to peculiar properties, such as Big Rip scenario [@Caldwell], the black hole mass decreasing by phantom energy accretion [@Babichev]. Therefore, the dark energy plays an important role in cosmology naturally makes us search for local astrophysical manifestation of it. In the present work we consider wormhole solution containing dark energy as equation of state. The gravitational lensing (GL) is a very useful tool of probing a number of interesting phenomena of the universe. Particular it can provide rich information for the structure of compact astrophysical objects like e.g., black holes, exotic matter, super-dense neutron stars, wormholes etc. Out of this, the observation of Einstein ring and the double or multiple mirror images are the powerful examples for gravitational lensing effect [@Hewitt]. In earlier works GL phenomenon has been studied in the weak field (see [@Schneider]), but success leads to explore other extreme regime, namely, the GL effect in the strong gravitational field has been studied by [@Virbhadra]. Out of various intriguing objects mentioned above, recently it was proposed that wormholes can act as gravitational lenses and induce a microlensing signature on a background source studied by Kim and Sung [@Kim] and Cramer *et al.,* [@Cramer] and lensing by negative mass wormholes have been studied by Safonova *et al.,* [@Safonova]. Related with the issue of GL effects on wormholes have been studied [@Nandi]. Recently, the possibility of detection of traversable wormholes in noncommutative-geometry is studied by Kuhfittig [@Peter; @K.; @F.; @Kuhfittig] in the outer regions of galactic halos by means of gravitational lensing. The possible existence of wormholes in the outer regions of the halo was discussed in Ref [@Rahaman(2014)], based on the NFW density profile. One of the aims of the current paper is to study the effect of lensing phenomenon for the wormhole solutions in the presence of exotic matter such as phantom fields admitting conformal motion. The present work has been considered in more systematic approach to find the exact solutions and study the natural relationship between geometry and matter. For instance, one may adopt a more systematic approach (see Ref. [@Herrera]) by assuming spherical symmetry and the existence of a non-static conformal symmetry. Suppose that a conformal Killing vector $\xi$ is defined on the metric tensor field g defined by the action of the Lie infinitesimal operator $\mathcal{L}_{\xi}$, which leads to the following relationship: $$\mathcal{L}_\xi g_{ik}=\psi g_{ik},$$ where $\mathcal{L}$ is the Lie derivative operator and $\psi$ is the conformal factor. Here the vector $\xi$ generates the conformal symmetry in such a way that the metric g is conformally mapped onto itself along $\xi$. For an interesting observation neither $\xi$ nor $\psi$ need to be static even though one considers a static metric. For $\psi = 0$ then Eq. (1) gives the killing vector, for $\psi = constant$ Eq. (1) gives a homothetic vector and if $\psi = \psi(x,t)$ then it gives conformal vectors. Further note that when $\psi = 0$ the underlying spacetime is asymptotically flat which implies that the Weyl tensor will also vanish. Thus we can develop a more vivid idea about the spacetime geometry by studying the conformal killing vectors. Recently, Bohmer et al. [@Bohmer] have studied the traversable wormholes under the assumption of spherical symmetry and the existence of a non-static conformal symmetry. The outline of the present paper is as follows: In Sec. **II.** we give a brief outline of the conformal killing vectors for spherically symmetric metric while in Sec. **III.** we present the structural equation of phantom energy traversable wormholes and discuss the physical properties of our solution in the outer region of the halo by recalling the movement of a test particles. In Sec. **IV.** we present the stability of wormholes under the different forces where the total gravitational energy for the exotic matter distribution in the wormhole discuss in Sec. **V.** In Sec. **VI.** gravitational lensing has been studied and the angle of surplus are calculated. In Sec. **VII.** the interior wormhole geometry is matched with an exterior Schwarzschild solution at the junction interference. Finally, in Sec. **VIII.** we discuss some specific comments regarding the results obtained in the study. Einstein field equations and conformal killing vector ===================================================== The spacetime metric representing a static and spherically symmetric line eleminent is given by $$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$$ where $\lambda$ and $\nu$ are functions of the radial coordinate, r. We shall assume that our source is filled with an anisotropic fluid distribution and using the Einstein field equation $G_{\mu\nu}= 8\pi T_{\mu\nu}$, for the above metric, which in our case read (with c = G = 1) $$\begin{aligned} e^{-\lambda}\left[\frac{\lambda'}{r}-\frac{1}{r^{2}} \right]+\frac{1}{r^{2}}=8\pi \rho,\\ e^{-\lambda}\left[\frac{1}{r^{2}}+\frac{\nu'}{r} \right]-\frac{1}{r^{2}}=8\pi p_r,\\ \frac{1}{2}e^{-\lambda}\left[ \frac{1}{2}\nu'^{2}+\nu''-\frac{1}{2}\lambda'\nu'+\frac{1}{r}(\nu'-\lambda')\right]=8\pi p_t,\end{aligned}$$ where $\rho$, $p_r$ and $p_t$ denotes the matter density, radial and transverse pressure respectively of the underlying fluid distribution. and ‘$\prime$’ denotes differentiation with respect to the radial coordinate r. Applying a systematic approach in order to get exact solutions, we demand that the interior spacetime admits conformal motion (but neither $\xi$ nor $\psi$ need to be static even though for a static metric) and therefore Eq. (1) provides the following relationship: $$\mathcal{L}_\xi g_{ik}=\xi_{i;k}+\xi_{k;i}=\psi g_{ik},$$ with $ \xi_{i}=g_{ik}\xi^{k}$. The above equation gives the following set of expressions as $$\xi^{1}\nu'=\psi,~~~\xi^{4}=C_1,~~~\xi^{1}=\frac{\psi r}{2}~~~and~~~\xi^{1}\lambda'+2\xi^{1},_1=\psi,$$ where $C_1$ is a constant and the conformal factor is independent of time i.e., $\psi =\psi(r)$. Now, the metric (2), and using the Eq. (6-7) provides the following results: $$\begin{aligned} e^{\nu}=C_2^{2}r^{2},\\ e^{\lambda}=\left(\frac{C_3}{\psi}\right)^{2},\\ \xi^{i}=C_1\delta_{4}^{i}+\left( \frac{\psi r}{2}\right)\delta_1^{i},\end{aligned}$$ where $C_2$ and $C_3$ are constants of integrations. An important note of this solutions that immediately ruled out, is that the conformal factor is zero by taking into account Eq. (9), at the throat of the wormhole i.e., $\psi(r_0) = 0$, where $r_0$ stands for location of the throat of the wormhole. Now, using Eqs. $(8)-(10)$, one can obtain the expression for Einstein field equations as $$\begin{aligned} \frac{1}{r^{2}}\left[1-\frac{\psi^{2}}{C_3^{2}}\right]-\frac{2\psi \psi'}{r C_3^{2}}=8\pi \rho,\\ \frac{1}{r^{2}}\left[\frac{3\psi^{2}}{C_3^{2}}-1\right]=8\pi p_r,\\ \frac{\psi^{2}}{C_3^{2}r^{2}}+\frac{2\psi \psi'}{r C_3^{2}}=8\pi p_t.\end{aligned}$$ Observing the Eqs. $(11)-(13)$, we have three equations with four unknowns namely $\rho$, $p_r$, $p_t$ and $\psi(r)$ respectively. In order to solve the system of equations, we need an equation of state relating matter and density by the following simplest relation $p= p(\rho)$. Solution for phantom wormhole and physical analysis =================================================== According to Morris and Throne [@Morris], for constructing a wormhole solution one require an unusual form of matter known as ‘extotic matter’, which is the fundamental ingredient to sustain traversable wormhole. The characteristic of such matter is that the energy density $\rho$ may be positive or negative but the radial pressure $p_r$ must be negative. Theoretical advances shows that the expansion of our present universe is accelerating and dark energy is a suitable candidate to explain this cosmic expansion. In this context, we study the construction of traversable wormholes, using the phantom energy equation of state by the following relationship $$p_r=\omega \rho~~~~~~~~~~with~~\omega<-1,$$ by taking into account Eqs. (11) and (12), with the help of equation (14) we obtain $$\psi^{2}=\left(\frac{\omega+1}{\omega+3}\right)c_3^{2}+\psi_0r^{-\left(\frac{\omega+3}{\omega}\right)},$$ where $\psi_0$ is the constant of integration. For convenience we rewrite the Eqs. (11)-(13), using Eq. (15) with new dimensionless parameters $\tilde{\psi_0}$=$\frac{\psi_0}{c_3^2}$, we obtain the expression of matter density, radial and transverse pressure as $$\begin{aligned} \rho&=&\frac{1}{8\pi}\left[\frac{2}{r^{2}\left(\omega+3\right)}-\tilde{\psi_0}\frac{\left(2\omega+3\right)} {\omega}r^{\frac{-3\left(\omega+1\right)}{\omega}}\right],\\ p_r&=&\frac{1}{8\pi}\left[\frac{2\omega}{r^{2}\left(\omega+3\right)}-\tilde{\psi_0}\left(2\omega+3\right)r^{\frac{-3\left(\omega+1\right)}{\omega}}\right],\\ p_t&=&\frac{1}{8\pi}\left[\frac{\omega+1}{r^{2}\left(\omega+3\right)}-\frac{3\tilde{\psi_0}}{\omega}r^{\frac{-3\left(\omega+1\right)}{\omega}}\right].\end{aligned}$$ Plugging the expression for $\psi^{2}$ given in Eq. (9) with the dimensionless parameter the expression for metric potential is obtained as $$e^{-\lambda}=\frac{\omega+1}{\omega+3}+\tilde{\psi_0}r^{-\left(\frac{\omega+3}{\omega}\right)}.$$ Thus, taking into account the relation between metric potential and the shape function of the wormhole by the relation $e^{\lambda}=\frac{1}{1-b(r)/r}$, we obtain form of shape function as $$b(r)=\frac{2r}{\omega+3}-\tilde{\psi_0}r^{-\frac{3}{\omega}}.$$ From the expression of $b(r)$, we see that $\frac{b(r)}{r}$ tends to a finite value as $r\rightarrow \infty $ and the redshift function does not approach zero as $r\rightarrow \infty $, so spacetime is not asymptotically flat due to the conformal symmetry. Now, we will concentrate to verify whether the obtained expression for the shape function $b(r)$ satisfies all the physical requirements to maintain a wormhole solution. For this purpose we are trying to describe fundamental property of wormholes with help of graphical representation. The profile of shape function $b(r)$ is plotted in Fig. 1 for the values of $\tilde{\psi_0}=0.09$ and $\omega=-1.58$, where the flaring out condition has been checked in Fig. 1 (right panel). We observe that shape function is decreasing with increase of the radius, and $\frac{db(r)}{dr}<0$ for $r>10.45$. From the left panel of Fig. 2, we observe that the throat of the wormhole occurs where $ b(r)-r$ cuts the r axis at a distance $r=5.39$. Therefore the throat of the wormhole occurs at $r = 5.39$ Km. for our present model. Consequently, we observe that $b'(5.39) = 0.632 < 1$ and for $r > r_0$ we see that $b(r )-r <0$, which implies $\frac{b(r)}{r}<1$ for $r > r_0$, strongly indicate that our solution satisfy all the physical criteria for wormhole solution. The slope of $b(r)$ is positive upto $r=10.45$, which concludes that the wormhole can not be arbitrarily large. The same situation occurred in the previous work [@bhar]. Moreover, we consider the energy conditions and the violation of the null energy condition (NEC) i.e., $\rho+p_r <~0$, is a necessary property for a static wormhole to exist. In Fig. 2 (right panel), we have studied all types of energy condition (using Eqs.(16)-(18)), graphically and observed that our solution violated the NEC to hold a wormhole open. ----------------------------- ------------------------------ {width="6"} {width="6"} ----------------------------- ------------------------------ ----------------------------- ----------------------------- {width="6"} {width="6"} ----------------------------- ----------------------------- Since, the wormhole space-time is non-asymptotically flat and hence the wormhole spacetime should match at some junction radius $r=R$, to the exterior schwarzschild spacetime given by the following metric $$\begin{aligned} ds^{2}&=&-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2} \nonumber\\ &&+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right).\end{aligned}$$ Here the matching occurs at a radius greater than the event horizon which gives $$\left(\frac{R}{b_0}\right)^{2}=1-\frac{2M}{R}~~~ and ~~~ 1-\frac{b(R)}{R}=1-\frac{2M}{R},$$ and using the Eq. (8) with the expression $e^{\nu(R)}=c_2^{2}R^{2}$, we determine the values of the constants $c_2^{2}$, $b_{0}$ and total mass M as follows $$\begin{aligned} M&=&\frac{R}{\omega+3}-\frac{\tilde{\psi_0}}{2}R^{-\frac{3}{\omega}},\\ c_2^{2}&=&\frac{1}{R^{2}}\left[\frac{\omega+1}{\omega+3}+\tilde{\psi_0}R^{-\frac{3+\omega}{\omega}}\right],\\ b_0&=&\frac{R}{\frac{\omega+1}{\omega+3}+\tilde{\psi_0}R^{-\frac{3+\omega}{\omega}}}.\end{aligned}$$ TOV Equation ============ An important step is to examine the stability of our present model under the different forces namely gravitational, hydrostatics and anisotropic forces. This is simply by considering the generalized Tolman-Oppenheimer-Volkov (TOV) equation according to Ponce de Le$\acute{o}$n [@leon] $$-\frac{M_G(r)(\rho+p_r)}{r}e^{\frac{\nu-\mu}{2}}-\frac{dp_r}{dr}+\frac{2}{r}(p_t-p_r)=0,$$ where $M_G(r) $ represents the effective gravitational mass within the radius $r$, which can derived from the Tolman-Whittaker formula and the explicit expression is given by $$M_G(r)=\frac{1}{2}re^{\frac{\mu-\nu}{2}\nu'}.$$ Substituting the above expression in Eq. (26), we obtain the simple expression as $$-\frac{\nu'}{2}(\rho+p_r)-\frac{dp_r}{dr}+\frac{2}{r}(p_t-p_r)=0.$$ ![Variation of different forces acting on the wormhole are plotted against r with the same values of parameters as stated earlier in Fig. 1. []{data-label="fig:1"}](tov.eps) Therefore, one can write it in a more suitable form to generate the simpler equation $$F_g+F_h+F_a=0,$$ where $F_g =-\frac{\nu'}{2}(\rho+p_r)$, $F_h =-\frac{dp_r}{dr}$ and $F_a=\frac{2}{r}(p_t-p_r)$ represents the gravitational, hydrostatics and anisotropic forces, respectively. Using the Eqs. (16-18), the above expression can be written as $$\begin{aligned} F_g=-\frac{(\omega+1)}{8\pi r}\left[\frac{2}{r^2\left(\omega+3\right)}-\frac{\tilde{\psi_0}\left(2\omega+3\right)} {\omega}r^{-\frac{3\left(\omega+1\right)}{\omega}}\right],\\ F_h=\frac{1}{8\pi}\left[\frac{4\omega}{r^3\left(\omega+3\right)}-\frac{3\tilde{\psi_0}\left(2\omega+3\right)\left(\omega+1\right)}{\omega }r^{-\frac{\left(4\omega+3\right)}{\omega}}\right],\\ F_a=\frac{1}{4\pi r}\left[\frac{1-\omega}{r^2\left(\omega+3\right)}+\frac{\tilde{\psi_0}\left(2\omega^2+3\omega-3\right)} {3}r^{-\frac{3\left(\omega+1\right)}{\omega}}\right].\end{aligned}$$ The profiles of $F_g, F_h,$ and $ F_a$ for our present model of wormhole are shown in Fig. $3$, by assigning the same value of $\omega$ = -1.58 and $\tilde{\psi_0}= 0.09$ as we used in Fig. $1$. It is clear from the Fig. 3, that the hydrostatics force ($F_h$) is dominating compare to gravitational ($F_g$) and anisotropic forces ($F_a$), respectively. The interesting feature is that $F_h$ takes the negative value while $F_g$ and $F_a$ are positive, which clearly indicate that hydrostatics force is counterbalanced by the combine effect of gravitational and anisotropic forces to hold the system in static equilibrium. There exist many excellent reviews on this topic have been studied in-depth by Rahaman et al. [@rah16] and Rani & Jawad [@jawad]. [ |p[2cm]{}||p[2.5cm]{}| ]{}\      r &       $E_g$\      6 &   2.777766660\     6.5 &   2.711356872\      7 &   2.618496680\     7.5 &   2.506836836\      8 &   2.380794614\     8.5 &   2.243339402\      9 &   2.096641966\     9.5 &   1.942376899\     10 &   1.781885447\    10.45 &   1.633036240\ Active mass function and Total Gravitational Energy =================================================== The active mass function for our wormhole ranging from $r_0+$ ($r_0$ is the throat of the wormhole) up to the radius R can be found as $$M_{active}=\int_{r_0+}^{R}4\pi\rho r^{2}dr =\left[\frac{r}{\omega+3}+\frac{\tilde{\psi_0}\left(2\omega+3\right)}{6}r^{\frac{-3}{\omega}}\right]_{r_0+}^{R}.$$ The active gravitational mass function of the wormhole is plotted in of Fig. 4 (left panel). From the Fig. 4, we see that $M_{\mathrm{active}}$ is positive outside the wormhole throat and monotonic increasing function of the radial co-ordinate, r. ---------------------------------- ----------------------------- {width="6"} {width="6"} ---------------------------------- ----------------------------- For the study of total gravitational energy of the exotic matter inside a static wormhole configuration we use the procedure adopted by Lyndell-Bell et al. and Nandi et al. [@lyn; @Amrita; @nandi] for calculating the total gravitational energy $E_g$ of the wormhole, can be written in the form $$E_g=Mc^{2}-E_M,$$ where the total mass-energy within the region from the throat $r_0$ up to the radius R can be provided as $$Mc^{2}=\frac{1}{2}\int_{r_0^{+}}^{R}T_{0}^{0}r^{2}dr+\frac{r_0}{2},$$ and the energy in other forms like kinetic energy, rest energy, internal energy etc. are defined by $$E_M=\frac{1}{2}\int_{r_0^{+}}^{R}\sqrt{g_{rr}}\rho r^{2}dr.$$ Note that here $\frac{4\pi}{8\pi}$ yields the factor $\frac{1}{2}$. By taking into account Eqs. (34 - 36), we obtain $$E_g=\frac{1}{2}\int_{r_0^{+}}^{R}[1-(g_{rr})^{\frac{1}{2}}]\rho r^{2} dr +\frac{r_0}{2},$$ where $g_{rr}=\left(1-\frac{b(r)}{r}\right)^{-1}$ and $r_0$ is the throat of the wormhole. Now to find out the expression of total gravitational energy $E_g$, we have performed the integral of Eq. (37). Due to the complexity of the coefficients $g_{rr}$ and $\rho$ we cannot extract analytical solution, for that we solve the integral numerically. The numerical values of $E_g$ are obtained by taking $r_0^{+}=5.6$ Km. as a lower limit and by changing the upper limits, which are given in Table. I. Gravitational Lensing ===================== We know that a photon follows a null geodesic $ds^2=0$, when external forces are absent. Then the equation of motion of a photon can be written as : $$\dot{r}^2 +e^{-\lambda}r^2\dot{\phi}^2=e^{\nu-\lambda}c^2\dot{t}^2,$$ where the dot represents derivative with respect to the arbitrary affine parameter. Since, neither $t$ nor $\phi$ appear explicitly in the variation principle, their conjugate momenta yield the following constants of motion : $$e^{\nu}c^2\dot{t}=E=\text{constant},$$ $$r^2\dot{\phi}=L=\text{constant},$$ where $ E$ and $ L$ are related with the conservation of energy and angular momentum, respectively. Using these two constants of motion in Eq. (38), we get $$\dot{r}^2+e^{-\lambda}\frac{L^2}{r^2}=\frac{E^2}{c^2}e^{-\nu-\lambda}.$$ Now, using $r=\frac{1}{u}$ and eliminating the derivatives with respect to the affine parameter by the help of the conservation equations, we obtain $$\label{eq:sqdu} \left(\frac{du}{d\phi}\right)^2+u^2=f(u).u^2+ \frac{1}{c^2}\frac{E^2}{L^2}e^{-\nu-\lambda}\equiv P(u),$$ where $e^{-\lambda}=1-f(u)$, while from Eq. (19) yields $$f(u)=\frac{2}{\omega+3}-\tilde\psi_0u^{\frac{\omega+3}{\omega}}.$$ Moreover, from Eq. (42), we get $$P(u)=\frac{u^2}{\omega+3}\left(2+\frac{E^2b_0^2(\omega+1)}{c^2L^2}\right) +\tilde\psi_0u^{\frac{3(\omega+1)}{\omega}}\left(\frac{E^2b_0^2}{c^2L^2}-1\right).$$ Let us proceed to discuss at the turning points [@lake], the derivative of the radial vector with respect to the affine parameter vanishes, which in turn leads to $\frac{du}{d\phi}=0$. Consequently the turning point is denoted by $r_\Sigma=1/u_\Sigma$ and given by $$\label{eq:turningpt} r_\Sigma=\left(-\tilde{\psi_0}\frac{\omega+3}{\omega+1}\right)^{\frac{\omega}{\omega+3}}.$$ [ |p[2cm]{}||p[2.3cm]{}|p[2.5cm]{}| ]{}\    $\omega$ &        $\phi$ &    $\delta$\ -1.38 & 1.126922 & -.887746\ -1.4 & 1.1130334357 & -.915523128\ -1.58 & 1.0061484 & -1.1292932\ -1.8 & 0.903820943 & -1.333948114\ -2.1 & 0.7825234 & -1.5765432\ -2.3 & 0.696393 & -1.748804\ -2.4 & 0.98541384 & -1.17076232\ -2.7 & 0.406932726 & -2.327724548\ Differentiating Eq. (42) with respect to $\phi$, we get $$\frac{d^2u}{d\phi^2}+u=Q(u),$$ where $Q(u)=\frac{1}{2}\frac{dP(u)}{du}$, and define by $$\begin{aligned} Q(u)&=& \frac{u}{\omega+3}\left(2+\frac{E^2b_0^2(\omega+1)}{c^2L^2}\right) \nonumber \\ &\;& +\frac{3\tilde\psi_0(\omega+1)}{2\omega}u^{\frac{2\omega+3}{\omega}}\left(\frac{E^2b_0^2}{c^2L^2}-1\right).\end{aligned}$$ Now, if the deflective source are absent, then the equation (46) modified to $$u=\frac{cos(\phi)}{R},$$ which is a straight line with R is the distance of closest approach to the wormhole. This solution can treated as first approximation. Furthermore, we use this solution as the first approximation to get the general solution. This yields the following form of Eq. (47) as $$\frac{d^2u}{d\phi^2}+u=A\cos\phi+B(\cos\phi)^{\frac{2\omega+3}{\omega}},$$ where [$$A=\frac{2}{R(\omega+3)}+\frac{E^2b_0^2(\omega+1)}{Rc^2L^2(\omega+3)}, \mathrm{and} ~~ B=\frac{3\tilde\psi_0(\omega+1)}{2\omega}\left(\frac{E^2b_0^2}{c^2L^2}-1\right).$$]{} With the aid of Eq. (49), the general solution is given by $$\begin{aligned} \label{eq:trajectory} u & = & \frac{\cos\phi}{R}+\frac{A}{2}\left(\cos\phi+\phi \sin\phi\right)+\nonumber\\ &\;& B\sin\phi\int\cos^{\frac{3(\omega+1)}{\omega}}\phi~~d\phi +B\frac{\omega}{3(\omega+1)}\cos^{\frac{4\omega+3}{\omega}}\phi.\nonumber\\.\end{aligned}$$ The light ray approaches from infinity at an asymptotic angle $\phi=-\left(\frac{\pi}{2}+\epsilon\right)$ and goes back to infinity at an asymptotic angle $\phi=\left(\frac{\pi}{2}+\epsilon\right)$. However the point of transition of the light ray from the Schwarzschild spacetime to the phantom spacetime is given by the turning points defined in (\[eq:turningpt\]). The solution of the equation $~~u\left(\frac{\pi}{2}+\epsilon\right)=0$ yields the angle $\epsilon$. The total deflection angle of the light ray can be obtained as $\delta=2\epsilon$. In case of our wormhole, we have calculated the deflection angles for different values of $\omega$ that are tabulated in Table II. One can note that rather finding angle of deficit, we have found angle of surplus. Lastly we must remember that the phantom spacetime under consideration, unlike Schwarzschild, is basically non flat. Hence strictly speaking, an asympotically straight line trajectory where $r\rightarrow\infty$ does not make sense. To resolve this issue we can consider the angles which the tangent to the light trajectory makes with the coordinate planes at a given point. Following Rindler and Ishak [@Rindler] we can find $$\tan (\Psi)=\frac{r\left[e^{\nu(r)}\right]^{1/2}}{\left|\frac{dr}{d\phi}\right|}~,$$ using Eq. \[\[eq:sqdu\]\] we have $$\tan(\Psi)=\sqrt{\frac{\frac{1}{R^2}\left[\frac{\omega+1}{\omega+3}+ \tilde\psi_0R^{-(3+\omega)/\omega}\right]}{\frac{AR-1}{r^2}+\frac{2B\omega}{3(\omega+1)}r^{-3(\omega+1)/\omega}}}.$$ Assuming the bending angle to be small we can take $\tan(\Psi)\rightarrow\Psi$, $\sin(\phi)\rightarrow\phi$ and $\cos(\phi)\rightarrow1$. Thus from Eq. \[\[eq:trajectory\]\] we can write the actual light deflection angle given by $|\epsilon|=\left|\Psi-\phi\right|$ as follows : [$$|\epsilon|=\left|\sqrt{\frac{\frac{1}{R^2}\left[\frac{\omega+1}{\omega+3}+\tilde\psi_0R^{-(3+\omega)/\omega}\right]} {\frac{AR-1}{r^2}+\frac{2B\omega}{3(\omega+1)}r^{-3(\omega+1)/\omega}}} -\left[\frac{\frac{1}{r}-\frac{1}{R}-\frac{A}{2}-\frac{B\omega}{3(\omega+1)}}{B+\frac{A}{2}}\right]\right|.$$]{} Working along the lines of Bhadra et. al. [@Bhadra] we can then calculate the total deflection angle in terms of location of the source ($d_{LS}, \phi_S$) and the observer ($d_{OL}, \phi_O$) as [$$\begin{aligned} |\epsilon| &= & \left|\sqrt{\frac{\frac{1}{R^2}\left[\frac{\omega+1}{\omega+3}+\tilde\psi_0R^{-(3+\omega)/\omega}\right]} {(AR-1)\left(\frac{1}{d_{LS}^2}+\frac{1}{d_{OL}^2}\right)+\frac{2B\omega}{3(\omega+1)}\left(d_{LS}^{-3(\omega+1)/\omega} +d_{OL}^{-3(\omega+1)/\omega}\right)}} \right.\nonumber\\ &\;&\left.-\left[\frac{\frac{1}{d_{LS}}+\frac{1}{d_{OL}}-\frac{1}{R}-\frac{A}{2}-\frac{B\omega} {3(\omega+1)}}{B+\frac{A}{2}}\right]~\right|\nonumber.\\\end{aligned}$$]{} Junction Condition ================== In previous section we matched our interior wormhole spacetime with the Schwarzschild exterior spacetime at the boundary $r=r_{\Sigma}$. Since the wormhole spacetime is not asymptotically flat we use the Darmois–Israel [@dm1; @dm2] formation to determine the surface stresses at the junction boundary. The intrinsic surface stress energy tensor $S_{ij}$ is given by Lancozs equations in the following form $$S^{i}_{j}=-\frac{1}{8\pi}(\kappa^{i}_j-\delta^{i}_j\kappa^{k}_k).$$ The second fundamental form is presented by $$K_{ij}^{\pm}=-n_{\nu}^{\pm}\left[\frac{\partial^{2}X_{\nu}}{\partial \xi^{i}\partial\xi^{j}}+\Gamma_{\alpha\beta}^{\nu}\frac{\partial X^{\alpha}}{\partial \xi^{i}}\frac{\partial X^{\beta}}{\partial \xi^{j}} \right]|_S,$$ and the discontinuity in the second fundamental form is written as, $$\kappa_{ij}=K_{ij}^{+}-K_{ij}^{-},$$ where $n_{\nu}^{\pm}$ are the unit normal vectors defined by, $$n_{\nu}^{\pm}=\pm\left|g^{\alpha\beta}\frac{\partial f}{\partial X^{\alpha}}\frac{\partial f}{\partial X^{\beta}} \right|^{-\frac{1}{2}}\frac{\partial f}{\partial X^{\nu}},$$ with $n^{\nu}n_{\nu}=1$. Where $\xi^{i}$ is the intrinsic coordinate on the shell. $+$ and $-$ corresponds to exterior i.e., Schwarzschild spacetime and interior (our) spacetime respectively. Considering the spherical symmetry of the spacetime surface stress energy tensor can be written as $S^{i}_j=diag(-\sigma,\mathcal{P})$. Where $\sigma$ and $\mathcal{P}$ is the surface energy density and surface pressure respectively. The expression for surface energy density $\sigma$ and the surface pressure $\mathcal{P}$ at the junction surface $r = r_{\Sigma}$ are obtained as, $$\begin{aligned} \sigma &=&-\frac{1}{4\pi r_{\Sigma}}\left[\sqrt{e^{-\lambda}} \right]_{-}^{+} \nonumber\\ &&=-\frac{1}{4\pi r_{\Sigma}}\left[\sqrt{1-\frac{2M}{r_{\Sigma}}}-\sqrt{\frac{\omega+1}{\omega+3}+\tilde{\psi_0}r_{\Sigma}^{-\frac{3+\omega}{\omega}}}\right],\end{aligned}$$ and $$\begin{aligned} \mathcal{P}&=&\frac{1}{8\pi r_{\Sigma}}\left[\left\{1+\frac{a\nu'}{2}\right\}\sqrt{e^{-\lambda}}\right]_{-}^{+}\nonumber\\ &&=\frac{1}{8\pi r_{\Sigma}}\left[\frac{1-\frac{M}{r_{\Sigma}}}{\sqrt{1-\frac{2M}{r_{\Sigma}}}}-\sqrt{\frac{\omega+1}{\omega+3}+\tilde{\psi_0} r_{\Sigma}^{-\frac{3+\omega}{\omega}}}\right].\end{aligned}$$ Hence we have matched our interior Wormhole solution to the exterior Schwarzschild spacetime in presence of thin shell. Discussion ========== By the confirmation of various observational data that the Universe is undergoing a phase of accelerated expansion and dark energy models have been proposed for this expansion. In the framework of GR, the violation of NEC namely ‘exotic matter’, is a fundamental ingredient of static traversable wormholes. In this work, we investigated some of the characteristics needed to support a traversable wormhole specific exotic form of dark energy, denoted phantom energy, admitting conformal motion of Killing Vectors. We analyzed the physical properties and characteristics of these wormholes with help of graphical representation. In the plots of Fig. 2 (left panel). we obtain the throat of wormholes where $b(r) - r$ cuts the r-axis, is located at r = $r_0$ = 5.39 Km., which implies that $b(r)/r < 1$, met the flare-out condition. Another fundamental property of wormhole is the violation of the null energy condition (NEC), also satisfy for our model given in Fig. 2 (right panel), which provide a natural scenario for the existence of traversable wormholes. We discuss the possibility of detection of wormholes by means of gravitational lensing and we have found the angle of deflection to be negative i.e. angle of surplus. Additionally in order to give a physically feasible meaning to light deflection in a nonflat phantom spacetime, we have computed the angles that the tangent to the light trajectory makes with the coordinate planes in terms of the location of the observer and the source. Further we investigate, total gravitational energy content in the interior of exotic matter distribution for the wormholes by Lyndell-Bell et al. [@lyn] and Nandi et al’s.[@nandi] perception. This lensing phenomena i.e. deflection of light by the wormhole offers a good possibility to detect the presence of wormhole. 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--- abstract: | Due to popularity surge social networks became lucrative targets for spammers and guerilla marketers, who are trying to game ranking systems and broadcast their messages at little to none cost. Ranking systems, for example Twitter’s Trends, can be gamed by scripted users also called bots, who are automatically or semi-automatically twitting essentially the same message. Judging by the prices and abundance of supply from PR firms this is an easy to implement and widely used tactic, at least in Russian blogosphere. Aggregative analysis of social networks should at best mark those messages as spam or at least correctly downplay their importance as they represent opinions only of a few, if dedicated, users. Hence bot detection plays a crucial role in social network mining and analysis. In this paper we propose technique called RepRank which could be viewed as Markov chain based model for reputation propagation on graphs utilizing simultaneous trust and anti-trust propagation and provide effective numerical approach for its computation. Comparison with another models such as [TrustRank]{}$\quad$and some of its modifications on sample of 320000 Russian speaking Twitter users is presented. The dataset is presented as well. **Keywords.** Antispam, social graph, Markov chain, trustrank. author: - | G.V. Ovchinnikov [^1]\ D.A. Kolesnikov [^2]\ - 'I.V. Oseledets [^3]' bibliography: - 'references.bib' title: 'Algebraic reputation model RepRank and its application to spambot detection [^4]' --- Introduction ============ While concepts of ’good’ and ’bad’ are quite complex and may be considered subjective, applied to spam filtration problems they are mapped to ’signal’ and ’noise’ correspondingly and became objective enough to build algorithms upon. The following assumption, proposed in [@TR] was successfully used in graph based antispam algorithms: Good graph vertices rarely link to the bad ones. \[ass1\] This assumption leads to trust propagation scheme proposed in [@TR]: pages linked from good ones are almost good, the pages linked from those are slightly worse and so on. There is a similar approach to mistrust propagation [@ATR]: pages linking to bad pages are almost bad, the pages linking to those pages may be slightly better and so on. The main difference being trust propagates forward (by graph edges direction) and mistrust propagates backward. [TrustRank]{} and other similar models for signal propagation on graphs can be viewed in random walker framework with random walkers carrying signal. In case of [TrustRank]{} it is trust charge equal to current vertex trust value. For some vertices marked by external verification process (also called oracle function) walkers with a priory set probability $\alpha$ take charge equal to $1$ and with probability $1-\alpha$ take charge equal to current charge of the vertex. Stationary distribution of such process with initial distribution vector $d$ with $d_i=1$ if $i$-th vertex marked by the oracle as a good one and $d_i = 0$ otherwise is gives us [TrustRank]{} scores for whole graph: $$t = \alpha F t + (1 - \alpha) d,$$ here $F$ is forward transition matrix which is column normalized adjacency matrix. The logical continuation of [TrustRank]{}and anti-[TrustRank]{}is the combination of both models. Major advantage of combined trust and mistrust propagation approach is the ability to use both positive and negative signals. [@TDR] uses this approach by penalizing trust and distrust propagation form from not trustworthy and trustworthy vertices correspondingly. [@GBR] is analogous to [@TDR], but stated in probabilistic framework. In this paper we further pursue the idea of simultaneous trust and mistrust propagation. In contrast with above-mentioned papers we combine trust and mistrust providing unified reputation score, called RepRank. Proposed model ============== To each graph vertex we attach a trust score witch is negative for bad vertices and positive for a good ones. Denote by $t_{+}$ and $t_{-}$ vectors obtained by zeroing negative and positive components in vector $t$ correspondingly. Then, we search trust distribution $t$ to satisfy $$\label{reprank:eq1} t = \alpha_1 F t_{+} + \alpha_2 B t_{-} + \alpha_3 d,$$ where $B$ is backward transition matrix which is column normalized transposed adjacency matrix. The solution of (\[reprank:eq1\]) we will call RepRank. It exists, unique and continuously dependent on the initial distribution $d$ (see the Theorem \[theorem\] for more details). While usefulness of existence and uniqueness of the solution to the equation (\[reprank:eq1\]) solution is beyond doubt we want to point out, that continuous dependence on oracle provided labeling $d$ is very nice as well. It protects RepRank from sudden “everything you knew is wrong” changes caused be addition of small portion of new data, mistakes and typos (a manual labeling is very tedious and error prone process). This follows natural intuition that once one have an idea of everyone’s reputation a small changes should only correct, not shake the foundations of his worldview. \[theorem\] Let $F, B$ be $n \times n$ stochastic matrices from (\[reprank:eq1\]), $0 < \alpha_1, \alpha_2, \alpha_3 < 1$, $\mathcal{P}_{+}$ be an operator which replaces all negative component of the $n \times 1$ vector by zeroes and $\mathcal{P}_{-}$ be operator which replaces all positive components of the $n \times 1$ vector by zeroes. Then mapping $\mathcal{R}$ an $n \times 1$ vector $d$ to the $n \times 1$ vector $t$ that the solution of the equation $$\label{reprank:eq2} t = \alpha_1 F \mathcal{P}_{+}(t) + \alpha_2 B \mathcal{P}_{-}(t) + \alpha_3 d,$$ has the next properties: 1. $R(d)$ exists for any vector $d$ from $\mathbf{R}^n$. 2. $R(d)$ is bijection mapping $\mathbf{R}^n$ to $\mathbf{R}^n$. 3. $R(d$) is Lipschitz continuous mapping. Proof:\ One can use the following iterative process to find $t$: $$\label{reprank:iter} \begin{split} & t^{(k+1)} = I(t^{(k)}),\\ &I(t) = \alpha_1 F \mathcal{P}_{+}(t) + \alpha_2 B \mathcal{P}_{-}(t) + \alpha_3 d, \end{split}$$ with any initial vector $t_0$. The mapping $I(t)$ is contractive on the metric space ($\mathbf{R}^n, \Vert \cdot \Vert_1$): $$\begin{split} \Vert I(t_1) - I(t_2) \Vert_1 &=\\ = \Vert \alpha_1 F (\mathcal{P}_{+}(t_1) - \mathcal{P}_{+}(t_2)) \\+\alpha_2 B (\mathcal{P}_{-}(t_1) - \mathcal{P}_{-}(t_2)) \Vert_1 \leq \\ \leq \alpha_1 \Vert F \Vert_1 \Vert \mathcal{P}_{+}(t_1) - \mathcal{P}_{+}(t_2)\Vert_1 + \\ \alpha_2 \Vert B \Vert_1 \Vert \mathcal{P}_{-}(t_1) - \mathcal{P}_{-}(t_2) \Vert_1 \leq \\ \leq \alpha_1 \Vert \mathcal{P}_{+}(t_1) - \mathcal{P}_{+}(t_2)\Vert_1 + \\ \alpha_2 \Vert \mathcal{P}_{-}(t_1) - \mathcal{P}_{-}(t_2) \Vert_1 \leq \\ \leq \max(\alpha_1 , \alpha_2) \Vert t_1 - t_2\Vert_1&, \end{split}$$ where we used that $\Vert F\Vert_1 = \Vert B \Vert_1 = 1$ and $$\label{1norm:eq} \Vert t_1 - t_2\Vert_1 = \Vert\mathcal{P}_{+}(t_1) - \mathcal{P}_{+}(t_2)\Vert_1 + \Vert \mathcal{P}_{-}(t_1) - \mathcal{P}_{-}(t_2)\Vert_1.$$ So $I(t)$ is contractive mapping with coefficient less or equal to $\max(\alpha_1 , \alpha_2) < 1$ and Banach fixed-point theorem guarantees the existence and uniqueness of fixed point for it. Let us denote that fixed point by $R(d)$ and notice that it is the solution for equation (\[reprank:eq1\]). Also $R(d)$ is Lipschitz continuous mapping with coefficient $$\frac{\alpha_3}{1- \max(\alpha_1, \alpha_2)}.$$ It can be proven in the following way: $$\begin{split} R(d_1) - R(d_2) &= \\ =\alpha_1 F (\mathcal{P}_{+}(R(d_1)) - \mathcal{P}_{+}(R(d_2)))& + \\ +\alpha_2 B (\mathcal{P}_{-}(R(d_1)) - \mathcal{P}_{-}(R(d_2))) &+\alpha_3 (d_1 - d_2) \\ \Vert R(d_1) - R(d_2) \Vert_1 &\leq \\ \leq \alpha_1 \Vert F \Vert_1 \Vert (\mathcal{P}_{+}(R(d_1)) - \mathcal{P}_{+}(R(d_2))) \Vert_1& +\\ +\alpha_2 \Vert B \Vert_1 \Vert (\mathcal{P}_{-}(R(d_1)) - \mathcal{P}_{-}(R(d_2))) \Vert_1 + \\ +\alpha_3 \Vert d_1 - d_2 \Vert d_2 \leq \\ \leq \max(\alpha_1, \alpha_2) \Vert R(d_1) - R(d_2) \Vert_1 &+ \alpha_3 \Vert d_1 - d_2 \Vert_1, \end{split}$$ therefore $$\Vert R(d_1) - R(d_2) \Vert_1 \leq \frac{\alpha_3}{1- \max(\alpha_1, \alpha_2)} \Vert d_1 - d_2 \Vert_1 .$$ The mapping $R(d)$ is injection because equality $R(d_1) = R(d_2)$ causes $d_1 = d_2$ in equation (\[reprank:eq1\]). It is also a surjection because for any $t \in \mathbf{R}^n$ exists $d_t$ such that $$d_t = \frac{1}{\alpha_3} (t - \alpha_1 F \mathcal{P}_{+}(t) - \alpha_2 B \mathcal{P}_{-}(t)),$$ that equality $R(d_t) = t$ holds. Experimental evaluation ======================= For our experiments we recursively crawled twitter using as seeds Russian speaking users we found in Twitter’s Streaming API. For each user we downloaded all people he follows, ’friends’ in Twitter terminology. This friends graph has 326130 vertices and 2713369 nodes. We manually labeled 3124 users as spammers or a good ones. [^5] Algorithm Accuracy ---------------- ---------- RepRank 0.8833 TrustRank 0.851 anti-TrustRank 0.8636 : Experimental results[]{data-label="tb:cmp"} We did a cross-validation with random subsampling splitting data set in two halves to test our algorithm against other single-score trust propagation algorithms, namely TrustRank and anti-TrustRank. The parameters for each algorithm were chosen to maximize accuracy. The results provided in the Table \[tb:cmp\]. As can be seen our algorithm outperforms both [TrustRank]{}$\quad$ and anti-[TrustRank]{}. [0.49]{} ![Connections between 300 vertices with highest in-degree (left) and 300 most reputable vertexes (right). Vertices not having connections within this group are omitted. []{data-label="fig:top300"}](top300deg.pdf "fig:"){width="90.00000%"} [0.49]{} ![Connections between 300 vertices with highest in-degree (left) and 300 most reputable vertexes (right). Vertices not having connections within this group are omitted. []{data-label="fig:top300"}](top300rep.pdf "fig:"){width="90.00000%"} It is interesting to note, that Russian-speaking part of Twitter is dominated by bots and spammers. According to our algorithm, out of 326130 accounts only 59691 (around 18%) are managed by humans. Among those only 375 (around 0.1%) correspond to high-reputation, profiles of prominent public figures, government officials and organizations, reputable press and so on (see reputation distribution on Figure \[fig:rephist\]). Spammers are more active at following each other than a normal people (see Figure \[fig:top300\]). A serious anti-spam effort is required if one wants to make use of Twitter’s data, otherwise he will analyze noise, or worse, some botmaster’s political views. Conclusion and further work =========================== The contribution of this paper is twofold. First, we proposed new reputation propagation algorithm which allows to use both bad and good vertices in its starting set and outperforms analogues. Second, we gathered a sample Russian Twitter social graph with manual labeling of good and bad seeds. Different normalization strategies along with regularizations (pagerank-style teleportations) could be used to further improve performance of proposed method. ![Distribution of reputation. RepRank score on the x-axis and logarithm of number of accounts on the y-axis. The gap is due to separation between masses and a handful of celebrities who are following each other. []{data-label="fig:rephist"}](reprankhist.eps){width="49.00000%"} [^1]: Skolkovo Institute of Science and Technology, Skolkovo 143025, Russia and Institute of Design Problems in Microelectronics, Zelenograd Sovetskaya 3, Moscow, 124365 (e-mail: ovgeorge@yandex.ru) [^2]: Skolkovo Institute of Science and Technology, Skolkovo 143025, Russia (e-mail: d4kolesnikov@yandex.ru) [^3]: Skolkovo Institute of Science and Technology, Skolkovo 143025, Russia and Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119333 Russia. (e-mail: ivan.oseledets@gmail.com) [^4]: This work was supported by Russian Science Foundation Grant 14-11-00659 [^5]: The dataset can be obtained from https://bitbucket.org/ovchinnikov/rutwitterdataset

--- abstract: 'We discuss the horizon problem in a universe dominated by fluid with negative pressure. We show that for generally accepted value of nonrelativistic matter energy density parameter $\Omega_{m0}<1$, the horizon problem can be solved only if the fluid influencing negative pressure (the so-called “X” component) violates the point-wise strong energy condition and if its energy density is sufficiently large $(\Omega_{X 0}>1)$. The calculated value of the $\Omega_{X0}$ parameter allowing for the solution of the horizon problem is confronted with some recent observational data. Assuming that $p_X/\rho_X<-0.6$ we find that the required amount of the “X” component is not ruled out by the supernova limits. Since the value of energy density parameter $\Omega_{v0}$ for cosmological constant larger than 1 is excluded by gravitational lensing observations the value of the ratio $p_X/\rho_X$ should lie between the values $-1$ and $-0.6$ if the model has to be free of the horizon problem beeing at the same time consistent with observations. The value of $\Omega_{X0}+\Omega_{m0}$ in the model is consistent with the constraints $0.2<\Omega_{\text{tot}}<1.5$ following from cosmic microwave background observations provided that $\Omega_{m0}$ is low ($<0.2$).' address: 'Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland' author: - 'Jerzy Stelmach [^1]' title: | Horizon problem in a closed universe\ dominated by fluid with negative pressure --- Horizon problem in standard cosmology ===================================== According to the standard scenario, about 300 000 years after the Big Bang the Universe cooled down to the level that atoms could form. Electrons were captured by nuclei and photons, which until that time were in thermal equilibrium with plasma, lost charged partners for interaction and started their free travel across the Universe. These photons are detected today and form the so-called cosmic microwave background (CMB). The radiation has a black-body spectrum and corresponds to the temperature $T=2.73\pm 0.01$ K, independently on the direction it comes from. Extreme isotropy of the radiation is its most puzzling property because the regions from which two hitting us today antipodal photons come, could never communicate with each other. Hence the same temperature of these regions cannot be explained in the standard scenario. This is called a horizon problem. Its inevitability in the standard cosmological model is usually illustrated by non-intersection of two past light cones of these regions at the recombination epoch, i.e. at the epoch when the relic radiation appeared (Fig. \[fig1\]). This non-intersection means that there were no events in the history of the universe which could influe the recombination process at the points $A$ and $B$ simultaneously. Some attempts to solve the horizon problem ========================================== One of the first conclusions which follow even from superficial analysis of the picture (Fig. \[fig1\]) is that the horizon problem exists because the recombination of atoms took place so early comparing to the age of the universe ($t_R/t_0<10^{-4}$). If $t_R$ was large enough the appropriate light cones would intersect and the problem would not appear. Unfortunately in the framework of the standard scenario the recombination epoch cannot be shifted significantly into the future. What we can do instead is to assume that the last scattering of relic photons did not take place at $t_R$ but more recently, for example, due to the Compton scattering on free electrons filling up the whole universe [@wein72]. Detailed calculations show, however, that for generally accepted values of cosmological parameters the horizon problem still exists [@stel90]. In other words shifting the last scattering surface into the future does not help to solve the problem. Careful glance at the picture suggests another solution to the problem: an appropriate bending of past light cones of the points $A$ and $B$ (concave form instead of convex one) can cause that the horizons would overlap (Fig. \[fig2\]). This is realized in an inflationary scenario [@guth81] where a large cosmological term gives the required form of the light cones. Inspite of the fact that there is no unique physical model of the inflation the idea of rapid exponential or power law expansion of the early universe is regarded as very attractive because it solves additionally several other problems of the standard model. We shall not discuss this topic now. In inflationary scenarios the horizon problem does not appear because the past light cones of points $A$ and $B$ bend radically towards the time axis (Fig. \[fig2\]) due to the accelerating expansion. In the present paper we discuss the possibility of solving the horizon problem without very effective inflationary epoch in the early universe and without shifting the last scattering surface into the future. The possibility arises in the closed universe, in which relic photons may come from geometrical antipode. This is of course trivial fact, but straightforward calculation shows that without fluid with negative pressure such universe should be extremely dense, what cannot be accepted. Inspite of the fact that many observations favor open or flat universe, the closed one is still not ruled out [@white96]. Since the universe in our model, apart from the relativistic and nonrelativistic matter, is also filled with the fluid influencing negative pressure (simulating repulsive force) the evolution of resulting closed cosmological models do not end up with the Big Crunch. Contrary, the universe is ever expanding reaching the size consistent with observations. Solution of the horizon problem in such models was discussed by several authors. Especially two kinds of fluid, in the context of the horizon problem, were in the past taken into account: the so-called string-like fluid, described by the equation of state $p_s=-\rho_s/3$ [@dav87; @stel94; @kam96], and the cosmological constant [@white96]. It seems that the first kind of fluid (string-like one) is already ruled out by supernova observations [@garn98] which indicate that the ratio $\alpha_X$ of the pressure $p_X$ to the energy density $\rho_X$ of the unknown, the so-called “X” component of the universe must be less than $-0.6$ (95% confidence). As regards the second kind of fluid, most of recent astronomical observations suggest existence of positive cosmological constant [@fort98; @filip98; @cooray98], but the upper limit for its density is slightly too low in order to solve the horizon problem, especially if the nonrelativistic matter content of the universe is large [@baum98]. The purpose of the present paper is discussion of the horizon problem in a universe dominated by a more general kind of fluid influencing negative pressure. In the next section we shortly motivate dealing with the fluid with negative pressure. In Section IV we discuss the horizon problem in the FRW model with such fluid. In Section V we show that the horizon problem does not appear if the fluid violates the point-wise strong energy condition and if its contribution to the total energy density of the universe is predominant, closing the universe. Two special cases: vacuum and string-like matter dominated models, are discussed in more detail. In the last section we compare the models with observations and summarize the results. Motivation for dealing with fluid with negative pressure ======================================================== Introduction of fluid with negative pressure in cosmology has a long history. Einstein introduced it in form of the $\Lambda$-term in order to get static cosmological model by compensating gravitational atraction with repulsive affect of the cosmological constant. In recent years cosmological models with $\Lambda$-term have been intensively investigated in the context of large scale structure formation [@klyp97] as well as in the context of the age-of-the-universe problem [@kraus97; @sper97]. In both cases vacuum energy forms a smooth component of the dark matter. Visser considered the age-of-the-universe problem by treating it, as far as possibly, in a model independent way, without assuming any particular equation of state, and he showed that if the Hubble parameter is high enough, in order to solve the problem, the point-wise strong energy condition (SEC) must be violated between the epoch of galaxy formation and the present [@vis97]. Another reason for dealing with the “X” component follows from high redshift supernovae observations which indicate that the universe is accelerating [@kim98; @riess98]. The simplest way to explain this is to assume that the universe is dominated by some fluid with negative pressure. Moreover the observed number of gravitational lensing events cannot be explained without existence of such form of matter [@cooray98]. Physical interpretation can be attached to the fluid with negative pressure according to different physical models. The cosmological constant interpreted as an energy density of physical vacuum (satisfying the equation of state $p_v=-\rho_v$) is the most obvious example. The equation of state $p_s=-\rho_s/3$ corresponds to the string-like matter, which can be regarded as a network of cosmic strings conformally stretched by the expansion [@vil84; @kam96], global texture [@dav87] or decaying cosmological constant [@silv94]. Intermediate equations of state different from $p_v=-\rho_v$ and $p_s=-\rho_s/3$ can also be achieved in models with fundamental fields (scalar, vector, or tensor) forming on average some “not normal” fluid [@cald98]. In the present paper we consider fluid with negative pressure by assuming the equation of state $$p_X=\alpha_X\rho_X,$$ where $$-1\leq\alpha_X<-{1\over 3},$$ hence violating SEC. Cosmological constant corresponds to $\alpha_v=-1$ and is, in some sense, an extreme possibility. For the purposes of the present paper we admit, however, the upper bound of the interval ($\alpha_s=-1/3$) as well. Summarizing, we assume that the pressure of the fluid fulfills the inequality $$-\rho_X\leq p_X\leq -{1\over 3}\rho_X.$$ Following Visser [@vis97] we shall call the fluid satisfying the inequality (3) – “abnormal”. Horizon problem in the FRW model with “abnormal” fluid ====================================================== We assume that the universe is filled with relativistic matter (e.g. relic radiation), nonrelativistic matter (e.g. galaxies) and the “abnormal” fluid (e.g. cosmological constant, stringlike matter, etc.) – the fluid satisfying the inequality (3). Let us define (cf. Fig. \[fig1\]): $\displaystyle r_0\equiv c\int_0^{t_0}{dt\over R(t)}$ – comoving radius of the observer’s particle horizon, $\displaystyle r_R\equiv c\int_0^{t_R}{dt\over R(t)}$ – comoving radius of the particle horizon at the recombination epoch, $\displaystyle \chi_R\equiv r_0-r_R=c\int_{t_R}^{t_0}{dt\over R(t)}$ – comoving coordinate of the regions from which hitting us today relic photons were emitted, $R(t)$ – scale factor. It follows that the horizon problem does not appear if $$2r_R>r_0, ~~\text{or ~equivalently ~if} ~~ r_R>\chi_R.$$ In other words comoving radius of the particle horizon at the recombination epoch must be larger than the present comoving distance to the scattering surface taking place at that epoch. In the model under consideration the expressions for $r_R$ and $\chi_R$ may be rewritten in the form (c.f. [@stel90]) $$\begin{aligned} r_R=\left({\Omega_{r0}+\Omega_{m0}+\Omega_{X0}-1\over k}\right)^{1/2} \int_{z_R+1}^\infty\biggl[x^4\left(\Omega_{r0}+ {\Omega_{m0}\over z_R+1}\right) &+& (1-\Omega_{r0}-\Omega_{m0}- \Omega_{X0})x^2 \nonumber \\ &+& \Omega_{X0}x^{3(\alpha_X+1)}\biggr]^{-1/2}dx,\end{aligned}$$ $$\begin{aligned} \chi_R=\left({\Omega_{r0}+\Omega_{m0}+\Omega_{X0}-1\over k}\right)^{1/2} \int_{1}^{z_R+1}\Bigl[\Omega_{r0}x^4 + \Omega_{m0}x^3 &+&(1-\Omega_{r0}- \Omega_{m0}-\Omega_{X0})x^2 \nonumber \\ &+&\Omega_{X0}x^{3(\alpha_X+1)}\Bigr]^{-1/2}dx,\end{aligned}$$ where $z_R$ – redshift corresponding to the recombination epoch, -6mm$\Omega_{r0}, \Omega_{m0}$ – relativistic and nonrelativistic matter energy density parameters, $\Omega_{X0}$ – “abnormal” fluid energy density parameter. Since the integrands in both cases are rapidly decreasing functions of $x$, and since $z_R$ is relatively large ($\approx 1200$) even rough estimate of numerical values of the above integrals leads to the conclusion that $r_R$ cannot be larger than $\chi_R$ for any reasonable values of $\Omega_{r0}$, $\Omega_{m0}$ and $\Omega_{X0}$. Numerical integration confirms this estimate. E.g. for $\Omega_{r0}=0.00004,~\Omega_{m0}=0.3$ and $\Omega_{v0}\approx 0.7$ ($\alpha_X=-1$ – nearly flat model with cosmological constant) we get the value $r_R/\chi_R\approx 0.0150$, much too low to solve the problem. For $\Omega_{m0}=0.1$ and $\Omega_{v0}=0.9$ we get $r_R/\chi_R\approx 0.0153$, and the problem still remains. Considering other forms of “abnormal” fluid ($\alpha_X > -1$) does not change the ratio significantly. Hence the conclusion could be drawn that the horizon problem cannot be solved in the standard (noninflationary) cosmological scenario independently of whether the “abnormal” fluid is involved or not. In the next part of the paper I will show that this conclusion is not quite correct. Horizon problem in a closed “abnormal” fluid-dominated universe =============================================================== Almost constant and small value of the ratio $r_R/\chi_R\approx 0.015$ suggests that the horizon problem cannot be solved in the standard scenario even with some form of “abnormal” fluid. Very small value of this ratio means that the angle $\theta$ at which we observe today causally connected region at the recombination epoch – the so-called particle horizon [@rind56] is of order of only few degrees ($\theta\leq 3^\circ)$ [@wein72]. It is known, however, that the above statement must be revised in a closed cosmological model. Now let us assume that the universe is closed ($k=1$), i.e. the total energy density of matter filling up the universe, including radiation, nonrelativistic matter and the “abnormal” fluid is larger than the critical density. In this case the expressions for $r_R$ and $\chi_R$ are $$\begin{aligned} r_R=(\Omega_{r0}+\Omega_{m0}+\Omega_{X0}-1)^{1/2}\int_{z_R+1} ^\infty\biggl[x^4\left(\Omega_{r0}+{\Omega_{m0}\over z_R+1}\right) &-&(\Omega_{r0}+\Omega_{m0}+\Omega_{X0}-1)x^2 \nonumber \\ &+&\Omega_{X0}x^{3(\alpha_X+1)} \biggr]^{-1/2}dx,\end{aligned}$$ $$\begin{aligned} \chi_R=(\Omega_{r0}+\Omega_{m0}+\Omega_{X0}-1)^{1/2}\int_1^{z_R+1} \Bigl[\Omega_{r0}x^4+\Omega_{m0}x^3&-&(\Omega_{r0}+\Omega_{m0}+\Omega_{X0} -1)x^2 \nonumber \\ &+&\Omega_{X0}x^{3(\alpha_X+1)}\Bigr]^{-1/2}dx,\end{aligned}$$ and play role of angular coordinates (cf. Fig. \[fig3\]). As we know from the earlier discussion and also see from the picture the horizon problem appears (shaded regions do not overlap) if $\chi_R>r_R$ what is always the case for reasonable values of the parameters $\Omega_{r0}, \Omega_{m0}$ and $\Omega_{X0}$. However, closdeness of the universe gives chance to solve the horizon problem even if $\chi_R\gg r_R$. Note that if $\chi_R$ was very close to $\pi$ then even for small value of $r_R$ the shaded regions could overlap. Hence what we need is to fulfill the condition (Fig. \[fig4\]) $$\vert\pi-\chi_R\vert<r_R.$$ Since $\Omega_{r0}$ and $\Omega_{m0}$ are more or less fixed (by observations) we can only vary the “abnormal” fluid energy density parameter $\Omega_{X0}$. Even without performing explicit integrations we notice that only for $$-1\leq\alpha_X\leq -{1\over 3}$$ $r_R$ as well as $\chi_R$ are growing functions of $\Omega_{X0}$, what is necessary to approach $\pi$ by $\chi_R$ in order to fulfill the triangle inequality (10) since $r_R$ is always relatively small. Numerical calculations show that the growth is relatively fast. Moreover, the fastest growth is achieved for cosmological constant ($\alpha_v=-1$) which corresponds to the lower bound of the interval (10), and the slowest one for the string-like matter ($\alpha_s=-1/3$) – upper bound of the interval. Since the cosmological constant acts more effectively let us focus on this case. Starting from $\chi_R\approx 0.02$ and $\chi_R\approx 0.0003$ for $\Omega_{m0}=0.3$ and $\Omega_{v0}=0.7$ we reach $\chi_R\approx 3.11$ and $r_R\approx 0.04$ for $\Omega_{v0}=1.311$. Note that in the latter case the condition (9) is fulfilled, and the horizon problem is solved. Further increasing $\Omega_{v0}$ causes violation of the condition (9) starting from $\Omega_{v0}=1.327$. In consequence horizon problem appears again. Summarizing, horizon problem does not appear in the vacuum-dominated closed universe for $\Omega_{r0}=0.00004,~\Omega_{m0}=0.3$, and $\Omega_{v0}$ from rather narrow interval $$1.310<\Omega_{v0}<1.327.$$ It is worth mentioning that the condition (10), up to the equality sign ($\alpha_s=-1/3$), is just the violation of the strong energy condition. We remind that according to Visser [@vis97] the same condition had to be violated in order to solve the age-of-the-universe problem. As it might have been expected the age of the universe in this case is relatively large $$t_0=12.4\times h^{-1}\times 10^9 ~ \text{years,}$$ where $h\in (1/2,1)$ is a normalized Hubble constant. For other values of $\Omega_{m0}$ (0.015, 0.1 and 0.2) corresponding minimum and maximum values of $\Omega_{v0}$, for which the horizon problem is solved, are given in Table \[table1\]. The value $\Omega_{m0}=0.015$ in the table is justified by the paper of Hoell [*et al.*]{} [@hoell94] in which it is argued that observations of absorption lines of the Lyman $\alpha$ forests of quasars would suggest that $\Omega_{m0}\approx 0.014$ and $\Omega_{v0}\approx 1.08$. Note that the value $\Omega_{v0}\approx 1.06$ solving the horizon problem in our model is close to the value of Hoell [*et al.*]{} Since the energy density parameters $\Omega_{r}, \Omega_{m}$ and $\Omega_v$ are not constant in time and the horizon problem is solved only for the value of $\Omega_{v0}$ belonging to some special interval, it follows that even if we now lived in the “isotropic era” (e.g. $\Omega_{m0}=0.3, \Omega_{v0}\approx 1.32$) it would not mean that the era would last forever. In order to see how the horizon problem evolves in time we evaluate the coordinate $\chi_R(z)$ for a hypothetical observer living not at the present epoch, but at the epoch determined by the redshift $z$. The coordinate reads $$\chi_R(z)=(\Omega_{r0}+\Omega_{m0}+\Omega_{v0}-1)^{1/2} \int_{z+1}^{z_R+1}\bigl[\Omega_{r0}x^4+\Omega_{m0}x^3 -(\Omega_{r0}+\Omega_{m0}+\Omega_{v0}-1)x^2+\Omega_{v0}\bigr]^{-1/2}dx,$$ If we calculate this integral for different values of $z$ starting with $z_R\approx 1200$ we realize that the horizon problem did not appear until $z\approx 540$ (for $\Omega_{m0}=0.3$). Nonexistence of the horizon problem just after the recombination epoch is not surprising of course, because the observer detects relic photons coming from his closest neighbourhood. The problem arises some time after the recombination when the observer starts to detect photons coming from more distant regions, the regions that could not ever be in thermal equilibrium. And this happens for $z\approx 540$, i.e. relatively soon after the recombination. In our model the recombination ($z_R\approx 1200$) takes place about $250\times 10^3\times h^{-1}$ years after the Big Bang and $z\approx 540$ corresponds to $880\times 10^3 \times h^{-1}$ years. In the long interval of time between $z\approx 540$ and $z\approx 0.05$ the horizon problem exists. It dissapears again about $500\times h^{-1}$ mln of years before the present epoch and will last for the next $500\times h^{-1}$ mln of years ($z\approx -0.04$). Before entering the “isotropic era” we would observe vanishing of fluctuations of the microwave background radiation first at the smallest angular scale and then due to the expansion fluctuations at larger scales would die out. While leaving the “isotropic era” first fluctuations at largest angular scales ($\theta\approx 180^\circ$) would come into existence. As we mentioned before effectiveness of the string-like fluid is lower than effectiveness of the $\Lambda$-term and larger value of $\Omega_{s0}$ is required in order to solve the horizon problem. In Table \[table2\] minimum and maximum values of $\Omega_{s0}$ (for which the horizon problem is solved), for different contents of nonrelativistic matter are presented [@stel94]. Note that the age of the universe in the string-like fluid dominated model is remarkably lower than in the vacuum dominated case. This might lead to the age-of-the-universe problem if it turned out that the Hubble constant is large. The discussion presented so far concerned two extreme cases of the fluid with negative pressure solving the horizon problem in a closed cosmological model. These cases bound the interval of values of the $\alpha_X$ – parameter allowing for the solution of the horizon problem, from below ($\alpha_v=-1$ – cosmological constant) and from above ($\alpha_s=-1/3$ – string-like matter). In Fig. \[fig5\] we present admissible values of “abnormal” fluid energy density parameter $\Omega_{X0}$ solving the horizon problem for various values of the $\alpha_X$ parameter. There are four pairs of lines corresponding to four values of the $\Omega_{m0}$ parameter. In each pair the lower line corresponds to the minimum value of $\Omega_{X0}$ parameter solving the horizon problem, while the upper line – to the maximum one. The left bound of the diagram ($\alpha_v=-1$) corresponds to the cosmological constant and the right bound ($\alpha_s=-1/3$) – to the string-like matter. Note that the interval of admissible values of $\Omega_{X0}$ grows with $\alpha_X$, i.e. it is narrow for the cosmological constant and relatively large for the string-like matter. Summary, constraints from observations and conclusions ====================================================== One of the possibilities of solving the horizon problem in the standard cosmological scenario (without inflation in the early universe) arises when we assume that the universe is closed. In such a model two relic photons reaching us from opposite directions could be in thermal equilibrium at the recombination epoch if since that time they travelled almost one-half of the circumference of the universe. Explicit calculation shows that in a matter-dominated model (without any form of “not normal” matter) with a last scattering surface of relic photons taking place at the recombination epoch ($z_R\approx 1200$) this is possible only in an extremaly dense universe, which of course cannot be accepted. However, the situation drastically changes if we admit existence of some form of matter influencing negative pressure (e.g. cosmological constant, textures, strings, etc.). Using some estimations it was shown by Davies [@dav87] that in a closed universe with global texture the horizon problem in fact does not exist. It was also shown [@stel94] that not only textures but any form of the so-called string-like matter (satisfying the equation of state $p_s=-\rho_s/3$) solves the horizon problem. In the present paper we showed that any fluid violating the point-wise strong energy condition, hence described by the equation of state satisfying the inequality $-\rho_X\leq p_X\leq -\rho_X/3$, is relevant for solving the horizon problem as well. Peculiar attention was paid to the lower bound of the interval (cosmological constant). For four values of the matter energy density parameter $\Omega_{m0}$ (0.015, 0.1, 0.2 and 0.3) and for the ratio $p_X/\rho_X$ from the interval $\langle -1, -1/3\rangle$ we calculated numerical values of the fluid energy density parameter $\Omega_{X0}$ necessary to solve the problem. The solution of the horizon problem in the model is not forever. It depends on the epoch of observations. For example the value $\Omega_{v0} \approx 1.32$ (for $\Omega_{m0}=0.3$) is necessary for observers living today. They would observe isotropic microwave background radiation for about $10^9\times h^{-1}$ years. If $\Omega_{v0}$ was larger the “isotropic era” would occur earlier; if smaller it would occur in future. So far we considered theoretical possibility of solving the horizon problem in a closed universe dominated by fluid with negative pressure not relating the values of $\Omega_{X0}$ and $\alpha_X$ to observations. Now we would like to discuss observational constraints on $\Omega_{X0}$ and $\alpha_X$. We remind that for our purposes the value of $\alpha_X$ must be less than $-1/3$, and the value of $\Omega_{X0}$ must be larger than 1. There are at least three various methods of measurements which in recent years are being applied for estimates of global curvature of the universe. These concern: high-redshift supernovae, gravitational lensing events and CMB. Especially the first two give strong evidence that the Universe is accelerating. The simplest way to explain this is assuming existence of some form of matter with negative pressure. A candidate which is being taken by many authors most seriously into account is cosmological constant. However, for our purposes this candidate is not the best one because gravitational lensing events in the Hubble Deep Field impose strong upper limit on $\Omega_{v0}$ which should be remarkably lower than $1$ [@cooray98]. The value of this limit is also confirmed by quasar statistics [@koch96]. Another candidate examined in this context few years ago was string-like matter [@dav87; @stel94], e.g. network of intercommuting cosmic strings, globally wound texture or decaying $\Lambda$-term. All of them are described by the equation of state $\alpha_s=-1/3$. Inspite of the fact that theoretically the string-like matter is relevant for solving the horizon problem, it is ruled out by high redshift supernovae observations which indicate that $\alpha_X<-0.6$ with 95 % confidence [@garn98]. Summarizing, if we want to solve the horizon problem with the aid of the fluid with negative pressure we must assume that the $\alpha_X$ parameter is more negative than $-0.6$ and less negative than $-1$. All values between them are so far consistent with observations. As regards upper constraints on $\Omega_{X0}$, evidences definitely ruling out the possibility $\Omega_{X0}>1$ (necessary for solving the horizon problem) are not known to us. Gravitational lensing events provide such constraints but only in the case of cosmological constant. Recent estimate of the position of a Doppler peak in the angular power spectrum of CMB fluctuations indicates that $0.2<\Omega_{\text{tot}}<1.5$ [@han98], what is consistent with the presented model provided $\Omega_{m0}$ is not too large (because $\Omega_{X0}=\Omega_{\text{tot}}-\Omega_{m0}$ must be larger than 1). There is a hope that new instruments exploring CMB such that VSA, MAP and Planck Surveyor satellite will provide more precise constraints on $\Omega_{\text{tot}}$ and $\Omega_{X0}$ and will show whether the presented scenario can be considered as a realistic physical model. I thank C. van de Bruck, B. Carr, W. Kundt, A. Liddle, W. Priester and R. Tavakol for valuable discussions. S. Weinberg, [*Gravitation and Cosmology*]{} (Wiley,New York, 1972). J. Stelmach, R. Byrka and M.P. Dabrowski, , 2434 (1990). A. H. Guth, , 347 (1981). M. White and D. Scott, , 415 (1996). R. L. 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[**294**]{}, L1 (1998). --------------- ------------------- ------------------- ------- $\Omega_{m0}$ $\Omega_{v0~min}$ $\Omega_{v0~max}$ $t_0$ 0.015 1.0567 1.0572 22.5 0.1 1.165 1.172 16.1 0.2 1.246 1.259 13.9 0.3 1.310 1.327 12.6 --------------- ------------------- ------------------- ------- : Admissible values of $\Omega_{v0}$ for which the horizon problem does not exist. $t_0$ is the age of the universe in units $10^9\times h^{-1}$ years. \[table1\] --------------- ------------------- ------------------- ------- $\Omega_{m0}$ $\Omega_{s0~min}$ $\Omega_{s0~max}$ $t_0$ 0.015 1.379 1.418 9.5 0.1 1.623 1.685 8.8 0.2 1.798 1.875 8.3 0.3 1.933 2.022 8.0 --------------- ------------------- ------------------- ------- : Admissible values of $\Omega_{s0}$ for which the horizon problem does not exist. $t_0$ is the age of the universe in units $10^9\times h^{-1}$ years. \[table2\] [^1]: E-mail: Jerzy\_Stelmach@univ.szczecin.pl

--- abstract: 'Motivated by the prevalence of high dimensional low sample size datasets in modern statistical applications, we propose a general nonparametric framework, Direction-Projection-Permutation (DiProPerm), for testing high dimensional hypotheses. The method is aimed at rigorous testing of whether lower dimensional visual differences are statistically significant. Theoretical analysis under the non-classical asymptotic regime of dimension going to infinity for fixed sample size reveals that certain natural variations of DiProPerm can have very different behaviors. An empirical power study both confirms the theoretical results and suggests DiProPerm is a powerful test in many settings. Finally DiProPerm is applied to a high dimensional gene expression dataset.' author: - Susan Wei - Chihoon Lee - Lindsay Wichers - Gen Li - 'J.S. Marron' bibliography: - 'master\_bibtex.bib' title: 'Direction-Projection-Permutation for High Dimensional Hypothesis Tests' --- Keywords: Behrens-Fisher problem; Distance Weighted Discrimination; Fisher’s Linear Discrimination; high dimensional hypothesis test; high dimensional low sample size; linear binary classification; Maximal Data Piling; permutation test; Support Vector Machine; two-sample problem. Introduction {#Introduction} ============ We propose a nonparametric procedure for testing high dimensional hypotheses that is especially practical in high dimensional low sample size (HDLSS) settings. HDLSS data sets arise in many modern applications of statistics, including genetics, chemometrics, and image analysis. An intuitive approach to looking for differences between two high dimensional distributions is by looking for differences between their one dimensional projections onto some appropriate direction. DiProPerm is a three-stepped procedure based on this idea. The procedure is as follows: 1. Direction — take the normal vector to the separating hyperplane of a binary linear classifier trained on the class labels. 2. Projection — project data from both samples onto this direction and calculate a univariate two-sample statistic. An illustration of this can be seen in the first panel of Figure \[introFig\]. 3. Permutation — assess the significance of this univariate statistic by a permutation test. Namely, (a) pool the two samples and permute the class labels; (b) take the normal vector to the binary linear classifier retrained on the permuted class labels; (c) project data onto this direction and re-calculate the univariate two-sample statistic. An illustration of this can be seen in the last three panels of Figure \[introFig\]. For a level $\alpha$ test, we reject the null if the original test statistic is among the $100\alpha\%$ largest of the permuted statistics. DiProPerm is not a single test but a general hypothesis testing framework. The number of combinations of direction and univariate statistic is large. We will focus on a select few in this paper but more options are discussed in detail in Section \[impl\] of the Supplement. In general, we are interested in testing the hypotheses: 1) equality of two distributions and 2) equality of means. That any DiProPerm test is an exact level $\alpha$ test for equality of distributions follows immediately from general permutation test theory. A perhaps surprising point is that for testing equality of means, validity does not hold for some natural versions of DiProPerm. In this paper we study the theoretical properties of two particular DiProPerm tests. We will show that one is valid for testing equality of means while the other is not. A Motivating Example {#ACE} -------------------- Lower dimensional projections in directions of interest are often used to understand structure in high dimensional data. One example is the directions found by applying Principal Component Analysis (PCA), see [@Jolliffe2002] for an excellent introduction, which yields directions maximizing variation. When there are two classes however, as in the case we are studying, additional insights come from directions based on binary linear classifiers, where a binary classification decision is based on the value of a linear combination of the data features. In very high dimensions many linear classifiers over-fit. Here is a simple example illustrating this. Draw two independent samples, each of size 50, from the 1000-variate standard Gaussian distribution. We use the Distance Weighted Discrimination (DWD) direction in step 1 of DiProPerm [@Marron2007]. DWD is a binary linear classifier similar to the Support Vector Machine (SVM) with certain advantages in high dimensions, see [@Cortes1995] for an introduction to SVM. The first panel of Figure \[introFig\] shows the one dimensional projection of the data onto the DWD direction trained on the original class labels. Colors are used to represent original class membership and are thus constant throughout the first three panels. The projections are jittered on the y-axis to allow easy visualization. A kernel density estimate of the projections is plotted in the background (solid black line). We see that the projections in the first panel of Figure \[introFig\] are very well separated despite the fact that the samples arise from the same underlying distribution. This clear over-fitting artifact common in HDLSS data is a strong motivation for DiProPerm. ![The data are standard 1000-variate Guassian. In the first panel, the DWD direction is trained on the original class labels, represented by colors (same in all panels). In the second and third panels, the DWD directions are trained on realizations of randomly permuted class labels, represented by symbols (different in each panel). The separation in the first panel is comparable to that in the second and third panels. One hundred permutation statistics resulting from a DiProPerm test are shown in the last panel which confirms the separation in the first panel is not significant.[]{data-label="introFig"}](ip10b-eps-converted-to){width=".9\textwidth"} \[introFigAnswer\] The middle two panels of Figure \[introFigAnswer\] show projections of the data onto re-trained DWD directions, each based on a realization of randomly permuted class labels. Symbols are used to represent permuted class labels and are thus different in the first three panels. We find the projections here to be well separated with respect to the symbols. Relative to the second and third panels, the original separation we observed in the first panel is quite unremarkable, suggesting that the two underlying distribution are not different. The last panel in Figure \[introFig\] confirms this observation. We perform a DiProPerm test with 100 permutations and display the statistic, chosen here to be the difference of sample means, calculated for each permutation. The vertical line is the original statistic calculated on the unprojected data. We see that based on the DiProPerm test, the null hypothesis of equal distributions should not be rejected. The Hypotheses -------------- Let $X_1,\ldots,X_{m}$ and $Y_1,\ldots, Y_{n}$ be independent random samples of $\mathbb{R}^d$-valued random vectors, $d \ge 1$ with distributions $F_1$ and $F_2$, respectively. We are interested in testing the null hypothesis of equality of distributions $$\label{strongNull} H_0: F_1=F_2 \quad \text{ versus } \quad H_1: F_1\ne F_2$$ Let $\mu(F)$ denote the mean of a distribution $F$. Another item of interest is to test the weaker null hypothesis of equality of means $$\label{weakNull} H_0: \mu(F_1) = \mu(F_2) \quad \text{ versus } \quad H_1: \mu(F_1) \ne \mu(F_2)$$ Note that the multivariate Behrens-Fisher problem concerns testing under normality. Overview -------- The outline for the paper is as follows. A review of related work is presented in Section \[relwork\]. In Section \[alg\], two DiProPerm tests are closely examined. HDLSS asymptotics are used to investigate the validity of these two tests for the weaker null hypothesis of equality of means in Section \[validity\]. In Section \[power\] we perform a Monte Carlo power study comparing DiProPerm to other methods. Finally in Section \[App\], DiProPerm is applied to a real microarray dataset. Related work {#relwork} ============ There is extensive literature on testing equality of distributions for two multivariate distributions under the classical setting of sample size larger than dimension. For the more challenging HDLSS setting, several methods have been developed and we discuss two of them here. First, there are nearest neighbor tests [@Bickel1983; @Henze1988; @Schilling1986] which are based on nearest neighbor coincidences - the number of neighbors around a data point that belong to the same sample. The null distribution of the test statistic can be derived parametrically using normal theory or nonparametrically using a permutation test. A more recent contribution to testing equality of distributions under HDLSS settings is Szekely and Rizzo’s nonparametric energy test [@Szekely2004]. The energy test statistic is based on the Euclidean distance between pairs of sample elements. Here significance is accessed through permutation testing. The nearest neighbor test and the energy test require calculation of all pairwise distances between sample elements. The computational complexity of both tests is independent of dimension, and is thus suitable for the HDLSS setting. In Section \[empStudyDist\] we perform an empirical power study comparing DiProPerm to the energy test. For testing equality of means for two multivariate distributions, the classical Hotelling $T^2$ test is often used in the setting of sample size larger than dimension. However, the Hotelling $T^2$ statistic is not computable in HDLSS situations because the covariance matrix is not of full rank. To address this issue, the methods in [@Bai1996],[@Chen2010], and [@Srivastava2008] replace the covariance matrix in the Hotelling $T^2$ statistic by a diagonalized version. Taking a different approach, the method proposed by Lopes et al. projects the high dimensional data onto a random subspace of low enough dimension so that the traditional Hotelling $T^2$ statistic may be used [@Lopes2011]. All of these tests have the disadvantage that equal covariances are assumed, which is not a restriction we place on DiProPerm. In Section \[empStudyMeans\] we perform an empirical power study comparing DiProPerm to the Random Projection test proposed in [@Lopes2011]. The Choice of The Univariate Statistic {#alg} ====================================== Here, we study the difference between two particular choices of the univariate statistic in Step 2 of DiProPerm. First, let the Mean Difference (MD) direction be the vector connecting the centroids of each sample. For simplicity, we will use this particular direction to compare two natural statistics of the projections: 1) the Mean Difference (MD) statistic — the difference of sample means, and 2) the two-sample t-statistic (t) — difference of sample means divided by $\{s_1/m + s_2/n\}^{1/2}$ where $s_1$ and $s_2$ are sample standard deviations of each class, sized $m$ and $n$ respectively. Henceforth we specify different DiProPerm tests by concatenating the direction name and two sample univariate statistic name. Following this convention, the DiProPerm test that uses the MD direction and the MD statistic will be referred to as the **MD-MD** test and the DiProPerm test that uses the MD direction and the two-sample t statistic as the **MD-t** test. We provide a toy example to contrast the difference between the MD and t statistic. We draw independent samples, each of size 50, where the first sample arises from the 1000-variate standard Gaussian distribution and the second the 1000-variate distribution with iid marginal $t(5)$ distributions. Note that the samples arise from *different* distributions that have the *same* means. Figure \[algProjs\] shows the one dimensional projection of the data onto various MD directions and the MD and t statistic applied to these projections. ![The first sample arises from a standard 1000-variate Gaussian distribution and the second sample arises from the 1000-variate distribution with iid $t(5)$ marginals. In the first panel, the MD direction is trained on the original class labels, represented by colors. In subsequent panels, the MD direction is trained on realizations of permuted class labels, represented by symbols. Note that the MD-MD statistic is similar across the first three panels while the MD-t statistic is much larger in the first panel. One thousand permutation MD-t statistics are shown in the last panel. The empirical p-value is small suggesting the test that uses MD-t would reject the null.[]{data-label="algProjs"}](ip11-eps-converted-to){width="\textwidth"} The lengths of the longer horizontal black bars represent the MD statistics while the lengths of the shorter horizontal bars represent the sample standard deviations of the projected data in each permuted group. The MD statistic and two-sample t statistic calculated on the projected data are displayed towards the top of each panel. We see that the t-statistic in the first panel is much higher than the permuted t-statistics in the second and third panels. On the other hand, the MD statistic is about the same between the original and permuted worlds. We confirm this is a systematic pattern by looking at 1000 permutations and calculating the MD-t statistic. The distribution of the permuted MD-t statistics can be seen in the last panel of Figure \[algProjs\]. We see that the original MD-t statistic, represented as a vertical line, is among the larger permutation statistics, leading us to reject the null hypothesis. The distribution of the MD-MD permutation statistics, not shown here, looks very similar to the last panel of Figure \[introFigAnswer\], where the original statistic is close to the middle of the permutation distribution. Thus under this setting the MD-t test rejects the null while the MD-MD does not. This apparent inconsistency is due to the fact that the MD and t statistics are actually testing different hypotheses. The former is testing the weak hypothesis of equality of means while the latter is testing the strong hypothesis of equality of distributions. In light of this, each test is correct in its decision. This phenomenon is studied in detail in the next section. Hypothesis Test Validity {#validity} ======================== In this section, we study the validity of the MD-MD and the MD-t for testing 1) equality of distributions and 2) equality of means. We work with the MD direction because it is most amenable to theoretical analysis. Future work will include other directions such as DWD, SVM, etc. High dimensional geometric representation of SVM and DWD described in [@Bolivar-Cime2013] could provide the basis for this endeavor. That both the MD-MD and the MD-t are exact tests for equality of distributions follows from standard theory on permutation tests. We will discuss how an exact level $\alpha$ test can be constructed by a permutation test. Let $N=m+n$ and write $ Z=(Z_1,\ldots,Z_N) = (X_1,\ldots,X_m,Y_1,\ldots,Y_n) $ for the pooled sample. Let $\{\pi(1),\ldots,\pi(N)\}$ be a permutation of $\{1,\ldots,N\}.$ Write $ Z_\pi=(Z_{\pi(1)},\ldots,Z_{\pi(N)}) $ for the permuted sample. Let $G_N$ denote the set of all permutations $\pi$ of $\{1,\ldots,N\}$. Then for any test statistic $V_{m,n}=V_{m,n}(Z_1,\ldots,Z_N)$, we can calculate $V_{m,n}(Z_{\pi(1)},\ldots,Z_{\pi(N)})$ for all $\pi \in G_{N}$. The test that rejects the null if the original statistic $V_{m,n}(Z_1,\ldots,Z_N)$ is larger than $(1-\alpha) 100 \%$ of the permuted statistics $V_{m,n}(Z_{\pi(1)},\ldots,Z_{\pi(N)})$ is an exact level $\alpha$ test. The exactness comes from the fact that the unconditional distribution and the permutation distribution of the statistic coincide under the null of equal distributions. It follows that the MD-MD test and the MD-t test, and any other DiProPerm test, are exact for testing equality of distributions. The matter of establishing validity for testing equality of means is not as straightforward on the other hand. In general, permutation tests cannot be expected to be valid for testing weaker hypotheses such as equality of means. For instance, if the covariances are *not* the same, we have to be very careful with our choice of direction and two-sample statistic. The signal in the covariances may confound our interpretation of tests that are sensitive to both the signal in the mean and the signal in the variances. This is consistent with our results which show that under normality and balanced sample sizes, the MD-MD remains valid for testing equality of means under heterogeneous covariances. On the other hand, the MD-t is invalid when the covariances are not the same. MD-MD {#valMDMD} ----- In this section, we establish that the MD-MD test is an exact test for equality of means under normality and balanced sample sizes. The MD-MD test statistic, $T_{m,n}(Z)$, is the mean of the projections of the $X$’s onto the unit vector in the direction of $\bar X - \bar Y$ minus the mean of the projections of the $Y$’s onto the unit vector in the direction of $\bar X - \bar Y$: $$\begin{aligned} T_{m,n}(Z) &= T_{m,n}(X_1,\ldots,X_m,Y_1,\ldots,Y_n) \\ &=\frac{1}{m} \sum_{i=1}^m X_i' \frac{(\bar X - \bar Y)}{||\bar X - \bar Y||} - \frac{1}{n} \sum_{j=1}^n Y_j' \frac{(\bar X - \bar Y)}{||\bar X - \bar Y||} \\ &= || \bar X - \bar Y || \label{MD MD}\end{aligned}$$ Let $X_1, \ldots, X_m$ be an iid sample from the d-variate Gaussian distribution $N(\mu_X, \Sigma_x)$ and $Y_1,\ldots,Y_n$ be an independent sample drawn iid from the d-variate Gaussian distribution $N(\mu_Y,\Sigma_y)$ where $\Sigma_X \ne \Sigma_Y$. If $m=n$ then the unconditional distribution and the permutation distribution of $T_{m,n}(Z)$ are equal under the null $\mu_X = \mu_Y$. \[MDMDthm\] Under $\mu_X=\mu_Y$, $\bar X - \bar Y$ is distributed as $$\begin{aligned} \label{norm1} N(0, {\Sigma_x }/{m} + {\Sigma_y }/{n}) \end{aligned}$$ and the permutation distribution of $\bar X - \bar Y$ is $$\label{norm2} \sum_{r=0}^{m} \frac{{ m \choose r} {n \choose r}}{{ N \choose m} } N\left (0, \frac{(m-r) \Sigma_x +r \Sigma_y }{m^2} +\frac{r \Sigma_x+(n-r) \Sigma_y }{n^2} \right)$$ If $m=n$, the expressions in and are the same, in which case the unconditional and permutation distribution of $T_{m,n}(Z)$ are also the same. MD-t {#valMDt} ---- The MD-t statistic, denoted by $U_{m,n}(Z)$, is the result of applying the unbalanced sample sizes, unequal variance two-sample t-test statistic (also known as Welch’s t-test [@Welch1947]) to the projections onto the MD direction. Let $a \cdot b$ denote the standard dot product between two vectors in $\mathbb R^d$. The sample variances of the projected data can be expressed as $$\begin{aligned} s_{\tilde X}^2 = \frac{1}{m-1} \sum_{i=1}^m [ (X_i - \bar X) \cdot (\bar X - \bar Y) ]^2 \end{aligned}$$ and $$\begin{aligned} s_{\tilde Y}^2 = \frac{1}{n-1} \sum_{i=1}^n [ (Y_i - \bar Y) \cdot (\bar X - \bar Y) ]^2 .\end{aligned}$$ Define $S_{m,n}(Z) = S_{m,n}(X_1,\ldots,X_m,Y_1,\ldots,Y_n) = { {s_{\tilde{X}}^2}/{m} + {s_{\tilde{Y}}^2}/{n} }$. The MD-t statistic is $$U_{m,n}(Z) = U_{m,n}(X_1,\ldots,X_m,Y_1,\ldots,Y_n) = {T_{m,n}(Z)^2}/{ \{ S_{m,n}(Z) \}^{1/2} }$$ where $T_{m,n}(Z)$ is as in Section \[valMDMD\]. We use the term “projected" rather loosely here since we have not normalized by $|| \bar X - \bar Y ||$. This is of no actual consequence since the two-sample t-statistic is scale invariant. Under equal means the numerator in the MD-t statistic behaves similarly in the permutation world and the original world. However, we will see that the denominator of the MD-t statistic has very different behavior. We find that the denominator of the MD-t is larger in the permuted world, as seen in Figure \[algProjs\]. This has the effect of making the unconditional distribution of the MD-t statistic larger than the permutation distribution. To gain some intuition, consider the following toy HDLSS example. Suppose we observe $X_1,X_2 \sim F_1$ and $Y_1,Y_2 \sim F_2$ where $F_1 = N(0,I_d)$ and $F_2 = N(0, \sigma^2 I_d)$, $\sigma^2 \ne 1$. The points $X_1,X_2,Y_1,Y_2$ form the vertices of a tetrahedron in three dimensional space. The two-dimensional plane generated by $Y_1, Y_2$ and $\bar X$ is shown in Figure \[triangle\]. Distances between elements of interest are calculated using standard HDLSS asymptotics, see [@Hall2005] for examples of this type of calculation. All distances have an additional $O_P(1)$ term that is not shown to avoid clutter. The geometric configuration in Figure \[triangle\] has the implication that $s_{\tilde Y}^2$ is small. To see this, note the projections of $Y_1$ and $Y_2$ onto the MD direction $\bar X-\bar Y$ is close to the projection of $\bar Y$ itself. A similar argument can be applied to show $s_{\tilde X}^2$ is small. ![ Plane generated by $Y_1$, $Y_2$ and $\bar X$ where $X_1,X_2 \sim F_1 = N(0,I_d)$ and $Y_1, Y_2 \sim F_2 = N(0, \sigma^2 I_d)$ for $\sigma^2 \ne 1$. Note that the projections of $Y_1$ and $Y_2$ onto $\bar X - \bar Y$ is close to the projection of $\bar Y$ onto $\bar X - \bar Y$. This has the implication that $s_{\tilde{Y}}^2$ will be small.[]{data-label="triangle"}](diproperm_unblinded-1-eps-converted-to){width=".9\textwidth"} Now let’s look at what happens in the permutation world. Figure \[permTriangle\] shows the two-dimensional plane generated by the realization of a random permutation where $X_1^*=X_2$, $X_2^* = Y_2$ and $Y_1^* = X_1$ and $Y_2^* = Y_1$. Notice that the distance between $Y_1^* $ and $\bar{ X^*}$ is different than the distance between $Y_2^*$ and $\bar{ X^*}$. This has the effect of making $s_{\tilde Y^*}^2$, the sample variance of the the projections of $Y_1^*$ and $Y_2^*$, large. To see this, note the projections of $Y_1^*$ and $Y_2^*$ onto the permuted MD direction are not close to the projection of $\bar Y^*$. A similar argument can be applied to show $s_{\tilde X^*}^2$, the sample variance of the projections of $X_1^*$ and $X_2^*$, is large. The derivations for the distances shown in Figures \[triangle\] and \[permTriangle\] can be found in the supplement. ![Plane generated by a particular permutation realization of $X_1,X_2,Y_1,$ and $Y_2$. Note that the projections of $Y_1^*$ and $Y_2^*$ onto $\bar X^* - \bar Y^*$ is not close to the projection of $\bar Y^*$ onto $\bar X^* - \bar Y^*$. This has the implication that $s_{\tilde Y^*}^2$ may be large.[]{data-label="permTriangle"}](diproperm_unblinded-2-eps-converted-to){width=".9\textwidth"} The toy example above suggests the denominator of the MD-t statistic is larger in the permutation world than in the original world. The next result gives us a sense of just how far apart are the permutation and unconditional distributions of $S_{m,n}(Z)$. Let $X_1, \ldots, X_m$ be a sample from the d-variate Gaussian distribution $N(\mu_x, \sigma_x^2 I_d)$ and $Y_1,\ldots,Y_n$ be an independent sample from the d-variate Gaussian distribution $N(\mu_y,\sigma_y^2 I_d)$ where $\sigma_x^2 \ne \sigma_y^2$ are scalars. Under $\mu_x = \mu_y$, we have $$\frac{1}{d} S_{m,n}(Z) \convd (\frac{\sigma_x^2}{m} + \frac{\sigma_y^2}{n}) \left \{\frac{1}{m-1} \frac{\sigma_x^2}{m} \chi^2(m-1) + \frac{1}{n-1} \frac{\sigma_y^2}{n} \chi^2(n-1) \right \}$$ as $d$ goes to infinity. For the permuted version, we have for some non-zero constant $c$, $$\frac{1}{d^2} S_{m,n}(Z_\pi) \to c \text{ in probability.}$$ \[MDtTheorem\] The results of this theorem are surprising in that the denominator of the MD-t statistic is actually of different orders in the unconditional and permutation worlds. In particular, in the unconditional world $S_{m,n}(Z)$ grows like a random variable times $d$, while in the permutation world it grows like a constant times $d^2$. Let us revisit the toy example earlier and see what Theorem \[MDtTheorem\] can tell us. We make $50$ draws from $F_1 = N(0,I_d)$ and another $50$ independent draws from $F_2 = N(0,100 I_d)$. We show in Figure \[permVsOriginal\], using 1000 Monte Carlo realizations, the simulated permutation and unconditional distributions of the MD-t statistic for various dimensions. ![The unconditional and permutation distribution of the MD-t statistic for the distributions $F_1 = N(0,I_d)$ and $F_2 = N(0,100 I_d)$. The separation between the unconditional and permutation distribution increases with dimension.[]{data-label="permVsOriginal"}](permVsOriginal-eps-converted-to){width="\textwidth"} Under the conditions in Theorem \[MDtTheorem\], when $\mu_x = \mu_y$, the numerator of the MD-t statistic is proportional to a $\chi^2(d)$ variable for both the unconditional and permutation distribution. On the other hand, by the results in Theorem \[MDtTheorem\] $S_{m,n}(Z)$ is of the order $\sqrt d$ and $d$ for the unconditional and permuted distributions, respectively. Thus we should expect the MD-t statistic to be of the order $\sqrt d$ in the original unconditional world and $1$ in the permutation world. This is consistent with Figure \[permVsOriginal\] — the unconditional distribution is centered around $\sqrt d$ while the permutation distribution is not growing with $d$. As Figure \[permVsOriginal\] illustrates, the unconditional distribution quickly separates from the permutation distribution as dimension increases. Thus it is very important that the MD-t statistic not be used when the goal is to test for equality of means. On the other hand, this shows the MD-t test has some power for testing equality of distributions against equal means alternatives. Power surfaces {#powersufs} -------------- In this section, we study the power of the MD-MD and MD-t for testing equality of means. In the simulations that follow, we make $m$ draws from $F_1=N(\mu_1,\sigma_1^2 I_d)$, and $n$ independent draws from $F_2=N(0,I_d)$. We set $d=500$ and $m=n=50$ for balanced sample sizes and $m=50,n=100$ for unbalanced. The dimension $d$ and sample sizes $m$ and $n$ are chosen to reflect a HDLSS setting. The significance level is set at $\alpha = 0.05$. Power is estimated using $1000$ Monte Carlo simulations. Figure \[powerMdt\] displays a 3D surface of power versus $\mu_1$ versus $\sigma_1^2$, using a color spectrum from cool to warm corresponding to the range 0 to 1. We also show an image underneath the surface where each pixel corresponds to the point in the 3D surface above. Figure \[equalPowerMDMD\] displays the estimated power surface of MD-MD under balanced sample sizes. By Theorem \[MDMDthm\], MD-MD is an exact test for equality of means under balanced sample sizes and normality. This is consistent with what we see in Figure \[equalPowerMDMD\] — when the means are equal (i.e. $\mu_1=0$), the power is around $\alpha=0.05$, as indicated by the streak at $\mu_1=0$. When sample sizes are unbalanced, see Figure \[equalPower\] in Section \[MDsMD\] of the Supplement, the MD-MD is no longer an exact test and may not even be asymptotically valid as $d \to \infty$. In Section \[MDsMD\] of the Supplement, we propose a modification of MD-MD that should be used when sample sizes are unbalanced. Figures \[equalPowerMDt\] and \[unequalPowerMDt\] show that under heterogeneous covariances (when $\sigma_1^2 \ne 1$), the MD-t test of equal means does not attain the correct level for either balanced or unbalanced sample sizes. In the immediate region around $(\mu_1,\sigma_1^2)=(0,1)$, the power of the MD-t test is close to $\alpha$ as expected. However as we move away from $(\mu_1,\sigma_1^2)=(0,1)$, the power quickly increases. Thus if we use the MD-t test for equality of means, we will reject too often. On the other hand this shows that the MD-t test has some power for testing equality of distributions against alternatives where the means are equal but the distributions are not. Comparison with Other Methods {#power} ============================= In this section we compare DiProPerm to other methods in the simulation contexts described in Table \[databank1\]. First, for testing equality of distributions, we compare the DiProPerm tests DWD-t and MD-t to the energy test proposed by Szekely and Rizzo [@Szekely2004]. Next, for testing equality of means, we compare the DiProPerm tests DWD-MD and MD-MD to the Random Projection test proposed by Lopes, Jacob and Wainwright [@Lopes2011]. Our simulation results show that no test is universally most powerful. As such, our goal is to learn general lessons about the situations under which each method can be expected to do well. Simulation Sample 1 Sample 2 ------------ ---------------------------------------------------------- ------------------------------------------------------------ S1 $N(0,I_{d})$ $t(5)^{d}$ S2 $N(0,\Sigma_B)$ $N(\mu,\Sigma_B)$ S3 $N( [3,30,0,\ldots,0],I_d)$ $N( [3,-30,0,\ldots,0],I_d)$ $N( [-3,30,0,\ldots,0],I_d)$ $N( [-3,-30,0,\ldots,0],I_d)$ : Simulation settings. The notation $N(\mu,\Sigma)$ denotes a multivariate Gaussian distribution with mean $\mu$ and covariance $\Sigma$. In S1, the notation $t(5)^d$ denotes the $d$-variate distribution with iid marginal distribution $t(5)$. In S2, the first 25% of the coordinates in $\mu$ are zero and the rest are set to $1/\sqrt{n}$. The covariance matrix $\Sigma_B$ has a block structure (described further in the text). In S3, each distribution is an equally weighted Gaussian Mixture of the components listed.[]{data-label="databank2"} \[databank1\] Simulation S1 in Table \[databank1\] was taken from Szekely and Rizzo. Simulation S2 is a modification of a simulation found in Lopes, Jacob, and Wainwright. Following their simulation setting, we let the covariance matrix $\Sigma_B$ be block-diagonal with identical blocks $B \in R^{5 \times 5}$ along the diagonal. The matrix $B$ has diagonal entries equal to 1 and off-diagonal entries equal to $0.2$. The mean vector is set to the zero vector in sample 1. In the second sample, the mean vector is set to zero in the first 25% of the coordinates and the rest is set to $1/\sqrt n$. Simulation S3 looks at data arising from equally weighted Gaussian mixtures with the components listed in Table \[databank1\]. All DiProPerm tests are implemented using $1000$ permutations. Power is estimated through 1000 Monte Carlo simulations at $0.1$ significance level. In Figures \[testEqualDist\] and \[EqualMeans1\] we display the power against a range of dimensions. Equality of distributions {#empStudyDist} ------------------------- The energy statistic is based on the Euclidean distance between pairs of sample elements. The two-sample test statistic is $$\epsilon_{m,n} = \frac{mn}{N} \left( \frac{2}{mn} \sum_{i=1}^m \sum_{j=1}^n ||X_i - Y_j|| - \frac{1}{m^2} \sum_{i=1}^m \sum_{j=1}^m ||X_i - X_j|| - \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n ||Y_i - Y_j|| \right)$$ The first term measures the average distance between the samples and the last two terms measure the average distance within each sample. The significance of the energy test statistic is assessed using a permutation test. In our implementation of the energy test, we used $1000$ permutations. For all simulations in Figure \[testEqualDist\], the sample sizes are set to be unbalanced: $m=50, n=150$. Figure \[testEqualDist\] compares the power of MD-t, DWD-t, and the energy test for testing equality of distributions. The first panel shows the result of simulation S1. The standard Gaussian and $t(5)^d$ both have mean zero but different covariances. Note that the signal in the covariance grows stronger with dimension. In light of this, it is not surprising that the MD-t and DWD-t do not perform as well as the energy test which is more attuned to variance effects. However, as the dimension increases all three tests attain full power. The second panel of Figure \[testEqualDist\] shows the results for simulation S2. All three tests perform well with power increasing to 1 with dimension. Note that the mean effect is along the 45 degree line. The structure of $\Sigma_B$ has the implication that the directions with highest variation are for some constant $c$, $(c,c,c,c,c,0,\ldots,0)$, $(0,0,0,0,0,c,c,c,c,c,0,\ldots,0)$, and etc. Thus the mean effect is further exaggerated by the covariance structure making this a rather unchallenging setting for all three methods. The result of simulation S3 is shown in the last panel of Figure \[testEqualDist\]. Here, both the DiProPerm DWD-t and MD-t test are seen to be more powerful than the energy test. This is not surprising since by way of its construction, the energy test can be expected to have difficulty in separating Gaussian mixture data types. The MD-t has good performance but DWD-t has the best power because DWD was developed to handle Gaussian mixture data types. Equality of means {#empStudyMeans} ----------------- In the RP test proposed by Lopes, Jacob and Wainwright, the data is first projected down to a dimension low enough so that the regular Hotelling $T^2$ statistic may be applied [@Lopes2011]. The projection matrix is a $k \times d$ matrix with iid $N(0,1)$ entries where $k$ is the dimension of the lower dimensional subspace. In our implementation of the RP method, we follow the authors’ recommendation and set the tuning parameter $k = \lfloor{n/2}\rfloor$. The samples are assumed to arise from Gaussian distributions with equal covariances. The resulting statistic then follows an $F$ distribution under the null of equal means. For all simulations in Figure \[EqualMeans1\], the sample sizes are set to be balanced: $m=50, n=50$. The standard multivariate Gaussian and the multivariate $t(5)^d$ both have mean zero, and thus the power of MD-MD and RP should be around $\alpha =0.1$ in simulation S1. The first panel of Figure \[EqualMeans1\] shows this is indeed the case. Note that if MD-MD or RP were to be used for testing equality of distributions, neither would have power against alternatives such as in S1. In simulation S2, the RP method does not perform as well as MD-MD or DWD-MD. This is perhaps due to the DiProPerm tests being able to pick up the mean effect more efficiently than the RP method which tries to sense random directions in very high dimensions. Note that simulation S2 is a setting in which the MD statistic is powerful for either direction DWD or MD as the mean effect is strong. Re-examining Figure \[testEqualDist\], we see that the DWD-MD is more powerful than the DWD-t and the MD-MD more powerful than the MD-t for simulation S2. Recall that the covariance structure in S2 amplifies the mean effect. The DiProPerm tests that use the two-sample t-statistic may have lower power than their MD counterpart because the standardization in the t-statistic cancels out some of the effect. In the Gaussian mixture S3 simulation, the DWD-MD and the RP test are both substantially more powerful than the MD-MD test. In this setting, the direction of discrimination is in the first coordinate direction but the direction of most variation is along the second coordinate. Not surprisingly, MD-MD has trouble in this setting. The RP test, which uses the Mahalanobis distance, is able to correct for this false signal in the second coordinate direction. DWD-MD is seen to perform slightly better than the RP test. Again, DWD is designed to work well in discriminating Gaussian mixture data types and this result matches our expectation. \[EqualMeans2\] Application: Microarray data analysis {#App} ====================================== The first application of DiProPerm to a real dataset can be found in [@Wichers2007]. DiProPerm was applied to an HDLSS dataset and used to find a statistically significant difference between heart rates of rats among different treatment groups. In this section we will apply DiProPerm to a different kind of HDLSS data — gene expression microarray data. Two HDLSS datasets are examined. The first dataset is denoted UNCGEO and the second UNCUP, following the naming convention of their source which can be found at `http://peroulab.med.unc.edu/`. The UNCGEO datasets consists of gene expression data of 9674 genes measured on 50 breast cancer patients at UNC. The UNCUP dataset looks at the same set of genes measured on 80 breast cancer patients in another study at UNC. We performed many different hypotheses of interest within each dataset. We highlight two particular comparisons here which highlight the main point that formal hypothesis testing is an important component of visualization in high dimensions. The UNCGEO patients are divided into standard breast cancer subtypes: 1) Luminal A versus 2) Luminal B and the UNCUP data into the groups: 1) Luminals (Luminal A and Luminal B) versus 2) HER and Basal. Luminals have a very different gene expression signature from HER and Basal. On the other hand, the difference between Luminal A and Luminal B is less clear cut. For each dataset, we use DWD-t to test equality of distributions between the gene expression in group 1 and group 2. Note that we have a HDLSS setting here since the number of genes well exceeds the sample sizes in each subgroup. Figure \[projMicroarray\] shows the data projected onto DWD directions. The projections in the left panel do not overlap at all whereas the projections in the right panel have a small amount of overlap. These projection plots suggest that the separation is better for Luminal A vs. Luminal B in the UNCGEO dataset than for Luminals vs. HER & Basal in the UNCUP dataset. However as previously seen in the toy example in Section \[ACE\], great care is needed before drawing conclusions of this type. ![ One dimensional projection plots onto DWD directions for the UNCGEO dataset and the UNCUP dataset. The separation in the projection plot for the UNCGEO dataset is more visually pronounced than in the UNCUP dataset. We will rigorously assess this visual result using DiProPerm.[]{data-label="projMicroarray"}](ip12-eps-converted-to){width=".75\textwidth"} Figure \[permMicroarray\] displays the DiProPerm test results. Each dot represents the test statistic resulting from a single permutation in the permutation test. We mark the position of the original univariate t-statistic with a vertical dashed line. The empirical p-values show the difference in the UNCGEO dataset is not significant while the difference in the UNCUP dataset is very significant. (We also display the Guasisan fit p-value and Gaussian fit z-score, two other types of “p-values" described in Section \[Perm\] of the Supplement). This result on a real world dataset parallels what we saw on the simulated toy dataset in Section \[ACE\] — what may seem to be a visually striking separation in lower dimensional visualizations could well be an artifact of over-fitting or sampling variation. ![DWD-t test result for the UNCGEO (left) and UNCUP (right) datasets. In the UNCGEO study (left), the difference between the Luminal A and Luminal B subgroups is not significant. In the UNCUP study (right), the difference between the Luminals and HER & Basal subgroups is very significant. This is surprising because the projection plots in Figure \[projMicroarray\] suggest the contrary.[]{data-label="permMicroarray"}](ip13-eps-converted-to){width=".75\textwidth"} Matlab Software =============== Matlab software for DiProPerm is available at `http://www.unc.edu/~marron/marron_software.html`. Acknowledgements {#acknowledgements .unnumbered} ================ The work presented in this paper was supported in part by the NSF Graduate Fellowship, and NIH grant T32 GM067553-05S1. Proofs ====== Let $X_1, \ldots, X_m$ be a sample from the d-variate Gaussian distribution $N(\mu_x, \sigma_x^2 I_d)$ and $Y_1,\ldots,Y_n$ be an independent sample from the d-variate Gaussian distribution $N(\mu_y,\sigma_y^2 I_d)$ where $\sigma_x^2 \ne \sigma_y^2$. Let $\tilde X_k = X_k' (\bar X - \bar Y)$. Let $\overline{\tilde X_{1:k-1}}$ be the sample mean of $\tilde X_1,\ldots \tilde X_{k-1}$. Under $\mu_x = \mu_y$, we have, for $k=2,\ldots, m$ $$\frac{d^{-1/2} ( (\tilde{X}_{k} - \overline{\tilde X_{1:k-1}}) )}{\{\frac{k}{k-1} \sigma_x^2 (\sigma_x^2/m + \sigma_y^2/n)\}^{1/2}} \convd N(0,1) \text{ as } d \to \infty.$$ Similarly we have $$\frac{d^{-1/2} ( (\tilde{Y}_{k} - \overline{\tilde Y_{1:k-1}}) )}{\{\frac{k}{k-1} \sigma_y^2 (\sigma_x^2/m + \sigma_y^2/n)\}^{1/2}} \convd N(0,1) \text{ as } d \to \infty$$ $k=2,\ldots, n$. \[projDiff\] We can write $\tilde{X}_{k} - \overline{\tilde X_{1:k-1}} $ as a sum of products $$\begin{aligned} \tilde{X}_{k} - \overline{\tilde X_{1:k-1}} = \sum_{p=1}^d (X_k - \bar X_{1:k-1})^{(p)} (\bar X - \bar Y)^{(p)} \label{componentDef}\end{aligned}$$ where $X^{(p)}$ simply refers to the $p$-th component in the $d$-dimensional vector $X$. The expectation of the summands in is zero: $$\begin{aligned} E (X_k - \bar X_{1:k-1})^{(p)} (\bar X - \bar Y)^{(p)} &= E (X_k^{(p)} \bar X^{(p)}) - E(\bar X_{1:k-1}^{(p)} \bar X^{(p)}) \\ &- E(X_k^{(p)} \bar Y^{(p)}) + E(\bar X_{1:k-1}^{(p)} \bar Y^{(p)}) \\ &=0\end{aligned}$$ Next we look at the variance of the summands. Recall for Gaussian data, zero covariance is equivalent to independence. We know the covariance between $(X_k - \bar X_{1:k-1})^{(p)}$ and $(\bar X - \bar Y)^{(p)}$ is zero since the expectation of the latter is zero and the expectation of the product was shown above to be zero as well. Thus each summand in is the product of two independent variables. The variance of a product of independent variables (see \[varProd\] for a derivation), $U$ and $V$, is $$(EU)^2 Var(V) + (EV)^2 Var(U) + Var(U) Var(V). \label{prodVar}$$ Thus we have $$\begin{aligned} Var (X_k - \bar X_{1:k-1})^{(p)} (\bar X - \bar Y)^{(p)} &= Var (X_k - \bar X_{1:k-1})^{(p)}Var (\bar X - \bar Y)^{(p)} \\ &= \frac{k}{k-1} \sigma_x^2 (\sigma_x^2/m + \sigma_y^2/n)\end{aligned}$$ By the Central Limit Theorem, we have $$\frac{d^{1/2} ( \frac{1}{d} (\tilde{X}_{k} - \overline{\tilde X_{1:k-1}}) )}{\{\frac{k}{k-1} \sigma_x^2 (\sigma_x^2/m + \sigma_y^2/n)\}^{1/2}} \convd N(0,1) \text{ as } d \to \infty$$ Let $X_1, \ldots, X_m$ be a sample from the d-variate Gaussian distribution $N(\mu_x, \sigma_x^2 I_d)$ and $Y_1,\ldots,Y_n$ be an independent sample from the d-variate Gaussian distribution $N(\mu_y,\sigma_y^2 I_d)$ where $\sigma_x^2 \ne \sigma_y^2$. Let $\pi$ be a permutation of $\{1,\ldots,N=m+n\}$. Let $\bar Z_\pi = ( \bar Z_{\pi(1:m)} - \bar Z_{\pi(m+1:N)} )$ be the MD direction trained on the permuted labels determined by $\pi$. We have for $i=1,\ldots,m$, $$E(({Z}_{\pi(i)} - \overline{Z_{\pi(1:m)}})^{(k)} \bar Z_\pi^{(k)})$$ is non-zero. Similarly, for $i=m+1,\ldots,N$, we have $$E( ({Z}_{\pi(i)} - \overline{Z_{\pi(m+1:N)}})^{(k)} \bar Z_\pi^{(k)})$$ is non-zero. \[nonzero\] We prove the first statement. The second can be shown in a similar fashion. Let $P(n,k)$ denote the number of $k$ permutations of $n$, i.e. $$P(n,k) = n \cdot (n-1) \cdot (n-2) \cdots (n-k+1)$$ We have for $i=1,\ldots,m$ and $k=1,\ldots,d$, $$\begin{aligned} E(({Z}_{\pi(i)} - \overline{Z_{\pi(1:m)}})^{(k)} \bar Z_\pi^{(k)}) &= E ((Z_{\pi(i)} - \bar Z_{\pi(1:m)})^{(k)} ( \bar Z_{\pi(1:m)} - \bar Z_{\pi(m+1:N)} )^{(k)}) \\ &= EZ_{\pi(i)}^{(k)} \bar Z_{\pi(1:m)}^{(k)} - EZ_{\pi(i)}^{(k)} \bar Z_{\pi(m+1:N)}^{(k)} - E( \bar Z_{\pi(1:m)}^{(k)} )^2 + E \bar Z_{\pi(1:m)}^{(k)} \bar Z_{\pi(m+1:N)}^{(k)} \\ &= \frac{(EZ_{\pi(i)}^{(k)})^2}{m} +\frac{m}{m-1} \mu^2 - \mu^2 - E( \bar Z_{\pi(1:m)}^{(k)} )^2 + \mu^2\\ &= \frac{ var(Z_{\pi(i)}^{(k)}) + \mu^2}{m} + \frac{m}{m-1} \mu^2 - ( var (\bar Z_{\pi(1:m)}^{(k)} ) + \mu^2) \\ &= \frac{ var(Z_{\pi(i)}^{(k)}) }{m} - var (\bar Z_{\pi(1:m)}^{(k)} ) \\ &=\frac{m}{N} \{ \frac{\sigma_x^2}{m} - \frac{1}{m^2} \frac{1}{w_1} \sum_{r=0}^{m-1} P(m-1,r) P(n,m-r) [r \sigma_x^2 + (m-r)\sigma_y^2] \} \\ &+\frac{n}{N} \{ \frac{\sigma_y^2}{m} - \frac{1}{m^2} \frac{1}{w_2} \sum_{r=0}^{n-1} P(n-1,r) P(m,m-r) [r \sigma_y^2 + (m-r)\sigma_x^2] \} \end{aligned}$$ where $w_1$ and $w_2$ are the weights $$w_1 := \sum_{r=0}^{m-1} P(m-1,r) P(n,m-r) \quad \text{ and } \quad w_2:= \sum_{r=0}^{n-1} P(n-1,r) P(m,m-r)$$ Thus if $\sigma_x^2 \ne \sigma_y^2$, we have $E(({Z}_{\pi(i)} - \overline{Z_{\pi(1:m)}})^{(k)} \bar Z_\pi^{(k)})$ is nonzero. Let $Z_1,Z_2$ be two random variables in $\mathbb R^d$ such that $Z_1^{(k)}Z_2^{(k)}$ are i.i.d. for $k=1,\ldots,d$ and $E(Z_1^{(k)}Z_2^{(k)})$ exists and is finite. Then $$\frac{1}{d^2} ( Z_1 \cdot Z_2 )^2 \to [E(Z_1^{(k)}Z_2^{(k)})]^2 \text{ in probability}$$ \[dot\] By the Law of Large Numbers, we have $$\frac{ 1}{d} (Z_1 \cdot Z_2) \to E(Z_1^{(k)}Z_2^{(k)}) \text{ in probability.}$$ By Continuous Mapping Theorem, we have $$\frac{ 1}{d^2} (Z_1 \cdot Z_2)^2 \to [E(Z_1^{(k)}Z_2^{(k)})]^2 \text{ in probability.}$$ Now we have all the necessary ingredients to prove Theorem \[MDtTheorem\] in Section \[valMDt\]. To prove the first part of Theorem \[MDtTheorem\], we decompose $s_{\tilde X}^2$ and $s_{\tilde Y}^2$ into a sum of independent variables. Let $\overline{\tilde{X}}_{k-1}$ be the sample mean of the first $k-1$ projections $\tilde X_1,\ldots \tilde X_{k-1}$. We will write $s_{\tilde X}^2$ in a recursive fashion. Define $s_1^2 = 0$. We will use the following recursive formula to define $s_k^2$ for $k = 2,\ldots,m$ $$(k-1) s_k^2 = (k-2) s_{k-1}^2 + \frac{k-1}{k} ( \tilde{X}_k - \overline{\tilde{X}}_{k-1})^2 \label{recur}$$ Since $s_{k-1}^2$ is independent of $( \tilde{X}_k - \overline{\tilde{X}}_{k-1})^2$, this recursive viewpoint allows us to decompose $s_{\tilde X}^2 = s_m^2$ into a sum of independent terms. Using the result in Lemma \[projDiff\] and the second-order Delta method, we have $$\frac{ \frac{1}{d} (\tilde{X}_{k} - \overline{\tilde X_{1:k-1}})^2 }{\frac{k}{k-1} \sigma_x^2 (\sigma_x^2/m + \sigma_y^2/n)} \convd \chi^2(1) \text{ as } d \to \infty \label{inc}$$ Inputting expression into the recursion defined in and exploiting the independence of the individual terms in $s_{\tilde{X}}^2$, we get $$\frac{1}{d} s_{\tilde{X}}^2 \convd \frac{1}{m-1} \sigma_x^2 (\frac{\sigma_x^2}{m} + \frac{\sigma_y^2}{n}) \chi^2(m-1) \text{ as } d \to \infty$$ Similarly, we can show for the sample of projections $\tilde Y_1,\ldots, \tilde Y_n$, $$\frac{1}{d} s_{\tilde{Y}}^2 \convd \frac{1}{n-1} \sigma_y^2 (\frac{\sigma_x^2}{m} + \frac{\sigma_y^2}{n}) \chi^2(n-1) \text{ as } d \to \infty$$ Thus we have $$\begin{aligned} \frac{1}{d} S_{m,n}(Z) &= \frac{1}{d} \left( \frac{s_{\tilde{X}}^2 }{m} + \frac{s_{\tilde{Y}}^2 }{n} \right) \\ & \convd \frac{1}{m-1} \frac{\sigma_x^2}{m} (\frac{\sigma_x^2}{m} + \frac{\sigma_y^2}{n}) \chi^2(m-1) + \frac{1}{n-1} \frac{\sigma_y^2}{n} (\frac{\sigma_x^2}{m} + \frac{\sigma_y^2}{n}) \chi^2(n-1)\end{aligned}$$ For the second part in Theorem \[MDtTheorem\], we expand the sample variance of the projected values in the permuted group as follows: $$\begin{aligned} s_{{\tilde Z_{\pi(1:m)}}}^2 &= \frac{1}{m-1} \sum_{i=1}^m (\tilde{Z}_{\pi(i)} - \overline{\tilde Z_{\pi(1:m)}})^2 \\ &= \frac{1}{m-1} \sum_{i=1}^m ( ({Z}_{\pi(i)} - \overline{Z_{\pi(1:m)}}) \cdot ( \bar Z_{\pi(1:m)} - \bar Z_{\pi(m+1:N)} ) )^2 \\\end{aligned}$$ Lemma \[nonzero\] shows $E (Z_{\pi(i)} - \bar Z_{\pi(1:m)})^{(k)} ( \bar Z_{\pi(1:m)} - \bar Z_{\pi(m+1:N)} )^{(k)}$ is nonzero. Now apply Lemma \[dot\] with $Z_1 = ({Z}_{\pi(i)} - \overline{Z_{\pi(1:m)}}) $ and $Z_2 = ( \bar Z_{\pi(1:m)} - \bar Z_{\pi(m+1:N)} )$ to see that $\frac{1}{d^2} s_{{\tilde Z_{\pi(1:m)}}}^2 $ converges in probability to a nonzero constant. A similar argument can be applied to $s_{{\tilde Z_{\pi(m+1:N)}}}^2$. Combining these results, it immediately follows that $\frac{1}{d^2} S_{m,n}(Z_\pi)$ converges in probability to a nonzero constant. MD-scaled MD {#MDsMD} ============ We established in Section \[valMDMD\] that under certain conditions, the MD-MD test is valid when sample sizes are balanced. Under these same conditions, MD-MD is no longer a valid test however when sample sizes are unbalanced. Here we propose a modification of MD-MD, called MD-scaled MD, that is asymptotically valid, as $m,n \to \infty$ for fixed $d$, for equality of means when covariances are unequal and sample sizes are unbalanced. We have chosen the classical asymptotic regime here to take advantage of the following results. Janssen proved the permutation test for equality of means based on the studentized statistic, $${ m^{1/2} (\bar X - \bar Y)}/{\{ s_x^2 + \frac{m}{n} s_y^2\}^{1/2} } \label{MDstud}$$ where $s_x^2$ and $s_y^2$ are the standard unbiased estimators of $\sigma_x^2$ and $\sigma_y^2$, is asymptotically valid as $m,n \to \infty$ for the univariate case [@Janssen1997]. Janssen’s result easily extends to the multivariate case if we assume a spherical covariance structure. Let $X_1, \ldots, X_m$ be a sample from a d-variate distribution with mean and covariance $(\mu_X, \sigma_x^2 I_d)$ and $Y_1,\ldots,Y_n$ be an independent sample with mean and covariance $(\mu_Y,\sigma_y^2 I_d)$. We propose the MD-scaled MD DiProPerm test whereby the MD direction is used in Step 1 of DiProPerm and a scaled MD statistic as in Equation is used in Step 2. The MD-scaled MD statistic is $${T_{m,n}(Z)}/{\{ {s_x^2}/{m} + {s_y^2}/{n} \}^{1/2}} \label{MDsMDeq}$$ where $T_{m,n}(Z)$ is as in Section \[valMDMD\]. The asymptotic validity of the MD-scaled MD statistic (as $m,n \to \infty$) follows immediately from Janssen’s result. Note that normality is not an assumption here. We study the empirical power of the MD-MD and MD-scaled MD for testing equality of means when sample sizes are unbalanced. We set $m = 50, n=100$ and make $m$ draws from $F_1 = N(\mu_1,\sigma_1^2 I_d)$ and $n$ draws from $F_2 = N(0,I_d)$ for $d=500$. The sample sizes and dimension are chosen to reflect a HDLSS setting. The significance level is set at $\alpha = 0.05$. Power is estimated using 1000 Monte Carlo simulations and displayed using a color spectrum from cool to warm, corresponding to the range 0 to 1. Figure \[equalPower\], as in the figures in Section \[powersufs\], displays the simulated power surface of MD-MD and MD-scaled MD. We see that when sample sizes are unbalanced and covariances unequal $(\sigma_1^2 \ne 1)$, MD-MD does not attain the correct level. Indeed MD-MD will reject increasingly often as the signal in $\sigma_1^2$ grows. On the other hand, we see from Figure \[powerMDsMD\] that the MD-scaled MD test attains the correct level under unbalanced sample sizes. This simulated power study also suggests that the asymptotics for the MD-scaled MD test is in effect for relatively small sample sizes and a much larger dimension. Additional Implementation Options {#impl} ================================= Direction {#Di} --------- The following binary linear classifiers are among many possible choices for the direction vector used in Step 1 of DiProPerm and all are implemented in the DiProPerm software: 1. The Mean Difference method is a simple binary linear classifier, also called the centroid method [@Hastie2003], where points are assigned to the class whose centroid is closest. The normal vector to the separating hyperplane is the unit vector in the direction of the line segment connecting the centroids of each class, $(\bar X-\bar Y)$. 2. Fisher Linear Discrimination (FLD) was an early binary linear classification method, see Chapter 11 of [@Mardia1979] for an introduction. FLD seeks a separation that maximizes the between sum-of-squares of the two classes while minimizing the within sum-of-squares of each class. The normal vector to the separating hyperplane is the unit vector in the direction of $W^{-1} (\bar X-\bar Y)'$ where $W$ is the $d \times d$ matrix $$W = \sum_{i=1}^m (X_i - \bar X)(X_i - \bar X)' + \sum_{j=1}^n (Y_j - \bar Y)(Y_j - \bar Y)'$$ 3. Support Vector Machine (SVM) is a popular binary linear classification method that minimizes training error while maximizing the margin between the two classes. See [@Hastie2003] for a good introduction. 4. Distance Weighted Discrimination (DWD) is a binary linear classifier similar to SVM except each data point has some weight in the final classifier [@Marron2007]. DWD better avoids the data piling problem exhibited by SVM in high dimensions. 5. Maximal Data Piling (MDP) is a binary linear classifier such that the projections of the data points from each class onto its normal direction vector have two distinct values [@Ahn2010]. Notice that we have not included any PCA directions on this list. This is because PCA is tailored to find directions that show maximal variation, which is different from our objective of finding directions that show separation between the two-samples. A more serious disadvantage to using PCA as the direction in step 1 of DiProPerm is that the univariate two-sample test statistic calculated in step 2 of DiProPerm would be invariant under relabelings. Projection and univariate statistic {#Pro} ----------------------------------- In the second step of DiProPerm, we project the data onto the direction in step one and compute a univariate two-sample statistic on the projected values. Large values of the test statistic indicate departure from the null hypothesis. The following univariate two-sample statistics are among many reasonable choices for the DiProPerm procedure and all are implemented in the DiProPerm software. 1. Two Sample t statistic 2. Difference of sample means 3. Difference of sample means scaled, as in Equation 4. Difference of sample medians 5. Difference of sample medians, divided by the median absolute deviation. 6. Area Under the Curve (AUC), from Receiver Operating Characteristic (ROC) curve 7. Paired sampling t-statistic It is of interest to note that the classical Hotelling $T^2$ statistic is a special case of the general DiProPerm framework. The FLD direction vector and the difference of sample means combination gives the statistic $(\bar X - \bar Y) W^{-1} (\bar X - \bar Y)'$. This is in fact the Hotelling $T^2$ test statistic scaled by a factor of $\frac{1}{n-2}\frac{n}{n_1 n_2}$. To see this, recall the Hotelling $T^2$ statistic is $$T^2 = \frac{n_1 n_2}{n} (\bar X - \bar Y) S_u^{-1} (\bar X - \bar Y)'$$ where $$S_u = \frac{\sum_{i=1}^m (X_i - \bar X)(X_i - \bar X)' + \sum_{j=1}^n (Y_j - \bar Y)(Y_j - \bar Y)'}{n-2} = \frac{W}{n-2}$$ The MD-FLD statistic is $\frac{1}{n-2}\frac{n}{n_1 n_2} T^2$ . Permutation {#Perm} ----------- In the final step of DiProPerm, an approximate permutation test is conducted to assess the significance of the test statistic in step two. Our permutation test is approximate because we perform a large number of random rearrangements of the labels on the observed data points, rather than all possible rearrangements. There are three kinds of indicators we commonly use and all are implemented in the DiProPerm software: 1. Empirical p-value: this is calculated as the proportion of the rearrangement test statistics that exceed the original test statistic. The empirical $p$-value has the disadvantage of often being zero. We may wish to compare two separations to see which is more significant. This motivates the next quantity. 2. Gaussian fit p-value: we fit a Gaussian distribution to the permutation test statistics and based on this calculate the percentage of rearrangement test statistics that exceed the original test statistic. (The term p-value is used loosely here). We do this not because we believe the permutation statistics are actually Gaussian, but because this provides a basis on which we can compare two DiProPerm results. In certain settings where the Gaussian fit $p$-value may suffer from round-off error, we use the next quantity as an alternative. 3. $z$-score: we fit a Gaussian distribution to the permutation test statistics and calculate the corresponding $z$-score of the original test statistic with respect to the fitted distribution. When interpreting the results of DiProPerm tests, it is generally useful to print all three indicators. When it is non-zero, the empirical $p$-value is the most interpretable. When it is zero we next look to the Gaussian fit p-value. Finally if the Gaussian fit p-value suffers from round-off error, the z-score is preferable. HDLSS calculations ================== Let $X \sim N(0,\sigma_x^2 I_d)$ and $Y \sim N(0,\sigma_y^2 I_d)$. We will study the asymptotic behavior of the distance between $X$ and $Y$. We have simply by definition $$|| X - Y||^2 / (\sigma_x^2 + \sigma_y^2) \sim \chi^2(d).$$ Then by the Central Limit Theorem, $$\sqrt d \left( \frac{|| X - Y||^2 / (\sigma_x^2 + \sigma_y^2)}{\sqrt 2 d} - \frac{1}{\sqrt 2} \right) \to N(0,1)$$ as $d \to \infty$. Applying the Delta Method, we get $$\sqrt d \left( \frac{|| X - Y ||}{2^{1/4} \sqrt{ (\sigma_x^2 + \sigma_y^2) d }} - \frac{1}{2^{1/4}} \right) = O_P(1)$$ and thus $$|| X - Y || = \sqrt{ (\sigma_x^2+\sigma_y^2) d } + O_P(1)$$

--- abstract: '[UTF8]{}[gbsn]{} Speech recognition in mixed language has difficulties to adapt end-to-end framework due to the lack of data and overlapping phone sets, for example in words such as “one” in English and “wàn” in Chinese. We propose a CTC-based end-to-end automatic speech recognition model for intra-sentential English-Mandarin code-switching. The model is trained by joint training on monolingual datasets, and fine-tuning with the mixed-language corpus. During the decoding process, we apply a beam search and combine CTC predictions and language model score. The proposed method is effective in leveraging monolingual corpus and detecting language transitions and it improves the CER by 5%.' address: | Center for Artificial Intelligence Research (CAiRE)\ Department of Electronic and Computer Engineering\ Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong\ {giwinata, amadotto, cwuak}@connect.ust.hk, pascale@ece.ust.hk bibliography: - 'strings.bib' - 'refs.bib' title: 'Towards End-to-end Automatic Code-Switching Speech Recognition' --- code-switch, end-to-end speech recognition, transfer learning, joint training, bilingual Introduction {#sec:intro} ============ Code-switching is the linguistic phenomenon of a person speaking or writing in one language and switches to another in the same sentence. It is very common among bilingual communities. With the advent of globalization, code-switching is becoming more common in predominantly monolingual societies as speakers use a second language in professional contexts. Speakers code-switch to empathize with each other, to express themselves better [@lowi2005codeswitching], and very often are not fully aware of using mixed codes in their language [@shay2015switch]. Code-switching poses a significant challenge for automatic speech recognition (ASR) systems even as the latter reach higher and higher performance within the new paradigm of neural network speech recognition [@graves2006connectionist; @amodei2016deep; @Toshniwal2018MultilingualSR]. The main reason lies in the unpredictability of points of code-switching in an utterance. Since speech recognition needs to be accomplished in real-time in most applications, it is not feasible to carry out language identification at each time step before transcribing the speech into text. Moreover, language identification of a single speaker within the same utterance is challenging as the speaker can carry over their pronunciation habits from the primary language to the foreign language in context. In the statistical ASR framework, acoustic modeling, pronunciation modeling and language modeling of code-switching speech are carried out separately and assumed to be independent of each other. The hypothesis $\hat{Y}$ is normally calculated as the following: $$\hat{Y} = \operatorname*{argmax}_Y{P(X|Y)P(Y)}$$ where $X$ is the input signal, $Y$ is the target sequence, $P(X|Y)$ is the acoustic observations and $P(Y)$ is the language model. The acoustic model consists of multiple Hidden Markov phoneme models, is trained from both languages bilingually, where the pronunciation of phones from both languages are mapped to each other. Each shared or mapped phoneme model is then trained from speech samples in both languages. Variations of this type of bilingual acoustic models abound, but they are not difficult to train. Another challenge is how to train the code-switching language model, $P(Y)$. Reliable statistical language models are derived from large amounts of text. In the case of code-switching speech, the data is often not enough to be generalizable. Previous work attempted to solve this problem by either generating more code-switching data by allowing code-switch at every point in the utterance, or at the phrasal boundaries [@li2012code] or even by phrasal alignment of parallel data according to linguistic constraints [@Li2014CodeSL]. In each case, the code-switching points are not learned automatically. The generalizability of language models derived from this data is therefore doubtful. In this paper, we propose and investigate an entirely different approach of automatic code-switching speech recognition, using an end-to-end neural network framework to recognize speech from input spectrogram to output text, dispensing with the pipelined architecture of acoustic modeling followed by language modeling. We apply a joint-training in CTC-based speech recognition [@amodei2016deep] and finetune the model on code-switching dataset to learn the language transitions between them. We show that our proposed approach learns how to distinguish signals from different languages and achieves better results compared to the training only with a code-switching corpus. We compare the results by adding different amounts of the mixed-language corpus and test the effectiveness of the joint-training. In the decoding step, we rescore our generated sequences with an n-gram language model. Related Work {#sec:related-work} ============ **Code-switching speech recognition:** The prior study on code-switching ASR is to incorporate a tri-phone HMMs as an acoustic model with an equivalence constraint in the language model [@li2012code]. [@adel2013recurrent] combined recurrent neural networks and factored language models to rescore the n-best hypothesis. [@Li2014CodeSL] proposed a lattice-based parsing to restrict the sequence paths to those permissible under the Functional Head Constraint. [@vu2012first] applied different phone merging approaches and combination with discriminative training. An extensive study on Hindi-English phone set sharing and gains improvement in WER [@sivasankaran2018phone]. In another line of work, multi-task learning approaches in code-switching had been used for learning a shared representation of two or more different tasks on language modeling [@W18-3207] and acoustic model [@seltzer2013multi]. **End-to-end approaches:** [@graves2006connectionist] presented the earliest implementation of CTC to end-to-end speech recognition. A sequence-to-sequence model with attention [@luong2015effective] that learned to transcribe speech utterances to characters has been introduced by [@chan2016listen] as Listen-Attend-Spell (LAS). A multi-lingual approach was proposed by [@Toshniwal2018MultilingualSR] using Seq2Seq approach using a union of language-specific grapheme sets and train a grapheme-based sequence-to-sequence model jointly on data from all languages. [@Hori2017AdvancesIJ] proposed to train a bilingual ASR for spontaneous Japanese and Chinese speech by using an end-to-end Seq2Seq model. Methodology =========== [UTF8]{}[gbsn]{} In this section, we describe the joint training on our proposed end-to-end approach including the learning strategies. During the decoding stage, we add an external language model for rescoring. We denote our training sets, English monolingual dataset $\{(X_1^{en}, Y_1^{en}),..., (X_n^{en}, Y_n^{en})\}$ and Mandarin Chinese monolingual dataset $\{(X_1^{zh}, Y_1^{zh}),..., (X_n^{zh}, Y_n^{zh})\}$, and a code-switching dataset $\{(X_1^{cs}, Y_1^{cs}),..., (X_n^{cs}, Y_n^{cs})\}$. The labels $Y$ are graphemes and the character set is the concatenation of English and Simplified Chinese characters [{a-z, space, apostrophe, 祥, 舌, ..., 底 }]{.nodecor}. \[sec:methodology\] [UTF8]{}[gbsn]{} Connectionist Temporal Classification Model ------------------------------------------- A CTC network uses an error criterion that optimizes the prediction of transcription by aligning the input signals with the hypothesis. The loss function is defined as the negative log likelihood: $$L_{CTC} = - \textnormal{log } P(I|X) = - \textnormal{ln}\sum_{\pi\in B^{-1}(Y)}{P(\pi|X)}$$ where we denote $\pi$ as the CTC path, $I$ as the transcription, and $B^{-1}(Y)$ as the mapping of all possible CTC paths $\pi$ resulting from $I$. Comparing to other end-to-end models such as LAS, CTC is more stable in training; thus it is easier to converge. Our model consists of a multi-layer Convolutional Neural Network [@lecun1998gradient] to generate a rich input representation, followed by a multi-layer Recurrent Neural Network (RNN) with Gated Recurrent Unit (GRU) [@cho2014learning] to learn the temporal information of the audio frame sequences. We are using the CTC-based architecture similar to [@amodei2016deep]. The input of the network is a sequence of log spectrograms and followed by a normalization step to regularize the parameters and avoid internal covariate shift. The CTC probabilities are used to find a better alignment between hypothesis and the input signals. To scale up the training process, batch normalization is employed to the recurrent layers to reduce the batch training time. The spectrograms are passed to a convolutional neural network (CNN) to encode the speech representation. The frame-wise posterior distribution $P(Y|X)$ is conditioned on the input $X$ and calculated by applying a fully-connected layer and a softmax function. $$P(Y|X) = \textnormal{Softmax}(\textnormal{Linear}(h))$$ where $h$ is the hidden state from the GRU. Next, it passes to a multi-layer bidirectional GRU. ![Connectionist Temporal Classification Model[]{data-label="fig:ctc"}](ctc.pdf){width="0.92\linewidth"} Joint Training -------------- We start the training as a joint training using monolingual dataset as a pretraining to learn the individual language. However, the CTC-based model can easily suffer catastrophic forgetting, where the model is not capable of remembering two distant languages such as English and Mandarin Chinese in separated supervision. It tends to keep one language and forget the other. Thus, we propose to train iteratively between two datasets; taking English and Chinese speeches in the batch. Thus, the model learns how to differentiate the characters. After the pretraining, we tune the model with code-switch data. Language Modeling ----------------- To improve the quality of the decoded sequence, we train a 5-gram language model with Kneser Ney smoothing [@heafield2013scalable] on our code-switching training data using KenLM[^1]. We use a prefix beam search with a beam width of $w$. We rescore the probability of the sequence $p_lm{(Y)}$ [@amodei2016deep] and find the maximum score of $Q(Y)$: $$Q(Y) = \textnormal{log}(P_{ctc}(Y|X)) + \alpha \textnormal{ log}(p_{lm}(Y)) + \beta \textnormal{ } wc(Y)$$ where $wc(Y)$ is the word count of sequence $Y$, $\alpha$ controls the contribution of language model and $\beta$ controls the number of word to be generated. In the beam search process, the decoder computes a score of each partial hypothesis and interpolates the result with the probability from the language model. Experiment ========== In this section, we describe our datasets and code-switching ASR experiments with our proposed end-to-end system. -- -- -- -- -- -- -- -- : Data Statistics of SEAME Phase II [@W18-3207].[]{data-label="data-statistics-phase-2"} **\# Duration (hr)** **\# Samples** -- ---------------------- ---------------- --------- **Train** 241.21 195,372 **Dev** 4.99 4,065 **Test** 4.94 3,986 **Train** 168.88 873 **Dev** 4.81 24 : Data Statistics of Common Voice and HKUST Corpus ------ We use speech data from SEAME Phase II (South East Asia Mandarin-English), a conversational Mandarin-English code-switching speech corpus consists of spontaneously spoken interviews and conversations [@SEAME2015]. We tokenize words using Stanford NLP toolkit [@manning-EtAl:2014:P14-5] and follow the same preprocessing step as [@W18-3207]. For the monolingual datasets, we use HKUST [@liu2006hkust], a spontaneous Mandarin Chinese telephone speech recordings and Common Voice, an open accented English dataset collected by Mozilla[^2]. Table \[data-statistics-phase-2\] shows the statistics of SEAME Phase II dataset and Table \[data-statistics-cv-hkust\] shows the statistics of Common Voice and HKUST datasets. **Model** **Dev** **Test** -------------------------------------------- ----------- ----------- **SEAME Phase II** - training (10% data, 10.13 hr) 49.81 46.23 - training (50% data, 50.41 hr) 38.08 32.10 - training (100% data, 100.58 hr) 36.18 29.82 []{.nodecor} + LM (5-gram, $\alpha$ = 0.2) 35.77 29.14 **Joint training** - fine tuning (10% data, 10.13 hr) 38.44 43.86 - fine tuning (50% data, 50.41 hr) 34.24 27.97 - fine tuning (100% data, 100.58 hr) 32.06 25.54 []{.nodecor} + LM (5-gram, $\alpha$ = 0.2) **31.35** **24.61** : Character Error Rate (CER %) for single dataset training, joint training, and rescoring with LM on SEAME Phase II.[]{data-label="results"} [UTF8]{}[gbsn]{} \[generated-sentences\] **Model** **Generated Sequence** --------------- ---------------------------------------------------------------------------------------------------------------------------- -------- reference 因为friendshyfriendnot reallyshysian - baseline 为 我的 [friend wa]{.nodecor} 很 s 我们 在friend 他 这not ra 能 s 他 就 要 人家 pa 他们 唱 因为 他们 觉得 一个 人 唱 很 闲 23.65% + fine-tuning 因为friend 很shy 我friend not re 15.05% + LM 因为 friendshy我friend他not real s s 11.82% reference thenbefore what kind of job - baseline 你whatdro 48.57% + fine-tuning 因for what kid of job 22.85% + LM 你for what kind of job 20.00% Experimental Setup ------------------ We convert the inputs into normalized frame-wise spectrograms. We take 20 ms width with a stride of 20 ms. The audio is down-sampled to a single channel with a sample rate of 8 kHz. The CNN encoder is described as the following: $$\begin{aligned} &\textnormal{Conv2d}(\textnormal{in}=1, \textnormal{out}=32, \textnormal{filter}=41\times11)\\ &\textnormal{Hardtanh(BatchNorm2d}(32))\\ &\textnormal{Conv2d}(\textnormal{in}=32, \textnormal{out}=32, \textnormal{filter}=21\times11)\\ &\textnormal{Hardtanh(BatchNorm2d}(32))\end{aligned}$$ It is followed by a 4-layer bidirectional GRU with a hidden size of 400 and a fully connected layer with a hidden size of 400 is inserted afterward. In the joint-training, we combine both monolingual datasets and sorted by their audio length, and groups them in buckets. We shuffle the buckets and take 20 samples in a batch. Then, we take every batch for the training. We start our training with different initial learning rates $\{1e-4, 3e-4\}$ and optimize our model by using Stochastic Gradient Descent (SGD) with momentum and Nesterov accelerated gradient [@nesterov1983method]. In the sequence decoding stage, we take to run a prefix beam-search a beam size of 100 to find the best sequence. The best hyperparameters for rescoring are $\alpha = 0.2$ and $\beta=1$. We first perform our fine-tuning experiment with smaller training sets to calculate the effectiveness of joint training. We take 10% (10.13 hr) and 50% (50.41 hr) from code-switching dataset. We also train our language model with 3-gram, 4-gram, and 5-gram models. Results {#sec:results} ======= Table \[results\] demonstrates speech recognition and language modeling results. The joint training with fine tuning improves the results 5% CER compared to training only with a code-switching corpus. Some generated characters are separated with excessive spaces. The joint training captures the sounds from monolingual corpus and transfer learns the information during the fine-tuning step. The baseline is not able to capture the word transition between Chinese and English. The joint training captures the sounds from the monolingual corpus and transfers learning to the code-switching sequences. **Joint training + Fine tuning:** Joint training helps the model to learn different sounds the perspective of single language. It is also an excellent way to initialization of our model. However, from the training, we still suffer an issue, where it does not keep the information of both languages, and we need to solve this issue in the future work. As shown in Table \[generated-sentences\], we can see that the results are getting better, it can generate a better mapping of similar sounds in English and Chinese to the corresponding characters. We test our model with smaller training data. According to our observation, by fine-tuning with only 50% code-switching data, we can achieve a comparable result to the whole training only with code-switching dataset. **Applying language model:** The external language model constraints the decoder to generate more grammatical sentences. In general, language model effects positively to the decoder and achieves an additional 1% reduction by adding a 5-gram language model. From Table \[generated-sentences\], it clearly shows that some misspelled words in the baseline are fixed. **Intra-sentential code-switching:** The model we trained can predict some English words correctly between Chinese words testing on code-switching dataset. It can still predict the code-switching points, unlike the work by [@Toshniwal2018MultilingualSR]. One of the possible reason is our CTC model is not constrained by the language information like the Seq2Seq-based model with language identifiers. In spite of that, the model is still predicting words with similar sound in the code-switching points such as “before" and “for". Conclusion {#sec:conclusion} ========== We propose a new direction on automatic code-switching speech recognition by applying end-to-end approach. Our training method can be adapted to any languages pair. We evaluate our model on English-Mandarin corpus and achieve a significant gain through a combination of joint training and fine-tuning. It can handle code-switching transitions and recognize both English and Chinese characters. The rescoring using an external language model improves the decoding result and fixes the spelling mistakes. Our proposed model achieves a 5% reduction in CER and the joint-training procedure allows the model to learn distant languages. For future work, we are going to mitigate further the catastrophic forgetting in our joint training network, which degrades the performance of our bilingual model. [^1]: The code can be found at https://github.com/kpu/kenlm [^2]: The dataset is available at https://voice.mozilla.org/

--- abstract: 'We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkale’s geometric method, but assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood-Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated by current research in computational complexity theory and effective algorithms.' address: - 'Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique' - 'Institut de Mathématiques de Jussieu, Paris Rive Gauche' - | Korteweg-de Vries Institute for Mathematics, University of Amsterdam & QuSoft\ Stanford Institute for Theoretical Physics, Stanford University author: - Nicole Berline - Michèle Vergne - Michael Walter bibliography: - 'belkale.bib' title: The Horn inequalities from a geometric point of view --- Introduction {#sec:intro} ============ The possible eigenvalues of Hermitian matrices $X_1,\dots,X_s$ such that $X_1+\dots+X_s = 0$ form a convex polytope. They can thus be characterized by a finite set of linear inequalities, most famously so by the inductive system of linear inequalities conjectured by Horn [@horn1962eigenvalues]. The very same inequalities give necessary and sufficient conditions on highest weights $\lambda_1,\dots,\lambda_s$ such that the tensor product of the corresponding irreducible $\operatorname{GL}(r)$-representations $L(\lambda_1),\dots,L(\lambda_s)$ contains a nonzero invariant vector, i.e., $c(\vec\lambda) := \dim (L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)} > 0.$ For $s=3$, the multiplicities $c(\vec\lambda)$ can be identified with the *Littlewood-Richardson coefficients*. Since the Horn inequalities are linear, $c(\vec\lambda) > 0$ if and only if $c(N \vec\lambda) > 0$ for any integer $N > 0$. This is the celebrated *saturation property* of $\operatorname{GL}(r)$, first established combinatorially by Knutson and Tao [@knutson1999honeycomb] building on work by Klyachko [@klyachko1998stable]. Some years after, Belkale has given an alternative proof of the Horn inequalities and the saturation property [@belkale2006geometric]. His main insight is to ‘geometrize’ the classical relationship between the invariant theory of $\operatorname{GL}(r)$ and the intersection theory of Schubert varieties of the Grassmannian. In particular, by a careful study of the tangent space of intersections, he shows how to obtain a geometric basis of invariants. The aim of this text is to give a self-contained exposition of the Horn inequalities, assuming only linear algebra and some basic representation theory and algebraic geometry, similar in spirit to the approach taken in [@vergne2014inequalities]. We also discuss a proof of Fulton’s conjecture which asserts that $c(\vec\lambda)=1$ if and only if $c(N\vec\lambda)=1$ for any integer $N\geq1$. We follow Belkale’s geometric method [@belkale2004invariant; @belkale2006geometric; @belkale2007geometric], as recently refined by Sherman [@sherman2015geometric], and do not claim any originality. Instead, we hope that our text might be useful by providing a more accessible introduction to these topics, since we tried to give simple and concrete proofs of all results. In particular, we do not use the Littlewood-Richardson rule for determining $c(\vec\lambda)$, and we do not discuss the relation of a basis of invariants to the integral points of the hive polytope [@knutson1999honeycomb]. Instead, we describe a basis of invariants that can be identified with the Howe-Tan-Willenbring basis, which is constructed using determinants associated to Littlewood-Richardson tableaux, as we explained in [@vergne2014inequalities]. We will come back to this subject in the future. We note that Derksen and Weyman’s work [@derksen2000semi] can be understood as a variant of the geometric approach in the context of quivers. For alternative accounts we refer to the work by Knutson and Tao [@knutson1999honeycomb] and Woodward [@knutson2004honeycomb], Ressayre [@ressayre2010geometric; @ressayre2009short] and to the expositions by Fulton and Knutson [@fulton2000eigenvalues; @knutson1999symplectic]. The desire for concrete approaches to questions of representation theory and algebraic geometry is also motivated by recent research in computational complexity and the interest in efficient algorithms. Indeed, the saturation property implies that deciding the nonvanishing of a Littlewood-Richardson coefficient can be decided in polynomial time [@mulmuley2012geometric]. In contrast, the analogous problem for the Kronecker coefficients, which are not saturated, is NP-hard, but believed to simplify in the asymptotic limit [@ikenmeyer2015vanishing; @burgisser2015membership]. We refer to [@mulmuley2011p; @burgisser2011overview] for further detail. These notes are organized as follows: In \[sec:easy\], we start by motivating the triple role of the Horn inequalities characterizing invariants, eigenvalues, and intersections. Then, in \[sec:linalg\], we collect some useful facts about positions and flags. This is used in \[sec:horn necessary,sec:horn sufficient\] to establish Belkale’s theorem characterizing intersecting Schubert varieties in terms of Horn’s inequalities. In \[sec:invariants\], we explain how to construct a geometric basis of invariants from intersecting Schubert varieties. This establishes the Horn inequalities for the Littlewood-Richardson coefficients, and thereby the saturation property, as well as for the eigenvalues of Hermitian matrices that sum to zero. In \[sec:fulton\], we sketch how Fulton’s conjecture can be proved geometrically by similar techniques. Lastly, in the , we have collected the Horn inequalities for three tensor factors and low dimensions. ### Notation {#notation .unnumbered} We write $[n] := \{1,\dots,n\}$ for any positive integer $n$. For any group $G$ and representation $M$, we write $M^G$ for the linear subspace of $G$-invariant vectors. For any subgroup $H\subseteq G$, we denote by $G/H = \{ gH \}$ the right coset space. If $F$ is an $H$-space, we denote by $G \times_H F$ the quotient of $G\times F$ by the equivalence relation $(g,f) \sim (gh^{-1}, hf)$ for $g\in G$, $f\in F$, $h\in H$. Note that $G \times_H F$ is a $G$-space fibered over $G/H$, with fiber $F$. If $F$ if a subspace of a $G$-space $X$, then $G\times_H F$ is identified by the $G$-equivariant map $[g,f] \mapsto (gH,gf)$ with the subspace of $G/H \times X$ (equipped with the diagonal $G$-action) consisting of the $(gH,x)$ such that $g^{-1}x\in F$. A panorama of invariants, eigenvalues, and intersections {#sec:easy} ======================================================== In this section we give a panoramic overview of the relationship between invariants, eigenvalues, and intersections. Our focus is on explaining the intuition, connections, and main results. To keep the discussion streamlined, more difficult proofs are postponed to later sections (in which case we use the numbering of the later section, so that the proofs can easily be found). The rest of this article, from \[sec:linalg\] onwards, is concerned with developing the necessary mathematical theory and giving these proofs. We start by recalling the basic representation theory of the general linear group $\operatorname{GL}(r) := \operatorname{GL}(r,{\mathbb C})$. Consider ${\mathbb C}^r$ with the ordered standard basis $e(1),\dots,e(r)$ and standard Hermitian inner product. Let $H(r)$ denote the subgroup of invertible matrices $t \in \operatorname{GL}(r)$ that are diagonal in the standard basis, i.e., $t\,e(i) = t(i)\cdot e(i)$ with all $t(i)\neq0$. We write $t = (t(1),\dots,t(r))$ and thereby identify $H(r) \cong ({\mathbb C}^*)^r$. To any sequence of integers $\mu = (\mu(1),\dots,\mu(r))$, we can associate a character of $H(r)$ by $t \mapsto t^\mu := t(1)^{\mu(1)}\cdots t(r)^{\mu(r)}$. We say that $\mu$ is a weight and call $\Lambda(r) = {\mathbb Z}^r$ the weight lattice. A weight is dominant if $\mu(1)\geq\dots\geq\mu(r)$, and the set of all dominant weights form a semigroup, denoted by $\Lambda_+(r)$. We later also consider antidominant weights $\omega$, which satisfy $\omega(1)\leq\dots\leq\omega(r)$. For any dominant weight $\lambda \in \Lambda_+(r)$, there is an unique irreducible representation $L(\lambda)$ of $\operatorname{GL}(r)$ with highest weight $\lambda$. That is, if $B(r)$ denotes the group of upper-triangular invertible matrices (the standard Borel subgroup of $\operatorname{GL}(r)$) and $N(r) \subseteq B(r)$ the subgroup of upper-triangular matrices with all ones on the diagonal (i.e., the corresponding unipotent), then $L(\lambda)^{N(r)} = {\mathbb C}v_\lambda$ is a one-dimensional eigenspace of $B(r)$ of $H(r)$-weight $\lambda$. We say that $v_\lambda$ is a highest weight vector of $L(\lambda)$. In \[subsec:borel-weil\] we describe a concrete construction of $L(\lambda)$ due to Borel and Weil. Now let $U(r)$ denote the group of unitary matrices, which is a maximally compact subgroup of $\operatorname{GL}(r)$. We can choose an $U(r)$-invariant Hermitian inner product $\braket{\cdot,\cdot}$ (by convention complex linear in the second argument) on each $L(\lambda)$ so that the representation $L(\lambda)$ restricts to an irreducible unitary representation of $U(r)$. Any two such representations of $U(r)$ are pairwise inequivalent, and, by Weyl’s trick, any irreducible unitary representation can be obtained in this way. Let us now decompose their Lie algebras as ${\mathfrak{gl}}(r) = \mathfrak u(r) {\oplus}i \mathfrak u(r)$, where $i = \sqrt{-1}$, and likewise ${\mathfrak h}(r) = {\mathfrak t}(r) {\oplus}i {\mathfrak t}(r)$, where we write ${\mathfrak t}(r)$ for the Lie algebra of $T(r)$, the group of diagonal unitary matrices, and similarly for the other Lie groups. Here, $i \mathfrak u(r)$ denotes the space of Hermitian matrices and $i {\mathfrak t}(r)$ the subspace of diagonal matrices with real entries. We freely identify vectors in ${\mathbb R}^r$ with the corresponding diagonal matrices in $i{\mathfrak t}(r)$ and denote by $(\cdot,\cdot)$ the usual inner product of $i{\mathfrak t}(r)\cong{\mathbb R}^r$. For a subset $J\subseteq[r]$, we write $T_J$ for the vector (diagonal matrix) in $i{\mathfrak t}(r)$ that has ones in position $J$, and otherwise zero. Now let ${\mathcal O}_\lambda$ denote the set of Hermitian matrices with eigenvalues $\lambda(1)\geq\dots\geq\lambda(r)$. By the spectral theorem, ${\mathcal O}_\lambda$ is a $U(r)$-orbit with respect to the adjoint action, $u \cdot X := u X u^*$, and so ${\mathcal O}_\lambda = U(r) \cdot \lambda$, where we identify $\lambda$ with the diagonal matrix with entries $\lambda(1)\geq\dots\geq\lambda(r)$. On the other hand, recall that any invertible matrix $g \in \operatorname{GL}(r)$ can be written as a product $g = u b$, where $u \in U(r)$ is unitary and $b \in B(r)$ upper-triangular. Since $v_\lambda$ is an eigenvector of $B(r)$, it follows that, in projective space ${\mathbb P}(L(\lambda))$, the orbits of $[v_\lambda]$ for $\operatorname{GL}(r)$ and $U(r)$ are the same! Moreover, it is not hard to see that the $U(r)$-stabilizers of $\lambda$ and of $[v_\lambda]$ agree, so we obtain a $U(r)$-equivariant diffeomorphism $$\label{eq:coadjoint orbit} {\mathcal O}_\lambda \to U(r) \cdot [v_\lambda] = \operatorname{GL}(r) \cdot [v_\lambda] \subseteq {\mathbb P}(L(\lambda)), \quad u \cdot \lambda \mapsto u \cdot [v_\lambda] = [u \cdot v_\lambda]$$ which also allows us to think of the adjoint orbit ${\mathcal O}_\lambda$ as a complex projective $\operatorname{GL}(r)$-variety. An important observation is that $$\label{eq:moment map equivariance} \operatorname{tr}\bigl((u \cdot \lambda) A\bigr) = \frac {\braket{u \cdot v_\lambda, \rho_\lambda(A) (u \cdot v_\lambda)}} {\lVert v_\lambda \rVert^2}$$ for all complex $r \times r$-matrices $A$, i.e., elements of the Lie algebra $\mathfrak{gl}(r)$ of $\operatorname{GL}(r)$; $\rho_\lambda$ denotes the Lie algebra representation on $L(\lambda)$. To see that  holds true, we may assume that $\lVert v_\lambda \rVert = 1$ as well as that $u = 1$, the latter by $U(r)$-equivariance. Now $\operatorname{tr}(A \lambda) = \braket{v_\lambda, \rho_\lambda(A) v_\lambda}$ is easily be verified by decomposing $A = L + H + R$ with $L$ strictly lower triangular, $R$ strictly upper triangular, and $H \in {\mathfrak h}(r)$ diagonal and comparing term by term. These observations lead to the following fundamental connection between the eigenvalues of Hermitian matrices and the invariant theory of the general linear group: \[prp:kempf-ness\] Let $\lambda_1,\dots,\lambda_s$ be dominant weights for $\operatorname{GL}(r)$ such that $(L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)} \neq \{0\}$. Then there exist Hermitian matrices $X_k \in \mathcal O_{\lambda_k}$ such that $\sum_{k=1}^s X_k = 0$. Let $0 \neq w \in (L(\lambda_1){\otimes}\cdots{\otimes}L(\lambda_s))^{\operatorname{GL}(r)}$ be a nonzero invariant vector. Then, $P(v) := \braket{w, v}$ is a nonzero linear function on $L(\lambda_1){\otimes}\cdots{\otimes}L(\lambda_s)$ that is invariant under the diagonal action of $\operatorname{GL}(r)$; indeed, $\braket{w, g \cdot v} = \braket{g^* \cdot w, v} = \braket{w, v}$. Since the $L(\lambda_k)$ are irreducible, they are spanned by the orbits $U(r) v_{\lambda_k}$. Thus we can find $u_1, \dots, u_s \in U(r)$ such that $P(v) \neq 0$ for $v = (u_1 \cdot v_{\lambda_1}) {\otimes}\cdots {\otimes}(u_s \cdot v_{\lambda_s})$. Consider the class $[v]$ of $v$ in the corresponding projective space ${\mathbb P}(L(\lambda_1){\otimes}\cdots{\otimes}L(\lambda_s))$. The orbit of $[v]$ under the diagonal $\operatorname{GL}(r)$-action is contained in the $\operatorname{GL}(r)^s$-orbit, which is the closed set $[U(r) \cdot v_{\lambda_1}{\otimes}\cdots{\otimes}U(r) \cdot v_{\lambda_s}]$ according to the discussion preceding . It follows that $\operatorname{GL}(r) \cdot v$ and its closure, $\overline{\operatorname{GL}(r) \cdot v}$ (say, in the Euclidean topology), are contained in the closed set $\{ \kappa (u'_1 \cdot v_{\lambda_1}) {\otimes}\cdots {\otimes}(u'_s \cdot v_{\lambda_s}) \}$ for $\kappa\in{\mathbb C}$ and $u'_1,\dots,u'_s\in U(r)$. Since $P$ is $\operatorname{GL}(r)$-invariant, $P(v') = P(v) \neq 0$ for any vector $v'$ in the diagonal $\operatorname{GL}(r)$-orbit of $v$. By continuity, this is also true in the orbits’ closure, $\overline{\operatorname{GL}(r) \cdot v}$. On the other hand, $P(0) = 0$. It follows that $0\not\in\overline{\operatorname{GL}(r) \cdot v}$, i.e., the origin does not belong to the orbit closure. Consider then a nonzero vector $v'$ of minimal norm in $\overline{\operatorname{GL}(r) \cdot v}$. By the discussion in the preceding paragraph, this vector is of the form $v' = \kappa (u'_1 \cdot v_{\lambda_1}) {\otimes}\cdots {\otimes}(u'_s \cdot v_{\lambda_s})$ for some $0 \neq \kappa \in {\mathbb C}$ and $u'_1,\dots,u'_s \in U(r)$. By rescaling $v$ we may moreover assume that $\kappa = 1$, so that $v'$ is a unit vector. The vector $v'$ is by construction a vector of minimal norm in its own $\operatorname{GL}(r)$-orbit. It follows that, for any Hermitian matrix $A$, $$\begin{aligned} 0 &= \frac12 \partial_{t=0} \lVert (e^{At}{\otimes}\cdots{\otimes}e^{At}) \cdot v' \rVert^2 \\ &= \braket{v', \bigl(\rho_{\lambda_1}(A) {\otimes}I{\otimes}\cdots{\otimes}I + \dots + I{\otimes}\cdots{\otimes}I{\otimes}\rho_{\lambda_s}(A)\bigr) v'} \\ &= \sum_{k=1}^s \braket{u'_k \cdot v_{\lambda_k}, \rho_{\lambda_k}(A) (u'_k \cdot v_{\lambda_k})} = \sum_{k=1}^s \operatorname{tr}\bigl(A (u'_k \cdot \lambda_k)\bigr) = \sum_{k=1}^s \operatorname{tr}(A X_k), \end{aligned}$$ where we have used \[eq:moment map equivariance\] and set $X_k := u'_k \cdot \lambda_k$ for $k\in[s]$. This implies at once that $\sum_{k=1}^s X_k = 0$. The adjoint orbits ${\mathcal O}_\lambda = U(r) \cdot \lambda$ (but not the map ) can be defined not only for dominant weights $\lambda$ but in fact for arbitrary Hermitian matrices. Conversely, any Hermitian matrix is conjugate to a unique element $\xi \in i{\mathfrak t}(r)$ such that $\xi(1)\geq\dots\geq\xi(r)$. The set of all such $\xi$ is a convex cone, known as the positive Weyl chamber $C_+(r)$, and it contains the semigroup of dominant weights. Throughout this text, we only ever write ${\mathcal O}_\xi = U(r)\cdot\xi$ for $\xi$ that are in the positive Weyl chamber. For example, if $\xi \in C_+(r)$ then $-\xi \in {\mathcal O}_{\xi^*}$, where $\xi^* = (-\xi(r),\dots,-\xi(1)) \in C_+(r)$. If $\lambda$ is a dominant weight then $\lambda^*=(-\lambda(d),\dots,-\lambda(1))$ is the highest weight of the dual representation of $L(\lambda)$, i.e., $L(\lambda^*) \cong L(\lambda)^*$. Using the inner product $(A,B) := \operatorname{tr}(AB)$ on Hermitian matrices we may also think of $\lambda$ as an element in $i{\mathfrak t}(r)^*$ and of ${\mathcal O}_\lambda$ as a *co*adjoint orbit in $i\mathfrak u(r)^*$. From the latter point of view, the map $(X_1,\dots,X_s)\mapsto\sum_{k=1}^s X_k$ is the moment map for the diagonal $U(r)$-action on the product of Hamiltonian manifolds ${\mathcal O}_{\lambda_k}$, $k\in[s]$. thus relates the existence of nonzero invariants to the statement that the zero set of the corresponding moment map is nonempty. This is a general fact of Mumford’s geometric invariant theory. \[def:kirwan cone\] The *Kirwan cone* $\operatorname{Kirwan}(r,s)$ is defined as the set of $\vec\xi = (\xi_1,\dots,\xi_s) \in C_+(r)^s$ such that there exist $X_k\in{\mathcal O}_{\xi_k}$ with $\sum_{k=1}^s X_k=0$. Using this language, \[prp:kempf-ness\] asserts that if the generalized Littlewood-Richardson coefficient $c(\vec\lambda) := \dim (L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)} > 0$ is nonzero then $\vec\lambda$ is a point in the Kirwan cone $\operatorname{Kirwan}(r,s)$. We will see in \[sec:invariants\] that, conversely, if $\vec\lambda\in\operatorname{Kirwan}(r,s)$, then $c(\vec\lambda)>0$ (by constructing an explicit nonzero invariant). As a consequence, it will follow that $c(\vec\lambda) > 0$ if and only if $c(N\vec\lambda) > 0$ for some integer $N > 0$. This is the remarkable saturation property of the Littlewood-Richardson coefficients. In fact, we will show that the Horn inequalities give a complete set of conditions for nonvanishing $c(\vec\lambda)$ as well as for $\vec\xi\in\operatorname{Kirwan}(r,s)$, which in particular establishes that $\operatorname{Kirwan}(r,s)$ is indeed a convex polyhedral cone. We will come back to these points at the end of this section. If there exist permutations $w_k$ such that $\sum_{k=1}^s w_k \cdot \xi_k = 0$ then $\vec\xi \in \operatorname{Kirwan}(r,s)$ (choose each $X_k$ as the diagonal matrix $w_k \cdot \xi_k$). This suffices to characterize the Kirwan cone for $s\leq2$: For $s=1$, it is clear that $\operatorname{Kirwan}(r,1) = \{0\}$. When $s=2$, then $\operatorname{Kirwan}(r,2) = \{(\xi,\xi^*)\}$. Indeed, if $X_1 \in {\mathcal O}_{\xi_1}$ and $X_2 \in {\mathcal O}_{\xi_2}$ with $X_1 + X_2 = 0$, then $X_2 = -X_1 \in {\mathcal O}_{\xi_1^*}$. In general, however, it is quite delicate to determine if a given $\vec\xi \in C_+(r)^s$ is in $\operatorname{Kirwan}(r,s)$ or not. Clearly, one necessary condition is that $\sum_{k=1}^s \lvert \xi_k \rvert = 0$, where we have defined $\lvert \mu \rvert := \sum_{j=1}^r \mu(j)$ for an arbitrary $\mu\in{\mathfrak h}(r)$. This follows by taking the trace of the equation $\sum_{k=1}^s X_k = 0$. In fact, it is clear that by adding or subtracting appropriate multiples of the identity matrix we can always reduce to the case where each $\lvert \xi_k \rvert = 0$. Let $X_k \in {\mathcal O}_{\xi_k}$ such that $\sum_{k=1}^s X_k = 0$. For each $k$, let $v_k$ denote a unit eigenvector of $X_k$ with eigenvalue $\xi_k(1)$. Then we have $$\label{eq:triangle ieqs} 0 = \braket{v_k, (\sum_{l=1}^s X_l) v_k} = \xi_k(1) + \sum_{l\neq k} \braket{v_k, X_l v_k} \geq \xi_k(1) + \sum_{l\neq k} \xi_l(r)$$ since $\xi_l(r) = \min_{\lVert v \rVert = 1} \braket{v, X_l v}$ by the variational principle for the minimal eigenvalue of a Hermitian matrix $X_l$. These inequalities, together with $\sum_{k=1}^s \lvert\xi_k\rvert = 0$, characterize the Kirwan cone for $r=2$, as can be verified by brute force. There is also a pleasant geometric way of understanding these inequalities in the case $r=2$. As discussed above, we may assume that the $X_k$ are traceless, i.e., that $\xi_k = (j_k,-j_k)$ for some $j_k\geq0$. Recall that the traceless Hermitian matrices form a three-dimensional real vector space, spanned by the Pauli matrices. Thus each $X_k$ identifies with a vector $x_k \in {\mathbb R}^3$, and the condition that $X_k \in {\mathcal O}_{\xi_k}$ translates into $\lVert x_k \rVert = j_k$. Thus we seek to characterize necessary and sufficient conditions on the lengths $j_k$ of vectors $x_k$ that sum to zero, $\sum_{k=1}^s x_k = 0$. By the triangle inequality, $j_k = \lVert x_k \rVert \leq \sum_{l\neq k} \lVert x_l \rVert = \sum_{l\neq k} j_l$, which is equivalent to the above. It is instructive to observe that $j_k \leq \sum_{l\neq k} j_l$ is precisely the Clebsch-Gordan rule for $\operatorname{SL}(2)$ when the $j_k$ are half-integers. The proof of \[eq:triangle ieqs\], which was valid for any $s$ and $r$, suggests that a more general variational principle for eigenvalues might be useful to produce linear inequalities for the Kirwan cones. \[def:flag and adapted basis\] A *(complete) flag* $F$ on a vector space $V$, $\dim V = r$, is a chain of subspaces $$\{0\} = F(0) \subset F(1) \subset \cdots \subset F(j) \subset F({j+1}) \subset \dots \subset F(r) = V,$$ such that $\dim F(j) = j$ for all $j=0,\dots,r$. Any ordered basis $f = (f(1),\dots,f(r))$ of $V$ determines a flag by $F(j) = \operatorname{span}\{ f(1), \dots, f(j) \}$. We say that $f$ is *adapted* to $F$. Now let $X \in {\mathcal O}_\xi$ be a Hermitian matrix with eigenvalues $\xi(1)\geq\dots\geq\xi(r)$. Let $(f_X(1),\dots,f_X(r))$ denote an orthonormal eigenbasis, ordered correspondingly, and denote by $F_X$ the corresponding *eigenflag* of $X$, defined as above. Note that $F_X$ is uniquely defined if the eigenvalues $\xi(j)$ are all distinct. We can quantify the position of a subspace with respect to a flag in the following way: \[def:schubert pos\] The *Schubert position* of an $d$-dimensional subspace $S \subseteq V$ with respect to a flag $F$ on $V$ is the strictly increasing sequence $J$ of integers defined by $$J(b) := \min \{ j \in [r], \,\dim F(j) \cap S = b \}$$ for $b \in [d]$. We write $\operatorname{Pos}(S,F) = J$ and freely identify $J$ with the subset $\{ J(1) < \dots < J(d) \}$ of $[r]$. In particular, $\operatorname{Pos}(S,F)=\emptyset$ for $S=\{0\}$ the zero-dimensional subspace. The upshot of these definitions is the following variational principle: \[lem:variational\] Let $\xi\in C_+(r)$, $X \in {\mathcal O}_\xi$ with eigenflag $F_X$, and $J \subseteq [r]$ a subset of cardinality $d$. Then, $$\min_{S : \operatorname{Pos}(S, F_X) = J} \operatorname{tr}(P_S X) = \sum_{j \in J} \xi(j) = (T_J, \xi),$$ where $P_S$ denotes the orthogonal projector onto an $d$-dimensional subspace $S \subseteq {\mathbb C}^r$. Recall that $F_X(j) = \operatorname{span}\{ f_X(1), \dots, f_X(j) \}$, where $(f_X(1),\dots,f_X(r))$ is an orthonormal eigenbasis of $X$, ordered according to $\xi(1)\geq\dots\geq\xi(r)$. Given a subspace $S$ with $\operatorname{Pos}(S, F_X) = J$, we can find an ordered orthonormal basis $(s(1),\dots,s(d))$ of $S$ where each $s(a) \in F_X(J(a))$. Therefore, $$\operatorname{tr}(P_S X) = \sum_{a=1}^d \braket{s(a), X s(a)} \geq \sum_{a=1}^d \xi(J(a)) = \sum_{j \in J} \xi(j).$$ The inequality holds term by term, as the Hermitian matrix obtained by restricting $X$ to the subspace $F_X(J(a))$ has smallest eigenvalue $\xi(J(a))$. Since $\operatorname{tr}(P_S X) = \sum_{j \in J} \xi(j)$ for $S = \operatorname{span}\{ f_X(j) : j \in J \}$, this establishes the lemma. Recall that the Grassmannian $\operatorname{Gr}(d,V)$ is the space of $d$-dimensional subspaces of $V$. We may partition $\operatorname{Gr}(d,V)$ according to the Schubert position with respect to a fixed flag: \[def:schubert cell and variety\] Let $F$ be a flag on $V$, $\dim V = r$, and $J \subseteq [r]$ a subset of cardinality $d$. The *Schubert cell* is $$\Omega^0_J(F) = \{ S \subseteq V : \dim S = d, \, \operatorname{Pos}(S, F) = J \}.$$ The *Schubert variety* $\Omega_J(F)$ is defined as the closure of $\Omega^0_J(F)$ in the Grassmannian $\operatorname{Gr}(d,V)$. The closures in the Euclidean and Zariski topology coincide; the $\Omega_J(F)$ are indeed algebraic varieties. Using these definitions, \[lem:variational\] asserts that $\min_{S \in \Omega^0_J(F_X)} \operatorname{tr}(P_S X) = \sum_{j \in J} \xi(j)$ for any $X \in {\mathcal O}_\xi$. Since the orthogonal projector $P_S$ is a continuous function of $S \in \operatorname{Gr}(d,V)$ (in fact, the Grassmannian is homeomorphic to the space of orthogonal projectors of rank $d$), it follows at once that $$\label{eq:variational closure} \min_{S \in \Omega_J(F_X)} \operatorname{tr}(P_S X) = \sum_{j \in J} \xi(j) = (T_J,\xi).$$ As a consequence, intersections of Schubert varieties imply linear inequalities of eigenvalues of matrices summing to zero: \[lem:eigenflag ieqs\] Let $X_k \in {\mathcal O}_{\xi_k}$ be Hermitian matrices with $\sum_{k=1}^s X_k = 0$. If $J_1,\dots,J_s \subseteq [r]$ are subsets of cardinality $d$ such that $\bigcap_{k=1}^s \Omega_{J_k}(F_{X_k}) \neq \emptyset$, then $\sum_{k=1}^s (T_{J_k},\xi_k) \leq 0$. Let $S \in \bigcap_{k=1}^s \Omega_{J_k}(F_{X_k})$. Then, $0 = \sum_{k=1}^s \operatorname{tr}(P_S X_k) \geq \sum_{k=1}^s (T_{J_k}, \xi_k)$ by . Remarkably, we will find that it suffices to consider only those $J_1,\dots,J_s$ such that $\bigcap_{k=1}^s \Omega_{J_k}(F_k)\neq\emptyset$ for all flags $F_1,\dots,F_s$. We record the corresponding eigenvalue inequalities, together with the trace condition, in \[cor:klyachko kirwan\] below. Following [@belkale2006geometric], we denote $s$-tuples by calligraphic letters, e.g., ${\mathcal J}= (J_1,\dots,J_s)$, ${\mathcal F}=(F_1,\dots,F_s)$, etc. In the case of Greek letters we continue to write $\vec\lambda=(\lambda_1,\dots,\lambda_s)$, etc., as above. \[def:intersecting\] We denote by $\operatorname{Subsets}(d,r,s)$ the set of $s$-tuples ${\mathcal J}$, where each $J_k$ is a subset of $[r]$ of cardinality $d$. Given such a ${\mathcal J}$, let ${\mathcal F}$ be an $s$-tuple of flags on $V$, with $\dim V = r$. Then we define $$\Omega^0_{\mathcal J}({\mathcal F}) := \bigcap_{k=1}^s \Omega^0_{J_k}(F_k), \qquad \Omega_{\mathcal J}({\mathcal F}) := \bigcap_{k=1}^s \Omega_{J_k}(F_k).$$ We shall say that ${\mathcal J}$ is *intersecting* if $\Omega_{\mathcal J}({\mathcal F})\neq\emptyset$ for every $s$-tuple of flags ${\mathcal F}$, and we denote denote the set of such ${\mathcal J}$ by $\operatorname{Intersecting}(d,r,s) \subseteq \operatorname{Subsets}(d,r,s)$. \[cor:klyachko kirwan\] If $\vec\xi \in \operatorname{Kirwan}(r,s)$ then $\sum_{k=1}^s \lvert \xi_k \rvert = 0$, and for any $0<d<r$ and any $s$-tuple ${\mathcal J}\in \operatorname{Intersecting}(d,r,s)$ we have that $\sum_{k=1}^s (T_{J_k}, \xi_k) \leq 0$. If $J = \{1,\dots,d\} \subseteq [r]$ then $\Omega^0_J(F) = \{ F(d) \}$ is a single point. On the other end, if $J = \{r-d+1,\dots,r\}$ then $\Omega^0_J(F)$ is dense in $\operatorname{Gr}(r, V)$, so that $\Omega_J(F) = \operatorname{Gr}(r, V)$. It follows that ${\mathcal J}= (J_1, \{r-d+1,\dots,r\},\dots,\{r-d+1,\dots,r\}) \in \operatorname{Intersecting}(d, r, s)$ is intersecting for any $J_1$ (and likewise for permutations of the $s$ factors). For $d=1$, this means that $\Omega_{\{r\}}(F) = {\mathbb P}(V)$, so that  reduces to the variational principle for the minimal eigenvalue, $\xi(r) = \min_{\lVert v \rVert = 1} \braket{v, X v}$, which we used to derive  above. Indeed, since $(\{a\},\{r\},\dots,\{r\})$ is intersecting for any $a$, we find that  is but a special case of \[cor:klyachko kirwan\]. In order to understand the linear inequalities in \[cor:klyachko kirwan\], we need to understand the sets of intersecting tuples. In the remainder of this section we thus motivate Belkale’s inductive system of conditions for an $s$-tuple to be intersecting. For reasons that will become clear shortly, we slightly change notation: $E$ will be a complete flag on some $n$-dimensional vector space $W$, $I$ will be a subset of $[n]$ of cardinality $r$, and hence $\Omega^0_I(E)$ will be a Schubert cell in the Grassmannian $\operatorname{Gr}(r,W)$. We will describe $\operatorname{Gr}(r,W)$ and $\Omega^0_I(E)$ in detail in \[sec:linalg\]. For now, we note that the dimension of $\operatorname{Gr}(r,W)$ is $r(n-r)$. In fact, $\operatorname{Gr}(r,W)$ is covered by affine charts isomorphic to ${\mathbb C}^{r(n-r)}$. The dimension of a Schubert cell and the corresponding Schubert variety (its Zariski closure) is given by [ $$\tag{\ref{eq:dim schubert cell}} \dim\Omega^0_I(E) = \dim\Omega_I(E) = \sum_{a=1}^r \bigl( I(a) - a \bigr) =: \dim I.$$]{} Indeed, $\Omega^0_I(E)$ is contained in an affine chart ${\mathbb C}^{r(n-r)}$ and is isomorphic to a vector subspace of dimension $\dim I$. So locally $\Omega^0_I(E)$ is defined by $r(n-r)-\dim I$ equations. This is easy to see and we give a proof in \[sec:linalg\]. \[def:edim\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. The *expected dimension* associated with ${\mathcal I}$ is $$\operatorname{edim}{\mathcal I}:= r(n-r) - \sum_{k=1}^s \bigl( r(n-r) - \dim I_k \bigr).$$ This definition is natural in terms of intersections, as the following lemma shows: \[lem:dim intersection\] Let ${\mathcal E}$ be an $s$-tuple of flags on $W$, $\dim W = n$, and ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. If $\Omega^0_{{\mathcal I}}({\mathcal E}) \neq \emptyset$ then its irreducible components (in the sense of algebraic geometry) are all of dimension at least $\operatorname{edim}{\mathcal I}$. Each Schubert cell $\Omega^0_{I_k}(E_k)$ is locally defined by $r(n-r) - \dim I_k$ equations. It follows that any irreducible component ${\mathcal Z}\subseteq \Omega^0_{{\mathcal I}}({\mathcal E}) = \bigcap_{k=1}^s \Omega^0_{I_k}(E_k)$ is locally defined by $\sum_{k=1}^s (r(n-r) - \dim I_k)$ equations. These equations, however, are not necessarily independent. Thus the codimension of ${\mathcal Z}$ is at most that number, and we conclude that $\dim {\mathcal Z}\geq \operatorname{edim}{\mathcal I}$. Belkale’s first observation is that the expected dimension of an intersecting tuple ${\mathcal I}\in \operatorname{Intersecting}(r, n, s)$ is necessarily nonnegative, [ $$\tag{\ref{eq:edim nonnegative}} \operatorname{edim}{\mathcal I}= r(n-r) - \sum_{k=1}^s (r(n-r) - \dim I_k)\geq0.$$]{} This inequality, as well as some others, will be proved in detail in \[sec:horn necessary\]. For now, we remark that the condition is rather natural from the perspective of Kleiman’s moving lemma. Given ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$, it not only implies that the intersection of the Schubert *cells*, $\Omega^0_{{\mathcal I}}({\mathcal E}) = \bigcap_{k=1}^s \Omega^0_{I_k}(E_k)\neq \emptyset$, is nonempty for generic flags, but in fact transverse, so that the dimensions of its irreducible components are exactly equal to the expected dimension; hence, $\operatorname{edim}{\mathcal I}\geq0$. We now show that  gives rise to an inductive system of conditions. Given a flag $E$ on $W$ and a subspace $V\subseteq W$, we denote by $E^V$ the flag obtained from the distinct subspaces in the sequence $E(i)\cap V$, $i=0,\dots,n$. Given subsets $I \subseteq [n]$ of cardinality $r$ and $J \subseteq [r]$ of cardinality $d$, we also define their *composition* $IJ$ as the subset $IJ = \{ I(J(1)) < \dots < I(J(d)) \} \subseteq [n]$. (For $s$-tuples ${\mathcal I}$ and ${\mathcal J}$ we define ${\mathcal I}{\mathcal J}$ componentwise.) Then we have the following ‘chain rule’ for positions: If $S \subseteq V \subseteq W$ are subspaces and $E$ is a flag on $W$ then [ $$\tag{\ref{eq:chain rule pos}} \operatorname{Pos}(S, E) = \operatorname{Pos}(V, E) \operatorname{Pos}(S, E^V).$$]{} We also have the following description of Schubert varieties in terms of Schubert cells: [ $$\tag{\ref{eq:schubert variety characterization}} \Omega_I(E) = \bigcup_{I' \leq I} \Omega^0_{I'}(E),$$]{} where the union is over all subsets $I' \subseteq [n]$ of cardinality $r$ such that $I'(a) \leq I(a)$ for $a\in[r]$. Both statements are not hard to see; we will give careful proofs in \[sec:linalg\] below. We thus obtain a corresponding chain rule for intersecting tuples: \[lem:chain rule intersecting\] If ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$ and ${\mathcal J}\in \operatorname{Intersecting}(d,r,s)$, then we have ${\mathcal I}{\mathcal J}\in \operatorname{Intersecting}(d,n,s)$. Let ${\mathcal E}$ be an $s$-tuple of flags on $W={\mathbb C}^n$. Since ${\mathcal I}$ is intersecting, there exists $V \in \Omega_{\mathcal I}({\mathcal E})$. Let ${\mathcal E}^V$ denote the $s$-tuple of induced flags on $V$. Likewise, since ${\mathcal J}$ is intersecting, we can find $S \in \Omega_{\mathcal J}({\mathcal E}^V)$. In particular, $\operatorname{Pos}(V, E_k)(a) \leq I_k(a)$ for $a\in[r]$ and $\operatorname{Pos}(S, E_k^V) \leq J_k(b)$ for $b\in[d]$ by . Thus  shows that $\operatorname{Pos}(S, E_k)(b) = \operatorname{Pos}(V, E_k)\bigl(\operatorname{Pos}(S, E_k^V)(b)\bigr) \leq \operatorname{Pos}(V, E_k)(J_k(b)) \leq I_k(J_k(b))$. Using  one last time, we conclude that $S \in \Omega_{{\mathcal I}{\mathcal J}}({\mathcal E})$. As an immediate consequence of Inequality  and \[lem:chain rule intersecting\] we obtain the following set of necessary conditions for an $s$-tuple ${\mathcal I}$ to be intersecting: \[cor:belkale inductive\] If ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$ then for any $0<d<r$ and any $s$-tuple ${\mathcal J}\in \operatorname{Intersecting}(d,r,s)$ we have that $\operatorname{edim}{\mathcal I}{\mathcal J}\geq 0$. Belkale’s theorem asserts that these conditions are also sufficient. In fact, it suffices to restrict to intersecting ${\mathcal J}$ with $\operatorname{edim}{\mathcal J}= 0$: \[def:horn\] Let $\operatorname{Horn}(r,n,s)$ denote the set of $s$-tuples ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ defined by the conditions that $\operatorname{edim}{\mathcal I}\geq0$ and, if $r>1$, that $$\operatorname{edim}{\mathcal I}{\mathcal J}\geq 0$$ for all ${\mathcal J}\in \operatorname{Horn}(d, r, s)$ with $0<d<r$ and $\operatorname{edim}{\mathcal J}=0$. For $r\in[n]$ and $s\geq2$, $\operatorname{Intersecting}(r, n, s) = \operatorname{Horn}(r, n, s)$. We will prove \[thm:belkale\] in \[sec:horn sufficient\]. The inequalities defining $\operatorname{Horn}(r,n,s)$ are in fact tightly related to those constraining the Kirwan cone $\operatorname{Kirwan}(r,s)$ and the existence of nonzero invariant vectors. To any $s$-tuple of dominant weights $\vec\lambda$ for $\operatorname{GL}(r)$ such that $\sum_{k=1}^s \lvert\lambda_k\rvert=0$, we will associate an $s$-tuple ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ for some $[n]$ such that $\operatorname{edim}{\mathcal I}=0$. Furthermore, if $\vec\lambda$ satisfies the inequalities in \[cor:klyachko kirwan\] then ${\mathcal I}\in\operatorname{Horn}(r,n,s)$. In \[sec:invariants\] we will explain this more carefully and show how Belkale’s considerations allow us to construct a corresponding nonzero $\operatorname{GL}(r)$-invariant in $L(\lambda_1){\otimes}\cdots{\otimes}L(\lambda_s)$. By \[prp:kempf-ness\], we will thus obtain at once a characterization of the Kirwan cone as well as of the existence of nonzero invariants in terms of Horn’s inequalities: \(a) *Horn inequalities:* The Kirwan cone $\operatorname{Kirwan}(r,s)$ is the convex polyhedral cone of $\vec\xi\in C_+(r)^s$ such that $\sum_{k=1}^s \lvert \xi_k \rvert = 0$, and for any $0<d<r$ and any $s$-tuple ${\mathcal J}\in \operatorname{Horn}(d,r,s)$ with $\operatorname{edim}{\mathcal J}=0$ we have that $\sum_{k=1}^s (T_{J_k},\xi_k) \leq 0$. \(b) *Saturation property:* For a dominant weight $\vec\lambda\in \Lambda_+(r)^s$, the space of invariants $(L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)}$ is nonzero if and only if $\vec\lambda \in \operatorname{Kirwan}(r,s)$. In particular, $c(\vec\lambda) := \dim (L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)} > 0$ if and only if $c(N \vec\lambda) > 0$ for some integer $N > 0$. The proof of \[cor:horn and saturation\] will be given in \[sec:invariants\]. In \[app:examples horn,app:examples kirwan\], we list the Horn triples as well as the Horn inequalities for the Kirwan cones up to $r=4$. Subspaces, flags, positions {#sec:linalg} =========================== In this section, we study the geometry of subspaces and flags in more detail and supply proofs of some linear algebra facts used previously in \[sec:easy\]. Schubert positions ------------------ We start with some remarks on the Grassmannian $\operatorname{Gr}(r,W)$, which is an irreducible algebraic variety on which the general linear group $\operatorname{GL}(W)$ acts transitively. The stabilizer of a subspace $V \in \operatorname{Gr}(r, W)$ is equal to the parabolic subgroup $P(V, W) = \{ \gamma \in \operatorname{GL}(W) : \gamma V \subseteq V \}$, with Lie algebra ${\mathfrak p}(V, W) = \{ x \in {\mathfrak{gl}}(W) : x V \subseteq V \}$. Thus we obtain that $$\operatorname{Gr}(r,W) = \operatorname{GL}(W) \cdot V \cong \operatorname{GL}(W)/P(V,W),$$ and we can identify the tangent space at $V$ with $$T_V \operatorname{Gr}(r,W) = {\mathfrak{gl}}(W) \cdot V \cong {\mathfrak{gl}}(W)/{\mathfrak p}(V,W) \cong \operatorname{Hom}(V,W/V).$$ If we choose a complement $Q$ of $V$ in $W$ then $$\label{eq:affine chart of grassmannian} \operatorname{Hom}(V,Q)\to\operatorname{Gr}(r,W), \quad \phi\mapsto(\operatorname{id}+\phi)(V)$$ parametrizes a neighborhood of $V$. This gives a system of affine charts in $\operatorname{Gr}(r,W)$ isomorphic to ${\mathbb C}^{r(n-r)}$. In particular, $\dim\operatorname{Gr}(r,W)=r(n-r)$, a fact we use repeatedly in this article. We now consider Schubert positions and the associated Schubert cells and varieties in more detail (\[def:schubert pos,def:schubert cell and variety\]). For all $\gamma \in \operatorname{GL}(W)$, we have the following equivariance property: $$\label{eq:position equivariance} \operatorname{Pos}(\gamma^{-1} V, E) = \operatorname{Pos}(V, \gamma E),$$ which in particular implies that $$\label{eq:schubert cell equivariance} \gamma \Omega^0_I(E) = \Omega^0_I(\gamma E).$$ Thus $\Omega^0_I(E)$ is preserved by the Borel subgroup $B(E) = \{ \gamma\in\operatorname{GL}(W) : \gamma E(i)\subseteq E(i) \; (\forall i) \}$, which is the stabilizer of the flag $E$. We will see momentarily that $\Omega^0_I(E)$ is in fact a single $B(E)$-orbit. We first state the following basic lemma, which shows that adapted bases (\[def:flag and adapted basis\]) provide a convenient way of computing Schubert positions: \[lem:adapted basis\] Let $E$ be a flag on $W$, $\dim W = n$, $V \subseteq W$ an $r$-dimensional subspace, and $I \subseteq [n]$ a subset of cardinality $r$, with complement $I^c$. The following are equivalent: 1. \[item:adapted basis i\] $\operatorname{Pos}(V, E) = I$. 2. \[item:adapted basis ii\] For any ordered basis $(f(1),\dots,f(n))$ adapted to $E$, there exists a (unique) basis $(v(1),\dots,v(r))$ of $V$ of the form $$v(a) \in f(I(a)) + \operatorname{span}\{ f(i) : i \in I^c, i < I(a) \}.$$ 3. \[item:adapted basis iii\] There exists an ordered basis $(f(1),\dots,f(n))$ adapted to $E$ such that $\{f(I(1)),\dots,f(I(r))\}$ is a basis of $V$. The proof of \[lem:adapted basis\] is left as an exercise to the reader. Clearly, $B(E)$ acts transitively on the set of ordered bases adapted to $E$. Thus, \[lem:adapted basis\], \[item:adapted basis iii\] shows that $\Omega^0_I(E)$ is a single $B(E)$-orbit. That is, just like Grassmannian itself, each Schubert cell is a homogeneous space. In particular, $\Omega^0_I(E)$ and its closure $\Omega_I(E)$ (\[def:schubert cell and variety\]) are both irreducible algebraic varieties. Consider the flag $E$ on $W={\mathbb C}^4$ with adapted basis $(f(1),\dots,f(4))$, where $f(1) = e(1) + e(2) + e(3)$, $f(2) = e(2) + e(3)$, $f(3) = e(3) + e(4)$, $f(4) = e(4)$. If $V = \operatorname{span}\{ e(1), e(2) \}$ then $\operatorname{Pos}(V, E) = \{2,4\}$, while $\operatorname{Pos}(V, E_0) = \{1,2\}$ for the standard flag $E_0$ with adapted basis $(e(1),e(2),e(3),e(4))$. Note that the basis $(v(1),v(2))$ of $V$ given by $v(1) = f(2) - f(1) = e(1)$ and $v(2) = f(4) - f(3) + f(1) = e(1) + e(2)$ satisfies the conditions in \[lem:adapted basis\], \[item:adapted basis ii\]. It follows that $(f(1),v(1),f(3),v(2))$ is an adapted basis of $E$ that satisfies the conditions in \[item:adapted basis iii\]. The following lemma characterizes each Schubert variety explicitly as a union of Schubert cells: \[lem:schubert variety characterization\] Let $E$ be a flag on $W$, $\dim W = n$, and $I \subseteq [n]$ a subset of cardinality $r$. Then, [ $$\label{eq:schubert variety characterization} \Omega_I(E) = \bigcup_{I' \leq I} \Omega^0_{I'}(E),$$]{} where the union is over all subsets $I' \subseteq [n]$ of cardinality $r$ such that $I'(a) \leq I(a)$ for $a\in[r]$. Recall that $\Omega_I(E)$ can be defined as the Euclidean closure of $\Omega^0_I(E)$. Thus let $(V_k)$ denote a convergent sequence of subspaces in $\Omega^0_I(E)$ with limit some $V \in \operatorname{Gr}(r, W)$. Then $\dim E(I(a)) \cap V \geq \dim E(I(a)) \cap V_k$ for sufficiently large $k$, since intersections can only become larger in the limit, but $\dim E(I(a)) \cap V_k = a$ for all $k$. It follows that $\operatorname{Pos}(V, E)(a) \leq I(a)$. Conversely, suppose that $V' \in \Omega^0_{I'}(E)$, where $I'(a) \leq I(a)$ for all $a$. Let $a'$ denote the minimal integer such that $I'(a) = I(a)$ for $a=a'+1,\dots,r$. We will show that $V' \in \Omega_I(E)$ by induction on $a'$. If $a' = 0$ then $I' = I$ and there is nothing to show. Otherwise, let $(f'(1),\dots,f'(n))$ denote an adapted basis for $E$ such that $v'(a) = f'(I'(a))$ is a basis of $V'$ (as in \[item:adapted basis iii\] of \[lem:adapted basis\]). For each $\varepsilon>0$, consider the subspace $V_\varepsilon$ with basis vectors $v_\varepsilon(a) = v'(a)$ for all $a\neq a'$ together with $v_\varepsilon(a') := v'(a') + \varepsilon f'(I(a'))$. Then the space $V_\varepsilon$ is of dimension $r$ and in position $\{ I'(1),\dots,I'(a'-1),I(a'),\dots,I(r) \}$ with respect to $E$. By the induction hypothesis, $V_\varepsilon \in \Omega_I(E)$ for any $\varepsilon > 0$, and thus $V' \in \Omega_I(E)$ as $V_\varepsilon\to V'$ for $\varepsilon\to0$. We now compute the dimensions of Schubert cells and varieties. This is straightforward from \[lem:adapted basis\], however it will be useful to make a slight detour and introduce some notation. This will allow us to show that we can exactly parametrize $\Omega^0_I(E)$ by a unipotent subgroup of $B(E)$, which in particular shows that it is an affine space. Choose an ordered basis $(f(1),\dots,f(n))$ that is adapted to $E$. Then $V := \operatorname{span}\{ f(i) : i \in I \} \in \Omega^0_I(E)$. By , \[item:adapted basis ii\] any $V\in\Omega^0_I(E)$ is of this form. Now define $$\begin{aligned} &\quad\; \operatorname{Hom}_E(V,W/V) := \{ \phi\in\operatorname{Hom}(V,W/V) : \phi(E(i)\cap V) \subseteq (E(i)+V)/V \text{~for~}i\in[n] \} \\ &= \{ \phi\in\operatorname{Hom}(V,W/V) : \phi(f(I(a))) \subseteq \operatorname{span}\{ f(I^c(b)) + V : b \in [I(a)-a] \} \text{~for~}a\in[r] \}\end{aligned}$$ where the $f(j) + V$ for $j\in I^c$ form a basis of $W/V$. In particular, $\operatorname{Hom}_E(V,W/V)$ is of dimension $\sum_{a=1}^r (I(a) - a)$. Using this basis, we can identify $W/V$ with $Q := \operatorname{span}\{ f(j) : j\in I^c \}$. Then $W = V {\oplus}Q$ and we can identify $\operatorname{Hom}_E(V,W/V)$ with $$H_E(V,Q) := \{ \phi\in\operatorname{Hom}(V,Q) : \phi(f(I(a))) \subseteq \operatorname{span}\{ f(I^c(b)) : b \in [I(a)-a] \} \text{~for~}a\in[r] \}.$$ , \[item:adapted basis ii\] shows that for any $\phi\in H_E(V,Q)$, we obtain a distinct subspace $(\operatorname{id}+\phi)(V)$ in $\Omega^0_I(E)$, and that all subspaces in $\Omega^0_I(E)$ can obtained in this way. Thus, $\Omega^0_I(E)$ is contained in the affine chart $\operatorname{Hom}(V,Q)$ of the Grassmannian described in  and isomorphic to the linear subspace $H_E(V,Q)$ of dimension $\dim I$. We define a corresponding unipotent subgroup, $$U_E(V,Q) := \{ u_\phi = \operatorname{id}+ \phi = \begin{pmatrix}\operatorname{id}_V & 0 \\ \phi & \operatorname{id}_Q \end{pmatrix} \in \operatorname{GL}(W) : \phi\in H_E(V,Q) \}.$$ Thus we obtain the following lemma: \[lem:dim schubert cell\] Let $E$ be a flag on $W$, $\dim W = n$, $I \subseteq [n]$ a subset of cardinality $r$, $V\in\Omega^0_I(E)$, and $Q$ as above. Then we can parametrize $H_E(V,Q) \cong U_E(V,Q) \cong \Omega^0_I(E) = U_E(V,Q)V$, hence $H_E(V,Q)\cong T_V \Omega^0_I(E)$ and [ $$\label{eq:dim schubert cell} \dim\Omega^0_I(E) = \dim\Omega_I(E) = \dim H_E(V,Q)= \sum_{a=1}^r \bigl( I(a) - a \bigr) =: \dim I.$$]{} It will be useful to rephrase the above to obtain a parametrization of $\Omega^0_I(E)$ in terms of the fixed subspaces $$\label{eq:V_0 and Q_0} \begin{aligned} V_0 &:= \operatorname{span}\{ f(1), \dots, f(r) \} = E(r), \\ Q_0 &:= \operatorname{span}\{ \bar f(1), \dots, \bar f(n-r) \}, \end{aligned}$$ where the $\bar f(i) := f(r+i)$ for $i\in[n-r]$ form a basis of $Q_0$. Then $W = V_0 {\oplus}Q_0$. \[def:shuffle permutation\] Let $I \subseteq [n]$ be a subset of cardinality $r$. The *shuffle permutation* $\sigma_I \in S_n$ is defined by $$\sigma_I(a) = \begin{cases} I(a) & \text{for $a=1,\dots,r$}, \\ I^c(a-r) & \text{for $a=r+1,\dots,n$}. \end{cases}$$ and $w_I \in \operatorname{GL}(W)$ is the corresponding permutation operator with respect to the adapted basis $(f(1),\dots,f(n))$, defined as $w_I \, f(i) := f(\sigma_I^{-1}(i))$ for $i\in[n]$. Then $V_0 = w_I V$, where $V = \operatorname{span}\{ f(i) : i\in I \} \in \Omega^0_I(E)$ as before, and so $$ V_0 \in w_I \Omega^0_I(E) = \Omega^0_I(w_I E)$$ using . The translated Schubert cell can be parametrized by $$\begin{aligned} H_{w_I E}(V_0, Q_0) = \{ \phi\in\operatorname{Hom}(V_0,Q_0) : \phi(f(a)) \subseteq \operatorname{span}\{ \bar f(1), \dots, \bar f(I(a) - a)\} \text{~for~} a\in[r]\},\end{aligned}$$ where we identify $Q_0\cong W/V_0$. We thus obtain the following consequence of \[lem:dim schubert cell\]: \[cor:schubert cell parametrization\] Let $E$ be a flag on $W$, $\dim W = n$, $I \subseteq [n]$ of cardinality $r$, and $V\in\Omega^0_I(E)$. Moreover, define $w_I$ as above for an adapted basis. Then, $$\Omega^0_I(E) = w_I^{-1} \Omega^0_I(w_I E) = w_I^{-1} U_{w_I E}(V_0, Q_0) V_0.$$ Let $I=\{1,3,4\}$ and $E_0$ the standard flag on $W={\mathbb C}^4$, with its adapted basis $(e(1),\dots,e(4))$. Then $\sigma_I = \bigl(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 1 & 3 & 4 & 2\end{smallmatrix}\bigr)$, $$\begin{aligned} w_I^{-1} &= \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{pmatrix} \\ \end{aligned}$$ and $V = w_I^{-1} V_0 = \operatorname{span}\{ e(1), e(3), e(4) \}$ is indeed in position $I$ with respect to $E_0$, in agreement with the preceding discussion. Moreover, $$\begin{aligned} H_{w_I E_0}(V_0,Q_0) &= \{ \begin{pmatrix} 0 & * & * \end{pmatrix} \} \subseteq \operatorname{Hom}({\mathbb C}^3, {\mathbb C}^1), \\ U_{w_I E_0}(V_0,Q_0) &= \{ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & * & * & 1 \end{pmatrix} \} \subseteq \operatorname{GL}(4), \end{aligned}$$ and so \[cor:schubert cell parametrization\] asserts that $$\Omega^0_I(E_0) = w_I^{-1} U_{w_I E_0}(V_0,Q_0) \operatorname{span}\{ e(1), e(2), e(3) \} = \operatorname{span}\{ \begin{pmatrix}1 \\ 0 \\ 0 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ * \\ 1 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ * \\ 0 \\ 1\end{pmatrix} \},$$ which agrees with \[lem:adapted basis\]. Induced flags and positions --------------------------- The space $\operatorname{Hom}_E(V,W/V)$ can be understood more conceptually as the space of homomorphisms that respect the filtrations $E(i) \cap V$ and $(E(i) + V)/V$ induced by the flag $E$. Here we have used the following concept: A *(complete) filtration* $F$ on a vector space $V$ is a chain of subspaces $$\{0\} = F(0) \subseteq F(1) \subseteq \cdots \subseteq F(i) \subseteq F({i+1}) \subseteq \dots \subseteq F(l) = V,$$ such that the dimensions increase by no more than one, i.e., $\dim F(i+1)\leq\dim F(i)+1$ for all $i=0,\dots,l-1$. Thus distinct subspaces in a filtration determine a flag. Given a flag $E$ on $W$ and a subspace $V\subseteq W$, we thus obtain an induced flag $E^V$ on $V$ from the distinct subspaces in the sequence $E(i) \cap V$, $i=0,\dots,n$. We may also induce a flag $E_{W/V}$ on the quotient $W/V$ from the distinct subspaces in the sequence $(E(i) + V)/V$. These flags can be readily computed from the Schubert position of $V$: \[lem:induced flags\] Let $E$ be a flag on $W$, $\dim W = n$, and $V \subseteq W$ an $r$-dimensional subspace in position $I = \operatorname{Pos}(V, E)$. Then the induced flags $E^V$ on $V$ and $E_{W/V}$ on $W/V$ are given by $$\begin{aligned} E^V(a) &= E(I(a)) \cap V, \\ E_{W/V}(b) &= (E(I^c(b)) + V)/V \end{aligned}$$ for $a\in[r]$ and $b\in[n-r]$, where $I^c$ denotes the complement of $I$ in $[n]$. Using an adapted basis as in \[lem:adapted basis\], \[item:adapted basis iii\], it is easy to see that $\dim E(i) \cap V = \lvert [i]\cap I \rvert$ and therefore that $\dim (E(i) + V)/V = \lvert [i]\cap I^c \rvert$. Now observe that $\lvert [i]\cap I\rvert = a$ if and only if $I(a) \leq i < I(a+1)$, while $\lvert [i]\cap I^c \rvert = b$ if and only if $I^c(b) \leq i < I^c(b+1)$. Thus we obtain the two assertions. We can use the preceding result to describe $\operatorname{Hom}_E(V,W/V)$ in terms of flags rather than filtrations and without any reference to the ambient space $W$. \[def:H\_I for flags\] Let $V$ and $Q$ be vector spaces of dimension $r$ and $n-r$, respectively, $I \subseteq[n]$ a subset of cardinality $r$, $F$ a flag on $V$ and $G$ a flag on $Q$. We define $$H_I(F,G) := \{\phi\in\operatorname{Hom}(V,Q) : \phi(F(a)) \subseteq G(I(a)-a) \},$$ which we note is well-defined by $$\label{eq:nondecreasing} 0 \leq I(a) - a \leq I(a+1) - (a+1) \leq n-r \qquad (a=1,\dots,r-1).$$ It now easily follows from \[lem:dim schubert cell,lem:induced flags\] that $$\label{eq:tangent space schubert cell} T_V \Omega^0_I(E) \cong \operatorname{Hom}_E(V, W/V) = H_I(E^V, E_{W/V}).$$ As a consequence: $$\label{eq:H_wIE} H_{w_I E}(V_0, Q_0) = H_I((w_I E)^{V_0}, (w_I E)_{Q_0}) = H_I(E^{V_0}, E_{Q_0})$$ We record the following equivariance property: \[lem:H\_I borel invariance\] Let $F$ be a flag on $V$ and $G$ a flag on $Q$. If $\phi\in H_I(F,G)$, $a\in GL(V)$ and $d\in GL(Q)$, then $d\phi a^{-1}\in H_I(aF,dG)$. In particular, $H_I(F, G)$ is stable under right multiplication by the Borel subgroup $B(F)$ and left multiplication by the Borel subgroup $B(G)$. We now compute the position of subspaces and subquotients with respect to induced flags. Given subsets $I \subseteq [n]$ of cardinality $r$ and $J \subseteq [r]$ of cardinality $d$, we recall that we had defined their *composition* $IJ$ in \[sec:easy\] as the subset $$IJ = \bigl\{ I(J(1)) < \dots < I(J(d)) \bigr\} \subseteq [n].$$ We also define their *quotient* to be the subset $$I/J = \bigl\{ I(J^c(b)) - J^c(b) + b \;:\; b \in [r-d] \bigr\} \subseteq [n-d],$$ where $J^c$ denotes the complement of $J$ in $[r]$. It follows from  that $I/J$ is indeed a subset of $[n-d]$. The following lemma establishes the ‘chain rule’ for positions: \[lem:chain rule\] Let $E$ be a flag on $W$, $S \subseteq V \subseteq W$ subspaces, and $I = \operatorname{Pos}(V, E)$, $J = \operatorname{Pos}(S, E^V)$ their relative positions. Then there exists an adapted basis $(f(1),\dots,f(n))$ for $E$ such that $\{f(I(a))\}$ is a basis of $V$ and $\{f(IJ(b))\}$ a basis of $S$. In particular, [ $$\label{eq:chain rule pos} \operatorname{Pos}(S, E) = IJ= \operatorname{Pos}(V, E) \operatorname{Pos}(S, E^V).$$]{} According to \[lem:adapted basis\], \[item:adapted basis iii\], there exists an adapted basis $(f(1),\dots,f(n))$ for $E$ such that $(f(I(1)), \dots, f(I(r)))$ is a basis of $V$, where $r=\dim V$. By \[lem:induced flags\], this ordered basis is in fact adapted to the induced flag $E^V$. Thus we can apply \[lem:adapted basis\], \[item:adapted basis ii\] to $E^V$ and the subspace $S \subseteq V$ to obtain a basis $(v(1),\dots,v(s))$ of $S$ of the form $$v(b) \in f(IJ(b)) + \operatorname{span}\{ f(I(a)) : a \in J^c, a < J(b) \}.$$ It follows that the ordered basis $(f'(1),\dots,f'(n))$ obtained from $(f(1),\dots,f(n))$ by replacing $f(IJ(b))$ with $v(b)$ has all desired properties. We now obtain the chain rule, $\operatorname{Pos}(S,E)=IJ$, as a consequence of \[lem:adapted basis\], \[item:adapted basis iii\] applied to $f'$ and $S\subseteq W$. We can visualize the subsets $IJ, IJ^c \subseteq [n]$ and $I/J \subseteq [n-d]$ as follows. Let $L$ denote the string of length $n$ defined by putting the symbol $s$ at the positions in $IJ$, $v$ at those in $I \setminus IJ = IJ^c$, and $w$ at all other positions. This mirrors the situation in the preceding \[lem:chain rule\], where the adapted basis $(f(1),\dots,f(n))$ can be partitioned into three sets according to membership in $S$, $V \setminus S$, and $W \setminus V$. Now let $L'$ denote the string of length $n-d$ obtained by deleting all occurrences of the symbol $s$. Thus the remaining symbols are either $v$ or $w$, i.e., those that were at locations $(IJ)^c$ in $L$. We observe that the $b$-th occurrence of $v$ in $L$ was at location $IJ^c(b)$, where it was preceded by $J^c(b) - b$ occurrences of $s$. Thus the occurrences of $v$ in $L'$ are given precisely by the quotient position, $(I/J)(b) = IJ^c(b) - (J^c(b) - b)$. If $n=6$, $I = \{1,3,5,6\}$ and $J = \{2,4\}$, then $IJ = \{3,6\}$ and $L = (v,w,s,w,v,s)$. It follows that $L' = (v,w,w,v)$ and hence the symbols $v$ appear indeed at positions $I/J = \{1,4\}$. We thus obtain the following recipe for computing positions of subquotients: \[lem:quotient rule\] Let $E$ be a flag on $W$ and $S\subseteq V\subseteq W$ subspaces. Then, $$\operatorname{Pos}(V/S, E_{W/S}) = \operatorname{Pos}(V, E) / \operatorname{Pos}(S, E^V).$$ Let $I = \operatorname{Pos}(V, E)$ and $J = \operatorname{Pos}(S, E^V)$. According to \[lem:chain rule\], there exists an adapted basis $(f(1),\dots,f(n))$ of $E$ such that $\{f(I(a))\}$ is a basis of $V$ and $\{f(IJ(b))\}$ a basis of $S$. This shows not only that $\{f(IJ^c(b))\}$ is a basis of $V/S$, but also, by \[lem:induced flags\], that $(f((IJ)^c(b)))$ is an adapted basis for $E_{W/S}$. Clearly, $IJ^c \subseteq (IJ)^c$, and the preceding discussion showed that the location of the $IJ^c$ in $(IJ)^c$ is exactly equal to the quotient position $I/J$. Thus we conclude from \[lem:adapted basis\], \[item:adapted basis iii\] that $\operatorname{Pos}(V/S, E_{W/S}) = I/J$. One last consequence of the preceding discussion is the following lemma: \[lem:IJ\^c\] Let $E$ be a flag on $W$, $\dim W = n$, $S \subseteq V \subseteq W$ subspaces, and $I = \operatorname{Pos}(V, E)$, $J = \operatorname{Pos}(S, E^V)$. Then $F(i) := \bigl((E(i) \cap V) + S\bigr)/S$ is a filtration on $V/S$, and $$IJ^c(b) = \min \{ i \in [n] : \dim F(i) = b \}$$ for $b=1,\dots,\dim V/S$. As in the preceding proof, we use the adapted basis $(f(1),\dots,f(n))$ from \[lem:chain rule\]. Then $\{f(IJ^c(b))\}$ is a basis of $V/S$ and $F(i) = \operatorname{span}\{ f(IJ^c(b)) : b\in[q], IJ^c(b) \leq i \}$, and this implies the claim. The following corollary uses \[lem:IJ\^c\] to compare filtrations for a space that is isomorphic to a subquotient in two different ways, $(S_1+S_2)/S_2 \cong S_1/(S_1\cap S_2)$. \[cor:slope preliminary\] Let $E$ be a flag on $W$, $\dim W = n$, and $S_1,S_2\subseteq W$ subspaces. Furthermore, let $J=\operatorname{Pos}(S_1,E)$, $K=\operatorname{Pos}(S_1\cap S_2, E^{S_1})$, $L=\operatorname{Pos}(S_1+S_2,E)$, and $M=\operatorname{Pos}(S_2, E^{S_1+S_2})$. Then both $JK^c$ and $LM^c$ are subsets of $[n]$ of cardinality $q := \dim S_1/(S_1\cap S_2) = \dim (S_1+S_2)/S_2$, and $$JK^c(b) \leq LM^c(b)$$ for $b\in[q]$. Consider the filtration $F(j) := \bigl( (E(j)\cap S_1) + (S_1\cap S_2) \bigr)/(S_1\cap S_2)$ of $S_1/(S_1\cap S_2)$ and the filtration $F'(j) := \bigl( (E(j)\cap (S_1+S_2)) + S_2 \bigr)/S_2$ of $(S_1+S_2)/S_2$. If we identify $S_1/(S_1\cap S_2) \cong (S_1+S_2)/S_2$, then $F(j)$ gets identified with the subspace $\bigl((E(j)\cap S_1) + S_2\bigr)/S_2$ of $F'(j)$. It follows that $$JK^c(b) = \min \{ j\in[n]: \dim F(j) = b \} \geq \min \{ j\in[n]: \dim F'(j) = b \} = LM^c(b),$$ where we have used \[lem:IJ\^c\] twice. We now compute the dimension of quotient positions: \[lem:quotient dim\] Let $I \subseteq [n]$ be a subset of cardinality $r$ and $J \subseteq [r]$ a subset of cardinality $d$. Then: $$\dim I/J = \dim I + \dim J - \dim IJ$$ Straight from the definition of dimension and quotient position, $$\begin{aligned} &\quad \dim I/J = \sum_{b=1}^{r-d} I(J^c(b)) - \sum_{b=1}^{r-d} J^c(b) \\ &= \bigl( \sum_{a=1}^r I(a) - \sum_{b=1}^d I(J(b)) \bigr) - \bigl( \sum_{a=1}^r a - \sum_{b=1}^d J(b) \bigr) \\ &= \sum_{a=1}^r (I(a) - a) + \sum_{b=1}^d (J(b) - b) - \sum_{b=1}^d (I(J(b)) - b) \\ &= \dim I + \dim J - \dim IJ. \qedhere \end{aligned}$$ Lastly, given subsets $I\subseteq[n]$ of cardinality $r$ and $J\subseteq[r]$ of cardinality $d$, we define $$I^J = \bigl\{ I(J(b)) - J(b) + b \;:\; b \in [d] \bigr\} \subseteq [n-(r-d)].$$ Clearly, $I^J = I/J^c$, but we prefer to introduce a new notation to avoid confusion, since the role of $I^J$ will be quite different. Indeed, $I^J$ is related to composition, as is indicated by the following lemmas: \[lem:exp dim chain rule\] Let $I\subseteq[n]$ be a subset of cardinality $r$, $J\subseteq[r]$ a subset of cardinality $d$. Then, $$\dim I^J K - \dim K = \dim I(JK) - \dim JK$$ for any subset $K\subseteq[d]$. In particular, $\dim I^J = \dim IJ - \dim J$. Let $m$ denote the cardinality of $K$. Then: $$\begin{aligned} &\quad \dim I^J K - \dim K = \sum_{c=1}^m \bigl( I^J(K(c)) - K(c) \bigr) \\ &= \sum_{c=1}^m \bigl( I(J(K(c)) - J(K(c)) \bigr) = \dim I(JK) - \dim JK. \qedhere \end{aligned}$$ \[lem:exp vs composition\] Let $I\subseteq[n]$ be a subset of cardinality $r$, $\phi \in \operatorname{Hom}(V,Q)$, and $F$ a flag on $V$. Let $S=\ker\phi$ denote the kernel, $J:=\operatorname{Pos}(S,F)$ its position with respect to $F$, and $\bar\phi\in\operatorname{Hom}(V/S,Q)$ the corresponding injection. Then $\phi\in H_I(F, G)$ if and only if $\bar\phi\in H_{I/J}(F_{V/S}, G)$. In this case, we have for all $\psi\in H_J(F^S, F_{V/S})$ that $\bar\phi\psi\in H_{I^J}(F^S, G)$. For the first claim, note that if $\phi\in H_I(F,G)$ then $$\bar\phi(F_{V/S}(b)) = \phi(F(J^c(b))) \subseteq G(I(J^c(b)) - J^c(b)) = G((I/J)(b) - b).$$ Conversely, if $\bar\phi\in H_{I/J}(F_{V/S}, G)$, then this shows that $$\phi(F(a)) \subseteq G(I(a) - a)$$ for all $a = J^c(b)$, and hence for all $a$, since $\phi(F(J^c(b))) = \dots = \phi(F(J^c(b+1)-1))$. For the second, we use $H_J(F^S, F_{V/S}) = \operatorname{Hom}_F(S, V/S)$ (\[eq:tangent space schubert cell\]) and compute $$\begin{aligned} &\quad \bar\phi\psi(F^S(a)) = \bar\phi\psi(F(J(a)) \cap S) \subseteq \bar\phi((F(J(a)) + S)/S) \\ &= \phi(F(J(a))) \subseteq G(I(J(a)) - J(a)) = G(I^J(a) - a). \qedhere \end{aligned}$$ The flag variety ---------------- The Schubert cells of the Grassmannian were defined by fixing a flag and classifying subspaces according to their Schubert position. As we will later be interested in intersections of Schubert cells for different flags, it will be useful to also consider variations of the flag for a fixed subspace. Let $\operatorname{Flag}(W)$ denote the *(complete) flag variety*, defined as the space of (complete) flags on $W$. It is a homogeneous space with respect to the transitive $\operatorname{GL}(W)$-action, so indeed an irreducible variety. Let $V \subseteq W$ be a subspace, $\dim V=r$, $\dim W=n$, and $I \subseteq [n]$ a subset of cardinality $r$. We define $$\operatorname{Flag}^0_I(V,W) = \{ E \in \operatorname{Flag}(W) : \operatorname{Pos}(V, E) = I \},$$ and $\operatorname{Flag}_I(V,W)$ as its closure in $\operatorname{Flag}(W)$ (in either the Euclidean or the Zariski topology). We have the following equivariance property as a consequence of : For all $\gamma \in \operatorname{GL}(W)$, $$\label{eq:flag equivariance} \gamma \operatorname{Flag}^0_I(V, W) = \operatorname{Flag}^0_I(\gamma V,W).$$ In particular, $\operatorname{Flag}^0_I(V,W)$ and $\operatorname{Flag}_I(V,W)$ are stable under the action of the parabolic subgroup $P(V, W) = \{ \gamma \in \operatorname{GL}(W) : \gamma V \subseteq V \}$, which is the stabilizer of $V$. We will now show that $\operatorname{Flag}^0_I(V,W)$ is in fact a single $P(V,W)$-orbit. This implies that both $\operatorname{Flag}^0_I(V,W)$ and $\operatorname{Flag}_I(V,W)$ are irreducible algebraic varieties. \[def:G\_I\] Let $E$ be a flag on $W$, $\dim W=n$, $V_0 = E(r)$, and $I \subseteq [n]$ a subset of cardinality $r$. We define $$G_I(V_0,E) := \{ \gamma\in\operatorname{GL}(W) : \gamma E \in \operatorname{Flag}^0_I(V_0,W) \},$$ so that $\operatorname{Flag}^0_I(V_0,W) \cong G_I(V_0,E)/B(E)$. \[lem:G\_I coarse\] Let $E$ be a flag on $W$, $\dim W=n$, $V_0 = E(r)$, and $I \subseteq [n]$ a subset of cardinality $r$. Then, $G_I(V_0,E) = P(V_0,W) w_I B(E)$. In particular, $\operatorname{Flag}^0_I(V_0,W) = P(V_0,W) w_I E$. Let $\gamma\in\operatorname{GL}(W)$. Then, $$\begin{aligned} \gamma\in G_I(V_0,E) \Leftrightarrow V_0\in\Omega^0_I(\gamma E)=\gamma\Omega^0_I(E)=\gamma B(E)w_I^{-1}V_0 \Leftrightarrow \gamma \in P(V_0,W)w_I B(E), \end{aligned}$$ where we have used that $\Omega^0_I(E) = B(E) w_I^{-1}V_0$. We now derive a more precise parametrization of $\operatorname{Flag}^0_I(V_0,W)$. \[lem:G\_I precise\] Let $E$ be a flag on $W$, $\dim W=n$, $V_0$ and $Q_0$ as in , and $I \subseteq [n]$ a subset of cardinality $r$. Then we have that $G_I(V_0,E) = P(V_0,W) U_{w_I E}(V_0,Q_0) w_I$. Let $\gamma\in\operatorname{GL}(W)$. Then, $$\begin{aligned} &\qquad \gamma\in G_I(V_0,E) \;\Leftrightarrow\; V_0\in\Omega^0_I(\gamma E)=\gamma\Omega^0_I(E) = \gamma w_I^{-1} U_{w_I E}(V_0, Q_0) V_0 \\ &\Leftrightarrow \gamma\in P(V_0,W)U_{w_I E}(V_0,Q_0)w_I, \end{aligned}$$ since $\Omega^0_I(E) = w_I^{-1} U_{w_I E}(V_0, Q_0) V_0$ (\[cor:schubert cell parametrization\]). In coordinates, using $W=V_0{\oplus}Q_0$ and , we obtain that $$\begin{aligned} G_I(V_0,E) &= \{ \begin{pmatrix}a & b\\0 & d\end{pmatrix} \begin{pmatrix}1 & 0\\\phi & 1\end{pmatrix} : a\in\operatorname{GL}(V_0), d\in\operatorname{GL}(Q_0), \phi\in H_I(E^{V_0}, E_{Q_0}) \} w_I \\ &= \{ \begin{pmatrix}a & b\\c & d\end{pmatrix} : a-bd^{-1}c\in\operatorname{GL}(V_0), d\in\operatorname{GL}(Q_0), d^{-1}c\in H_I(E^{V_0}, E_{Q_0}) \} w_I.\end{aligned}$$ In particular, $\dim G_I(V_0,E) = \dim P(V_0,W) + \dim I$. This allows us to compute the dimension of the subvarieties $\operatorname{Flag}^0_I(V,W)$ and to relate their codimension to the codimension of the Schubert cells of the Grassmannian: Let $V\subseteq W$ be a subspace, $\dim W=n$, $\dim V=r$, and $I \subseteq [n]$ a subset of cardinality $r$. Then, $$\label{eq:dim flag^0_I} \dim \operatorname{Flag}^0_I(V,W) = \dim \operatorname{Flag}_I(V,W) = \dim \operatorname{Flag}(V) + \dim \operatorname{Flag}(Q) + \dim I$$ and $$\label{eq:codim flag vs grass} \dim \operatorname{Flag}(W) - \dim \operatorname{Flag}^0_I(V,W) = \dim\operatorname{Gr}(r,W) - \dim I.$$ Without loss of generality, we may assume that $V=V_0=E(r)$ for some flag $E$ on $W$. Then, $\operatorname{Flag}_I^0(V_0,W)\cong G_I(V_0,E)/B(E)$ and hence $$\begin{aligned} &\quad \dim \operatorname{Flag}_I^0(V_0,E) = \dim P(V_0,W) + \dim I - \dim B(E)\\ &= \dim\operatorname{GL}(W) - \dim\operatorname{Gr}(r,W) + \dim I - \dim B(E) \\ &= \dim\operatorname{Flag}(W) - \dim\operatorname{Gr}(r,W) + \dim I \end{aligned}$$ since $\operatorname{Gr}(r,W)\cong\operatorname{GL}(W)/P(V_0,W)$ and $\operatorname{Flag}(W)=\operatorname{GL}(W)/B(E)$. This establishes . On the other hand, a direct calculation shows that $$\dim\operatorname{Flag}(W)-\dim\operatorname{Gr}(r,W) = \dim\operatorname{Flag}(V)+\dim\operatorname{Flag}(Q),$$ so we also obtain . At last, we study the following set of flags on the target space of a given homomorphism: \[def:flags for injective morphism\] Let $V$, $Q$ be vector spaces of dimension $r$ and $n-r$, respectively, and $I\subseteq[n]$. Moreover, let $F$ be a flag on $V$ and $\phi\in\operatorname{Hom}(V,Q)$ an *injective* homomorphism. We define $$\operatorname{Flag}^0_I(F,\phi) := \{ G \in \operatorname{Flag}(Q) : \phi\in H_I(F,G) \}$$ where we recall that $H_I(F,G)$ was defined in \[def:H\_I for flags\]. It is clear that $I(a)\geq 2a$ is necessary and sufficient for $\operatorname{Flag}_I^0(F,\phi)$ to be nonempty. Let $V_0\cong{\mathbb C}^3$, with basis $e(1),\dots,e(3)$, and $Q_0\cong{\mathbb C}^5$, with basis $\bar e(1),\dots,\bar e(5)$. Take $\phi\colon V_0\to Q_0$ to be the canonical injection and let $F_0$ denote the standard flag on $V_0$. For $I=\{3,4,7\}$, $G\in\operatorname{Flag}^0_I(F_0,\phi)$ if and only if $${\mathbb C}\bar e(1) \subseteq G(2), \quad {\mathbb C}\bar e(1) {\oplus}{\mathbb C}\bar e(2) \subseteq G(2), \quad {\mathbb C}\bar e(1){\oplus}{\mathbb C}\bar e(2){\oplus}{\mathbb C}\bar e(3) \subseteq G(4).$$ For example, the standard flag $G_0$ on $Q_0$ is a point in $\operatorname{Flag}^0_I(F_0,\phi)$. On the other hand, if $I=\{2,3,7\}$ then we obtain the condition ${\mathbb C}\bar e(1) {\oplus}{\mathbb C}\bar e(2) \subseteq G(1)$ which can never be satisfied. Thus in this case $\operatorname{Flag}^0_I(F_0,\phi)=\emptyset$. In the following lemma we show that $\operatorname{Flag}_I^0(F,\phi)$ is a smooth variety and compute its dimension. \[lem:image flags\] Let $V$, $Q$ be vector spaces of dimension $r$ and $n-r$, respectively, and $I\subseteq[n]$ a subset of cardinality $r$. Moreover, let $F$ be a flag on $V$ and $\phi\in\operatorname{Hom}(V,Q)$ an *injective* homomorphism. If $\operatorname{Flag}_I^0(F,\phi)$ is nonempty, that is, if $I(a)\geq2a$ for all $a\in[r]$, then it is a smooth irreducible subvariety of $\operatorname{Flag}(Q)$ of dimension $$\dim \operatorname{Flag}_I^0(F,\phi) = \dim \operatorname{Flag}(Q_0) + \dim I - r(n-r).$$ Without loss of generality, we may assume that $V=V_0\cong{\mathbb C}^r$, $Q=Q_0\cong{\mathbb C}^{n-r}$, that $F=F_0$ is the standard flag on $V_0$ and $\phi$ the canonical injection ${\mathbb C}^r\to{\mathbb C}^{n-r}$. Then the standard flag $G_0$ on $Q_0$ is an element of $\operatorname{Flag}^0_I(F_0,\phi)$. We will show that $$M_I := \{ h\in\operatorname{GL}(Q_0) : h \, G_0 \in \operatorname{Flag}^0_I(F_0,\phi) \}$$ is a subvariety of $\operatorname{GL}(Q_0)$ and compute its dimension. Note that $h \in M_I$ if and only if $h^{-1}\phi\in H_I(F_0, G_0)$. We now identify $V_0$ with its image $\phi(V_0)$ and denote by $R_0\cong{\mathbb C}^{n-2r}$ its standard complement in $Q_0$. Thus $Q_0 \cong V_0 {\oplus}R_0$ and we can think of $h^{-1}\in\operatorname{GL}(Q_0)$ as a block matrix $$h^{-1} = \begin{pmatrix} A & B \end{pmatrix}$$ where $A\in\operatorname{Hom}(V_0,Q_0)$ and $B\in\operatorname{Hom}(R_0,Q_0)$. The condition $h^{-1}\phi\in H_I(F_0, G_0)$ amounts to demanding that $A\in H_I(F_0, G_0)$, while $B$ is unconstrained. Thus we can identify $M_I$ via $h\mapsto h^{-1}$ with the invertible elements in $$H_I(F_0, G_0) \times \operatorname{Hom}(R_0,Q_0),$$ which form a nonempty Zariski-open subset, and hence a smooth irreducible subvariety of $\operatorname{GL}(Q_0)$. It follows that $\operatorname{Flag}_I^0(F_0,\phi) = M_I / B(G_0)$ is likewise a smooth irreducible subvariety, and $$\begin{aligned} &\quad \operatorname{Flag}_I^0(F_0,\phi) = \dim M_I - \dim B(G_0) = \dim I + (n-r)(n-2r) - \dim B(G_0)\\ &= \dim\operatorname{Flag}(Q_0) + \dim I - (n-r)r, \end{aligned}$$ where we have used \[eq:dim schubert cell\] and that $\operatorname{Flag}(Q_0) \cong \operatorname{GL}(Q_0)/B(G_0)$. Intersections and Horn inequalities {#sec:horn necessary} =================================== In this section, we study intersections of Schubert varieties. Recall from \[def:intersecting\] that given an $s$-tuple ${\mathcal E}$ of flags on $W$, $\dim W = n$, and ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$, we had defined $$\Omega^0_{\mathcal I}({\mathcal E}) = \bigcap_{k=1}^s \Omega_{I_k}^0(E_k) \quad\text{and}\quad \Omega_{\mathcal I}({\mathcal E}) = \bigcap_{k=1}^s \Omega_{I_k}(E_k).$$ We are particularly interested in the intersecting ${\mathcal I}$, denoted ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$, for which $\Omega_{\mathcal I}({\mathcal E})\neq\emptyset$ for every ${\mathcal E}$. Coordinates {#subsec:coordinates} ----------- Without loss of generality, we may assume that $W={\mathbb C}^n$, and we shall do so for the remainder of this article. As before, we denote by $(e(1),\dots,e(n))$ the ordered standard basis of ${\mathbb C}^n$ and by $E_0$ the corresponding *standard flag*. Let $V_0 = E_0(r)$ be the standard $r$-dimensional subspace, with ordered basis $(e(1),\dots,e(r))$, and $Q_0$ the subspace with ordered basis $(\bar e(1),\dots,\bar e(n-r))$, where $\bar e(b) := e(r+b)$. Thus $W=V_0{\oplus}Q_0$. We denote the corresponding standard flags on $V_0$ and $Q_0$ by $F_0$ and $G_0$, respectively. Note that $F_0=E_0^{V_0}$ and, if we identify $Q_0\cong W/V_0$, then $G_0=(E_0)_{W/V_0}$. We further abbreviate the Grassmannian by $\operatorname{Gr}(r,n):=\operatorname{Gr}(r,{\mathbb C}^n)$, the parabolic by $P(r,n):=P(V_0,{\mathbb C}^n)$ and the Borel by $B(n):=B(E_0)$. We write $\operatorname{Flag}(n) := \operatorname{Flag}(W)$ and $\operatorname{Flag}^0_I(r,n) := \operatorname{Flag}^0_I(V_0, W)$ for the set of flags with respect to which $V_0$ has position $I$; $\operatorname{Flag}_I(r,n) := \operatorname{Flag}_I(V_0,W)$ is its closure. We recall from \[def:H\_I for flags\] that $$\begin{aligned} H_I(F_0, G_0) = \{ \phi\in\operatorname{Hom}(V_0,Q_0) : \phi(e(a))\subseteq\operatorname{span}\{\bar e(1),\dots,\bar e(I(a)-a)\} \},\end{aligned}$$ and \[lem:G\_I precise\] reads $$\label{eq:G_I precise concrete} G_I(r,n) = P(r,n) \bigl\{ \begin{pmatrix}\operatorname{id}_{V_0} & 0 \\ \phi & \operatorname{id}_{Q_0}\end{pmatrix} : \phi \in H_I(F_0, G_0) \bigr\} w_I,$$ where we have introduced $G_I(r,n) := G_I(V_0,E_0)$. Intersections and dominance --------------------------- We start by reformulating the intersecting property in terms of the dominance of certain morphisms of algebraic varieties. This allows us to give a simple proof of \[lem:edim nonnegative\], which states that the expected dimension of an intersecting tuple is necessarily nonnegative. We caution that while $\Omega^0_{\mathcal I}({\mathcal E}) \subseteq \Omega_{\mathcal I}({\mathcal E})$, the latter is *not* necessarily the closure of the former: Let $W = {\mathbb C}^2$, $I_1 = \{1\}$, $I_2 = \{2\}$, and $E_1 = E_2$ the same flag on $W$. Since the Schubert cells $\Omega^0_{I_k}({\mathcal E})$ partition the projective space ${\mathbb P}(W) = \operatorname{Gr}(1, W)$, $\Omega^0_{\mathcal I}({\mathcal E}) = \emptyset$ is empty, but $\Omega_{\mathcal I}({\mathcal E}) = \{ E_1(1) \}$ is a point. It is also possible that $\Omega^0_{\mathcal I}({\mathcal E})$ or $\Omega_{\mathcal I}({\mathcal E})$ are nonempty for some ${\mathcal E}$ but empty for generic $s$-tuples ${\mathcal E}$: Let $W = {\mathbb C}^2$, $I_1 = I_2 = \{1\}$. Then $\Omega^0_{\mathcal I}({\mathcal E}) = \Omega_{\mathcal I}({\mathcal E}) = E_1(1) \cap E_2(1)$, so the intersection is nonempty if and only if $E_1 = E_2$. We will later show the existence of a ‘good set’ of sufficiently generic ${\mathcal E}$ such that ${\mathcal I}$ is intersecting if and only if $\Omega^0_{\mathcal I}({\mathcal E})\neq\emptyset$ for any single ‘good’ ${\mathcal E}$ (\[lem:good set\]). Here is a more interesting example: \[ex:interesting\] Let $W = {\mathbb C}^6$, $s = 3$, and ${\mathcal I}= (I_1, I_2, I_3)$ where all $I_k = \{2,4,6\}$. The triple ${\mathcal I}$ is intersecting. Let $$f(t) := e_1 + t e_2 + \frac {t^2} {2!} e_3 + \frac {t^3} {3!} e_4 + \frac {t^4} {4!} e_5 + \frac {t^5} {5!} e_6$$ and consider the one-parameter family of flags $E(t)$ with adapted basis $(f(t), \frac d{dt}f(t),\dots,\frac {d^5}{dt^5} f(t))$. We consider the 3-tuple ${\mathcal E}= (E_1, E_2, E_3)$, where $E_1 := E(0)$ is the standard flag, $E_2 := E(1)$, and $E_3 := E(-1)$. Then the intersection $\Omega^0_{\mathcal I}({\mathcal E})$ consists of precisely two points: $$\begin{aligned} V_1 &= \operatorname{span}\{ e_2 + \sqrt 5 e_1, e_4 - 24 \sqrt 5 e_1 - 3 \sqrt 5 e_3, e_6 - 24 \sqrt 5 e_3 + \sqrt 5 e_5 \}, \\ V_2 &= \operatorname{span}\{ e_2 - \sqrt 5 e_1, e_4 + 24 \sqrt 5 e_1 + 3 \sqrt 5 e_3, e_6 + 24 \sqrt 5 e_3 - \sqrt 5 e_5 \}, \end{aligned}$$ and coincides with $\Omega_{\mathcal I}(\mathcal E)$. To study generic intersections of Schubert cells, it is useful to introduce the following maps: Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. We define $$\omega^0_{\mathcal I}\colon \begin{cases} \operatorname{GL}(n) \times \operatorname{Flag}^0_{I_1}(r,n)\times\dots\times\operatorname{Flag}^0_{I_s}(r,n) \to \operatorname{Flag}(n)^s \\ (\gamma, E_1, \dots, E_s) \mapsto (\gamma E_1, \dots, \gamma E_s) \end{cases}$$ and its extension $$\label{eq:delta_I} \omega_{\mathcal I}\colon \begin{cases} \operatorname{GL}(n) \times \operatorname{Flag}_{I_1}(r,n)\times\dots\times\operatorname{Flag}_{I_s}(r,n) \to \operatorname{Flag}(n)^s \\ (\gamma, E_1, \dots, E_s) \mapsto (\gamma E_1, \dots, \gamma E_s). \end{cases}$$ The following lemma shows that the images of $\omega^0_{\mathcal I}$ and $\omega_{\mathcal I}$, respectively, characterize the $s$-tuples ${\mathcal E}$ of flags for which the intersections $\Omega^0_{\mathcal I}({\mathcal E})$ and $\Omega_{\mathcal I}({\mathcal E})$ are nonempty: \[lem:images\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. Then, $$\begin{aligned} \operatorname{im}\omega^0_{\mathcal I}&= \{ {\mathcal E}\in \operatorname{Flag}(n)^s : \Omega^0_{\mathcal I}({\mathcal E})\neq\emptyset \}, \\ \operatorname{im}\omega_{\mathcal I}&= \{ {\mathcal E}\in \operatorname{Flag}(n)^s : \Omega_{\mathcal I}({\mathcal E})\neq\emptyset \}. \end{aligned}$$ In particular, ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$ if and only if $\omega_{\mathcal I}$ is surjective. If ${\mathcal E}\in \operatorname{im}\omega^0_{\mathcal I}$ then there exists $\gamma \in \operatorname{GL}(n)$ such that $E_k \in \gamma \operatorname{Flag}^0_{I_k}(r,n)$ for $k\in[s]$. But $$E_k \in \gamma \operatorname{Flag}^0_{I_k}(r,n) \Leftrightarrow \gamma^{-1} E_k \in \operatorname{Flag}^0_{I_k}(r,n) \Leftrightarrow V_0 \in \Omega^0_{I_k}(\gamma^{-1} E_k) \Leftrightarrow \gamma V_0 \in \Omega^0_{I_k}(E_k),$$ and therefore $\gamma V_0 \in \Omega^0_{\mathcal I}({\mathcal E})$. Conversely, if $V \in \Omega^0_{\mathcal I}({\mathcal E})$, then we write $V = \gamma V_0$ and obtain that $E_k \in \gamma \operatorname{Flag}^0_{I_k}(r,n)$ for all $k$, and hence that ${\mathcal E}\in\operatorname{im}\omega^0_{\mathcal I}$. The result for $\operatorname{im}\omega_{\mathcal I}$ is proved in the same way. We now use some basic algebraic geometry (see, e.g., [@perrin2007algebraic]). Recall that a morphism $f\colon{\mathcal X}\to{\mathcal Y}$ of irreducible algebraic varieties is called *dominant* if its image is Zariski dense. In this case, the image contains a nonempty Zariski-open subset ${\mathcal Y}_0$ such that the dimension of any irreducible component of the fibers $f^{-1}(y)$ for $y \in {\mathcal Y}_0$ is equal to $\dim {\mathcal X}- \dim {\mathcal Y}$. Furthermore, if ${\mathcal X}_0 \subseteq {\mathcal X}$ is a nonempty Zariski-open subset then $f$ is dominant if and only if its restriction $f$ to ${\mathcal X}_0$ is dominant. We also recall for future reference the following results: If ${\mathcal X}$ and ${\mathcal Y}$ are smooth (irreducible algebraic) varieties and $f\colon{\mathcal X}\to{\mathcal Y}$ is dominant then the set of regular values (i.e., the points $y$ such that $df_x$ is surjective for all preimages $x\in f^{-1}(y)$) contains a Zariski-open set. Also, if $df_x$ is surjective for every $x$ then the image by $f$ of any Zariski-open set in ${\mathcal X}$ is a Zariski-open set in ${\mathcal Y}$. In particular this is the case when $f\colon\mathcal V\to \mathcal B$ is a vector bundle. In the present context, the maps $\omega^0_{\mathcal I}$ and $\omega_{\mathcal I}$ are morphisms of irreducible algebraic varieties and so the preceding discussion applies. Furthermore, the domain of $\omega_{\mathcal I}$ is the closure of the domain of $\omega^0_{\mathcal I}$ in $\operatorname{GL}(n) \times \operatorname{Flag}(n)^s$. Therefore, $\omega_{\mathcal I}$ is dominant if and only if $\omega^0_{\mathcal I}$ is dominant. \[lem:dominant\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. Then ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$ if and only if $\omega_{\mathcal I}$ or $\omega_{\mathcal I}^0$ is dominant. On the one hand, \[lem:images\] shows that ${\mathcal I}$ is intersecting if and only if $\omega_{\mathcal I}$ is surjective. On the other hand, we have just observed that $\omega_{\mathcal I}^0$ is dominant if and only if $\omega_{\mathcal I}$ is dominant. Thus it remains to show that $\omega_{\mathcal I}$ is automatically surjective if it is dominant. For this, we observe that the space $\operatorname{Flag}_{I_1}(r,n)\times\dots\times\operatorname{Flag}_{I_s}(r,n)$ is left invariant by the diagonal action of the parabolic $P(r,n)$, as can be seen from . Thus $\omega_{\mathcal I}$ factors over a map $$\label{eq:delta I factorized} \bar\omega_{\mathcal I}\colon \begin{cases} \operatorname{GL}(n) \times_{P(r,n)} \operatorname{Flag}_{I_1}(r,n)\times\dots\times\operatorname{Flag}_{I_s}(r,n) \to \operatorname{Flag}(n)^s \\ [\gamma, E_1, \dots, E_s] \mapsto (\gamma E_1, \dots, \gamma E_s). \end{cases}$$ Clearly, $\omega_{\mathcal I}$ and $\bar\omega_{\mathcal I}$ have the same image. If $\bar\omega_{\mathcal I}$ is dominant, then its image contains a nonempty Zariski-open set and therefore is dense in the Euclidean topology. But the domain of $\bar\omega_{\mathcal I}$ is compact in the Euclidean topology and hence the image is also closed in the Euclidean topology. It follows that $\bar\omega_{\mathcal I}$ is automatically surjective if $\bar\omega_{\mathcal I}$ is dominant. A first, obvious condition for ${\mathcal I}$ to be intersecting is therefore that the dimension of the domain of $\omega_{\mathcal I}$ is no smaller than the dimension of the target space. If we apply this argument to the factored map , which has the same image, we obtain that the expected dimension introduced in \[def:edim\] is nonnegative: \[lem:edim nonnegative\] If ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$ then [ $$\label{eq:edim nonnegative} \operatorname{edim}{\mathcal I}= r(n-r) - \sum_{k=1}^s (r(n-r) - \dim I_k)\geq0.$$]{} Let ${\mathcal X}:= \operatorname{GL}(n) \times_{P(r,n)} \operatorname{Flag}_{I_1}(r,n)\times\dots\times\operatorname{Flag}_{I_s}(r,n)$ and ${\mathcal Y}:= \operatorname{Flag}(n)^s$. If ${\mathcal I}$ is intersecting then the map $\bar\omega_{\mathcal I}\colon{\mathcal X}\to{\mathcal Y}$ in  is dominant, hence $\dim{\mathcal X}\geq\dim{\mathcal Y}$. But $$\label{eq:edim as dim diff} \begin{aligned} &\quad \dim{\mathcal X}-\dim{\mathcal Y}= (\dim \operatorname{GL}(n)/P(r,n)) + \sum_{k=1}^s (\dim\operatorname{Flag}_{I_k}(r,n) - \dim \operatorname{Flag}(n)) \\ &= \dim \operatorname{Gr}(r,n) - \sum_{k=1}^s (\dim \operatorname{Gr}(r,n) - \dim I_k) = \operatorname{edim}{\mathcal I}\end{aligned}$$ where the first equality is obvious and the second is \[eq:codim flag vs grass\]. At this point, we have established all facts that we used in \[sec:easy\] to prove \[cor:belkale inductive\]. That is, the proof of \[cor:belkale inductive\] is now complete. We conclude this section by recording the following rules for the expected dimension, $$\begin{aligned} \label{eq:quotient edim} \operatorname{edim}{\mathcal I}/{\mathcal J}&= \operatorname{edim}{\mathcal I}+ \operatorname{edim}{\mathcal J}- \operatorname{edim}{\mathcal I}{\mathcal J}, \\ \label{eq:exp edim induction} \operatorname{edim}{\mathcal I}^{\mathcal J}{\mathcal K}- \operatorname{edim}{\mathcal K}&= \operatorname{edim}{\mathcal I}({\mathcal J}{\mathcal K}) - \operatorname{edim}{\mathcal J}{\mathcal K}, \\ \label{eq:exp edim} \operatorname{edim}{\mathcal I}^{\mathcal J}&= \operatorname{edim}{\mathcal I}{\mathcal J}- \operatorname{edim}{\mathcal J},\end{aligned}$$ which hold for all ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$, ${\mathcal J}\in\operatorname{Subsets}(d,r,s)$, and ${\mathcal K}\in\operatorname{Subsets}(m,d,s)$. are direct consequences of \[lem:quotient dim,lem:exp dim chain rule\]. in particular will play a crucial role in \[subsec:kernel recurrence\], as we will use it to show that if ${\mathcal I}$ satisfies the Horn inequalities and ${\mathcal J}$ is intersecting then so does ${\mathcal I}^{\mathcal J}$. This will be key to establishing Belkale’s theorem on the sufficiency of the Horn inequalities by induction (\[thm:belkale\]). Slopes and Horn inequalities ---------------------------- We are now interested in proving a strengthened version of \[cor:belkale inductive\] (see \[cor:belkale inductive stronger\] below). As a first step, we introduce the promised ‘good set’ of $s$-tuples of flags which are sufficiently generic to detect when an $s$-tuple ${\mathcal I}$ is intersecting: Define in analogy to  the map $$ \bar\omega^0_{\mathcal I}\colon \begin{cases} \operatorname{GL}(n) \times_{P(r,n)}\operatorname{Flag}^0_{I_1}(r,n)\times\cdots\times\operatorname{Flag}^0_{I_s}(r,n) \to \operatorname{Flag}(n)^s \\ [\gamma, E_1, \dots, E_s] \mapsto (\gamma E_1, \dots, \gamma E_s) \end{cases}.$$ \[lem:good set\] There exists a nonempty Zariski-open subset $\operatorname{Good}(n, s) \subseteq \operatorname{Flag}(n)^s$ that satisfies the following three properties for all $r\in[n]$: 1. \[item:good set a\] $\operatorname{Good}(n,s)$ consists of regular values (in the image) of $\bar\omega^0_{\mathcal I}$ for every ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$. 2. \[item:good set b\] For every ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$, the following are equivalent: 1. \[item:good set i\] ${\mathcal I}\in \operatorname{Intersecting}(r, n, s)$. 2. \[item:good set ii\] For all ${\mathcal E}\in \operatorname{Good}(n, s)$, $\Omega^0_{\mathcal I}({\mathcal E}) \neq \emptyset$. 3. \[item:good set iii\] There exists ${\mathcal E}\in \operatorname{Good}(n, s)$ such that $\Omega^0_{{\mathcal I}}({\mathcal E}) \neq \emptyset$. 3. \[item:good set c\] If ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$, then for every ${\mathcal E}\in\operatorname{Good}(n,s)$ the variety $\Omega^0_{\mathcal I}({\mathcal E})$ has the same number of irreducible components, each connected component is of dimension $\operatorname{edim}{\mathcal I}$, and $\Omega^0_{{\mathcal I}}({\mathcal E})$ is dense in $\Omega_{{\mathcal I}}({\mathcal E})$. Let us construct $\operatorname{Good}(n,s)$ satisfying the properties above. Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$, where $r\in[n]$. If ${\mathcal I}\not\in\operatorname{Intersecting}(r,n,s)$ then by \[lem:dominant\] the map $\omega^0_{\mathcal I}$ is not dominant, and we define $U_{\mathcal I}$ as the complement of the Zariski-closure of $\operatorname{im}\omega^0_{\mathcal I}$. Thus $U_{\mathcal I}$ is a nonempty Zariski-open subset of $\operatorname{Flag}(n)^s$. Otherwise, if ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$ then $\omega^0_{\mathcal I}$ is dominant by \[lem:dominant\]. The map $\bar\omega^0_{\mathcal I}$ has the same image as $\omega^0_{\mathcal I}$ and is therefore also a dominant map between smooth irreducible varieties. Thus its image contains a nonempty Zariski-open subset $U_{\mathcal I}$ of $\operatorname{Flag}(n)^s$ consisting of regular values, such that the fibers $(\bar\omega^0_{\mathcal I})^{-1}({\mathcal E})$ for ${\mathcal E}\in U_{{\mathcal I}}$ all have the same number of irreducible components, each of dimension equal to $\operatorname{edim}{\mathcal I}$, by the calculation in . We now define the good set as $$\operatorname{Good}(n,s) := \bigcap_{\mathcal I}U_{\mathcal I},$$ where the intersection is over all $s$-tuples ${\mathcal I}$, intersecting or not. As a finite intersection of nonempty Zariski-open subsets, $\operatorname{Good}(n,s)$ is again nonempty and Zariski-open. By construction, it satisfies property \[item:good set a\]. We now show that $\operatorname{Good}(n,s)$ satisfies \[item:good set b\]. To see that \[item:good set i\] implies \[item:good set ii\], note that for any ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$ and ${\mathcal E}\in\operatorname{Good}(n,s)$, ${\mathcal E}\in U_{\mathcal I}\subseteq \operatorname{im}\bar\omega_{\mathcal I}^0 = \operatorname{im}\omega_{\mathcal I}^0$. Thus \[lem:images\] shows that $\Omega^0_{\mathcal I}({\mathcal E})\neq\emptyset$. Clearly, \[item:good set ii\] implies \[item:good set iii\] since $\operatorname{Good}(n,s)$ is nonempty. Lastly, suppose that \[item:good set iii\] holds. By \[lem:images\], $\Omega^0_{\mathcal I}({\mathcal E})\neq\emptyset$ implies that ${\mathcal E}\in\operatorname{im}\omega_{\mathcal I}^0$. But ${\mathcal E}\in\operatorname{Good}(n,s) \subseteq (\operatorname{im}\omega_{\mathcal I})^c \subseteq (\operatorname{im}\omega_{\mathcal I}^0)^c$ unless ${\mathcal I}$ is intersecting; this establishes \[item:good set i\]. Lastly, we verify  \[item:good set c\]. Observe that, for any ${\mathcal E}\in\operatorname{Flag}(n)^s$, the fiber $(\bar\omega^0_{\mathcal I})^{-1}({\mathcal E})$ is equal to the set of $[\gamma, \gamma^{-1}E_1, \dots, \gamma^{-1}E_s]$ such that $\gamma^{-1} E_k \in \operatorname{Flag}^0_{I_k}(r,n)$ for all $k\in[s]$. It can therefore by $\gamma \mapsto \gamma V_0$ be identified with $\Omega^0_{\mathcal I}({\mathcal E})$. Now assume that ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$. As we vary ${\mathcal E}\in\operatorname{Good}(n,s)$, ${\mathcal E}\in U_{\mathcal I}$ and so $(\bar\omega^0_{\mathcal I})^{-1}({\mathcal E}) \cong \Omega^0_{\mathcal I}({\mathcal E})$ has the same number of irreducible components, each of dimension $\operatorname{edim}{\mathcal I}$. We still need to show that $\Omega_{\mathcal I}^0({\mathcal E})$ is dense in $\Omega_{\mathcal I}({\mathcal E})$. This will follow if we can show that $\Omega_{\mathcal I}^0({\mathcal E})$ meets any irreducible component ${\mathcal Z}$ of $\Omega_{\mathcal I}({\mathcal E})$. Let us assume that this is not the case, so that ${\mathcal Z}\subseteq \Omega_{\mathcal I}({\mathcal E}) \setminus \Omega_{\mathcal I}^0({\mathcal E})$. But $$\Omega_{\mathcal I}({\mathcal E}) \setminus \Omega_{\mathcal I}^0({\mathcal E}) = \bigcup_{k=1}^s \biggl( (\Omega_{I_k}(E_k) \setminus \Omega^0_{I_k}(E_k)) \cap \bigcap_{l\neq k} \Omega_{I_l}(E_l) \biggr) = \bigcup_{\substack{I'_1\leq I_1,\ \dots,\ I'_s\leq I_s \\ \exists k \in [s]: I'_k \neq I_k}} \Omega^0_{{\mathcal I}'}$$ by \[lem:schubert variety characterization\]. That is, $\Omega_{\mathcal I}({\mathcal E}) \setminus \Omega_{\mathcal I}^0({\mathcal E})$ is a union of varieties $\Omega^0_{{\mathcal I}'}({\mathcal E})$ with $\operatorname{edim}{\mathcal I}'<\operatorname{edim}{\mathcal I}$. If ${\mathcal I}'$ is intersecting then any irreducible component of $\Omega^0_{{\mathcal I}'}({\mathcal E})$ has dimension equal to $\operatorname{edim}{\mathcal I}'$. Otherwise, if ${\mathcal I}'$ is not intersecting, then $\Omega^0_{{\mathcal I}'}({\mathcal E}) = \emptyset$. It follows that any irreducible component of $\Omega_{\mathcal I}({\mathcal E}) \setminus \Omega_{\mathcal I}^0({\mathcal E})$ has dimension strictly smaller than $\operatorname{edim}{\mathcal I}$. But this is a contradiction, since the dimension of ${\mathcal Z}$ is equal to at least $\operatorname{edim}{\mathcal I}$. The following is a direct consequence of the equivalence between \[item:good set i\] and \[item:good set iii\] in \[lem:good set\]: $$\label{eq:intersecting via good} \operatorname{Intersecting}(r,n,s) = \{ \operatorname{Pos}(V,{\mathcal E}) : V \subseteq {\mathbb C}^n, \dim V = r \}$$ for every ${\mathcal E}\in\operatorname{Good}(n,s)$. We now study the numerical inequalities satisfied by intersecting $s$-tuples more carefully. Recall that a weight $\theta$ for $\operatorname{GL}(r)$ is *antidominant* if $\theta(1)\leq\dots\leq\theta(r)$. For example, given a subset $I\subseteq[n]$ of cardinality $r$, the weight $\theta(a) := I(a)-a$ is antidominant. It is convenient to introduce the following definition: \[def:slope\] Given an $s$-tuple $\vec\theta=(\theta_1,\dots,\theta_s)$ of antidominant weights for $\operatorname{GL}(r)$, we define the *slope* of a tuple ${\mathcal J}\in\operatorname{Subsets}(d,r,s)$ as $$\mu_{\vec\theta}({\mathcal J}) := \frac1d \sum_{k=1}^s \sum_{a \in J_k} \theta_k(a) = \frac1d \sum_{k=1}^s (T_{J_k}, \theta_k).$$ For any nonzero subspace $\{0\}\neq S\subseteq{\mathbb C}^r$ and $s$-tuple of flags ${\mathcal F}$ on ${\mathbb C}^r$, we further define $$\mu_{\vec\theta}(S,{\mathcal F}) := \mu_{\vec\theta}(\operatorname{Pos}(S, {\mathcal F})).$$ Here and in the following, we write $\operatorname{Pos}(S,{\mathcal F})$ for the $s$-tuple of positions $(\operatorname{Pos}(S,F_k))_{k\in[s]}$. Note that we can interpret $\mu_{\vec\theta}({\mathcal J})$ as a sum of averages of the nowhere decreasing functions $\theta_k$ for uniform choice of $a \in J_k$. The following lemma asserts that there is a unique slope-minimizing subspace of maximal dimension: \[lem:slope\] Let $\vec\theta$ be an $s$-tuple of antidominant weights for $\operatorname{GL}(r)$, and ${\mathcal F}\in\operatorname{Flag}(r)^s$. Let $m_* := \min_{\{0\}\neq S\subseteq{\mathbb C}^r} \mu_{\vec\theta}(S,{\mathcal F})$ and $d_* := \max \{ \dim S : \mu_{\vec\theta}(S,{\mathcal F}) = m_* \}$. Then there exists a unique subspace $S_* \subseteq {\mathbb C}^r$ such that $\mu_{\vec\theta}(S_*,{\mathcal F}) = m_*$ and $\dim S_* = d_*>0$. Existence is immediate, so it remains to show uniqueness. Thus suppose for sake of finding a contradiction that there are two such subspaces, $S_1\neq S_2$, such that $\mu_{\vec\theta}(S_j,{\mathcal F}) = m_*$ and $\dim S_j = d_*$ for $j=1,2$. We note that $d_*>0$ and that the inclusions $S_1\cap S_2\subsetneq S_1$ and $S_2 \subsetneq S_1+S_2$ are strict. Let ${\mathcal J}= \operatorname{Pos}(S_1,{\mathcal F})$ and ${\mathcal K}= \operatorname{Pos}(S_1\cap S_2, {\mathcal F}^{S_1})$. Then $\operatorname{Pos}(S_1\cap S_2, {\mathcal F}) = {\mathcal J}{\mathcal K}$ by the chain rule (\[lem:chain rule\]). Let us first assume that $S_1\cap S_2\neq\{0\}$, so that $\mu_{\vec\theta}({\mathcal J}{\mathcal K})$ is well-defined. Then, $$\mu_{\vec\theta}({\mathcal J}{\mathcal K}) = \mu_{\vec\theta}(S_1\cap S_2,{\mathcal F}) \geq m_* = \mu_{\vec\theta}(S_1,{\mathcal F}) = \mu_{\vec\theta}({\mathcal J}),$$ where the equalities hold by definition, and the inequality holds as $m_*$ is the minimal slope. On the other hand, note that $J_k = J_k K_k \cup J_k K_k^c$ for each $k\in[s]$, hence we can write $$\mu_{\vec\theta}({\mathcal J}) = \frac d {d_*} \mu_{\vec\theta}({\mathcal J}{\mathcal K}) + \frac {d_*-d} {d_*} \mu_{\vec\theta}({\mathcal J}{\mathcal K}^c),$$ where $d := \dim S_1\cap S_2 < \dim S_1 = d_*$. It follows that $$\label{eq:slope first ieq} m_* = \mu_{\vec\theta}({\mathcal J}) \geq \mu_{\vec\theta}({\mathcal J}{\mathcal K}^c).$$ If $S_1\cap S_2=\{0\}$ then ${\mathcal J}={\mathcal J}{\mathcal K}^c$ and so  holds with equality. Likewise, let ${\mathcal L}= \operatorname{Pos}(S_1 + S_2, {\mathcal F})$ and ${\mathcal M}= \operatorname{Pos}(S_2, {\mathcal F}^{S_1+S_2})$. Since $S_2\subsetneq S_1+S_2$, but $S_2$ was assumed to be a maximal-dimensional subspace with minimal slope, it follows that the slope of $S_1+S_2$ is strictly larger than $m_*$: $$\mu_{\vec\theta}({\mathcal L}{\mathcal M}) = \mu_{\vec\theta}(S_2, {\mathcal F}) = m_* < \mu_{\vec\theta}(S_1 + S_2, {\mathcal F}) = \mu_{\vec\theta}({\mathcal L}).$$ Just as before, we decompose $$\mu_{\vec\theta}({\mathcal L}) = \frac {d_*} {d'} \mu_{\vec\theta}({\mathcal L}{\mathcal M}) + \frac {d' - d_*} {d'} \mu_{\vec\theta}({\mathcal L}{\mathcal M}^c),$$ where now $d' := \dim S_1+S_2 > \dim S_2 = d_* > 0$. Thus we obtain the strict inequality $$\label{eq:slope second ieq} \mu_{\vec\theta}({\mathcal L}{\mathcal M}^c) > \mu_{\vec\theta}({\mathcal L}{\mathcal M}) = m_*.$$ At last, we apply \[cor:slope preliminary\], which shows that $J_k K_k^c(b) \geq L_k M_k^c(b)$ for all $b$ and $k$, and hence $$\mu_{\vec\theta}({\mathcal J}{\mathcal K}^c) \geq \mu_{\vec\theta}({\mathcal L}{\mathcal M}^c).$$ Together with  and , we obtain the desired contradiction: $$m_* \geq \mu_{\vec\theta}({\mathcal J}{\mathcal K}^c) \geq \mu_{\vec\theta}({\mathcal L}{\mathcal M}^c) > m_*. \qedhere$$ We will now use \[lem:good set,lem:slope\] to show that the conditions in \[cor:belkale inductive\] with $\operatorname{edim}{\mathcal J}= 0$ imply those for general intersecting ${\mathcal J}$. \[def:lambda\_I\] Let $I \subseteq [n]$ be a subset of cardinality $r$. We define $\lambda_I\in\Lambda_+(r)$ by $$\lambda_I(a) := a - I(a) \qquad (a\in[r]).$$ Any highest weight $\lambda$ with $\lambda(1)\leq 0$, $\lambda(r)\geq r-n$ can be written in this form. Moreover, if $I^c$ denotes the complement of $I$ in $[n]$ then the dominant weight $\lambda_{I^c}\in\Lambda_+(n-r)$ can be written as $$\label{eq:complement} \lambda_{I^c}(b) = b - I^c(b) = -\#\{ a \in [r] : I(a) < I^c(b) \} = -\#\{ a \in [r] : I(a) - a < b \}.$$ This equation has a pleasant interpretation in terms of Young diagrams. Consider the Young diagram $Y_I$ corresponding to $\lambda_I^*$, which has $I(r+1-a) - (r+1-a)$ boxes in its $a$-th row. By definition, its transpose $Y_I^t$ is the Young diagram such that the number of boxes in the $b$-th row is equal to the number of boxes in the $b$-th column of $T_I$. Thus  asserts that $Y_I^t = r{\mathbbm 1}_{n-r} + \lambda_{I^c}$, i.e., the two Young diagrams $Y_I^t$ and $Y_{I^c}$ (the latter with rows in reverse order) make up a rectangle of size $r\times(n-r)$. \[lem:I to lambdaprime\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. Set $\lambda_k = \lambda_{I_k} + (n-r){\mathbbm 1}_r$ for $k\in[s-1]$ and $\lambda_s = \lambda_{I_s}$. Then we have that $\operatorname{edim}{\mathcal I}=-\sum_{k=1}^s \lvert\lambda_k\rvert$. More generally, for every ${\mathcal J}\in\operatorname{Subsets}(d,r,s)$, $$\operatorname{edim}{\mathcal I}{\mathcal J}-\operatorname{edim}{\mathcal J}= -\sum_{k=1}^s (T_{J_k}, \lambda_k) = d \mu_{-\vec\lambda}({\mathcal J}),$$ where we recall that $(T_J,\xi)=\sum_{j\in J} \xi(j)$ for any $J\subseteq[r]$ and $\xi\in i\mathfrak t(r)$. It suffices to prove the second statement, which follows from $$\begin{aligned} &\quad \operatorname{edim}{\mathcal I}{\mathcal J}-\operatorname{edim}{\mathcal J}= d(n-r)(1-s) + \sum_{k=1}^s \sum_{a\in J_k} \bigl( I_k(a) - a \bigr) \\ &= d(n-r)(1-s) - \sum_{k=1}^s (T_{J_k}, \lambda_{I_k}) = - \sum_{k=1}^s (T_{J_k}, \lambda_k) = d \mu_{-\vec\lambda}({\mathcal J}). \qedhere \end{aligned}$$ It follows that minimizing $\mu_{-\vec\lambda}({\mathcal J})$ and $\frac1d\bigl(\operatorname{edim}{\mathcal I}{\mathcal J}- \operatorname{edim}{\mathcal J}\bigr)$ as a function of ${\mathcal J}$ are equivalent. We then have the following result: \[prp:belkale strong weak\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $\operatorname{edim}{\mathcal I}\geq0$ and, for any $0<d<r$ and ${\mathcal J}\in\operatorname{Intersecting}(d,r,s)$ with $\operatorname{edim}{\mathcal J}=0$ we have that $\operatorname{edim}{\mathcal I}{\mathcal J}\geq0$. Then we have for any $0<d<r$ and ${\mathcal J}\in\operatorname{Intersecting}(d,r,s)$ that $$\operatorname{edim}{\mathcal I}{\mathcal J}\geq\operatorname{edim}{\mathcal J}.$$ Suppose for sake of finding a contradiction that there exists ${\mathcal J}\in\operatorname{Intersecting}(d,r,s)$ with $0<d<r$ and $\operatorname{edim}{\mathcal I}{\mathcal J}< \operatorname{edim}{\mathcal J}$, so that $\mu_{-\vec\lambda}({\mathcal J}) < 0$ according to \[lem:I to lambdaprime\]. Fix some ${\mathcal F}\in\operatorname{Good}(r,s)$. Then $\Omega^0_{\mathcal J}({\mathcal F})\neq\emptyset$ by \[lem:good set\], \[item:good set ii\]. Thus there exists a subspace $\{0\}\neq S\subseteq{\mathbb C}^r$ such that $\mu_{-\vec\lambda}(S,{\mathcal F}) = \mu_{-\vec\lambda}({\mathcal J}) < 0$. Now let $S_*$ be the unique subspace of minimal slope $m_*<0$ and maximal dimension $d_*>0$ from \[lem:slope\] and denote by ${\mathcal J}_* := \operatorname{Pos}(S_*,{\mathcal F})$ its $s$-tuple of positions. The uniqueness statement implies that $\Omega^0_{{\mathcal J}_*}({\mathcal F}) = \{S_*\}$, since slope and dimension are fully determined by the position. Moreover, ${\mathcal J}_*$ is intersecting by , and therefore $\operatorname{edim}{\mathcal J}_* = \dim \Omega^0_{{\mathcal J}_*}({\mathcal F}) = 0$ by \[lem:good set\]. Thus we have found an $s$-tuple ${\mathcal J}_* \in \operatorname{Intersecting}(d_*,r,s)$ with $d_*>0$, $\operatorname{edim}{\mathcal J}_* = 0$, and $$\operatorname{edim}{\mathcal I}{\mathcal J}_* = \operatorname{edim}{\mathcal I}{\mathcal J}_* - \operatorname{edim}{\mathcal J}_* = d_* m_* < 0,$$ where we have used \[lem:I to lambdaprime\] once again in the last equality. Since $\operatorname{edim}{\mathcal I}\geq 0$, this also implies that $d_*<r$. This is the desired contradiction. will be useful to prove Belkale’s \[thm:belkale\] in \[sec:horn sufficient\] below, since it allows us to work with a larger set of inequalities. The proof of \[prp:belkale strong weak\] shows that we may in fact restrict to ${\mathcal J}$ such that $\Omega^0_{\mathcal J}({\mathcal F})$ is a point for all $s$-tuples of good flags ${\mathcal F}\in\operatorname{Good}(r,s)$ – or also to those for which $\Omega_{\mathcal J}({\mathcal F})$ is a point, which is equivalent by the last statement in \[lem:good set\]. See the remark after \[cor:horn and saturation\] for the implications of this on the description of the Kirwan cone. We also record the following corollary which follows together with and improves over \[cor:belkale inductive\]. \[cor:belkale inductive stronger\] If ${\mathcal I}\in \operatorname{Intersecting}(r,n,s)$ then for any $0<d<r$ and any $s$-tuple ${\mathcal J}\in \operatorname{Intersecting}(d,r,s)$ we have that $\operatorname{edim}{\mathcal I}{\mathcal J}\geq \operatorname{edim}{\mathcal J}$. We remark that for $d=r$ there is only one $s$-tuple, ${\mathcal J}= ([r],\dots,[r])$, and it is intersecting and satisfies $\operatorname{edim}{\mathcal J}=0$. In this case, $\operatorname{edim}{\mathcal I}{\mathcal J}-\operatorname{edim}{\mathcal J}=\operatorname{edim}{\mathcal I}$, and so we may safely allow for $d=r$ in \[cor:belkale inductive,prp:belkale strong weak,cor:belkale inductive stronger\]. We conclude this section with some simple examples of the Horn inequalities of \[cor:belkale inductive stronger\]. We refer to \[app:examples horn\] for lists of all Horn triples ${\mathcal I}=(I_1,I_2,I_3)$ up to $n=4$. \[ex:base case\] The only condition for ${\mathcal I}\in\operatorname{Intersecting}(1,n,s)$ is the dimension condition, $\operatorname{edim}{\mathcal I}\geq0$. Indeed, the Grassmannian $\operatorname{Gr}(1,n)$ is the projective space ${\mathbb P}({\mathbb C}^n)$, whose Schubert varieties are given by $\Omega_{\{i\}}(E) = \{ [v] \in {\mathbb P}({\mathbb C}^n) : v \in E(i) \}$. Thus ${\mathcal I}= (\{i_1\},\dots,\{i_s\})$ is intersecting if and only if for any $s$-tuple of flags ${\mathcal E}$, $E_1(i_1)\cap\dots\cap E_s(i_s)\neq\{0\}$. By linear algebra, it is certainly sufficient that $\sum_{k=1}^s (n - i_k) \leq n-1$, which is equivalent to $\operatorname{edim}{\mathcal I}\geq0$. This also establishes \[thm:belkale\] in the case $r=1$. Let ${\mathcal I}= (I_1,I_2)$. Then the condition $\operatorname{edim}{\mathcal I}\geq0$ is $I_1(1) + I_1(2) + I_2(1) + I_2(2) \geq 2n+2$. However, there are two additional conditions coming from the ${\mathcal J}\in\operatorname{Intersecting}(1,2,2)$ with $\operatorname{edim}{\mathcal J}=0$. By the preceding example, there are two such pairs, $(\{1\},\{2\})$ and $(\{2\},\{1\})$. The corresponding conditions are $I_1(1) + I_2(2) \geq n+1$ and $I_1(2) + I_2(1) \geq n+1$. For example, if $n=4$ then ${\mathcal I}= (\{1,4\},\{2,4\})$ satisfies all Horn inequalities. On the other hand, ${\mathcal I}= (\{1,4\},\{2,3\})$ fails one the Horn inequalities. Indeed, if we consider ${\mathcal J}=(\{1\},\{2\})$ then ${\mathcal I}{\mathcal J}=(\{1\},\{3\})$ is such that $\operatorname{edim}{\mathcal I}{\mathcal J}= -1 < 0$. Sufficiency of Horn inequalities {#sec:horn sufficient} ================================ In this section we prove that the Horn inequalities are also sufficient to characterize intersections of Schubert varieties. Tangent maps ------------ In \[lem:dominant\], we established that an $s$-tuple ${\mathcal I}$ is intersecting if and only if the corresponding morphism $\omega_{\mathcal I}$ defined in  is dominant. Now it is a general fact that a morphism $f\colon{\mathcal X}\to{\mathcal Y}$ between *smooth* and irreducible varieties is dominant if and only if there exists a point $p\in{\mathcal X}$ where the differential $T_pf$ is surjective. This will presently allow us to reduce the intersecting of Schubert varieties to an infinitesimal question about tangent maps. Later, in \[sec:invariants\], we will also use the determinant of the tangent map to construct explicit nonzero tensor product invariants and establish the saturation property. \[lem:differential concrete\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. Then ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$ if and only if there exist $\vec g=(g_1,\dots,g_s)\in\operatorname{GL}(V_0)^s$ and $\vec h=(h_1,\dots,h_s)\in\operatorname{GL}(Q_0)^s$ such that the linear map $$\label{eq:differential concrete} \Delta_{{\mathcal I},\vec g,\vec h}\colon\begin{cases} \operatorname{Hom}(V_0, Q_0)\times H_{I_1}(F_0, G_0)\times\dots\times H_{I_s}(F_0, G_0)\to \operatorname{Hom}(V_0,Q_0)^s \\ (\zeta,\phi_1,\dots,\phi_s)\mapsto(\zeta+h_1\phi_1g_1^{-1},\dots,\zeta+h_s\phi_s g_s^{-1}) \end{cases}$$ is surjective. Using the isomorphisms $\operatorname{Flag}^0_{I_k}(r,n) = G_{I_k}(r,n) E_0 \cong G_{I_k}(r,n)/B(n)$ (\[def:G\_I,subsec:coordinates\]) and $\operatorname{Flag}(n) \cong \operatorname{GL}(n)/B(n)$, we find that $\omega^0_{\mathcal I}$ is dominant if and only if $$\label{eq:dominant cover} \operatorname{GL}(n) \times G_{I_1}(r,n) \times \dots \times G_{I_s}(r,n) \to \operatorname{GL}(n)^s, (\gamma, \gamma_1, \dots, \gamma_s) \mapsto (\gamma \gamma_1, \dots, \gamma \gamma_s)$$ is dominant. This is again a morphism between smooth and irreducible varieties and thus dominance is equivalent to surjectivity of the differential at some point $(\gamma,\gamma_1,\dots,\gamma_s)$. The map  is $\operatorname{GL}(n)$-equivariant on the left and $B(n)^s$-equivariant on the right. By the former, we may assume that $\gamma=1$, and by the latter that $\gamma_k = p_k w_{I_k}$ for some $p_k = \bigl(\begin{smallmatrix}g_k & b_k \\ 0 & h_k\end{smallmatrix}\bigr)$, since $G_{I_k}(r,n) = P(r, n) w_{I_k} B(n)$ according \[lem:G\_I coarse\]. We now compute the differential. Thus we consider an arbitrary curve $1 + \varepsilon X$ tangent to $\gamma=1$, where $X \in {\mathfrak{gl}}(n)$, and curves $(1 + \varepsilon Y_k) p_k w_{I_k}$ through the $\gamma_k = p_k w_k$, where $Y_k \in {\mathfrak{gl}}(n)$. If we write $Y_k = \bigl(\begin{smallmatrix}A_k & B_k \\ C_k & D_k \end{smallmatrix}\bigr)$ with $A_k \in {\mathfrak{gl}}(r)$ etc., then we see from  that $(1 + \varepsilon Y_k) p_k w_{I_k}$ is tangent to $G_{I_k}(r,n)$ precisely if $h^{-1}_k C_k g_k \in H_{I_k}(F_0, G_0)$, that is, if $C_k \in h_k H_{I_k}(F_0,G_0) g_k^{-1}$. Lastly, the calculation $(1 + \varepsilon X)(1 + \varepsilon Y_k) \gamma_k = \gamma_k + \varepsilon (X + Y_k) \gamma_k + O(\varepsilon^2)$ shows that the differential of  at $(g,\gamma_1,\dots,\gamma_s)$ can be identified with $(X, Y_1, \dots, Y_s) \mapsto (X + Y_1, \dots, X + Y_s)$. We may check for surjectivity block by block. Since there are no constraints on the $A_k$, $B_k$, and $D_k$, it is clear that the differential is surjective on the three blocks corresponding to ${\mathfrak p}(r,n)$. Thus we only need to check surjectivity on the last block of the linear map, corresponding to $\operatorname{Hom}(V_0, Q_0)$. This block can plainly be identified with , since the $C_k$ are constrained to be elements of $h_k H_{I_k}(F_0,G_0) g_k^{-1}$. Thus we obtain that $\omega_{\mathcal I}$ is dominant if and only if  is surjective. \[rem:delta\^0\_I factorized differential\] The map $\Delta_{{\mathcal I},\vec g,\vec h}$ can be identified with the differential of $\bar\delta^0_{\mathcal I}$ at the point $[1, {\mathcal E}]$, where $E_k\in\operatorname{Flag}^0_{I_k}(r,n)$ is such that $(E_k)^{V_0} = g_k \cdot F_0$ and $(E_k)_{Q_0} = h_k \cdot G_0$ for $k\in[s]$. This follows from the proof of \[lem:differential concrete\] and justifies calling $\Delta_{{\mathcal I},\vec g,\vec h}$ a *tangent map*. By the rank-nullity theorem and using \[lem:dim schubert cell\], the kernel of the linear map $\Delta_{{\mathcal I},\vec g,\vec h}$ defined in  is of dimension at least $$\label{eq:rank nullity} \begin{aligned} &\quad \dim \bigl( \operatorname{Hom}(V_0,Q_0)\times H_{I_1}(F_0, G_0)\times\dots\times H_{I_s}(F_0, G_0) \bigr) - \dim \operatorname{Hom}(V_0,Q_0)^s \\ &= r(n-r)(1-s) + \sum_{k=1}^s \dim I_k = \operatorname{edim}{\mathcal I}, \end{aligned}$$ and $\Delta_{{\mathcal I},\vec g,\vec h}$ is surjective if and only if equality holds. On the other hand, it is immediate that $$\label{eq:ker Delta} \ker \Delta_{{\mathcal I},\vec g,\vec h} = \bigcap_{k=1}^s h_k H_{I_k}(F_0, G_0) g_k = \bigcap_{k=1}^s H_{I_k}(F_k, G_k),$$ where $F_k = g_k F_0$ and $G_k = h_k G_0$. As we vary $g_k$ and $h_k$, the $F_k$ and $G_k$ are arbitrary flags on $V_0$ and $Q_0$, respectively. Thus we obtain the following characterization: Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. We define the *true dimension* of ${\mathcal I}$ as $$\label{eq:tdim} \operatorname{tdim}{\mathcal I}:= \min_{{\mathcal F},{\mathcal G}} \dim H_{\mathcal I}({\mathcal F},{\mathcal G}) = \min_{\vec g, \vec h} \dim\ker\Delta_{{\mathcal I},\vec g,\vec h},$$ where the first side minimization is over all $s$-tuples of flags ${\mathcal F}$ on $V_0$ and ${\mathcal G}$ on $Q_0$, the second one over $\vec g\in\operatorname{GL}(r)^s$, $\vec h\in\operatorname{GL}(n-r)^s$, and where $$H_{\mathcal I}({\mathcal F},{\mathcal G}) := \bigcap_{k=1}^s H_{I_k}(F_k, G_k) \subseteq \operatorname{Hom}(V_0,Q_0).$$ \[cor:tdim edim\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. Then we have $\operatorname{tdim}{\mathcal I}\geq\operatorname{edim}{\mathcal I}$, with equality if and only if ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$. We note that for the purpose of computing true dimensions we may always assume that $F_1$ and $G_1$ are the standard flags on $V_0$ and $Q_0$, respectively (by equivariance). We verify the example at the end of \[sec:horn necessary\] by using \[cor:tdim edim\]. We first consider ${\mathcal I}=(\{1,4\},\{2,4\})$. Then $\operatorname{edim}{\mathcal I}=1$. To bound $\operatorname{tdim}{\mathcal I}$, we let ${\mathcal F}=(F_1,F_2)$ and ${\mathcal G}=(G_1,G_2)$, where $F_1$ is the standard flag on $V_0$, $F_2$ the flag with adapted basis $(e(1)+e(2), e(2))$, and $G_1=G_2$ the standard flags on $Q_0$. Then $$H_{\mathcal I}({\mathcal F},{\mathcal G}) = \{ \begin{pmatrix} 0 & * \\ 0 & * \end{pmatrix} \} \cap H_{I_2}(F_2, G_2) = \{ \begin{pmatrix} 0 & * \\ 0 & 0 \end{pmatrix} \}$$ is one-dimensional, which shows that $\operatorname{tdim}{\mathcal I}\leq1$. Since always $\operatorname{tdim}{\mathcal I}\geq\operatorname{edim}{\mathcal I}$, it follows that, in fact, $\operatorname{tdim}{\mathcal I}=\operatorname{edim}{\mathcal I}$ and so ${\mathcal I}$ is intersecting. We now consider ${\mathcal I}=(\{1,4\}, \{2,3\})$. Then $\operatorname{edim}{\mathcal I}=0$. Let ${\mathcal F}$ and ${\mathcal G}$ be pairs of flags on $V_0$ and $Q_0$, respectively. Without loss of generality, we shall assume that $F_1$ and $G_1$ are the standard flags. Then $$H_{\mathcal I}({\mathcal F},{\mathcal G}) = \{ \begin{pmatrix} 0 & * \\ 0 & * \end{pmatrix} \} \cap H_{I_2}(F_2,G_2) = {\mathbb C}\begin{pmatrix} 0 & x \\ 0 & y \end{pmatrix},$$ where ${\mathbb C}\bigl(\begin{smallmatrix}x \\ y\end{smallmatrix}\bigr) := G_2(1)$. Indeed, $H_{I_2}(F_2,G_2)$ consists of those linear maps that map any vector in $V_0$ into $G_2(1)$. In particular, $H_{\mathcal I}({\mathcal F},{\mathcal G})$ is one-dimensional for any choice of $F_2$ and $G_2$. Thus $\operatorname{tdim}{\mathcal I}=1>0=\operatorname{edim}{\mathcal I}$, and we conclude that ${\mathcal I}$ is not intersecting. Let ${\mathcal I}=(\{3,4,6\},\{2,4,5\})$. Then $\operatorname{edim}{\mathcal I}=3$. We now establish that ${\mathcal I}$ is intersecting by verifying that $\operatorname{tdim}{\mathcal I}=3$. Again we choose $F_1$ and $G_1$ to be the standard flags on $V_0$ and $Q_0$, respectively, while $F_2$ and $G_2$ are defined as follows in terms of adapted bases: $$\begin{aligned} F_2\colon \quad &e(1) + z_{21} e(2) + z_{31} e(3),\quad e(2) + z_{32} e(3),\quad e(3),\\ G_2\colon \quad &\bar e(1) + u_{21} \bar e(2) + u_{31} \bar e(3),\quad\bar e(2) + u_{32} \bar e(3),\quad\bar e(3). \end{aligned}$$ Then a basis for $H_{\mathcal I}({\mathcal F},{\mathcal G})$ is on the open set where $u_{31}u_{32}\neq0$ given by $$\begin{aligned} \phi_1 &= \begin{pmatrix} -z_{21} u_{32} & u_{32} & 0 \\ -z_{21} ( u_{32} u_{21} - u_{31} ) & u_{32} u_{21} - u_{31} & 0 \\ 0 & 0 & 0 \end{pmatrix}, \\ \phi_2 &= \begin{pmatrix} z_{31} u_{32} & 0 & 0 \\ z_{31} ( u_{32} u_{21} - u_{31} ) & 0 & u_{31} \\ 0 & 0 & u_{32} u_{31} \end{pmatrix}, \qquad \phi_3 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & u_{21} \\ 0 & 0 & u_{31} \end{pmatrix} \end{aligned}$$ as can be checked by manual inspection. Kernel dimension and position {#subsec:kdim kpos} ----------------------------- Let us consider a tuple ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$, where we always assume that $r\in[n]$. To prove sufficiency of the Horn inequalities, we aim to use \[cor:tdim edim\], which states that $\operatorname{tdim}{\mathcal I}\geq\operatorname{edim}{\mathcal I}$, with equality if and only if ${\mathcal I}$ is intersecting. If $\operatorname{tdim}{\mathcal I}=0$ then, necessarily, $\operatorname{tdim}{\mathcal I}=\operatorname{edim}{\mathcal I}=0$, since $\operatorname{edim}{\mathcal I}$ is nonnegative by assumption (part of the Horn inequalities). Hence in this case ${\mathcal I}$ is intersecting. Thus the interesting case is when $\operatorname{tdim}{\mathcal I}>0$. To study the spaces $H_{\mathcal I}({\mathcal F},{\mathcal G})$ in a unified fashion, we consider the space $${\mathrm P}({\mathcal I}) := \{ ({\mathcal F},{\mathcal G},\phi) \in \operatorname{Flag}(V_0)^s \times \operatorname{Flag}(Q_0)^s \times \operatorname{Hom}(V_0,Q_0) : \phi\in H_{\mathcal I}({\mathcal F},{\mathcal G}) \}.$$ We caution that ${\mathrm P}({\mathcal I})$ is not in general irreducible, as the following example shows: Let $s=2$, $n=3$, $r=1$, and consider $I_1 = I_2 = \{2\}$. There is only a single flag on $V_0 \cong {\mathbb C}$, while any flag $G$ on $Q_0 \cong {\mathbb C}^2$ is determined by a line $L = G(1)\in{\mathbb P}({\mathbb C}^2)$. Thus we can identify ${\mathrm P}({\mathcal I}) \cong \{ (L_1, L_2, \phi) \in {\mathbb P}({\mathbb C}^2)^2 \times \operatorname{Hom}({\mathbb C}^1,{\mathbb C}^2) : \phi(e(1)) \in L_1 \cap L_2 \}$. If we consider the map $(L_1,L_2,\phi)\mapsto(L_1,L_2)$, then the fiber for any $L_1=L_2$ is a one-dimensional line, while for any $L_1\neq L_2$ the fiber is just $\phi=0$. In particular, we note that ${\mathrm P}({\mathcal I})$ is not irreducible. We now restrict to those $({\mathcal F},{\mathcal G})$ such that the intersection $H_{\mathcal I}({\mathcal F},{\mathcal G})$ is of dimension $\operatorname{tdim}{\mathcal I}$. Thus we introduce $$\begin{aligned} {\mathrm P_{\mathrm t}}({\mathcal I}) &:= \{ ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P}({\mathcal I}) : \dim H_{\mathcal I}({\mathcal F},{\mathcal G}) = \operatorname{tdim}{\mathcal I}\}, \\ {\mathrm B_{\mathrm t}}({\mathcal I}) &:= \{ ({\mathcal F},{\mathcal G}) \in \operatorname{Flag}(V_0)^s \times \operatorname{Flag}(Q_0)^s : \dim H_{\mathcal I}({\mathcal F},{\mathcal G}) = \operatorname{tdim}{\mathcal I}\}.\end{aligned}$$ The subscripts in ${\mathrm P_{\mathrm t}}({\mathcal I})$ and ${\mathrm B_{\mathrm t}}({\mathcal I})$ stands for the true dimension, $\operatorname{tdim}{\mathcal I}$. We use similar subscripts throughout this section when we fix various other dimensions and positions. Since $\operatorname{tdim}{\mathcal I}$ is the minimal possible dimension, this is the generic case. Moreover, this restriction makes ${\mathrm P_{\mathrm t}}({\mathcal I})$ irreducible, as it is a vector bundle over ${\mathrm B_{\mathrm t}}({\mathcal I})$. We record this in the following lemma: \[lem:P\_t zopen\] The space ${\mathrm P}({\mathcal I})$ is a closed subvariety of $\operatorname{Flag}(V_0)^s\times\operatorname{Flag}(Q_0)^s\times\operatorname{Hom}(V_0,Q_0)$, and ${\mathrm P_{\mathrm t}}({\mathcal I})$ is a nonempty Zariski-open subset of ${\mathrm P}({\mathcal I})$. Moreover, ${\mathrm B_{\mathrm t}}({\mathcal I})$ is a nonempty Zariski-open subset of $\operatorname{Flag}(V_0)^s\times\operatorname{Flag}(Q_0)^s$, and the map $({\mathcal F},{\mathcal G},\phi)\mapsto({\mathcal F},{\mathcal G})$ turns ${\mathrm P_{\mathrm t}}({\mathcal I})$ into a vector bundle over ${\mathrm B_{\mathrm t}}({\mathcal I})$. In particular, ${\mathrm P_{\mathrm t}}({\mathcal I})$ is an irreducible and smooth variety. In particular: $$\label{eq:dim Pt} \dim {\mathrm P_{\mathrm t}}({\mathcal I}) = s \bigl( \dim \operatorname{Flag}(V_0) + \dim \operatorname{Flag}(Q_0) \bigr) + \operatorname{tdim}{\mathcal I}$$ Belkale’s insight is now to consider the behavior of generic kernels of maps $\phi\in H_{\mathcal I}({\mathcal F},{\mathcal G})$, where $({\mathcal F},{\mathcal G})\in {\mathrm B_{\mathrm t}}({\mathcal I})$. We start with the following definition: Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. We define the *kernel dimension* of ${\mathcal I}$ as $$\operatorname{kdim}{\mathcal I}:= \min \{ \dim\ker\phi : \phi\in H_{\mathcal I}({\mathcal F}, {\mathcal G}) \text{ where } ({\mathcal F},{\mathcal G}) \in {\mathrm B_{\mathrm t}}({\mathcal I}) \}$$ There are two special cases that we can treat right away. If $\operatorname{kdim}{\mathcal I}=r$ then any morphism in $H_{\mathcal I}({\mathcal F},{\mathcal G})$ for $({\mathcal F},{\mathcal G})\in {\mathrm B_{\mathrm t}}({\mathcal I})$ is zero, and hence $\operatorname{tdim}{\mathcal I}=0$. This is the case that we had discussed initially and we record this observation for future reference: \[lem:kdim r intersecting\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $\operatorname{edim}{\mathcal I}\geq0$. If $\operatorname{kdim}{\mathcal I}=r$ then $\operatorname{tdim}{\mathcal I}=\operatorname{edim}{\mathcal I}=0$, and hence ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$. Likewise, the case where $\operatorname{kdim}{\mathcal I}=0$ can easily be treated directly. The idea is to compute the dimension of ${\mathrm P_{\mathrm t}}({\mathcal I})$ in a second way and compare the result with . \[lem:dim P\_t zero\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. If $\operatorname{kdim}{\mathcal I}=0$ then $$\dim {\mathrm P_{\mathrm t}}({\mathcal I}) = s \bigl( \dim \operatorname{Flag}(V_0) + \dim \operatorname{Flag}(Q_0) \bigr) + \operatorname{edim}{\mathcal I}.$$ We first note that $\operatorname{kdim}{\mathcal I}=0$ implies that there exists an injective map $\phi\in H_{\mathcal I}({\mathcal F},{\mathcal G})$ for some $({\mathcal F},{\mathcal G}) \in {\mathrm B_{\mathrm t}}({\mathcal I})$. In particular, $I_k(a)-a\geq a$ for all $k\in[s]$ and $a\in[r]$ (a fact that we use further below in the proof). Now define $${\mathrm P_{\mathrm k}}:= \{ ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P}({\mathcal I}) : \dim\ker\phi=0 \}$$ Then ${\mathrm P_{\mathrm k}}$ is a nonempty Zariski-open subset of ${\mathrm P}({\mathcal I})$ that intersects ${\mathrm P_{\mathrm t}}({\mathcal I})$. By \[lem:P\_t zopen\], the latter is irreducible. Thus it suffices to show that ${\mathrm P_{\mathrm k}}$ is likewise irreducible and to compute its dimension. For this, we consider the map $$\pi\colon {\mathrm P_{\mathrm k}}\to {\mathrm M_{\mathrm k}}:=\operatorname{Flag}(V_0)^s\times\operatorname{Hom}^\times(V_0,Q_0), \quad ({\mathcal F},{\mathcal G},\phi)\mapsto({\mathcal F},\phi)$$ where we write $\operatorname{Hom}^\times(V_0,Q_0)$ for the Zariski-open subset of injective linear maps in $\operatorname{Hom}(V_0,Q_0)$. The fibers of $\pi$ are given by $$\pi^{-1}({\mathcal F},\phi) \cong \prod_{k=1}^s \operatorname{Flag}_{I_k}^0(F_k, \phi)$$ which according to \[lem:image flags\] are smooth irreducible varieties of dimension $s \dim \operatorname{Flag}(Q_0) - s r(n-r) + \sum_{k=1}^s \dim I_k$. It is not hard to see that $\pi$ gives ${\mathrm P_{\mathrm k}}$ the structure of a fiber bundle over ${\mathrm M_{\mathrm k}}$. Therefore, ${\mathrm P_{\mathrm k}}$ is irreducible. Moreover, the space ${\mathrm M_{\mathrm k}}$ has dimension $s \dim \operatorname{Flag}(V_0) + r(n-r)$. By adding the dimension of the fibers, we obtain that the dimension of ${\mathrm P_{\mathrm k}}$, and hence of ${\mathrm P_{\mathrm t}}({\mathcal I})$, is indeed the one claimed in the lemma. \[cor:kdim zero intersecting\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. If $\operatorname{kdim}{\mathcal I}=0$ then $\operatorname{tdim}{\mathcal I}=\operatorname{edim}{\mathcal I}$, and hence ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$. This follows directly by comparing \[eq:dim Pt,lem:dim P\_t zero\]. We now consider the general case, where $0<d:=\operatorname{kdim}{\mathcal I}<r$. We first note that the kernel dimension is attained generically. Thus we define $$\begin{aligned} {\mathrm P_{\mathrm{kt}}}({\mathcal I}) &:= \{ ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P_{\mathrm t}}({\mathcal I}) : \dim\ker\phi = \operatorname{kdim}{\mathcal I}\}, \\ {\mathrm B_{\mathrm{kt}}}({\mathcal I}) &:= \{ ({\mathcal F},{\mathcal G}) : \exists\phi \text{~s.th.~} ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P_{\mathrm{kt}}}({\mathcal I}) \} \subseteq {\mathrm B_{\mathrm t}}({\mathcal I}),\end{aligned}$$ where the subscripts denote that we fix both the true dimension as well as the kernel dimension. We have the following lemma: \[lem:P\_kt zopen\] The set ${\mathrm P_{\mathrm{kt}}}({\mathcal I})$ is a nonempty Zariski-open subset of ${\mathrm P_{\mathrm t}}({\mathcal I})$, hence also irreducible. Moreover, ${\mathrm B_{\mathrm{kt}}}({\mathcal I})$ is a nonempty Zariski-open subset of $\operatorname{Flag}(V_0)^s\times\operatorname{Flag}(Q_0)^s$. The first claim holds since ${\mathrm P_{\mathrm{kt}}}({\mathcal I})$ can be defined by the nonvanishing of certain minors. The second claim now follows as ${\mathrm B_{\mathrm{kt}}}({\mathcal I})$ is the image of the Zariski-open subset ${\mathrm P_{\mathrm{kt}}}({\mathcal I})$ of the vector bundle ${\mathrm P_{\mathrm t}}({\mathcal I})\to{\mathrm B_{\mathrm t}}({\mathcal I})$. Belkale’s insight is to consider the positions of generic kernels for an induction: \[def:kerpos\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$. Then we define the *kernel position* of ${\mathcal I}$ as the tuple ${\mathcal J}\in\operatorname{Subsets}(d,r,s)$ defined by $$J_k(b) := \min \{ \operatorname{Pos}(\ker\phi,F_k)(b) \;:\; ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P_{\mathrm{kt}}}({\mathcal I}) \}$$ for $b \in [d]$ and $k \in [s]$. We write $\operatorname{kPos}({\mathcal I}) = {\mathcal J}$. The goal in the remainder of this subsection is to prove the following equality: $$\operatorname{tdim}{\mathcal I}= \operatorname{edim}{\mathcal J}+ \operatorname{edim}{\mathcal I}/{\mathcal J},$$ where ${\mathcal J}=\operatorname{kPos}({\mathcal I})$. This will again be accomplished by computing the dimension of ${\mathrm P_{\mathrm t}}({\mathcal I})$ in a second way and comparing the result with . Specifically, we consider the spaces $$\begin{aligned} {\mathrm P_{\mathrm{kpt}}}({\mathcal I}) &:= \{ ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P_{\mathrm{kt}}}({\mathcal I}) : \operatorname{Pos}(\ker\phi, {\mathcal F}) = \operatorname{kPos}({\mathcal I}) \}, \\ {\mathrm B_{\mathrm{kpt}}}({\mathcal I}) &:= \{ ({\mathcal F},{\mathcal G}) : \exists\phi \text{~s.th.~} ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P_{\mathrm{kpt}}}({\mathcal I}) \} \subseteq {\mathrm B_{\mathrm{kt}}}({\mathcal I}).\end{aligned}$$ Then ${\mathrm P_{\mathrm{kpt}}}({\mathcal I})$ is Zariski-open in ${\mathrm P_{\mathrm{kt}}}({\mathcal I})$, since it can again be defined by demanding that certain minors are nonzero. We obtain the following lemma, the second claim in which is proved as before: \[lem:P\_kpt zopen\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $0<\operatorname{kdim}{\mathcal I}<r$. Then ${\mathrm P_{\mathrm{kpt}}}({\mathcal I})$ is a nonempty Zariski-open subset of ${\mathrm P_{\mathrm{kt}}}({\mathcal I})$, hence also irreducible. Moreover, ${\mathrm B_{\mathrm{kpt}}}({\mathcal I})$ is a nonempty Zariski-open subset of $\operatorname{Flag}(V_0)^s\times\operatorname{Flag}(Q_0)^s$. \[cor:kerpos intersecting\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $0<\operatorname{kdim}{\mathcal I}<r$. Then $\operatorname{kPos}({\mathcal I})\in\operatorname{Intersecting}(d,r,s)$. According to \[lem:P\_kpt zopen\], ${\mathrm B_{\mathrm{kpt}}}({\mathcal I})$ is a nonempty Zariski-open subset of $\operatorname{Flag}(V_0)^s\times\operatorname{Flag}(Q_0)^s$, hence Zariski-dense. It follows that its image under the projection $({\mathcal F},{\mathcal G})\mapsto{\mathcal F}$ is likewise Zariski-dense. For any such ${\mathcal F}$, there exists a ${\mathcal G}$ and $\phi$ such that $({\mathcal F},{\mathcal G},\phi) \in {\mathrm P_{\mathrm{kpt}}}({\mathcal I})$, and hence $\ker\phi \in \Omega^0_{\operatorname{kPos}({\mathcal I})}({\mathcal F})$; in particular, $\Omega^0_{\operatorname{kPos}({\mathcal I})}({\mathcal F})$ is nonempty. Thus \[lem:images,lem:dominant\] show that $\operatorname{kPos}({\mathcal I})$ is intersecting. We now compute the dimension of ${\mathrm P_{\mathrm{kpt}}}({\mathcal I})$. As in the proof of \[lem:dim P\_t zero\], it will be useful to consider an auxiliary space where we do *not* enforce the true dimension: $${\mathrm P_{\mathrm{kp}}}({\mathcal I}) := \{ ({\mathcal F},{\mathcal G},\phi) \in {\mathrm P}({\mathcal I}) : \operatorname{Pos}(\ker\phi,{\mathcal F}) = \operatorname{kPos}({\mathcal I}) \}$$ Note that constraint on the position of the kernel implies that its dimension is $\operatorname{kdim}{\mathcal I}$. \[lem:dim P\_kpt nonzero\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $0<\operatorname{kdim}{\mathcal I}<r$. Then ${\mathrm P_{\mathrm{kp}}}({\mathcal I})$ is nonempty, smooth, irreducible, and satisfies $$\dim {\mathrm P_{\mathrm{kp}}}({\mathcal I}) = s \bigl( \dim \operatorname{Flag}(V_0) + \dim \operatorname{Flag}(Q_0) \bigr) + \operatorname{edim}{\mathcal J}+ \operatorname{edim}{\mathcal I}/{\mathcal J},$$ where ${\mathcal J}:= \operatorname{kPos}({\mathcal I})$. Clearly, ${\mathrm P_{\mathrm{kp}}}({\mathcal I})$ is nonempty since it contains ${\mathrm P_{\mathrm{kpt}}}({\mathcal I})$. We now introduce $${\mathrm M_{\mathrm{kp}}}:= \{ ({\mathcal F},\phi) \in \operatorname{Flag}(V_0)^s \times \operatorname{Hom}(V_0,Q_0) : \operatorname{Pos}(\ker\phi,{\mathcal F}) = \operatorname{kPos}({\mathcal I}) \}$$ and consider the map $$\pi\colon {\mathrm P_{\mathrm{kp}}}({\mathcal I})\to{\mathrm M_{\mathrm{kp}}}, \quad ({\mathcal F},{\mathcal G},\phi)\mapsto({\mathcal F},\phi).$$ Its fibers are given by $$\pi^{-1}({\mathcal F},\phi) \cong \prod_{k=1}^s \{ G_k \in \operatorname{Flag}(Q_0) : \phi \in H_{I_k}(F_k, G_k) \}$$ To understand the right-hand side, define $S:=\ker\phi$ and let $\bar\phi\colon V_0/S\to Q_0$ the corresponding injective map. By \[lem:exp vs composition\], $\phi \in H_{I_k}(F_k, G_k)$ if and only if $\bar\phi \in H_{I_k/J_k}((F_k)_{V_0/S}, G_k)$, that is, $G_k \in \operatorname{Flag}_{I_k/J_k}^0((F_k)_{V_0/S}, \bar\phi)$ as introduced in \[def:flags for injective morphism\]. Thus we find that the fibers of $\pi$ can be identified as $$\pi^{-1}({\mathcal F},\phi) \cong \prod_{k=1}^s \operatorname{Flag}_{I_k/J_k}^0((F_k)_{V_0/S}, \bar\phi).$$ By \[lem:image flags\], the $k$-th factor on the right-hand side is a smooth irreducible variety of dimension $\dim \operatorname{Flag}(Q_0) - (r-d)(n-r) + \dim I_k/J_k$, where $d := \dim\ker\phi = \operatorname{kdim}{\mathcal I}$. It is not hard to see that $\pi$ is a fiber bundle, and we will show momentarily that ${\mathrm M_{\mathrm{kp}}}$ is irreducible. Hence $$\label{eq:dim P_kpt half} \dim {\mathrm P_{\mathrm{kp}}}({\mathcal I}) = \dim {\mathrm M_{\mathrm{kp}}}+ s \dim \operatorname{Flag}(Q_0) - s(r-d)(n-r) + \sum_{k=1}^s \dim I_k/J_k.$$ It remains to show that ${\mathrm M_{\mathrm{kp}}}$ is smooth and irreducible and to compute its dimension. For this, we consider the map $$\tau \colon {\mathrm M_{\mathrm{kp}}}\to \operatorname{Gr}(d,V_0), ({\mathcal F},\phi) \mapsto \ker\phi.$$ Since $\phi$ can be specified in terms of the kernel $S:=\ker\phi$ and the injection $\bar\phi\colon V_0/S\to Q_0$, it is clear that the fibers of $\tau$ are given by $$\tau^{-1}(S) = \operatorname{Hom}^\times(V_0/S, Q_0) \times \prod_{k=1}^s \operatorname{Flag}^0_{J_k}(S, V_0).$$ Since $\tau$ is likewise a fiber bundle, we obtain that ${\mathrm M_{\mathrm{kp}}}$ is smooth and irreducible and, using , that $$\begin{aligned} &\quad \dim {\mathrm M_{\mathrm{kp}}}= \dim \operatorname{Gr}(d,V_0) + (r-d)(n-r) + \sum_{k=1}^s \dim\operatorname{Flag}^0_{J_k}(S, V_0) \\ &= d(r-d) + (r-d)(n-r) + s \bigl( \dim \operatorname{Flag}(S) + \dim \operatorname{Flag}(V_0/S) \bigr) + \sum_{k=1}^s \dim J_k \\ &= d(r-d)(1-s) + (r-d)(n-r) + s \dim \operatorname{Flag}(V_0) + \sum_{k=1}^s \dim J_k. \end{aligned}$$ By plugging this result into  and simplifying, we obtain the desired result. Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $0<\operatorname{kdim}{\mathcal I}<r$, and ${\mathcal J}=\operatorname{kPos}({\mathcal I})$. Then, $$\label{eq:tdim via kpos} \operatorname{tdim}{\mathcal I}= \operatorname{edim}{\mathcal J}+ \operatorname{edim}{\mathcal I}/{\mathcal J}$$ Recall that ${\mathrm P_{\mathrm{kp}}}({\mathcal I}) \subseteq {\mathrm P}({\mathcal I}) \supseteq {\mathrm P_{\mathrm t}}({\mathcal I})$. Moreover, $${\mathrm P_{\mathrm{kpt}}}({\mathcal I}) = {\mathrm P_{\mathrm{kp}}}({\mathcal I})\cap{\mathrm P_{\mathrm t}}({\mathcal I}) \subseteq {\mathrm P}({\mathcal I}).$$ All three varieties ${\mathrm P_{\mathrm{kpt}}}({\mathcal I})$, ${\mathrm P_{\mathrm{kp}}}({\mathcal I})$, ${\mathrm P_{\mathrm t}}({\mathcal I})$ are irreducible (\[lem:P\_t zopen,lem:P\_kpt zopen,lem:dim P\_kpt nonzero\]). Moreover, ${\mathrm P_{\mathrm{kpt}}}({\mathcal I})$ is nonempty and Zariski-open in ${\mathrm P}({\mathcal I})$, hence in both ${\mathrm P_{\mathrm{kp}}}({\mathcal I})$ and ${\mathrm P_{\mathrm t}}({\mathcal I})$. It follows that $$\dim{\mathrm P_{\mathrm{kp}}}({\mathcal I}) = \dim{\mathrm P_{\mathrm{kpt}}}({\mathcal I}) = \dim{\mathrm P_{\mathrm t}}({\mathcal I}).$$ We now obtain  via \[lem:dim P\_kpt nonzero,eq:dim Pt\]. Purbhoo [@purbhoo2006two] asserts that if ${\mathcal J}$ denotes the kernel position of ${\mathcal I}$ then ${\mathcal I}/{\mathcal J}$ is intersecting. However, we believe that the proof given therein is incomplete, as it is not clear that the map $({\mathcal F},{\mathcal G},\phi)\mapsto({\mathcal F}_{V/S},{\mathcal G})$ is dominant (cf. the remark at [@purbhooweb]). The following argument suggests that the situation is somewhat more delicate. The kernel recurrence {#subsec:kernel recurrence} --------------------- To conclude the proof in the case that $0<\operatorname{kdim}{\mathcal I}<r$, we need to understand the right-hand side of  some more. We start with the calculation $$\label{eq:tdim minus edim first} \operatorname{tdim}{\mathcal I}- \operatorname{edim}{\mathcal I}= \operatorname{edim}{\mathcal J}- (\operatorname{edim}{\mathcal I}{\mathcal J}- \operatorname{edim}{\mathcal J}) = \operatorname{edim}{\mathcal J}- \operatorname{edim}{\mathcal I}^{\mathcal J},$$ where the first equality is due to \[eq:tdim via kpos,eq:quotient edim\] and the second is \[eq:exp edim\]. The last missing ingredient is to understand the expected dimension of the kernel position, $\operatorname{edim}{\mathcal J}$. \[lem:sherman upper bound\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $0<\operatorname{kdim}{\mathcal I}<r$, and let ${\mathcal J}:=\operatorname{kPos}({\mathcal I})$. Then we have $\operatorname{edim}{\mathcal J}\leq \operatorname{tdim}{\mathcal I}^{\mathcal J}$. For any $({\mathcal F},{\mathcal G},\phi)\in {\mathrm P_{\mathrm{kp}}}({\mathcal I})$, the space $H_{\mathcal J}({\mathcal F}^{\ker\phi}, {\mathcal F}_{V_0/\ker\phi})$ injects into $H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^{\ker\phi},{\mathcal G})$ by composition with the injective map $\bar\phi\colon V_0/\ker\phi\to Q_0$ induced by $\phi$ (\[lem:exp vs composition\]). Thus, $$\operatorname{edim}{\mathcal J}\leq\operatorname{tdim}{\mathcal J}\leq \dim H_{\mathcal J}({\mathcal F}^{\ker\phi},{\mathcal F}_{V_0/\ker\phi})\leq\dim H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^{\ker\phi},{\mathcal G}),$$ where the first inequality is always true (\[cor:tdim edim\]), the second holds by definition of the true dimension and the third follows from the injection. It thus suffices to prove that there exists $({\mathcal F},{\mathcal G},\phi)\in{\mathrm P_{\mathrm{kp}}}({\mathcal I})$ such that $\dim H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^S,{\mathcal G})\leq\operatorname{tdim}{\mathcal I}^{\mathcal J}$. For this, let $K(d,V_0)$ denote the fiber bundle over $\operatorname{Gr}(d,V_0)$ with fiber over $S\in\operatorname{Gr}(d,V_0)$ given by $\operatorname{Flag}(S)^s\times\operatorname{Flag}(Q_0)^s$. It is an irreducible algebraic variety and we denote its elements by $(S,\tilde{\mathcal F},{\mathcal G})$. We consider the morphism $$\pi\colon {\mathrm P_{\mathrm{kp}}}({\mathcal I})\to K(d,V_0), \quad ({\mathcal F},{\mathcal G},\phi)\mapsto(\ker\phi,{\mathcal F}^{\ker\phi},{\mathcal G}).$$ For any $({\mathcal F},{\mathcal G},\phi)\in{\mathrm P_{\mathrm{kp}}}({\mathcal I})$, $\dim\ker\phi=d$ and $\operatorname{Pos}(\ker\phi,{\mathcal F})={\mathcal J}$, hence $\pi$ is indeed a morphism. We first prove that $\pi$ is dominant. Note that, as a consequence of \[lem:P\_kpt zopen\], the map ${\mathrm P_{\mathrm{kp}}}({\mathcal I})\to\operatorname{Flag}(Q_0)^s, ({\mathcal F},{\mathcal G},\phi)\mapsto{\mathcal G}$ contains a nonempty Zariski-open subset $U \subseteq \operatorname{Flag}(Q_0)^s$. We now show that the image of $\pi$ contains all elements $(S,\tilde{\mathcal F},{\mathcal G})$ with $S\in\operatorname{Gr}(d,V_0)$, $\tilde{\mathcal F}\in\operatorname{Flag}(S)^s$ and ${\mathcal G}\in U$. For this, let $({\mathcal F}_0,{\mathcal G},\phi_0)\in{\mathrm P_{\mathrm{kp}}}({\mathcal I})$ be the preimage of some arbitrary ${\mathcal G}\in U$. Let $S_0 :=\ker\phi_0$ and choose some $g \in \operatorname{GL}(V_0)$ such that $g \cdot S_0 = S$. Using the corresponding diagonal action, ${\mathcal F}:=g\cdot{\mathcal F}_0$ and $\phi:=g\cdot\phi_0$, we obtain that $({\mathcal F},{\mathcal G},\phi)\in{\mathrm P_{\mathrm{kp}}}({\mathcal I})$ and $\ker\phi=S$. Given $\tilde{\mathcal F}\in\operatorname{Flag}(S)^s$, we now choose $\vec{h}\in\operatorname{GL}(V_0)^s$ such that $h_k S \subseteq S$, $h_k \cdot F_k^S = \tilde F_k$, and $h_k$ acts trivially on $V_0/S$ for all $k\in[s]$. Then $\operatorname{Pos}(S, \vec{h} \cdot {\mathcal F}) = \operatorname{Pos}(S, {\mathcal F}) = {\mathcal J}$, which shows that $(\vec{h} \cdot {\mathcal F})^S = \vec{h} \cdot {\mathcal F}^S = \tilde{\mathcal F}$. Moreover, $(\vec{h} \cdot {\mathcal F})_{V_0/S} = {\mathcal F}_{V_0/S}$. Thus $\phi\in H_{\mathcal I}({\mathcal F}, {\mathcal G})$ implies that $\phi\in H_{\mathcal I}(\vec{h} \cdot {\mathcal F}, {\mathcal G})$ by \[lem:exp vs composition\]. Together, we find that the triple $(\vec{h}\cdot{\mathcal F},{\mathcal G},\phi)$ is in ${\mathrm P_{\mathrm{kp}}}({\mathcal I})$ and mapped by $\pi$ to $(S,\tilde{\mathcal F},{\mathcal G})$. We thus obtain that $\pi$ is dominant. To conclude the proof, we note that the subset $W\subseteq K(d,V_0)$ consisting of those $(S,\tilde{\mathcal F},{\mathcal G})$ with $\dim H_{{\mathcal I}^{\mathcal J}}(\tilde{\mathcal F},{\mathcal G})=\operatorname{tdim}{\mathcal I}^{\mathcal J}$ is a nonempty Zariski-open subset, and hence Zariski-dense since $K(d,V_0)$ is irreducible. For each fixed choice of $S$, this is the claim in \[lem:P\_t zopen\] for $B_t({\mathcal I})$, with ${\mathcal I}^{\mathcal J}$ instead of ${\mathcal I}$. The ‘parametrized version’ is proved in the same way. Since $\pi$ is dominant, the preimage $\pi^{-1}(W)$ is a nonempty Zariski-open subset of ${\mathrm P_{\mathrm{kp}}}({\mathcal I})$. In particular, any $({\mathcal F},{\mathcal G},\pi)\in\pi^{-1}(W)\subseteq{\mathrm P_{\mathrm{kp}}}({\mathcal I})$ satisfies $\dim H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^S,{\mathcal G})\leq\operatorname{tdim}{\mathcal I}^{\mathcal J}$. We thus obtain the following fundamental recurrence relation, due to Sherman [@sherman2015geometric], as a consequence of \[eq:tdim minus edim first,lem:sherman upper bound\]: $$\label{eq:sherman recurrence} \operatorname{tdim}{\mathcal I}-\operatorname{edim}{\mathcal I}\leq \operatorname{tdim}{\mathcal I}^{\mathcal J}-\operatorname{edim}{\mathcal I}^{\mathcal J}$$ Now we have assembled all ingredients to prove Belkale’s theorem: \[thm:belkale\] For $r\in[n]$ and $s\geq2$, $\operatorname{Intersecting}(r, n, s) = \operatorname{Horn}(r, n, s)$. We proceed by induction on $r$. The base case, $r=1$, is \[ex:base case\]. Thus we have $\operatorname{Intersecting}(1,n,s) = \operatorname{Horn}(1,n,s)$ for all $n\geq1$. Now let $r>1$. By the induction hypothesis, $\operatorname{Horn}(d,n',s) = \operatorname{Intersecting}(d,n',s)$ for all $0<d<r$ and $d\leq n'$. In particular, $\operatorname{Horn}(r,n,s)$ from \[def:horn\] can be written in the following form: $$\begin{aligned} &\quad \operatorname{Horn}(r,n,s) \\ &= \{ {\mathcal I}: \operatorname{edim}{\mathcal I}\geq0, \;\;\forall {\mathcal J}\in\operatorname{Intersecting}(d,r,s),0<d<r, \;\operatorname{edim}{\mathcal J}=0: \operatorname{edim}{\mathcal I}{\mathcal J}\geq0\} \\ &= \{ {\mathcal I}: \operatorname{edim}{\mathcal I}\geq0, \;\;\forall {\mathcal J}\in\operatorname{Intersecting}(d,r,s),0<d<r, \;\operatorname{edim}{\mathcal I}{\mathcal J}\geq\operatorname{edim}{\mathcal J}\} \end{aligned}$$ where the second equality is due to \[prp:belkale strong weak\]. Hence it is a direct consequence of \[cor:belkale inductive\] that $\operatorname{Intersecting}(r,n,s) \subseteq \operatorname{Horn}(r,n,s)$. We now prove the converse. Thus let ${\mathcal I}\in\operatorname{Horn}(r,n,s)$. Let $d:=\operatorname{kdim}{\mathcal I}$. If $d=0$ or $d=r$ then we know from \[lem:kdim r intersecting,cor:kdim zero intersecting\], respectively, that ${\mathcal I}$ is intersecting. We now discuss the case where $0<d<r$. By \[eq:sherman recurrence\], we have that $$\operatorname{tdim}{\mathcal I}- \operatorname{edim}{\mathcal I}\leq \operatorname{tdim}{\mathcal I}^{\mathcal J}- \operatorname{edim}{\mathcal I}^{\mathcal J},$$ where ${\mathcal J}:= \operatorname{kPos}({\mathcal I})$ denotes the kernel position of ${\mathcal I}$. If we can show that ${\mathcal I}^{\mathcal J}$ is intersecting then the right-hand side is zero by \[cor:tdim edim\], hence so is the left-hand side, since $\operatorname{tdim}{\mathcal I}-\operatorname{edim}{\mathcal I}\geq 0$, and thus ${\mathcal I}$ is intersecting, which is what we set out to prove. To see that ${\mathcal I}^{\mathcal J}$ is intersecting, we note that $\operatorname{Intersecting}(d,n-r+d,s) = \operatorname{Horn}(d,n-r+d,s)$ by the induction hypothesis, hence it remains to verify that ${\mathcal I}^{\mathcal J}$ satisfies the Horn inequalities. Let ${\mathcal K}\in\operatorname{Horn}(m,d,s)=\operatorname{Intersecting}(m,d,s)$ for any $0<m\leq d$, where we have used the induction hypothesis one last time. Thus ${\mathcal J}{\mathcal K}\in\operatorname{Intersecting}(m,r,s)$ by \[cor:kerpos intersecting,lem:chain rule intersecting\]. It follows that $$\begin{aligned} \operatorname{edim}{\mathcal I}^{\mathcal J}{\mathcal K}- \operatorname{edim}{\mathcal K}= \operatorname{edim}{\mathcal I}({\mathcal J}{\mathcal K}) - \operatorname{edim}{\mathcal J}{\mathcal K}\geq 0 \end{aligned}$$ where the first step is  and the second step holds because by assumption ${\mathcal I}\in\operatorname{Horn}(r,n,s)$ and ${\mathcal J}{\mathcal K}\in\operatorname{Intersecting}(m,r,s)=\operatorname{Horn}(m,r,s)$, as explained above. We remark that these inequalities include $\operatorname{edim}{\mathcal I}^{\mathcal J}\geq0$ (corresponding to $m=d$). Thus we have shown that ${\mathcal I}^{\mathcal J}$ satisfies the Horn inequalities. This is what remained to be proved. Invariants and Horn inequalities {#sec:invariants} ================================ In this section, we show that the Horn inequalities not only characterize intersections, but also the existence of corresponding nonzero invariants and, thereby, the Kirwan cone for the eigenvalues of sums of Hermitian matrices. Borel-Weil construction {#subsec:borel-weil} ----------------------- For any dominant weight $\lambda\in\Lambda_+(r)$ there exists an irreducible representation $L(\lambda)$ of $\operatorname{GL}(r)$ with highest weight $\lambda$, unique up to isomorphism. Following Borel and Weil, it can be constructed as follows: For any weight $\mu\in\Lambda(r)$, let us denote by $\chi_\mu\colon B(r)\to{\mathbb C}^*$ the character of $B(r)$ such that $\chi_\mu(t) = t^\mu = t(1)^{\mu(1)}\cdots t(r)^{\mu(r)}$ for all $t\in H(r)\subseteq B(r)$. Here, we recall that $B(r)$ is the group of upper-triangular invertible matrices and $H(r) \subseteq B(r)$ the Cartan subgroup, which consists of invertible matrices $t\in\operatorname{GL}(r)$ that are diagonal in the standard basis, with diagonal entries $t(1),\dots,t(r)$. Lastly, we write ${\mathbbm 1}_r=(1,\dots,1)\in\Lambda(r)$ for the highest weight of the determinant representation of $\operatorname{GL}(r)$, denoted $\det_r$. It is clear that $L(\lambda + k {\mathbbm 1}_r) = L(\lambda) {\otimes}\det_r^k$ for any $\lambda\in\Lambda_+(r)$ and $k\in{\mathbb Z}$. \[def:borel weil\] Let $\lambda\in\Lambda_+(r)$. Then we define the *Borel-Weil realization* of $L(\lambda)$ as $$L_{BW}(\lambda) = \{ s\colon \operatorname{GL}(r) \to {\mathbb C}\text{ holomorphic} \;:\; s(gb) = s(g)\chi_{\lambda^*}(b) \quad \forall g\in\operatorname{GL}(r), b\in B(r) \}$$ with the action of $\operatorname{GL}(r)$ given by $(g \cdot s)(h) := s(g^{-1}h)$. We recall that $\lambda^*=(-\lambda(r),\dots,-\lambda(1))$. The Borel-Weil theorem asserts that $L_{BW}(\lambda)$ is an irreducible $\operatorname{GL}(r)$-representation of highest weight $\lambda$. Note that, by definition, a holomorphic function is in $L_{BW}(\lambda)$ if it is a highest weight vector of weight $\lambda^*$ with respect to the *right* multiplication representation, $(g \star s)(h) := s(h g)$. The space $L_{BW}(\lambda)$ can also be interpreted as the space of holomorphic sections of the $\operatorname{GL}(r)$-equivariant line bundle ${\mathcal L}_{BW}(\lambda) := \operatorname{GL}(r) \times_{B(r)} {\mathbb C}_{-\lambda^*}$ over $\operatorname{Flag}(r)\cong\operatorname{GL}(r)/B(r)$, where we write ${\mathbb C}_\mu$ for the one-dimensional representation of $B(r)$ given by the character $\chi_\mu$. It is useful to observe that we have a $\operatorname{GL}(r)$-equivariant isomorphism $$\label{eq:borel weil dual iso} L(\lambda)^* \to L_{BW}(\lambda^*), \quad f \mapsto (s_f \colon \operatorname{GL}(r)\to{\mathbb C}, \; g \mapsto f(g \cdot v_\lambda))$$ where $v_\lambda$ denotes a fixed highest weight vector in $L(\lambda)$. The tensor product of several Borel-Weil representations can again be identified with a space of functions. E.g., if $\lambda\in\Lambda_+(r)$ and $\lambda'\in\Lambda_+(r')$ then $$\begin{aligned} &L_{BW}(\lambda) {\otimes}L_{BW}(\lambda') \cong \{ s \colon \operatorname{GL}(r)\times\operatorname{GL}(r')\to{\mathbb C}\text{ holomorphic}, \\ &\qquad s(gb,g'b') = s(g,g')\chi_{\lambda^*}(b)\chi_{{\lambda'}^*}(b') \quad \forall g\in\operatorname{GL}(r),g'\in\operatorname{GL}(r'), b\in B(r),b'\in B(r') \}.\end{aligned}$$ We will use this below to obtain a nonzero vector in a tensor product space by exhibiting a corresponding holomorphic function with the appropriate equivariance properties. Invariants from intersecting tuples {#subsec:invariants from intersecting tuples} ----------------------------------- Let us consider the tangent map , $$\Delta_{{\mathcal I},\vec g,\vec h}\colon\begin{cases} \operatorname{Hom}(V_0,Q_0)\times H_{I_1}(F_0,G_0)\times\dots\times H_{I_s}(F_0,G_0)\to \operatorname{Hom}(V_0,Q_0)^s \\ (\zeta,\phi_1,\dots,\phi_s)\mapsto(\zeta+h_1\phi_1g_1^{-1},\dots,\zeta+h_s\phi_s g_s^{-1}) \end{cases}$$ If $\operatorname{edim}{\mathcal I}=0$ then  implies that the dimension of the domain and target space are the same. Thus we may consider the determinant of $\Delta_{{\mathcal I},\vec g,\vec\ d}$, as in the following definition: \[def:determinant\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $\operatorname{edim}{\mathcal I}=0$. Then we define the *determinant function* as the holomorphic function $$\delta_{\mathcal I}\colon\begin{cases} \operatorname{GL}(r)^s \times \operatorname{GL}(n-r)^s \rightarrow {\mathbb C}, \\ (\vec g, \vec h) \mapsto \det\Delta_{{\mathcal I},\vec g,\vec h} \end{cases}$$ where the determinant is evaluated with respect to two arbitrary bases. Here, and throughout the following, we identify $V_0\cong{\mathbb C}^r$ and $Q_0 \cong{\mathbb C}^{n-r}$, so that $\operatorname{GL}(V_0)\cong\operatorname{GL}(r)$ and $\operatorname{GL}(Q_0)\cong\operatorname{GL}(n-r)$ and the discussion in \[subsec:borel-weil\] is applicable. If ${\mathcal I}$ is intersecting then also $\operatorname{tdim}{\mathcal I}=\operatorname{edim}{\mathcal I}$ by \[cor:tdim edim\]. Hence by  there exist $\vec g,\vec h$ such that $\delta_{\mathcal I}(\vec g,\vec h)\neq 0$. That is, $\delta_{\mathcal I}$ is a nonvanishing holomorphic function of $\operatorname{GL}(r)^s \times \operatorname{GL}(n-r)^s$. Our goal is to show that $\delta_{\mathcal I}$ can be interpreted as an invariant in a tensor product of irreducible $\operatorname{GL}(r)\times\operatorname{GL}(n-r)$-representations. We now consider the representation of $\operatorname{GL}(r)\times\operatorname{GL}(n-r)$ on $\operatorname{Hom}(V_0,Q_0)$ given by $(a,d)\cdot\phi := d\phi a^{-1}$. Since $\operatorname{Hom}(V_0, Q_0) = V_0^* {\otimes}Q_0$, it is clear that for $g\in\operatorname{GL}(V_0)$, $g'\in\operatorname{GL}(Q_0)$, $$\label{eq:base change H} \det\Bigl( \operatorname{Hom}(V_0, Q_0)\ni\phi \mapsto g'\phi g^{-1}\in\operatorname{Hom}(V_0, Q_0) \Bigr) = \det(g)^{-(n-r)} \det(g')^r.$$ We now restrict to the subspaces $H_I(F_0, G_0)$: \[lem:base change H\_I\] Let $I \subseteq [n]$ be a subset of cardinality $r$. Then $H_I(F_0, G_0) \subseteq \operatorname{Hom}(V_0,Q_0)$ is $B(r) \times B(n-r)$-stable. Furthermore, for $b\in B(r)$ and $b'\in B(n-r)$ we have $$\det \Bigl( H_I(F_0, G_0)\ni\phi \mapsto b'\phi b^{-1} \in H_I(F_0, G_0) \Bigr) = \chi_{\lambda_I}(b) \chi_{\lambda_{I^c} + r{\mathbbm 1}_{n-r}}(b'),$$ where we recall that $\lambda_I$ was defined in \[def:lambda\_I\]. For the first claim, we use \[lem:H\_I borel invariance\]: Since the flag $F_0$ is stabilized by $B(r)$ and the flag $G_0$ is stabilized by $B(n-r)$, it is clear that $H_I(F_0, G_0)$ is stable under the action of $B(r)\times B(n-r)$. For the second claim, we note that unipotent elements always act by representation matrices of determinant one. Hence it suffices to verify the formula for the determinant for $t \in H(r)$ and $t' \in H(n-r)$. For this, we work in the weight basis of $H_I(F_0, G_0)$ given by the elementary matrices $E_{b,a}$ that send $e(a) \mapsto \bar e(b)$, where $a\in[r]$ and $b\in[I(a)-a]$, and all other basis vectors to zero. Then: $$\begin{aligned} &\quad \det (H_I(F_0, G_0)\ni\phi \mapsto t'\phi t^{-1} \in H_I(F_0, G_0)) \\ &= \prod_{a=1}^r \prod_{b=1}^{I(a)-a} t'(b) t(a)^{-1} = \left( \prod_{a=1}^r t(a)^{a-I(a)} \right) \left( \prod_{b=1}^{n-r} t'(b)^{r - \#\{ a : I(a)-a<b\}} \right) \\ &= t^{\lambda_I} {t'}^{r{\mathbbm 1}_{n-r} + \lambda_{I^c}}, \end{aligned}$$ where we have used  in the last step. We now show that the $\delta_{\mathcal I}$ can be interpreted as an invariant: \[thm:invariant\] Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $\operatorname{edim}{\mathcal I}=0$, and let $\delta_{\mathcal I}$ denote the corresponding determinant function (\[def:determinant\]). Then $\delta_{\mathcal I}$ belongs to $\bigotimes_{k=1}^s \bigl( L_{BW}(\lambda_{I_k}^*) {\otimes}L_{BW}(\lambda_{I_k^c}^* - r{\mathbbm 1}_{n-r}) \bigr)$. Moreover, it transforms under the diagonal action of $\operatorname{GL}(r)\times\operatorname{GL}(n-r)$ by the character $\det_r^{(n-r)(s-1)} {\otimes}\det_{n-r}^{r(1-s)}$. For the first claim, we note that if $\vec{g'}\in B(r)^s$, $\vec{h'}\in B(n-r)^s$ then we can write $\Delta_{{\mathcal I},\vec g\vec{g'},\vec h\vec{h'}}$ as a composition of $\Delta_{{\mathcal I},\vec g,\vec h}$ with the automorphisms on $H_{I_k}(F_0, G_0)$ that send $\phi_k \mapsto h'_k \phi_k (g'_k)^{-1}$. Using \[lem:base change H\_I\], we obtain $$\delta_{\mathcal I}(\vec g\vec{g'}, \vec h\vec{h'}) = \delta_{\mathcal I}(\vec g, \vec h) \prod_{k=1}^s \chi_{\lambda_I}(g'_k) \chi_{\lambda_{I_k^c} + r{\mathbbm 1}_{n-r}}(h'_k).$$ In view of the discussion at the end of \[subsec:borel-weil\] this establishes the first claim. For the second claim, let $g\in\operatorname{GL}(r)$ and $g'\in\operatorname{GL}(n-r)$. Thus $\Delta_{{\mathcal I},g^{-1} \vec g,{g'}^{-1} \vec h}$ maps $(\zeta, \phi_1, \dots, \phi_s)$ to $$\begin{aligned} &\quad(\zeta + g'^{-1} h_1 \phi_1 g_1^{-1} g, \dots, \zeta + g'^{-1} h_s \phi_s g_s^{-1} g) \\ &= g'^{-1} (g' \zeta g^{-1} + h_1 \phi_1 g_1^{-1}, \dots, g' \zeta g^{-1} + h_s \phi_s g_s^{-1}) g \end{aligned}$$ Thus we can write $\Delta_{{\mathcal I},g^{-1} \vec g,{g'}^{-1} \vec h}$ as a composition of three maps: The automorphism $\zeta \mapsto g' \zeta g^{-1}$ of $\operatorname{Hom}(V_0,Q_0)$, the map $\Delta_{{\mathcal I},\vec g,\vec h}$ and the automorphism $\vec\phi \mapsto g'^{-1}\vec\phi g$ on $\operatorname{Hom}(V_0,Q_0)^s$. Thus, using \[eq:base change H\], $$\begin{aligned} ((g,g') \cdot \delta_{\mathcal I})(\vec g, \vec h) = \delta_{\mathcal I}(g^{-1} \vec g, {g'}^{-1} \vec h) = \det(g)^{-(n-r)(1-s)} \det(g')^{r(1-s)} \delta_{\mathcal I}(\vec g, \vec h), \end{aligned}$$ which establishes the second claim. If ${\mathcal I}$ is intersecting then we had argued before that $\delta_{\mathcal I}$ is nonzero. By dualizing and simplifying, we obtain the following corollary of \[thm:invariant\]: \[cor:invariant\] Let ${\mathcal I}\in\operatorname{Intersecting}(r,n,s)$ and $\operatorname{edim}{\mathcal I}=0$. Then, $$\bigl( {\det}_r^{(s-1)(n-r)} {\otimes}\bigotimes_{k=1}^s L(\lambda_{I_k}) \bigr)^{\operatorname{GL}(r)} \neq 0 \quad\text{and}\quad \bigl( {\det}_{n-r}^r {\otimes}\bigotimes_{k=1}^s L(\lambda_{I^c_k}) \bigr)^{\operatorname{GL}(n-r)} \neq 0.$$ Let us correspondingly define $$\begin{aligned} c({\mathcal I}) := \dim \bigl( {\det}_r^{(s-1)(n-r)} {\otimes}\bigotimes_{k=1}^s L(\lambda_{I_k}) \bigr)^{\operatorname{GL}(r)}.\end{aligned}$$ Then \[cor:invariant\] states that, if ${\mathcal I}$ is intersecting and $\operatorname{edim}{\mathcal I}=0$ then $c({\mathcal I})>0$. This relationship between generic intersections of Schubert cells and tensor product multiplicities can be made quantitative. While we do not use this in the following \[subsec:kirwan\] to describe the Kirwan cone and prove the saturation property for tensor product multiplicities, we will give a brief sketch later on in \[subsec:invariants and intersections\] and use it to establish the Fulton conjecture. Kirwan cone and saturation {#subsec:kirwan} -------------------------- We now show that the existence of nonzero invariants is characterized by the Horn inequalities. For this, recall that we defined $c(\vec\lambda)$ as the dimension of the space of $\operatorname{GL}(r)$-invariants in the tensor product $\bigotimes_{k=1}^s L(\lambda_k)$. Thus, if we define $\lambda_k = \lambda_{I_k} + (n-r){\mathbbm 1}_r$ for $k\in[s-1]$ and $\lambda_s = \lambda_{I_s}$, then \[cor:invariant\] shows that $$\label{eq:tensor product invariant} c(\vec\lambda) = c({\mathcal I}) > 0$$ whenever ${\mathcal I}$ is intersecting and $\operatorname{edim}{\mathcal I}=0$. Here, we have somewhat arbitrarily selected the first $s-1$ highest weights $\lambda_1,\dots,\lambda_{s-1}$ to have nonnegative entries no larger than $n-r$, while $\lambda_s$ has nonpositive entries no smaller than $r-n$. Conversely, any $s$-tuple of highest weights $\vec{\lambda}$ with these properties can be obtained in this way from some ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ (recall discussion below \[def:lambda\_I\]). \[prp:horn implies invariant\] Let $\vec\lambda\in\Lambda_+(r)^s$ be an $s$-tuple such that $\sum_{k=1}^s \lvert\lambda_k\rvert = 0$, and for any $0<d<r$ and any $s$-tuple ${\mathcal J}\in\operatorname{Horn}(d,r,s)$ with $\operatorname{edim}{\mathcal J}=0$ we have that $\sum_{k=1}^s (T_{J_k},\lambda_k) \leq 0$. Then $c(\vec\lambda)>0$. By adding/removing suitable multiples of ${\mathbbm 1}_r$, the highest weight of the determinant representation, we may assume that $\lambda_1(r),\dots,\lambda_{s-1}(r)\geq0$ and $\lambda_s(1)\leq0$. Let $n:=r+q$, where $q:=\max\{\lambda_1(1),\dots,\lambda_{s-1}(1),-\lambda_s(r)\}$. Then $\vec\lambda$ is associated to an $s$-tuple ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ as in \[lem:I to lambdaprime\]. We now show that $\operatorname{edim}{\mathcal I}=0$ and that ${\mathcal I}$ is intersecting. The former follows from the first statement in \[lem:I to lambdaprime\], which gives that $\operatorname{edim}{\mathcal I}=-\sum_{k=1}^s \lvert\lambda_k\rvert=0$. To see that ${\mathcal I}$ is intersecting, we may use \[thm:belkale\] and show instead that ${\mathcal I}$ satisfies the Horn inequalities $\operatorname{edim}{\mathcal I}{\mathcal J}\geq0$ for any ${\mathcal J}\in\operatorname{Horn}(d,r,s)$ with $\operatorname{edim}{\mathcal J}=0$ and $0<d<r$. But the second statement in \[lem:I to lambdaprime\] implies that these are equivalent to the linear inequalities $\sum_{k=1}^s (T_{J_k},\lambda_k)\leq0$, which hold by assumption. Thus ${\mathcal I}$ is indeed intersecting and satisfies $\operatorname{edim}{\mathcal I}=0$. Now  shows that $c(\vec\lambda)=c({\mathcal I})>0$. At last we can prove the saturation property and characterize of the Kirwan cone in terms of Horn inequalities. \[cor:horn and saturation\] (a) *Horn inequalities:* The Kirwan cone $\operatorname{Kirwan}(r,s)$ is the convex polyhedral cone of $\vec\xi\in C_+(r)^s$ such that $\sum_{k=1}^s \lvert \xi_k \rvert = 0$, and for any $0<d<r$ and any $s$-tuple ${\mathcal J}\in \operatorname{Horn}(d,r,s)$ with $\operatorname{edim}{\mathcal J}=0$ we have that $\sum_{k=1}^s (T_{J_k},\xi_k) \leq 0$. \(b) *Saturation property:* For a dominant weight $\vec\lambda\in \Lambda_+(r)^s$, the space of invariants $(L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)}$ is nonzero if and only if $\vec\lambda \in \operatorname{Kirwan}(r,s)$. In particular, $c(\vec\lambda) := \dim (L(\lambda_1) {\otimes}\cdots {\otimes}L(\lambda_s))^{\operatorname{GL}(r)} > 0$ if and only if $c(N \vec\lambda) > 0$ for some integer $N > 0$. The two statements are closely interlinked. For clarity, we give separate proofs that do not refer to each other. \(a) Any $\vec\xi\in\operatorname{Kirwan}(r,s)$ satisfies the Horn inequalities (\[cor:klyachko kirwan\]). We now observe that $\operatorname{Kirwan}(r,s)$ is a closed subset of $C_+(r)^s$ which, moreover, is invariant under rescaling by nonnegative real numbers. Thus it suffices to prove the converse only for $\vec\lambda\in\Lambda_+(r)^s$. For this, we use that if $\vec\lambda$ satisfies the Horn inequalities then $c(\vec\lambda)>0$ by \[prp:horn implies invariant\], hence $\vec\lambda\in\operatorname{Kirwan}(r,s)$ by \[prp:kempf-ness\]. \(b) Let $\vec\lambda\in\Lambda_+(r)^s$. If $c(\vec\lambda)>0$ then $\vec\lambda\in\operatorname{Kirwan}(r,s)$ by \[prp:kempf-ness\]. Conversely, if $\vec\lambda\in\operatorname{Kirwan}(r,s)$ then it satisfies the Horn inequalities by \[cor:klyachko kirwan\], hence $c(\vec\lambda)>0$ by \[prp:horn implies invariant\]. \[rem:redundant\] As follows from the discussion below \[prp:belkale strong weak\], the Kirwan cone is in fact already defined by those ${\mathcal J}$ such that $\Omega_{\mathcal J}({\mathcal G})$ is a point for all ${\mathcal G}\in\operatorname{Good}(r,s)$. Ressayre has shown that the corresponding inequalities are irredundant and can be computed by an inductive algorithm [@ressayre2011cohomology]. Demanding that $\Omega_{\mathcal J}({\mathcal G})$ is a point for all good ${\mathcal G}$ is a more stringent requirement than $\operatorname{edim}{\mathcal J}=0$, and indeed the set of inequalities $\operatorname{edim}{\mathcal I}{\mathcal J}\geq0$ for ${\mathcal J}\in\operatorname{Horn}(d,r,s)$ with $\operatorname{edim}{\mathcal J}=0$ is in general still redundant. However, from a practical point of view we prefer the latter criterion since it is much easier to check numerically. Invariants and intersection theory {#subsec:invariants and intersections} ---------------------------------- We now explain how the relationship between generic intersections of Schubert cells and tensor product multiplicities can be made more quantitative. Specifically, we shall relate the dimension $c({\mathcal I})$ of the space of $\operatorname{GL}(r)$-invariants to the number of points in a generic intersection $\Omega_{\mathcal I}({\mathcal E})$, as in the following definition: Let ${\mathcal I}\in\operatorname{Subsets}(r,n,s)$ such that $\operatorname{edim}{\mathcal I}=0$. We define the corresponding *intersection number* as $${c_{\mathrm{int}}}({\mathcal I}) := \#\Omega^0_{\mathcal I}({\mathcal E}) = \#\Omega_{\mathcal I}({\mathcal E}),$$ where ${\mathcal E}$ is an arbitrary $s$-tuple of flags in $\operatorname{Good}(n,s)$. By \[lem:good set\], the right-hand side is finite and independent of the choice of ${\mathcal E}$ in $\operatorname{Good}(n,s)$. Moreover, ${c_{\mathrm{int}}}({\mathcal I})>0$ if and only if ${\mathcal I}$ is intersecting. In \[subsec:invariants from intersecting tuples\] we showed that if ${\mathcal I}$ is intersecting then $c({\mathcal I})>0$. Indeed, in this case the determinant function $\delta_{\mathcal I}$ on $\operatorname{GL}(r)^s\times\operatorname{GL}(n-r)^s$ is nonzero, so that for some suitable $\vec h\in\operatorname{GL}(n-r)^s$ the function $$\label{eq:invariant in one argument} \delta_{{\mathcal I},\vec h} \colon \operatorname{GL}(r)^s \to {\mathbb C}, \quad \delta_{{\mathcal I},\vec h}(\vec g) := \delta_{\mathcal I}(\vec g, \vec h)$$ is a nonzero vector in $\bigotimes_{k=1}^s L_{BW}(\lambda^*_{I_k})$ that transforms as the character $\det_r^{(n-r)(s-1)}$ with respect to the diagonal action of $\operatorname{GL}(r)$. In the following we show that, as we vary $\vec h$, the functions $\delta_{{\mathcal I},\vec h}$ span a vector space of dimension at least ${c_{\mathrm{int}}}({\mathcal I})$, which will imply that $c({\mathcal I})\geq{c_{\mathrm{int}}}({\mathcal I})$. More precisely, we shall construct elements $(\vec g_\alpha, \vec h_\alpha)\in\operatorname{GL}(r)^s\times\operatorname{GL}(n-r)^s$ for $\alpha\in[{c_{\mathrm{int}}}({\mathcal I})]$ such that $\delta_{\mathcal I}(\vec g_\alpha, \vec h_\alpha)\neq0$ while $\delta_{\mathcal I}(\vec g_\alpha, \vec h_\beta)=0$ if $\alpha\neq\beta$. The construction, due to Belkale [@belkale2004invariant], depends on a choice of good flags ${\mathcal E}$ and goes as follows. Let ${\mathcal E}$ be an $s$-tuple of good flags and consider the intersection $$\Omega^0_{\mathcal I}({\mathcal E}) = \{ V_1, \dots, V_{{c_{\mathrm{int}}}({\mathcal I})} \}.$$ Let $\gamma_\alpha\in\operatorname{GL}(n)$ such that $V_\alpha = \gamma_\alpha \cdot V_0$ for each $\alpha\in[{c_{\mathrm{int}}}({\mathcal I})]$, and consider the $s$-tuple of flags ${\mathcal E}_\alpha=(E_{\alpha,1},\dots,E_{\alpha,s})$ defined by $E_{\alpha,k} = \gamma_\alpha^{-1} \cdot E_k$. Then $\bar\omega^0_{\mathcal I}([\gamma_\alpha, {\mathcal E}_\alpha]) = {\mathcal E}$. According to \[lem:good set\], ${\mathcal E}$ is a regular value of $\bar\omega^0_{\mathcal I}$, since ${\mathcal I}$ is intersecting. Since $\operatorname{edim}{\mathcal I}=0$, this implies that the differential of $\bar\omega^0_{\mathcal I}$ is bijective at $[\gamma_\alpha, {\mathcal E}_\alpha]$, and, by equivariance, so is its differential at $[1, {\mathcal E}_\alpha]$. By \[rem:delta\^0\_I factorized differential\], its determinant is precisely $\delta_{\mathcal I}(\vec g_\alpha, \vec h_\alpha)$, where $\vec g_\alpha=(g_{\alpha,1},\dots,g_{\alpha,s})\in\operatorname{GL}(r)$ and $\vec h_\alpha=(h_{\alpha,1},\dots,h_{\alpha,s})\in\operatorname{GL}(n-r)$ are such that $g_{\alpha,k} \cdot F_0 = (E_{\alpha,k})^{V_0}$ and $h_{\alpha,k} \cdot G_0 = (E_{\alpha,k})_{Q_0}$ for all $\alpha$ and $k$. In particular, $\delta_{\mathcal I}(\vec g_\alpha, \vec h_\alpha)\neq0$. Using $\operatorname{edim}{\mathcal I}=0$, \[eq:rank nullity,eq:ker Delta\] imply that $$\label{eq:delta_I vs H_I} \delta_{\mathcal I}(\vec g,\vec h)\neq0 \;\Leftrightarrow\; \dim H_{\mathcal I}(\vec g \cdot F_0, \vec h \cdot G_0)=0.$$ Then we have the following lemma: \[lem:criss cross\] Let ${\mathcal I}$ be intersecting, $\operatorname{edim}{\mathcal I}=0$, and ${\mathcal E}\in\operatorname{Good}(n,s)$. As above, choose $\gamma_\alpha$, $\vec g_\alpha$ and $\vec h_\alpha$ for $\alpha\in[{c_{\mathrm{int}}}({\mathcal I})]$. Define $\delta_{{\mathcal I},\alpha}(\vec g) := \det\Delta_{{\mathcal I},\vec g,\vec h_\alpha}$. Then $\delta_{{\mathcal I},\alpha}(\vec g_\alpha)\neq0$ for all $\alpha$, while $\delta_{{\mathcal I},\beta}(\vec g_\alpha)=0$ for all $\alpha\neq\beta$. We only need to consider the case that $\alpha\neq \beta$. In view of , it suffices to show that $H_{\mathcal I}(({\mathcal E}_\alpha)^{V_0}, ({\mathcal E}_\beta)_{Q_0})\neq\{0\}$. For this, we define the map $$\phi_{\alpha,\beta}\colon V_0\to {\mathbb C}^n/V_0\cong Q_0, \; v \mapsto (\gamma_\alpha)^{-1} \gamma_\beta v + V_0,$$ which is nonzero since $\gamma_\alpha V_0=V_\alpha\neq V_\beta=\gamma_\beta V_0$. Then $\phi_{\alpha,\beta}$ is a nonzero element in $H_{\mathcal I}(({\mathcal E}_\alpha)^{V_0}, ({\mathcal E}_\beta)_{Q_0})$, since $$\phi_{\alpha,\beta}((E_{\beta,k})^{V_0}(a)) = \phi_{\alpha,\beta}(E_{\beta,k}(I_k(a))) = E_{\alpha,k}(I_k(a)) + V_0 = (E_{\alpha,k})_{Q_0}(I_k(a)-a)$$ for all $a\in[r]$ and $k\in[s]$, using that ${\mathcal I}=\operatorname{Pos}(V_0,{\mathcal E}_\alpha)$. shows that the functions $\delta_{{\mathcal I},1},\dots,\delta_{{\mathcal I},{c_{\mathrm{int}}}({\mathcal I})}$ are linearly independent. If we identify them with $\operatorname{GL}(r)$-invariants as before, we obtain the following corollary: \[cor:quantitative invariants\] Let $\operatorname{edim}{\mathcal I}=0$. Then, $c({\mathcal I}) \geq {c_{\mathrm{int}}}({\mathcal I})$. In fact, it is a classical result that $$\label{eq:c=cint} c({\mathcal I}) = {c_{\mathrm{int}}}({\mathcal I})$$ (see, e.g., [@fulton1997young]). Thus \[cor:quantitative invariants\] shows that we can produce a basis of the tensor product invariants from Belkale’s determinants $\delta_{{\mathcal I},\vec h}(\vec g) = \det\Delta_{{\mathcal I},\vec g,\vec h}$. These invariants can be identified with the construction of Howe, Tan and Willenbring [@howe2005basis], as described in [@vergne2014inequalities]. Proof of Fulton’s conjecture {#sec:fulton} ============================ We now revisit the conjecture by Fulton which states that if $c(\vec\lambda)=1$ for an $s$-tuple of highest weights then $c(N\vec\lambda)=1$ for all $N\geq1$. We note that its converse is also true and holds as a direct consequence of the saturation property and the bound $c(N\vec\lambda)\geq c(\vec\lambda)$, which follows from the semigroup property of the Littlewood-Richardson coefficients. Fulton’s conjecture was first proved by Knutson, Tao and Woodward [@knutson2004honeycomb]. We closely follow Belkale’s geometric proof [@belkale2004invariant; @belkale2006geometric; @belkale2007geometric], in its simplified form due to Sherman [@sherman2015geometric], which in turn was in part inspired by the technique of Schofield [@schofield1992general]. Nonzero invariants and intersections {#subsec:nonzero} ------------------------------------ Let $c(\vec\lambda)=1$. Equivalently, $c(\vec\lambda^*)=1$ and so there exists a nonzero $\operatorname{GL}(r)$-invariant holomorphic function $f$ in $L_{BW}(\lambda^*_1){\otimes}\dots{\otimes}L_{BW}(\lambda^*_s)$, which is unique up to rescaling. Suppose for a moment that there exists a nowhere vanishing function $g$ in $L_{BW}(\lambda^*_1){\otimes}\dots{\otimes}L_{BW}(\lambda^*_s)$ (not necessarily $\operatorname{GL}(r)$-invariant). In this case, if $g'$ is any other holomorphic function in $L_{BW}(\lambda^*_1){\otimes}\dots{\otimes}L_{BW}(\lambda^*_s)$, then $g'/g$ is right $B(r)^s$-invariant and therefore descends to a holomorphic function on $\operatorname{Flag}(r)^s$. But this is a compact space, hence any such function is constant. It then follows that each $L_{BW}(\lambda^*_k)$ is one-dimensional and hence that the $\lambda_k$ are just characters, i.e., $\lambda_k = m_k {\mathbbm 1}_r$ and $L(\lambda_k)=\det_r^{m_k}$ for some $m_k\in{\mathbb Z}$. In this case, Fulton’s conjecture is certainly true. We now consider the nontrivial case when $f$ has zeros. For any function in $L_{BW}(\lambda^*_1){\otimes}\dots{\otimes}L_{BW}(\lambda^*_s)$, the zero set is right $B(r)^s$-stable. Accordingly, we shall write $f({\mathcal F})=0$ for the condition that $f(\vec g)=0$, where ${\mathcal F}=\vec g \cdot F_0$, and consider $$Z_f := \{ {\mathcal F}\in \operatorname{Flag}(r)^s : f({\mathcal F}) = 0 \}.$$ Without loss of generality, we may assume that there exists an $s$-tuple ${\mathcal I}$ with $\operatorname{edim}{\mathcal I}=0$ that is related to $\vec\lambda$ as in \[lem:I to lambdaprime\], i.e., $$\label{eq:lambda vs CI} \lambda_k = \lambda_{I_k} + (n-r) {\mathbbm 1}_r \text{ for $k\in[s-1]$}, \quad \lambda_s = \lambda_{I_s}$$ (otherwise we may add/remove suitable multiples of ${\mathbbm 1}_r$, as in the proof of \[prp:horn implies invariant\]). Now recall from  that the functions $\delta_{{\mathcal I},\vec h} = \delta_{\mathcal I}(-, \vec h)$ are in $\bigotimes_{k=1}^s L_{BW}(\lambda^*_{I_k})$ and transform as the character $\det_r^{(n-r)(s-1)}$ with respect to the diagonal action of $\operatorname{GL}(r)$. It follows that each $\tilde\delta_{{\mathcal I},\vec h}(\vec g) := \det_r^{-(n-r)}(g_1) \cdots \det_r^{-(n-r)}(g_{s-1}) \delta_{{\mathcal I},\vec h}(\vec g)$ must be proportional to $f$. Hence, $$\label{eq:delta decomposed} \delta_{\mathcal I}(\vec g,\vec h) = {\det}_r^{(n-r)}(g_1) \cdots {\det}_r^{(n-r)}(g_{s-1}) f(\vec g) \hat f(\vec h),$$ for some function $\hat f\colon\operatorname{GL}(n-r)^s\to{\mathbb C}$, which is nonzero due to . In view of , we obtain the following lemma: \[lem:Z vs H\_I\] Let $f$, ${\mathcal I}$ as above. If ${\mathcal F}\in Z_f$ then $H_{\mathcal I}({\mathcal F},{\mathcal G})\neq\{0\}$ for all ${\mathcal G}\in\operatorname{Flag}(Q_0)^s$. Conversely, if $\hat f({\mathcal G})\neq0$ then $H_{\mathcal I}({\mathcal F},{\mathcal G})\neq\{0\}$ implies that ${\mathcal F}\in Z_f$. For sake of finding a contradiction, let us assume that $c(N\vec\lambda)>1$ for some $N$. Then there exists an invariant $f' \in L_{BW}(N\lambda_1^*){\otimes}\dots{\otimes}L_{BW}(N\lambda_s^*)$ that is linearly independent from $f^N$. \[lem:algebraic independence\] Let ${\mathcal L}$ be a holomorphic line bundle over a smooth irreducible variety. Then two linearly independent holomorphic sections $f_1$, $f_2$ are automatically algebraically independent. Let us suppose that $f_1$ and $f_2$ satisfy a nontrivial relation $\sum_{i,j} c_{i,j} f_1^i f_2^j=0$. Each $f_1^i f_2^j$ is a section of the line bundle ${\mathcal L}^{{\otimes}(i+j)}$. The relation holds degree by degree, and so we may assume that $i+j$ is the same for each nonzero $c_{i,j}$. But any homogeneous polynomial in two variables is a product of linear factors. Thus we have $\prod_i (a_i f_1 + b_i f_2) = 0$ for some $a_i, b_i \in {\mathbb C}$, and one of the factors has to vanish identically. This shows that $f_1$ and $f_2$ are linearly dependent, in contradiction to our assumption. implies that $f^N$ and $f'$, and therefore $f$ and $f'$ are algebraically independent. As a consequence, there exists a nonempty Zariski-open subset of ${\mathcal F}\in Z_f$ such that $f'({\mathcal F}) \neq 0$. Our strategy in the below will be as follows. As before, we consider the kernel position ${\mathcal J}$ of a generic map $0\neq\phi\in H_{\mathcal I}({\mathcal F},{\mathcal G})$, with now ${\mathcal F}$ varying in $Z_f$. Although ${\mathcal J}$ is not necessarily intersecting, the condition $f'({\mathcal F})\neq0$ will be sufficient to show that the tuple ${\mathcal I}^{\mathcal J}$ is intersecting. In \[subsec:sherman fulton\] we will then prove Sherman’s refined version of his recurrence relation , which will allow us to show that $H_{\mathcal I}({\mathcal F},{\mathcal G})=\{0\}$ for generic ${\mathcal F}\in Z_f$. In view of \[lem:Z vs H\_I\], this will give a contradiction. We first prove a general lemma relating semistable vectors and moment maps. Let $M$ be a complex vector space equipped with a $\operatorname{GL}(r)$-representation and $U(r)$-invariant Hermitian inner product $\braket{\cdot,\cdot}$, complex linear in the second argument, and denote by $\rho_M\colon{\mathfrak{gl}}(r)\to{\mathfrak{gl}}(M)$ the Lie algebra representation. We define the corresponding *moment map* $\Phi_M\colon{\mathbb P}(M)\to i\mathfrak u(r)$ by $$\operatorname{tr}\bigl(\Phi_M([m]) A\bigr) = \frac {\braket{m, \rho_M(A) m}} {\lVert m \rVert^2}$$ for all $A\in{\mathfrak{gl}}(r)$; cf. \[eq:moment map equivariance\]. \[lem:moma nonpositive\] Let $A\in i\mathfrak u(r)$ and $0\neq m\in M$. If $\exp(At) \cdot m\not\to0$ as $t\to-\infty$ then $$\lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_M([\exp(At) \cdot m]) A\bigr) \leq0.$$ Write $m=\sum_{i=1}^k m_i$ where the $m_i$ are nonzero eigenvectors of $\rho_M(A)$, with eigenvalues $\theta_1<\dots<\theta_k$. Then, $$\exp(At) \cdot m = \sum_{i=1}^k e^{\theta_it} m_i \not\to 0$$ as $t\to-\infty$ if and only if $\theta_1\leq0$. In this case, $$\lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_M([\exp(At) \cdot m]) A\bigr) = \lim_{t\to-\infty} \frac {\sum_i \theta_i e^{2\theta_it} \lVert m_i\rVert^2} {\sum_i e^{2\theta_it} \lVert m_i\rVert^2} = \theta_1\leq0. \qedhere$$ We now relate the position of subspaces to components of the moment map: \[lem:single slope\] Let $\lambda\in\Lambda_+(r)$, $F = g \cdot F_0$ a flag on $V_0$, $S$ a nonzero subspace of ${\mathbb C}^r$, and $P_S$ the orthogonal projector. Then, $$\lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_{L(\lambda)}([\exp(P_St) g \cdot v_\lambda]) P_S\bigr) = \braket{T_J, \lambda},$$ where $J=\operatorname{Pos}(S,F)$. Let $d=\#J$. We may assume that $S=S_0$ is generated by the first $d$ vectors $e(1),\dots,e(d)$ of the standard basis of $V_0$, and also that $g=u$ is unitary. Thus $P_{S_0}$ is the diagonal matrix with $d$ ones and $r-d$ zeros, and we need to show that $$\lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_{L(\lambda)}([\exp(P_{S_0}t) u \cdot v_\lambda]) P_{S_0}\bigr) = \braket{T_J, \lambda}.$$ Let $R_0$ denote the orthogonal complement of $S_0$ in $V_0$. The action of $U(S_0) \times U(R_0)$ commutes with $P_{S_0}$ and hence we can assume that $F^{S_0}$ is the standard flag on $S_0$, while $F_{V_0/S_0}$ has the adapted basis $e(J^c(b))+S_0$ for $b\in[r-d]$. Thus we see that $\lim_{t\to-\infty} \exp(P_{S_0}t) F = w_J F_0$. It follows that $\lim_{t\to-\infty} [\exp(P_{S_0}t) u \cdot v_\lambda] = [w_J \cdot v_\lambda]$ and hence, using , that $$\lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_{L(\lambda)}([\exp(P_{S_0}t) u \cdot v_\lambda]) P_{S_0}\bigr) = \frac {\braket{v_\lambda, \rho_\lambda(w_J^{-1} P_{S_0} w_J) v_\lambda}} {\lVert v_\lambda\rVert^2} = \braket{T_J, \lambda}. \qedhere$$ We now use the preceding lemma to obtain from any nonzero invariant an $s$-tuple of flags with nonnegative slope: \[lem:semistable slope\] Let $p \in (L(N\lambda_1){\otimes}\cdots{\otimes}L(N\lambda_s))^*$ a $\operatorname{GL}(r)$-invariant homogeneous polynomial such that $p(g_1\cdot v_{N\lambda_1}{\otimes}\cdots{\otimes}g_s\cdot v_{N\lambda_s})\neq0$, and define ${\mathcal F}=(g_1 F_0,\dots,g_s F_0)$. Then $\sum_{k=1}^s (T_{J_k}, \lambda_k)\leq0$ for all ${\mathcal J}=\operatorname{Pos}(S,{\mathcal F})$, where $S$ is an arbitrary nonzero subspace of ${\mathbb C}^r$. Consider the representation $M=L(N\lambda_1){\otimes}\cdots{\otimes}L(N\lambda_s)$ with its moment map $\Phi_M$, and $m := g_1\cdot v_{N\lambda_1}{\otimes}\cdots{\otimes}g_s\cdot v_{N\lambda_s}$. Let $P_S$ denote the orthogonal projector onto the subspace $S$. As $p$ is $\operatorname{GL}(r)$-invariant, $p(\exp(P_St) \cdot m)=p(m)\neq0$, which implies that $\exp(P_St) \cdot m\not\to0$ as $t\to-\infty$. Thus \[lem:moma nonpositive\] implies that $$\lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_M([\exp(P_St) \cdot m]) P_S\bigr) \leq 0.$$ On the other hand, \[lem:single slope\] shows that the left-hand side of this inequality is equal to $$ \sum_{k=1}^s \lim_{t\to-\infty} \operatorname{tr}\bigl(\Phi_{L(N\lambda_k)}([\exp(P_St) g_k \cdot v_{N\lambda_k}]) P_S\bigr) = \sum_{k=1}^s \lim_{t\to-\infty} (T_{J_k}, \lambda_k). \qedhere$$ \[cor:refined intersecting\] Let $\vec\lambda$ and ${\mathcal I}$ as in , $f' \in (L_{BW}(N\lambda_1^*) {\otimes}\cdots {\otimes}L_{BW}(N\lambda_s^*))^{\operatorname{GL}(r)}$. Let ${\mathcal F}\in\operatorname{Flag}(V_0)^s$, $\{0\}\neq S\subseteq{\mathbb C}^r$, and ${\mathcal J}=\operatorname{Pos}(S,{\mathcal F})$. If $f'({\mathcal F})\neq0$ then ${\mathcal I}^{\mathcal J}$ is intersecting. Write ${\mathcal F}=(g_1 \cdot F_0, \dots, g_s \cdot F_0)$ for suitable $g_1,\dots,g_s \in\operatorname{GL}(r)$. Then, using , there exists a $\operatorname{GL}(r)$-invariant homogeneous polynomial $p\in(L(N\lambda_1){\otimes}\cdots{\otimes}L(N\lambda_s))^*$ such that $p(g_1 \cdot v_{N\lambda_1} {\otimes}\cdots{\otimes}g_s \cdot v_{N\lambda_s})=f'(g_1,\dots,g_s)\neq0$. Thus the assumptions of \[lem:semistable slope\] are satisfied. We now show that ${\mathcal I}^{\mathcal J}$ is intersecting. For this, we use \[thm:belkale\] and verify the Horn inequalities. Thus let $0<m\leq d=\dim S$ and ${\mathcal K}\in\operatorname{Horn}(m,d,s)=\operatorname{Intersecting}(m,d,s)$: Since ${\mathcal K}$ is intersecting, there exists some subspace $S'\in\Omega_{\mathcal K}({\mathcal F}^S)$. Hence $S'\in\Omega_{{\mathcal J}{\mathcal K}}({\mathcal F})$ by the chain rule . According to \[lem:schubert variety characterization\], ${\mathcal J}'=\operatorname{Pos}(S',{\mathcal F})$ is such that $J'_k(a) \leq J_k K_k(a)$ for all $k\in[s]$ and $a\in[m]$. Thus we obtain the first inequality in $$\begin{aligned} \operatorname{edim}{\mathcal I}^{\mathcal J}{\mathcal K}- \operatorname{edim}{\mathcal K}= \operatorname{edim}{\mathcal I}({\mathcal J}{\mathcal K}) - \operatorname{edim}{\mathcal J}{\mathcal K}= -\sum_{k=1}^s (T_{J_k K_k}, \lambda_k) \geq -\sum_{k=1}^s (T_{J'_k}, \lambda_k) \geq 0; \end{aligned}$$ the first equality is , the second is \[lem:I to lambdaprime\], and the last inequality is \[lem:semistable slope\], applied to $S'$. This concludes the proof. Sherman’s refined lemma {#subsec:sherman fulton} ----------------------- We now study the behavior of $\dim H_{\mathcal I}({\mathcal F}, {\mathcal G})$ in more detail. We proceed as in \[sec:horn sufficient\], but for a *fixed* $s$-tuple of flags ${\mathcal F}\in\operatorname{Flag}(V_0)^s$. Specifically, we consider the following refinement of the true dimension  for fixed ${\mathcal F}$: $$\operatorname{tdim}_{\mathcal F}{\mathcal I}:= \min_{{\mathcal G}} \dim H_{\mathcal I}({\mathcal F},{\mathcal G})$$ Thus we study the variety $${\mathrm P_{\mathcal F}}({\mathcal I}) := \{ ({\mathcal G},\phi) \in \operatorname{Flag}(Q_0)^s \times \operatorname{Hom}(V_0,Q_0) : \phi\in H_{\mathcal I}({\mathcal F},{\mathcal G}) \}.$$ Restricting to those ${\mathcal G}$ such that $\dim H_{\mathcal I}({\mathcal F},{\mathcal G})=\operatorname{tdim}_{\mathcal F}{\mathcal I}$, we obtain open sets ${\mathrm B_{{\mathcal F},\mathrm t}}({\mathcal I})\subseteq\operatorname{Flag}(Q_0)^s$ and ${\mathrm P_{{\mathcal F},\mathrm t}}({\mathcal I})\subseteq{\mathrm P_{\mathcal F}}({\mathcal I})$. Let $\operatorname{kdim}_{\mathcal F}({\mathcal I})$ denote the minimal (and hence generic) dimension of $\ker\phi$ for $({\mathcal G},\phi)\in{\mathrm P_{{\mathcal F},\mathrm t}}({\mathcal I})$. The following lemma is proved just like \[cor:kdim zero intersecting\]: \[lem:kdim\_F zero\] If $\operatorname{kdim}_{\mathcal F}{\mathcal I}=0$ then $\operatorname{tdim}_{\mathcal F}{\mathcal I}=\operatorname{edim}{\mathcal I}$. Let us now assume that $\operatorname{kdim}_{\mathcal F}{\mathcal I}>0$. Let $\operatorname{kPos}_{\mathcal F}({\mathcal I})$ denote the kernel position, defined as in \[def:kerpos\] but for fixed ${\mathcal F}$. We thus obtain an irreducible variety ${\mathrm P_{{\mathcal F},\mathrm{kpt}}}({\mathcal I})$ over a Zariski-open subset ${\mathrm B_{{\mathcal F},\mathrm{kpt}}}({\mathcal I})$ of ${\mathrm B_{{\mathcal F},\mathrm t}}({\mathcal I})$. To compute its dimension, we again define ${\mathrm P_{{\mathcal F},\mathrm{kp}}}({\mathcal I})\subseteq{\mathrm P_{\mathcal F}}({\mathcal I})$, where we fix the kernel dimension and position, but *not* the dimension of $H_{\mathcal I}({\mathcal F},{\mathcal G})$. In contrast to \[lem:dim P\_kpt nonzero\], the variety ${\mathrm P_{{\mathcal F},\mathrm{kp}}}({\mathcal I})$ is in general neither smooth nor irreducible. However, we can describe it similarly as before: We first constrain $S=\ker\phi$ to be in $\Omega^0_{\mathcal J}({\mathcal F})$ (which may not be irreducible), then $\phi$ is determined by $\bar\phi\in\operatorname{Hom}^\times(V_0/S,Q_0)$ and ${\mathcal G}$ by $G_k\in\operatorname{Flag}^0_{I_k/J_k}((F_k)_{V_0/S},\bar\phi)$. Thus we obtain for each irreducible component $C \subseteq\Omega^0_{\mathcal J}({\mathcal F})$ a corresponding irreducible component ${\mathrm P_{{\mathcal F},\mathrm{kp},C}}({\mathcal I})$. In particular, there exists some component $C_{\mathcal F}$ such that ${\mathrm P_{{\mathcal F},\mathrm{kp},C_{\mathcal F}}}({\mathcal I})$ is the closure of ${\mathrm P_{{\mathcal F},\mathrm{kpt}}}({\mathcal I})$ in ${\mathrm P_{{\mathcal F},\mathrm{kp}}}({\mathcal I})$, namely the irreducible component containing the elements $S=\ker\phi$ for $(\phi,{\mathcal G})$ varying in the irreducible variety ${\mathrm P_{{\mathcal F},\mathrm{kpt}}}({\mathcal I})$. As a consequence, $\dim{\mathrm P_{{\mathcal F},\mathrm{kpt}}}({\mathcal I}) = \dim{\mathrm P_{{\mathcal F},\mathrm{kp},C_{\mathcal F}}}({\mathcal I})$, and so we obtain, using completely analogous dimension computations, the following refinement of : $$\label{eq:tdim minus edim refined} \operatorname{tdim}_{\mathcal F}{\mathcal I}- \operatorname{edim}{\mathcal I}= \dim C_{\mathcal F}- \operatorname{edim}{\mathcal I}^{\mathcal J}$$ Indeed, when we apply  to generic ${\mathcal F}\in\operatorname{Flag}(V_0)^s$ then ${\mathcal J}$ is intersecting and $\dim C_{\mathcal F}=\operatorname{edim}{\mathcal J}$, so we recover . We now instead apply the above to generic ${\mathcal F}$ in a component of the zero set $Z_f$ of the unique nonzero invariant $f$. Thus we obtain the following variant of the key recursion relation : \[lem:sherman refined\] Let $f,{\mathcal I}$ as above in \[subsec:nonzero\], and $Z\subseteq Z_f$ an irreducible component such that $\operatorname{kdim}_{\mathcal F}{\mathcal I}\neq0$ for all ${\mathcal F}\in Z$. Then there exists ${\mathcal J}$ and a nonempty Zariski-open subset of ${\mathcal F}\in Z$ such that $\operatorname{kPos}_{\mathcal F}{\mathcal I}={\mathcal J}$ and $$\operatorname{tdim}_{\mathcal F}{\mathcal I}- \operatorname{edim}{\mathcal I}\leq \operatorname{tdim}{\mathcal I}^{\mathcal J}- \operatorname{edim}{\mathcal I}^{\mathcal J}.$$ We choose $d$ and ${\mathcal J}$ as the kernel dimension and position for generic ${\mathcal F}\in Z$. We note that $d<r$, since $d=r$ would imply that $H_{\mathcal I}({\mathcal F},{\mathcal G})=\{0\}$, in contradiction to \[lem:Z vs H\_I\]. Let $U\subseteq Z$ denote the Zariski-open subset such that $\operatorname{kPos}_{\mathcal F}{\mathcal I}={\mathcal J}$ for all ${\mathcal F}\in U$. We proceed as in \[lem:sherman upper bound\]. Let $$\begin{aligned} {\mathcal X}&:= \{ ({\mathcal F},{\mathcal G},\phi) : {\mathcal F}\in U, ({\mathcal G},\phi) \in {\mathrm P_{{\mathcal F},\mathrm{kp},C_{\mathcal F}}}({\mathcal I}) \}, \\ {\mathcal Y}&:= \{ (S,\tilde{\mathcal F},{\mathcal G}) : S \in \operatorname{Gr}(d, V_0), \tilde{\mathcal F}\in \operatorname{Flag}(S)^s, {\mathcal G}\in\operatorname{Flag}(Q_0)^s \}. \end{aligned}$$ Both ${\mathcal X}$ and ${\mathcal Y}$ are irreducible varieties and we have a morphism $$\pi\colon{\mathcal X}\to{\mathcal Y}, ({\mathcal F},{\mathcal G},\phi) \mapsto (\ker\phi,{\mathcal F}^{\ker\phi},{\mathcal G}).$$ As before, we argue that $\pi$ is dominant. By construction, the image of ${\mathcal X}$ by the map $({\mathcal F},{\mathcal G},\phi)\mapsto{\mathcal G}$ contains a Zariski-open subset $U'$ of $\operatorname{Flag}(Q_0)^s$. We may also assume that $\hat f({\mathcal G})\neq0$ for all ${\mathcal G}\in U'$, where $\hat f$ is the map from . We now show that the image of $\pi$ contains all elements $(S,\tilde{\mathcal F},{\mathcal G})$ with $S\in\operatorname{Gr}(d,V_0)$, $\tilde{\mathcal F}\in\operatorname{Flag}(S)^s$, and ${\mathcal G}\in U'$. For this, let $({\mathcal F}_0,{\mathcal G},\phi_0)\in{\mathcal X}$ be the preimage of some arbitrary ${\mathcal G}\in U'$. Let $S_0 :=\ker\phi_0$ and choose some $g \in \operatorname{GL}(V_0)$ such that $g \cdot S_0 = S$. Using the corresponding diagonal action, define ${\mathcal F}:= g \cdot {\mathcal F}_0$ and $\phi := g \cdot \phi_0$. Then $({\mathcal F},{\mathcal G},\phi)\in{\mathcal X}$, since $Z$ is stable under the diagonal action of $\operatorname{GL}(V_0)$, and $\ker\phi=S$. Now consider the group $G \subseteq \operatorname{GL}(V_0)^s$ consisting of all elements $\vec h\in\operatorname{GL}(V_0)^s$ such that $h_k S\subseteq S$ and $h_k$ acts trivially on $V_0/S$ for all $k\in[s]$. Note that $G$ is an irreducible algebraic group. By construction, $\phi\in H_{\mathcal I}(\vec h\cdot{\mathcal F},{\mathcal G})$, while $d<r$ implies that $\phi\neq0$. This means that $H_{\mathcal I}(\vec h\cdot{\mathcal F},{\mathcal G})\neq 0$, and so we obtain from \[lem:Z vs H\_I\] that $\vec h\cdot{\mathcal F}\in Z_f$. It follows that, in fact, $\vec h\cdot{\mathcal F}\in Z$, as it is obtained by the action of the irreducible algebraic group $G$ on ${\mathcal F}\in Z$, and so stays in the same irreducible component. For given $\tilde{\mathcal F}\in\operatorname{Flag}(S)^s$, we now choose $\vec h\in G$ such that $h_k \cdot F^S_k = \tilde F_k$ for $k\in[s]$. Then $(\vec h\cdot{\mathcal F},{\mathcal G},\phi)\in{\mathcal X}$ is a preimage of $(S,\tilde{\mathcal F},{\mathcal G})$, and we conclude that $\pi$ is dominant. As before, the dominance implies that we can find a nonempty Zariski-open set of $({\mathcal F},{\mathcal G},\phi)\in{\mathcal X}$ such that $\dim H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^{\ker\phi},{\mathcal G})=\operatorname{tdim}{\mathcal I}^{\mathcal J}$. We may assume in addition that $\ker\phi \in C_{\mathcal F}$ is a smooth point. For any ${\mathcal F}$ in this set, $\bar\phi$ injects $H_{\mathcal J}({\mathcal F}^{\ker\phi}, {\mathcal F}_{V_0/\ker\phi})$ into $H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^{\ker\phi},{\mathcal G})$ (\[lem:exp vs composition\]). Thus, $$\dim C_{\mathcal F}= \dim T_{\ker\phi} C_{\mathcal F}\leq H_{\mathcal J}({\mathcal F}^{\ker\phi},{\mathcal F}_{V_0/\ker\phi})\leq\dim H_{{\mathcal I}^{\mathcal J}}({\mathcal F}^{\ker\phi},{\mathcal G}) = \operatorname{tdim}{\mathcal I}^{\mathcal J},$$ where in the first inequality we have used that the intersection $\Omega^0_{\mathcal J}({\mathcal F})$ is not necessarily transversal at $S$ and so the tangent space of the intersection is in general only a subspace of the intersection of the tangent spaces . In view of , we obtain the desired inequality. Let $c(\vec\lambda)=1$. Then $c(N\vec\lambda)=1$ for all $N>1$. Let $f$, ${\mathcal I}$ as in \[subsec:nonzero\] and recall that $\operatorname{edim}{\mathcal I}=0$. Assume for sake of finding a contradiction that $c(N\vec\lambda)\neq1$ for some $N>0$. Then there exists another invariant $f'$, as in \[subsec:nonzero\], such that $f'({\mathcal F})\neq0$ for a nonempty Zariski-open subset of some irreducible component $Z\subseteq Z_f$. If $\operatorname{kdim}_{\mathcal F}{\mathcal I}=0$ for some ${\mathcal F}\in Z$ then $\operatorname{tdim}_{\mathcal F}{\mathcal I}=0$ by \[lem:kdim\_F zero\]. Otherwise, we may apply \[lem:sherman refined\] to the component $Z$. We find that there exists some ${\mathcal J}$ and another Zariski-open subset of ${\mathcal F}$ in $Z$ such that $\operatorname{kPos}_{\mathcal F}{\mathcal I}={\mathcal J}$ and $$\label{eq:sherman for belkale} \operatorname{tdim}_{\mathcal F}{\mathcal I}- \operatorname{edim}{\mathcal I}\leq \operatorname{tdim}{\mathcal I}^{\mathcal J}- \operatorname{edim}{\mathcal I}^{\mathcal J}.$$ As a consequence, there exists some ${\mathcal F}\in Z_f$ for which all three of the properties $f'({\mathcal F})\neq0$, $\operatorname{kPos}_{\mathcal F}{\mathcal I}={\mathcal J}$ and  hold true. By \[cor:refined intersecting\], the first two properties imply that ${\mathcal I}^{\mathcal J}=0$, and hence the right-hand side of  is equal to zero by \[cor:tdim edim\]. This again implies that $\operatorname{tdim}_{\mathcal F}{\mathcal I}=0$. It follows that in either case there exist some ${\mathcal G}$ such that $H_{\mathcal I}({\mathcal F},{\mathcal G})=\{0\}$. According to \[lem:Z vs H\_I\], this can only be if ${\mathcal F}\not\in Z_f$. But ${\mathcal F}\in Z_f$. This is the desired contradiction. It likewise holds that $c(\vec\lambda)=2$ implies that $c(N\vec\lambda)=N+1$ [@ikenmeyer2012small; @sherman2015geometric]. However, in general it is *not* true that $c(\vec\lambda)=c$ implies $c(N\vec\lambda) = O(N^{c-1})$. Belkale has a found a counterexample for $c=6$. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Velleda Baldoni and Vladimir Popov for interesting discussions. We also thank Prakash Belkale and Cass Sherman for their helpful remarks. We would also like to thank an anonymous referee for their suggestions. MW acknowledges financial support by the Simons Foundation and AFOSR grant FA9550-16-1-0082. Horn triples in low dimensions {#app:examples horn} ============================== In this appendix, we list all Horn triples ${\mathcal J}\in\operatorname{Horn}(d,r,3)$ for $d<r\leq4$, as defined in \[def:horn\], as well as the expected dimensions $\operatorname{edim}{\mathcal J}$. The triples with $\operatorname{edim}{\mathcal J}=0$ are highlighted in bold. \[ex:horn 1\] As discussed in \[ex:base case\], only the dimension condition $\operatorname{edim}{\mathcal J}\geq0$ is necessary. The following are the triples in $\operatorname{Horn}(1,r,3)$ (up to permutations): ---------------------------------------------------------------------------------------------------------- r $J_1$ $J_2$ $J_3$ $\operatorname{edim}{\mathcal J}$ --- -------------------------------------------------- ------- ------- ----------------------------------- r $J_1$ $J_2$ $J_3$ $\operatorname{edim}{\mathcal J}$ 2 **{1} & **{2} & **{2} & **0\ & {2} & {2} & {2} & 1\ 3 & **{1} & **{3} & **{3} & **0\ & **{2} & **{2} & **{3} & **0\ & {2} & {3} & {3} & 1\ & {3} & {3} & {3} & 2\ 4 & **{1} & **{4} & **{4} & **0\ & **{2} & **{3} & **{4} & **0\ & {2} & {4} & {4} & 1\ & **{3} & **{3} & **{3} & **0\ & {3} & {3} & {4} & 1\ & {3} & {4} & {4} & 2\ & {4} & {4} & {4} & 3\ ************************************************ ---------------------------------------------------------------------------------------------------------- \[ex:horn 2\] The dimension condition $\operatorname{edim}{\mathcal J}\geq0$ reads $$\left( J_1(1) + J_1(2) \right) + \left( J_2(1) + J_2(2) \right) + \left( J_3(1) + J_3(2) \right) \geq 4r+1.$$ In addition, we have to satisfy three Horn inequalities, corresponding to ${\mathcal K}= (\{1\},\{2\},\{2\})$ and its permutations, which are the only elements in $\operatorname{Horn}(1,2,3)$ with $\dim{\mathcal K}=0$. The resulting Horn inequalities, $\operatorname{edim}{\mathcal J}{\mathcal K}\geq0$, are $$\begin{aligned} J_1(1) + J_2(2) + J_3(2) &\geq 2r+1, \\ J_1(2) + J_2(1) + J_3(2) &\geq 2r+1, \\ J_1(2) + J_2(2) + J_3(1) &\geq 2r+1. \end{aligned}$$ Thus we obtain the following triples in $\operatorname{Horn}(2,r,3)$ (up to permutations): -------------------------------------------------------------------------------------------------------------------------- r $J_1$ $J_2$ $J_3$ $\operatorname{edim}{\mathcal J}$ --- ------------------------------------------------------------------ ------- ------- ----------------------------------- r $J_1$ $J_2$ $J_3$ $\operatorname{edim}{\mathcal J}$ 3 **{1, 2} & **{2, 3} & **{2, 3} & **0\ & **{1, 3} & **{1, 3} & **{2, 3} & **0\ & {1, 3} & {2, 3} & {2, 3} & 1\ & {2, 3} & {2, 3} & {2, 3} & 2\ 4 & **{1, 2} & **{3, 4} & **{3, 4} & **0\ & **{1, 3} & **{2, 4} & **{3, 4} & **0\ & {1, 3} & {3, 4} & {3, 4} & 1\ & **{1, 4} & **{1, 4} & **{3, 4} & **0\ & **{1, 4} & **{2, 4} & **{2, 4} & **0\ & {1, 4} & {2, 4} & {3, 4} & 1\ & {1, 4} & {3, 4} & {3, 4} & 2\ & **{2, 3} & **{2, 3} & **{3, 4} & **0\ & **{2, 3} & **{2, 4} & **{2, 4} & **0\ & {2, 3} & {2, 4} & {3, 4} & 1\ & {2, 3} & {3, 4} & {3, 4} & 2\ & {2, 4} & {2, 4} & {2, 4} & 1\ & {2, 4} & {2, 4} & {3, 4} & 2\ & {2, 4} & {3, 4} & {3, 4} & 3\ & {3, 4} & {3, 4} & {3, 4} & 4\ **************************************************************** -------------------------------------------------------------------------------------------------------------------------- \[ex:horn 3\] We find the following triples in $\operatorname{Horn}(3,4,3)$ (up to permutations): ---------------------------------------------------------------------------------------------------------- r $J_1$ $J_2$ $J_3$ $\operatorname{edim}{\mathcal J}$ --- -------------------------------------------------- ------- ------- ----------------------------------- r $J_1$ $J_2$ $J_3$ $\operatorname{edim}{\mathcal J}$ 4 **{1, 2, 3} & **{2, 3, 4} & **{2, 3, 4} & **0\ & **{1, 2, 4} & **{1, 3, 4} & **{2, 3, 4} & **0\ & {1, 2, 4} & {2, 3, 4} & {2, 3, 4} & 1\ & **{1, 3, 4} & **{1, 3, 4} & **{1, 3, 4} & **0\ & {1, 3, 4} & {1, 3, 4} & {2, 3, 4} & 1\ & {1, 3, 4} & {2, 3, 4} & {2, 3, 4} & 2\ & {2, 3, 4} & {2, 3, 4} & {2, 3, 4} & 3\ ************************ ---------------------------------------------------------------------------------------------------------- It is not an accident that both $\operatorname{Horn}(1,4,3)$ and $\operatorname{Horn}(3,4,3)$ have the same number of elements. In fact, we can identify $\operatorname{Intersecting}(d,r,s)\cong\operatorname{Intersecting}(r-d,r,s)$ via $J_k \mapsto \{ r+1-a : a \in J_k^c \}$. This can be seen by using the canonical isomorphism $\operatorname{Gr}(d,{\mathbb C}^r)\cong\operatorname{Gr}(r-d,({\mathbb C}^r)^*)$. However, the corresponding Horn inequalities are distinct (see below). Kirwan cones in low dimensions {#app:examples kirwan} ============================== In this appendix, we list necessary and sufficient conditions on highest weights $\lambda,\mu,\nu\in\Lambda_+(r)$ such that $\left(L(\lambda){\otimes}L(\mu){\otimes}L(\nu) \right)^{U(r)}\neq\{0\}$, up to $r=4$. That is, these conditions describe the Kirwan cones as in \[cor:horn and saturation\]. We use the abbreviation $\operatorname{Horn}_0(d,r,s)$ for the set of Horn triples in ${\mathcal J}\in\operatorname{Horn}(d,r,s)$ such that $\operatorname{edim}{\mathcal J}=0$ (highlighted bold in \[app:examples horn\]). Clearly, the only condition is $\lambda(1)+\mu(1)+\nu(1)=0$. We always have the Weyl chamber inequalities $\lambda(1)\geq\lambda(2)$, $\mu(1)\geq\mu(2)$, and $\nu(1)\geq\nu(2)$, and the equation $$\left( \lambda(1)+\lambda(2) \right) + \left( \mu(1)+\mu(2) \right) + \left( \nu(1)+\nu(2) \right) = 0.$$ Using \[ex:horn 1\], we obtain three Horn inequalities, namely $$\lambda(1) + \mu(2) + \nu(2) \leq 0,$$ corresponding to the triple $(\{1\},\{2\},\{2\})\in\operatorname{Horn}_0(d,r,s)$, and its permutations. These are the well-known conditions for the existence of nonzero invariants in a triple tensor product of irreducible $U(2)$-representations. We remark that the Weyl chamber inequalities are redundant. In addition to the Weyl chamber inequalities and $\lvert\lambda\rvert+\lvert\mu\rvert+\lvert\nu\rvert=0$, we obtain the following two inequalities from $\operatorname{Horn}_0(1,3,3)$ and \[ex:horn 1\], $$\begin{aligned} \lambda(1) + \mu(3) + \nu(3) &\leq 0, \\ \lambda(2) + \mu(2) + \nu(3) &\leq 0, \end{aligned}$$ and the following from $\operatorname{Horn}_0(2,3,3)$ and \[ex:horn 2\], $$\begin{aligned} \left( \lambda(1) + \lambda(2) \right) + \left( \mu(2) + \mu(3) \right) + \left( \nu(2) + \nu(3) \right) &\leq 0, \\ \left( \lambda(1) + \lambda(3) \right) + \left( \mu(1) + \mu(3) \right) + \left( \nu(2) + \nu(3) \right) &\leq 0, \end{aligned}$$ as well as their permutations. Again we have the Weyl chamber inequalities and $\lvert\lambda\rvert+\lvert\mu\rvert+\lvert\nu\rvert=0$. We have the following two inequalities and their permutations from $\operatorname{Horn}_0(1,4,3)$ and \[ex:horn 1\], $$\begin{aligned} \lambda(1) + \mu(4) + \nu(4) &\leq 0, \\ \lambda(2) + \mu(3) + \nu(4) &\leq 0, \\ \lambda(3) + \mu(3) + \nu(3) &\leq 0, \end{aligned}$$ the following six and their permutations from $\operatorname{Horn}_0(2,4,3)$ and \[ex:horn 2\], $$\begin{aligned} \left( \lambda(1) + \lambda(2) \right) + \left( \mu(3) + \mu(4) \right) + \left( \nu(3) + \nu(4) \right) &\leq 0, \\ \left( \lambda(1) + \lambda(3) \right) + \left( \mu(2) + \mu(4) \right) + \left( \nu(3) + \nu(4) \right) &\leq 0, \\ \left( \lambda(1) + \lambda(4) \right) + \left( \mu(1) + \mu(4) \right) + \left( \nu(3) + \nu(4) \right) &\leq 0, \\ \left( \lambda(1) + \lambda(4) \right) + \left( \mu(2) + \mu(4) \right) + \left( \nu(2) + \nu(4) \right) &\leq 0, \\ \left( \lambda(2) + \lambda(3) \right) + \left( \mu(2) + \mu(3) \right) + \left( \nu(3) + \nu(4) \right) &\leq 0, \\ \left( \lambda(2) + \lambda(3) \right) + \left( \mu(2) + \mu(4) \right) + \left( \nu(2) + \nu(4) \right) &\leq 0, \end{aligned}$$ and the following three and their permutations from $\operatorname{Horn}_0(3,4,3)$ and \[ex:horn 3\], $$\begin{aligned} \left( \lambda(1) + \lambda(2) + \lambda(3) \right) + \left( \mu(2) + \mu(3) + \mu(4) \right) + \left( \nu(2) + \nu(3) + \nu(4) \right) &\leq 0, \\ \left( \lambda(1) + \lambda(2) + \lambda(4) \right) + \left( \mu(1) + \mu(3) + \mu(4) \right) + \left( \nu(2) + \nu(3) + \nu(4) \right) &\leq 0, \\ \left( \lambda(1) + \lambda(3) + \lambda(4) \right) + \left( \mu(1) + \mu(3) + \mu(4) \right) + \left( \nu(1) + \nu(3) + \nu(4) \right) &\leq 0. \end{aligned}$$ In low dimensions, all Horn triples with $\operatorname{edim}{\mathcal J}=0$ are such that the intersection is one point, i.e., $c({\mathcal I})={c_{\mathrm{int}}}({\mathcal I})=1$. This implies that the equations are irredundant [@belkale2006geometric; @ressayre2010geometric] (cf. \[rem:redundant\]), and it can also be explicitly checked in the examples above. In general, however, this is not the case, and so the Horn inequalities are still redundant. An example of such a Horn triple is the one given in \[ex:interesting\].

--- abstract: 'Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group $G$, a set $I$ and with two bijections $\lambda,\rho:I \to I.$ Using a clever construction on the ordinal sum of $(G^+)^I$ and $(G^-)^I,$ we can define a pseudo effect algebra which can be non-commutative even if $G$ is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.' author: - 'Anatolij Dvurečenskij$^{1,2}$, W. Charles Holland$^3$' title: Some Remarks on Kite Pseudo Effect Algebras --- [^1] Mathematical Institute, Slovak Academy of Sciences\ Štefánikova 49, SK-814 73 Bratislava, Slovakia\ $^2$ Depart. Algebra Geom., Palacký University\ 17. listopadu 12, CZ-771 46 Olomouc, Czech Republic\ $^3$ Department of Mathematics, University of Colorado\ 8323 Thunderhead Drive, Boulder, Colorado 80302, USA\ E-mail: [dvurecen@mat.savba.sk]{} Introduction ============ [*Effect algebras*]{} are partial algebras introduced in [@FoBe] in order to model quantum mechanical measurements. The primary notion is addition $+$ such that $a + b$ describes the disjunction of two mutually excluding events $a$ and $b.$ A basic model of effect algebras is the system $\mathcal E(H)$ of Hermitian operators of a Hilbert space $H$ which are between the zero and the identity operator. Effect algebras generalize many quantum structures like Boolean algebras, orthomodular lattices and posets, orthoalgebras, and MV-algebras. Effect algebras combine in an algebraic way both sharp and fuzzy features of a measurement process in quantum mechanics. Many important effect algebras are intervals $[0,u]$ in Abelian partially ordered groups (po-groups for short) with strong unit $u;$ we call them interval effect algebras. This is true, e.g. for $\mathcal E(H)$ when it is the interval $[O,I]$ in the po-group $\mathcal B(H),$ the system of all Hermitian operators of $H,$ where $O$ and $I$ are the zero and the identity operator. Another class of interval effect algebras is all of those with the Riesz Decomposition Property (RDP for short), see [@Rav]. This property enables a joint refinement of two decompositions of the unit element. This class contains all MV-algebras. For effect algebras, the partial operation $+$ was assumed to be commutative. This assumption was canceled in [@DvVe1; @DvVe2], where a non-commutative version of effect algebras, called [*pseudo effect algebras*]{}, is introduced. Using a stronger version of RDP, also some pseudo effect algebras are intervals in po-groups which are not necessarily Abelian. A physical motivation for introducing pseudo effect algebras with possible physical situations in quantum mechanics was presented in [@DvVe4]. We note the family of pseudo effect algebras contains also the family of pseudo MV-algebras, or equivalently, the family of generalized MV-algebras, [@GeIo; @Rac]. Recently, in [@JiMo] there was presented an interesting construction of a pseudo BL-algebra whose underlying set is an ordinal sum of $\mathbb Z^+$ and $\mathbb Z^-\times \mathbb Z^-,$ where $\mathbb Z$ denotes the group of integers; its shape resembles a kite. This construction was generalized in [@DvKo] for an arbitrary $\ell$-group $G,$ and kite BL-algebras were introduced. In the paper [@DvuK], instead of $\ell$-groups, we have used po-groups and [*kite pseudo effect algebras*]{}, as a special family of pseudo effect algebras, were defined. This family enriches a reservoir of interesting examples of pseudo effect algebras. In addition, it shows importance of po-groups and $\ell$-groups for the theory of pseudo effect algebras. Basic properties of kite pseudo effect algebras were presented in [@DvuK], in particular, a characterization of subdirectly irreducible kite pseudo effect algebras was given. In this paper, we continue with the study of kite pseudo effect algebras studying conditions when a kite pseudo effect algebra is a subdirect product of subdirectly irreducible kite pseudo effect algebras. In addition, we show that every kite pseudo effect algebra is always an interval in a partially ordered loop. The paper is organized as follows. Section 2 gathers basic facts of pseudo effect algebras, po-groups, and different forms of RDP’s. Section 3 redefines kite pseudo effect algebras and characterize subdirect product of subdirectly irreducible kite pseudo effect algebras. Finally, Section 4 shows that every kite pseudo effect algebra is an interval in a unital po-loop. In addition, some illustrating examples are given and some open problems are formulated. Elements of Pseudo Effect Algebras ================================== According to [@DvVe1; @DvVe2], we say that a [*pseudo effect algebra*]{} is a partial algebra $ E=(E; +, 0, 1)$, where $+$ is a partial binary operation and $0$ and $1$ are constants, such that for all $a, b, c \in E$, the following holds 1. $a+b$ and $(a+b)+c$ exist if and only if $b+c$ and $a+(b+c)$ exist, and in this case $(a+b)+c = a+(b+c)$; 2. there is exactly one $d \in E$ and exactly one $e \in E$ such that $a+d = e+a = 1$; 3. if $a+b$ exists, there are elements $d, e \in E$ such that $a+b = d+a = b+e$; 4. if $1+a$ or $a+1$ exists, then $a = 0$. If we define $a \le b$ if and only if there exists an element $c\in E$ such that $a+c =b,$ then $\le$ is a partial ordering on $E$ such that $0 \le a \le 1$ for any $a \in E.$ It is possible to show that $a \le b$ if and only if $b = a+c = d+a$ for some $c,d \in E$. We write $c = a {\hspace{0.3em}\raisebox{0.3ex}{\sl \tiny /}\hspace{0.3em}}b$ and $d = b {\hspace{0.3em}\raisebox{0.3ex}{\sl \tiny $\setminus $}\hspace{0.3em}}a.$ Then $$(b {\hspace{0.3em}\raisebox{0.3ex}{\sl \tiny $\setminus $}\hspace{0.3em}}a) + a = a + (a {\hspace{0.3em}\raisebox{0.3ex}{\sl \tiny /}\hspace{0.3em}}b) = b,$$ and we write $a^- = 1 {\hspace{0.3em}\raisebox{0.3ex}{\sl \tiny $\setminus $}\hspace{0.3em}}a$ and $a^\sim = a{\hspace{0.3em}\raisebox{0.3ex}{\sl \tiny /}\hspace{0.3em}}1$ for any $a \in E.$ Then $a^-+a=1=a+a^\sim$ and $a^{-\sim}=a=a^{\sim-}$ for any $a\in E.$ We note that an [*ideal*]{} of a pseudo effect algebra $E$ is any non-empty subset $I$ of $E$ such that (i) if $x,y \in I$ and $x+y$ is defined in $E,$ then $x+y \in I,$ and (ii) $x\le y \in I$ implies $x\in I.$ An ideal $I$ is [*normal*]{} if $x+I=I+x$ for any $x \in E,$ where $x+I:=\{x+y: y \in I,\ x+y $ exists in $E\}$ and in the dual way we define $I+x.$ For basic properties of pseudo effect algebras, we recommend [@DvVe1; @DvVe2], where unexplained notions and results can be found. We note that a pseudo effect algebra is an [*effect algebra*]{} iff $+$ is commutative. We note that a [*po-group*]{} (= partially ordered group) is a group $G=(G;+,-,0)$ endowed with a partial order $\le$ such that if $a\le b,$ $a,b \in G,$ then $x+a+y \le x+b+y$ for all $x,y \in G.$ (We note that we will use both additive and multiplicative form of po-groups.) We denote by $G^+:=\{g \in G: g \ge 0\}$ and $G^-:=\{g \in G: g \le 0\}$ the [*positive cone*]{} and the [*negative cone*]{} of $G.$ If, in addition, $G$ is a lattice under $\le$, we call it an $\ell$-group (= lattice ordered group). An element $u \in G^+$ is said to be a [*strong unit*]{} (or an [*order unit*]{}) if, given $g \in G,$ there is an integer $n \ge 1$ such that $g \le nu.$ The pair $(G,u),$ where $u$ is a fixed strong unit of $G,$ is said to be a [*unital po-group*]{}. We recall that the [*lexicographic product*]{} of two po-groups $G_1$ and $G_2$ is the group $G_1\times G_2,$ where the group operations are defined by coordinates, and the ordering $\le $ on $G_1 \times G_2$ is defined as follows: For $(g_1,h_1),(g_2,h_2) \in G_1 \times G_2,$ we have $(g_1,h_1)\le (g_2,h_2)$ whenever $g_1 <g_2$ or $g_1=g_2$ and $h_1\le h_2.$ A po-group $G$ is said to be [*directed*]{} if, given $g_1,g_2 \in G,$ there is an element $g \in G$ such that $g \ge g_1,g_2.$ We note every $\ell$-group or every po-group with strong unit is directed. For more information on po-groups and $\ell$-groups we recommend the books [@Fuc; @Gla]. Now let $G$ be a po-group and fix $u \in G^+.$ If we set $\Gamma(G,u):=[0,u]=\{g \in G: 0 \le g \le u\},$ then $\Gamma(G,u)=(\Gamma(G,u); +,0,u)$ is a pseudo effect algebra, where $+$ is the restriction of the group addition $+$ to $[0,u],$ i.e. $a+b$ is defined in $\Gamma(G,u)$ for $a,b \in \Gamma(G,u)$ iff $a+b \in \Gamma(G,u).$ Then $a^-=u-a$ and $a^\sim=-a+u$ for any $a \in \Gamma(G,u).$ A pseudo effect algebra which is isomorphic to some $\Gamma(G,u)$ for some po-group $G$ with $u>0$ is said to be an [*interval pseudo effect algebra*]{}. We recall that by an [*o-ideal*]{} of a directed po-group $G$ we mean any normal directed convex subgroup $H$ of $G;$ convexity means if $g,h\in H$ and $v\in G$ such that $g\le v\le h,$ then $v \in H.$ If $G$ is a po-group and $H$ an o-ideal of $G$, then $G/H,$ where $x/H \le y/H$ iff $x\le h_1+y$ for some $h_1\in H$ iff $x \le y+h_2$ for some $h_2 \in H,$ is also a po-group. The following kinds of the Riesz Decomposition properties were introduced in [@DvVe1; @DvVe2] and are crucial for the study of pseudo effect algebras. We say that a po-group $G$ satisfies 1. the [*Riesz Interpolation Property*]{} (RIP for short) if, for $a_1,a_2, b_1,b_2\in G,$ $a_1,a_2 \le b_1,b_2$ implies there exists an element $c\in G$ such that $a_1,a_2 \le c \le b_1,b_2;$ 2. $_0$ if, for $a,b,c \in G^+,$ $a \le b+c$, there exist $b_1,c_1 \in G^+,$ such that $b_1\le b,$ $c_1 \le c$ and $a = b_1 +c_1;$ 3.   if, for all $a_1,a_2,b_1,b_2 \in G^+$ such that $a_1 + a_2 = b_1+b_2,$ there are four elements $c_{11},c_{12},c_{21},c_{22}\in G^+$ such that $a_1 = c_{11}+c_{12},$ $a_2= c_{21}+c_{22},$ $b_1= c_{11} + c_{21}$ and $b_2= c_{12}+c_{22};$ 4. $_1$ if, for all $a_1,a_2,b_1,b_2 \in G^+$ such that $a_1 + a_2 = b_1+b_2,$ there are four elements $c_{11},c_{12},c_{21},c_{22}\in G^+$ such that $a_1 = c_{11}+c_{12},$ $a_2= c_{21}+c_{22},$ $b_1= c_{11} + c_{21}$ and $b_2= c_{12}+c_{22}$, and $0\le x\le c_{12}$ and $0\le y \le c_{21}$ imply $x+y=y+x;$ 5. $_2$ if, for all $a_1,a_2,b_1,b_2 \in G^+$ such that $a_1 + a_2 = b_1+b_2,$ there are four elements $c_{11},c_{12},c_{21},c_{22}\in G^+$ such that $a_1 = c_{11}+c_{12},$ $a_2= c_{21}+c_{22},$ $b_1= c_{11} + c_{21}$ and $b_2= c_{12}+c_{22}$, and $c_{12}\wedge c_{21}=0.$ If, for $a,b \in G^+,$ we have for all $0\le x \le a$ and $0\le y\le b,$ $x+y=y+x,$ we denote this property by $a\, \mbox{\rm \bf com}\, b.$ For Abelian po-groups, RDP, RDP$_1,$ RDP$_0$ and RIP are equivalent. By [@DvVe1 Prop 4.2] for directed po-groups, we have $${\mbox{\rm RDP}}_2 \quad \Rightarrow {\mbox{\rm RDP}}_1 \quad \Rightarrow {\mbox{\rm RDP}}\quad \Rightarrow {\mbox{\rm RDP}}_0 \quad \Leftrightarrow \quad {\mbox{\rm RIP}},$$ but the converse implications do not hold, in general. A directed po-group $G$ satisfies $_2$ iff $G$ is an $\ell$-group, [@DvVe1 Prop 4.2(ii)]. Finally, we say that a pseudo effect algebra $E$ satisfies the above types of the Riesz decomposition properties, if in the definition of RDP’s, we change $G^+$ to $E.$ A fundamental result of pseudo effect algebras which binds them with po-groups says the following, [@DvVe2 Thm 7.2]: \[th:2.2\] For every pseudo effect algebra $E$ with $_1,$ there is a unique (up to isomorphism of unital po-groups) unital po-group $(G,u)$ with $_1$ such that $E \cong \Gamma(G,u).$ In addition, the functor $\Gamma$ defines a categorical equivalence between the category of pseudo effect algebras with $_1$ and the category of unital po-groups with $_1.$ In particular, a pseudo effect algebra $E$ is a pseudo MV-algebra (for definition see [@GeIo]) iff $E$ satisfies RDP$_2,$ and in such a case, every pseudo MV-algebra is an interval in a unital $\ell$-group; for details see [@Dvu2] and [@DvVe2]. Subdirect Product of Subdirectly Irreducible Kite Pseudo Effect Algebras ======================================================================== The present section defines kite pseudo effect algebras and gives a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras. According to [@DvuK], we present kite pseudo effect algebras. Let $G=(G;\cdot,^{-1},e)$ be a multiplicatively-written po-group with an inverse $^{-1},$ identity element $e,$ and equipped with a partial order $\le.$ Then $G^+:=\{g \in G\colon g\ge e\}$ and $G^-:=\{g\in G \colon g \le e\}.$ Let $I$ be a set. Define an algebra whose universe is the set $(G^+)^I \uplus (G^-)^I,$ where $\uplus$ denotes a union of disjoint sets. We order its universe by keeping the original co-ordinatewise ordering within $(G^+)^I$ and $(G^-)^I$, and setting $x\leq y$ for all $x\in(G^+)^I$, $y\in(G^-)^I$. Then $\le$ is a partial order on $(G^+)^I \uplus (G^-)^I.$ Hence, the element $e^I:=\langle e: i \in I\rangle$ appears twice: at the bottom of $(G^+)^I$ and at the top of $(G^-)^I$. To avoid confusion in the definitions below, we adopt a convention of writing $a_i^{-1},b_i^{-1}, \dots$ for co-ordinates of elements of $(G^-)^I$ and $f_j,g_j,\dots$ for co-ordinates of elements of $(G^+)^I$. In particular, we will write $e^{-1}$ for $e$ as an element of $G^-,$ $e$ as an element of $G^+,$ and without loss of generality, we will assume that formally $e^{-1}\ne e.$ We also put $1$ for the constant sequence $(e^{-1})^I:=\langle e^{-1}\colon i \in I\rangle$ and $0$ for the constant sequence $e^I:=\langle e_j\colon j \in I\rangle$. Then $0$ and $1$ are the least and greatest elements of $(G^+)^I \uplus (G^-)^I.$ The following construction of kite pseudo effect algebras was presented in [@DvuK Thm 3.4]: \[th:3.4\] Let $G$ be a po-group and $\lambda,\rho:I\to I$ be bijections. Let us endow the set $(G^+)^I \uplus (G^-)^I$ with $0=e^I,$ $1=(e^{-1})^I$ and with a partial operation $+$ as follows, $$\langle a_i^{-1}\colon i\in I\rangle + \langle b_i^{-1}\colon i\in I\rangle \eqno(I)$$ is not defined; $$\langle a_i^{-1}\colon i\in I\rangle + \langle f_j\colon j\in I\rangle:= \langle a_i^{-1}f_{\rho^{-1}(i)}\colon i\in I\rangle \eqno(II)$$ whenever $f_{\rho^{-1}(i)}\le a_i,$ $i \in I;$ $$\langle f_j\colon j\in I\rangle+ \langle a_i^{-1}\colon i\in I\rangle := \langle f_{\lambda^{-1}(i)} a_i^{-1}\colon i\in I\rangle \eqno(III)$$ whenever $f_{\lambda^{-1}(i)}\le a_i,$ $i \in I,$ $$\langle f_j\colon j\in I\rangle + \langle g_j\colon j\in I\rangle:= \langle f_j g_j\colon j\in I\rangle \eqno(IV)$$ for all $\langle f_j\colon j\in I\rangle$ and $\langle g_j\colon j\in I\rangle.$ Then the partial algebra $K^{\lambda,\rho}_I(G)_{ea}:=((G^+)^I \uplus (G^-)^I; +,0,1)$ becomes a pseudo effect algebra. If $G$ is an $\ell$-group, then $K^{\lambda,\rho}_I(G)_{ea}$ is a pseudo effect algebra with $_2.$ According to [@DvuK], the pseudo effect algebra $(K^{\lambda,\rho}_I(G)_{ea};+,0,1)$ is said to be the [*kite pseudo effect algebra*]{} of a po-group $G.$ Let $\mathcal G$ be the category of po-groups, where objects are po-groups and morphisms are homomorphisms of po-groups. Similarly, let $\mathcal{PEA}$ be the category of pseudo effect algebras, where objects are pseudo effect algebras and morphisms are homomorphisms of pseudo effect algebras. Let us fix a set $I,$ bijective mappings $\rho,\lambda\colon I \to I,$ and define the mapping $K_{I}^{\lambda,\rho}\colon \mathcal G \to \mathcal{PEA}$ as follows: $$K_{I}^{\lambda,\rho}: G \mapsto K_{I}^{\lambda,\rho}(G)$$ and if $h$ is a homomorphism from a po-group $G_1$ into a po-group $G_2,$ then $$K_{I}^{\lambda,\rho}(h)(x) = \begin{cases} \langle h(a_i^{-1})\colon i \in I\rangle & \text{ if } x = \langle a_i^{-1}\colon i \in I\rangle,\\ \langle h(f_j)\colon j \in I\rangle & \text{ if } x = \langle f_j\colon j \in I\rangle. \end{cases}$$ Then $K_{I}^{\lambda,\rho}$ is a functor. We say that a pseudo effect algebra $E$ (or a po-group) is a [*subdirect product*]{} of a family $(E_t\colon t \in T)$ of pseudo effect algebras (po-groups), and we write $E \leq \prod_{t \in T}E_t$ if there is an injective homomorphism $h\colon E \to \prod_{t \in T}E_t$ such that $\pi_t\circ h(E)=E_t$ for all $t \in T,$ where $\pi_t$ is the $t$-th projection from $\prod_{t \in T}E_t$ onto $E_t.$ In addition, $E$ is [*subdirectly irreducible*]{} if whenever $E$ is a subdirect product of $(E_t: t \in T),$ there exists $t_0 \in T$ such that $\pi_{t_0}\circ h$ is an isomorphism of pseudo effect algebras. For pseudo effect algebras, the relation between congruences and ideals is more complicated. Fortunately, this is true for pseudo effect algebras with RDP$_1$, [@DvVe3], or for Riesz ideals of general pseudo effect algebras. Therefore, for pseudo MV-algebras we have: a non-trivial pseudo MV-algebra $E$(i.e. $0\not=1$) is subdirectly irreducible iff the intersection of all non-trivial normal ideals of $E$ is non-trivial, or equivalently, $E$ has the least non-trivial normal ideal. An analogous result is true also for pseudo effect algebras with RDP$_1$: \[le:7.1\] Every pseudo effect algebra with $_1$ is a subdirect product of subdirectly irreducible pseudo effect algebras with $_1.$ If $E$ is a trivial pseudo effect algebra, $E$ is subdirectly irreducible. Thus, suppose $E$ is not trivial. By [@DvVe2 Thm 7.2], RDP$_1$ entails that $E \cong \Gamma(G,u)$ for some unital po-group $(G,u),$ where $G$ satisfies RDP$_1$; for simplicity, we assume $E \subset \Gamma(G,u).$ Given a nonzero element $g\in G,$ let $N_g$ be an o-ideal of $G$ which is maximal with respect to not containing $g$; by Zorn’s Lemma, it exists. Then $\bigcap_{g\ne 0} N_g=\{0\}$ and $N_g\cap[0,u]$ is a normal ideal of $E.$ Therefore, $G \leq \prod_{g\ne 0}G/N_g$ and $E \le \prod_{g\ne 0}E/(N_g\cap [0,u]).$ In addition, for every $g\ne 0,$ the normal ideal of $G/N_g$ generated by $g/N_g$ is the least non-trivial o-ideal of $G/N_g$ which proves that every $G/N_g$ is subdirectly irreducible. Therefore, every $E/(N_g\cap [0,u])$ is a subdirectly irreducible pseudo effect algebra, and by [@185 Prop 4.1], the quotient pseudo effect algebra $E/(N_g\cap [0,u])$ also satisfies RDP$_1.$ \[le:7.2\] Let $G$ be a directed po-group with $_1$ subdirectly represented as $G\leq \prod_{s\in S}G_s,$ where each po-group $G_s,$ $s \in S,$ is directed and satisfies $_1.$ Then any kite $K_{I}^{\lambda,\rho}(G)_{ea}$ is subdirectly represented as $K_{I}^{\lambda,\rho}(G)_{ea}\leq \prod_{s\in S}K_{I}^{\lambda,\rho}(G_s)_{ea}$. Let $h\colon G \to \prod_{s\in S}G_s$ be an injective homomorphism of po-groups corresponding to the subdirect product $G \leq \prod_{s \in S}G_s,$ i.e. $\pi_s(h(G))=G_s,$ $s \in S.$ Then the mapping $h^{\lambda,\rho}_I\colon K_{I}^{\lambda,\rho}(G)_{ea} \to \prod_{s\in S}K_{I}^{\lambda,\rho}(G_s)_{ea},$ defined by $h^{\lambda,\rho}_I(\langle a_i^{-1}\colon i \in I\rangle) = \langle \langle \pi_s(h(a_i^{-1}))\colon i \in I\rangle\colon s \in S\rangle$ and $h^{\lambda,\rho}_I(\langle f_j\colon j \in I\rangle) = \langle \langle \pi_s(h(f_j))\colon j \in I\rangle\colon s \in S\rangle,$ shows $K_{I}^{\lambda,\rho}(G)_{ea}\leq \prod_{s\in S}K_{I}^{\lambda,\rho}(G_s)_{ea}$. \[le:7.3\] [(1)]{} Let $G$ be a directed po-group with $_1$ and $P$ be an o-ideal of $G.$ Then $G/P$ is a directed po-group with $_1.$ [(2)]{} Every directed po-group with $_1$ is a subdirect product of subdirectly irreducible po-groups with $_1.$ \(1) Let $G$ be a directed po-group with $_1$ and $P$ an o-ideal of $G.$ We denote by $g/P=[g]_P:=\{h\in G \colon g-h\in P\}$ the quotient class corresponding to an element $g \in G.$ Since every element of $G$ is expressible as a difference of two positive elements, hence, $G/P$ is a directed po-group. In addition, if $[g]_P \ge 0,$ there is an element $g_1 \in [g]_P$ such that $g_1 \in G^+.$ Let $[g_1]_P + [g_2]_P = [h_1]_P + [h_2]_P$ for some positive elements $g_1,g_2,h_1,h_2\in G^+.$ There are $e,f \in P^+$ such that $(g_1 +g_2)- e = (h_1 +h_2)- f\ge 0$, so that $g_1 +g_2 = ((h_1+h_2) -f) + e\ge 0.$ By RDP$_1$ holding in $G$, there are $c_{11}, c_{12}, c_{21}, c_{22}$ in $G^+$ such that $$\begin{aligned} g_1 = c_{11} + c_{12},& &(h_1+h_2)- f = c_{11} + c_{21},\\ g_2 = c_{21} + c_{22}, & & e = c_{12} + c_{22}.\end{aligned}$$ This gives $c_{11} + c_{21} + f = h_1 + h_2$. Again due to RDP$_1$, there are $d_{11}, d_{12}, d_{21},$ $ d_{22}, d_{31}, d_{31} \in G^+$ with $d_{12} \ {\mbox{\bf com}} \ (d_{21} +d_{31})$ such that $$\begin{aligned} c_{11} = d_{11} +d_{12},& & b_1 = d_{11} + d_{21} + d_{31},\\ c_{21} = d_{21} +d_{22}, & & b_2 = d_{12}+ d_{22} + d_{32},\\ f = d_{31} + d_{32}. & &\end{aligned}$$ It is clear that $c_{12}, c_{22}, d_{31}, d_{32} \in P$, which gives $[g_1]_P = [c_{11}]_P = [d_{11}]_P + [d_{12}]_P,$ $ [g_2]_P = [c_{21}]_P = [d_{21}]_P +[d_{22}]_P, $ $[h_1]_P = [d_{11}]_P +[d_{21}]_P,$ $[h_2]_P = [d_{12}]_P+[d_{22}]_P.$ Assume $[x]_P \le [d_{12}]_P$ and $[y]_P \le [d_{21}]_P.$ Then there are $x_1 \in [x]_P$ and $y_1 \in [y]_P$ such that $x_1 \le d_{12} $ and $y_1 \le d_{21}$, i.e., $[d_{12}]_P \ {\mbox{\bf com}} \ [d_{21}]_P$ which proves that $G/P$ is with RDP$_1$. \(2) It follows the same ideas as the proof of Proposition \[le:7.1\]. According to [@DvKo; @DvuK], we say that elements $i,j\in I$ are [*connected*]{} if there is an integer $m \ge 0$ such that $(\rho\circ\lambda^{-1})^m(i)= j$ or $(\lambda\circ \rho^{-1})^m(i)= j$; otherwise, $i$ and $j$ are said to be [*disconnected*]{}. We note that (i) every element $i\in I$ is connected to $i$ because $(\rho\circ\lambda^{-1})^0(i)= i,$ (ii) $i$ is connected to $j$ iff $j$ is connected to $i,$ and (iii) if $i$ and $j$ are connected and also $j$ and $k$ are connected, then $i$ and $k$ are connected, too. Indeed, let (a) $(\rho\circ\lambda^{-1})^m(i)= j$ and $(\rho\circ\lambda^{-1})^n(j)= k$ for some integers $m,n \ge 0.$ Then $(\rho\circ\lambda^{-1})^{m+n}(i)= k.$ (b) If $(\rho\circ\lambda^{-1})^m(i)= j$ and $(\lambda\circ\rho^{-1})^n(j)= k$ for some integers $m,n \ge 0,$ then $(\rho\circ \lambda^{-1})^n(k)=j.$ Then $(\rho\circ\lambda^{-1})^m(i)= (\rho\circ \lambda^{-1})^n(k).$ If $m=n,$ then $i=k$ and they are connected, otherwise, $n<m$ or $m<n.$ In the first case $(\rho\circ\lambda^{-1})^{m-n}(i) =k,$ in the second one $(\lambda\circ\rho^{-1})^{n-m}(i) =k,$ proving $i$ and $k$ are connected. In the same way we deal for the rest of transitivity. Consequently, the relation $i$ and $j$ are connected, is an equivalence on $I$, and every equivalence class defines a subset of indices such that every two distinct elements of it are connected and the equivalence class is maximal under this property. We call this equivalence class a [*connected component*]{} of $I.$ Another property: for each $i \in I,$ $\lambda(i)$ and $\rho(i)$ are connected. In a dual way we can say that $i,j\in I$ are [*dually connected*]{} if there is an integer $m \ge 0$ such that $(\rho^{-1}\circ\lambda)^m(i)= j$ or $(\lambda^{-1}\circ \rho)^m(i)= j$; otherwise, $i$ and $j$ are said to be [*dually disconnected*]{}. Similarly as above, this notion defines an equivalence on $I$ and any equivalence class is said to be a [*dual component*]{} of $I,$ and it is a set of mutually dually connected indices and maximal under this property. There holds: (i) for each $i \in I,$ $\lambda^{-1}(i)$ and $\rho^{-1}(i)$ are dually connected. (ii) If two distinct elements $i,j\in I$ are dually connected, then $\lambda(i)$ and $\rho(j)$ or $\rho(i)$ and $\lambda(j)$ are dually connected. (iii) If $i$ and $j$ are dually connected, then $\lambda(i)$ and $\rho(j)$ or $\rho(i)$ and $\lambda(j)$ are connected. (iv) If $i$ and $j$ are connected, then $\lambda^{-1}(i)$ and $\rho^{-1}(j)$ are dually connected and $\rho^{-1}(i)$ and $\rho^{-1}(j)$ are connected or $\rho^{-1}(i)$ and $\lambda^{-1}(j)$ are dually connected and $\lambda^{-1}(i)$ and $\lambda^{-1}(j)$ are connected. If $C$ is a connected component of $I,$ then $\lambda^{-1}(C)= \rho^{-1}(C).$ Indeed, let $i \in C$ and $k=\lambda^{-1}(i).$ Then $j=\rho (k)= \rho \circ \lambda^{-1}(i) \in C.$ Hence, $k=\rho^{-1}(j)$ which proves $\lambda^{-1}(C)\subseteq \rho^{-1}(C).$ In the same way we prove the opposite inclusion. In particular, we have $\lambda(\rho^{-1}(C))= \rho(\lambda^{-1}(C)) = C=\lambda(\lambda^{-1}(C))= \rho(\rho^{-1}(C)).$ We note that if $C$ is a connected component, then $\lambda^{-1}(C)$ is not necessarily $C.$ In fact, let $I=\{1,2\}$ and $\lambda(1)=2=\rho(1)$ and $\lambda(2)=1=\rho(2).$ Then $I$ has only two connected components $\{1\}$ and $\{2\}$, and $\lambda^{-1}(\{1\})=\{2\}$ and $\lambda^{-1}(\{2\})=\{1\}.$ It is possible to define kite pseudo effect algebras also in the following way. Let $J$ and $I$ be two sets, $\lambda,\rho\colon J \to I$ be two bijections, and $G$ be a po-group. We define $K^{\lambda,\rho}_{J,I}(G)= (G^+)^J \uplus (G^-)^I,$ where $\uplus$ denotes a union of disjoint sets. The elements $0=\langle e_j\colon j \in J\rangle$ and $1=\langle e^{-1}_i\colon i \in I\rangle$ are assumed to be the least and greatest elements of $ K^{\lambda,\rho}_{J,I}.$ The elements in $(G^+)^J$ and in $(G^-)^I$ are ordered by coordinates and for each elements $x \in (G^+)^J$ and $y \in (G^-)^I$ we have $x \le y.$ If we define a partial operation $+$ by Theorem \[th:3.4\], changing in formulas $(II)--(IV)$ the notation $j \in I$ to $j \in J,$ we obtain that $K_{J,I}^{\lambda,\rho}(G)_{ea} =(K^{\lambda,\rho}_{J,I};+,0,1) $ with this $+$ $0$ and $1$ is a pseudo effect algebra, called also a [*kite pseudo effect algebra.*]{} In particular, $ K^{\lambda,\rho}_{I}(G)_{ea} = K^{\lambda,\rho}_{I,I}(G)_{ea}.$ Since $J$ and $I$ are of the same cardinality, there is practically no substantial difference between kite pseudo effect algebras of the form $K^{\lambda,\rho}_{I}(G)_{ea}$ and $K^{\lambda,\rho}_{J,I}(G)_{ea}$ and all known results holding for the first kind are also valid for the second one. We note that in [@DvKo], the “kite" structure used two index sets, $J$ and $I.$ The second form will be useful for the following result. \[le:7.4\] Let $K_{I}^{\lambda,\rho}(G)_{ea}$ be a kite pseudo effect algebra of a directed po-group $G,$ where $G$ satisfies $_1.$ Then $K_{I}^{\lambda,\rho}(G)_{ea}$ is a subdirect product of the system of kite pseudo effect algebras $(K_{J',I'}^{\lambda',\rho'}(G)_{ea}\colon I')$, where $I'$ is a connected component of the graph $I$, $J'=\lambda^{-1}(I')=\rho^{-1}(I'),$ and $\lambda',\rho'\colon J'\to I'$ are the restrictions of $\lambda$ and $\rho$ to $J'\subseteq I.$ By the comments before this lemma, we see that $\lambda',\rho'\colon J'\to I'$ are bijections. Let $I'$ be a connected component of $I$. Let $N_{I'}$ be the set of all elements $f = \langle f_j: j \in I\rangle\in (G^+)^I$ such that $f_j = e$ whenever $j\in J'$. It is straightforward to see that $N_{I'}$ is a normal ideal of $K_{I}^{\lambda,\rho}(G)_{ea}$. It is also not difficult to see that $K_{I}^{\lambda,\rho}(G)_{ea}/N_{I'}$ is isomorphic to $K_{J',I'}^{\lambda',\rho'}(G)_{ea}$. Now, let $\mathcal{C}$ be the set of all connected components of $I$, and for each $I'\in\mathcal{C}$ let $N_{I'}$ be the normal filter defined as above. As connected components are disjoint, we have $\bigcap_{I'\in\mathcal{C}} N_{I'} = \{0\}.$ This proves $K_{I}^{\lambda,\rho}(G)_{ea}= K_{I,I}^{\lambda,\rho}(G)_{ea}\leq \prod_{I'} K_{J',I'}^{\lambda',\rho'}(G)_{ea}.$ Now we are ready to formulate an analogue of the Birkhoff representation theorem, [@BuSa Thm II.8.6] for kite pseudo effect algebras with RDP$_1.$ \[th:7.5\] Every kite pseudo effect algebra with $_1$ is a subdirect product of subdirectly irreducible kite pseudo effect algebras with $_1.$ Consider a kite $K_{I}^{\lambda,\rho}(G)$. By [@DvuK Thm 4.1], $G$ satisfies RDP$_1.$ If it is not subdirectly irreducible, then [@DvuK Thm 6.6] yields two possible cases: (i) $G$ is not subdirectly irreducible, or (ii) $G$ is subdirectly irreducible but there exist $i,j\in I$ such that, for every $m\in\mathbb N,$ we have $(\rho\circ\lambda^{-1})^m(i)\neq j$ and $(\lambda\circ\rho^{-1})^m(i)\neq j$. Observe that this happens if and only if $i$ and $j$ do not belong to the same connected component of $I$. Now, using Lemma \[le:7.3\] we can reduce (i) to (ii). So, suppose $G$ is subdirectly irreducible. Then, using Lemma \[le:7.4\], we can subdirectly embed $K_{I}^{\lambda,\rho}(G)_{ea}$ into $\prod_{I'} K_{J',I'}^{\lambda',\rho'}(G)_{ea}$, where $I'$ ranges over the connected components of $I,$ $J'=\lambda^{-1}(I')$ and $\lambda', \rho'$ are restrictions of $\lambda, \rho$ to $J'.$ But then, each $K_{J',I'}^{\lambda,\rho}(G)_{ea}$ is subdirectly irreducible by [@DvuK Thm 6.6] and by [@DvuK Thm 4.1], it satisfies RDP$_1.$ We note that if $G$ is an $\ell$-group, the corresponding kite pseudo effect algebra is a pseudo MV-algebra, and Theorem \[th:7.5\] can be reformulated as follows, see [@DvKo Cor 5.14]. \[th:7.6\] The variety of pseudo MV-algebras generated by kite pseudo MV-algebras is generated by all subdirectly irreducible kite pseudo MV-algebras. Kite Pseudo Effect algebras and po-loops ======================================== In this section, we show that every kite pseudo effect algebra is an interval in a po-loop which is not necessarily a po-group (i.e. it is not necessarily an associative po-loop). This construction is accompanied by illustrating examples. We remind that according to [@Fuc], a [*po-groupoid*]{} is an algebraic structure $(H;\cdot,\le~)$ such that (i) $(H;\cdot)$ is a groupoid, i.e. $H$ is closed under a multiplication $\cdot,$ and (ii) $H$ is endowed with a partial order $\le$ such that $a\le b$ entails $c\cdot a\cdot d \le c\cdot b\cdot d$ for all $c,d \in H.$ Sometimes for simplicity we will write $ab$ instead of $a\cdot b.$ If the multiplication $\cdot$ is associative, $(H;\cdot,\le)$ is said to be a [*po-semigroup*]{}. If $(H;\cdot)$ is a quasigroup, i.e. for each $a$ and $b$ in $H$, there exist unique elements $x$ and $y$ in $H$ such that $a\cdot x=b$ and $y\cdot a=b,$ then $(H;\cdot,\le)$ is said to be a [*po-quasigroup*]{}. If in addition, a po-quasigroup $(H;\cdot,\le)$ has an identity element (or a neutral element) $e\in H$ such that $x\cdot e=x=e\cdot x$ for each $x \in H,$ then $(H;\cdot,e,\le)$ is said to be a [*po-loop*]{}. Let $x \in H$ be given. A unique element $x^{-1_r} \in H$ such that $x \cdot x^{-1_r}=e$ is said to be the [*right inverse*]{} of $x.$ Similarly, a unique element $x^{-1_l}\in H$ such that $x^{-1_l}\cdot x = e$ is said to be the [*left inverse*]{} of $x.$ Given an element $a$ of a groupoid $H,$ we define (i) $a^1 :=a,$ and (ii) if $n\ge 1,$ then $a^{n+1}:=a^na.$ As in the case of po-groups, an element $u\ge e$ is said to be a [*strong unit*]{} of a po-groupoid $H$ if given an element $g \in H,$ there is an integer $n \ge 0$ such that $g \le u^n$; the couple $(H,u)$ is said to be a [*unital po-groupoid*]{}. For a unital po-loop $(H,u),$ we define an interval $\Gamma(H,u)=\{h \in H\colon e\le h\le u\}.$ We know that every pseudo effect algebra with RDP$_1,$ [@DvVe1; @DvVe2], or every pseudo MV-algebra, [@Dvu2], is an interval of a unital po-group with RDP$_1$ and of a unital $\ell$-group, respectively. A similar result, as we now show, is true also for every kite pseudo effect algebra, however, in this case it is an interval of a unital po-loop, which is not necessarily a po-group. In what follows, we show that there are unital po-loops $(H,u),$ which are not necessarily associative, such that $\Gamma(H,u)$ can be endowed with a partial operation $+$ such that $(\Gamma(H,u);+,e,u)$ is a pseudo effect algebra, where $a+b$ for $a,b \in \Gamma(H,u)$ is defined in $\Gamma(H,u)$ whenever $ab\le u$ and then $a+b:= ab.$ We note that no assumption on the Riesz Decomposition Property of the kite pseudo effect algebra will be assumed. \[th:8.1\] For every kite pseudo effect algebra $K_{I}^{\lambda,\rho}(G)_{ea}$, there is a unital po-loop $(H,u)$ such that $\Gamma(H,u)$ is a pseudo effect algebra and $K_{I}^{\lambda,\rho}(G)_{ea}$ is isomorphic to $\Gamma(H,u).$ Let $K_{I}^{\lambda,\rho}(G)_{ea}$ be a kite pseudo effect algebra of a po-group $G.$ We define $H:=W^{\lambda,\rho}_I(G):=\mathbb Z {\,\overrightarrow{\times}\,}G^I$ and let multiplication $*$ on $W^{\lambda,\rho}_I(G)$ be defined as follows $$(m_1,x_i)*(m_2,y_i):=(m_1+m_2, x_{\lambda^{-m_2}(i)}y_{\rho^{-m_1}(i)}).$$ Then $W^{\lambda,\rho}_I(G)$ is a po-groupoid ordered lexicographically such that $(0,(e))$ is the neutral element, $(m,x_i)^{-1_r}=(-m,x^{-1}_{(\rho^m\circ \lambda^m)(i)}),$ $(m,x_i)^{-1_l}=(-m,x^{-1}_{(\lambda^m\circ \rho^m)(i)}),$ and the element $u=(1,(e))$ is a strong unit. In fact, $W^{\lambda,\rho}_I(G)$ is a po-loop. We note that the po-loop $W^{\lambda,\rho}_I(G)$ is associative iff $\lambda \circ \rho = \rho \circ \lambda,$ and in such a case, $W^{\lambda,\rho}_I(G)$ is a po-group. An easy calculation shows that $(\Gamma(W^{\lambda,\rho}_I(G),u);+,e,u),$ where $e$ is an identity element of the po-loop $W^{\lambda,\rho}_I(G),$ is a pseudo effect algebra such that $$\begin{aligned} (1,a_i^{-1})^\sim &= (0,a_{\rho(i)}),\quad (1,a_i^{-1})^-= (0,a_{\lambda(i)}),\\ (0,f_i)^\sim &= (1,f^{-1}_{\lambda^{-1}(i)}),\quad (0,f_i)^-= (1,f^{-1}_{\rho^{-1}(i)}).\end{aligned}$$ In addition, the pseudo effect algebra embedding of $K_{I}^{\lambda,\rho}(G)_{ea}$ onto $\Gamma(W^{\lambda,\rho}_I(G),u)$ is defined as follows $\langle f_j\colon j \in I\rangle \mapsto (0,f_j)$ and $\langle a^{-1}_i\colon i \in I\rangle \mapsto (1,a^{-1}_i).$ Now we show an example such that $\lambda\circ \rho \ne \rho \circ \lambda,$ $W^{\lambda,\rho}_I(G)$ is a non-associative po-loop, but $K_{I}^{\lambda,\rho}(G)_{ea}\cong\Gamma(W^{\lambda,\rho}_I(G),u).$ \[ex:8.2\] Let $I=\{1,2,3,4\},$ $\lambda(1)=1,$ $\lambda(2)=3,$ $\lambda(3)= 2,$ $\lambda(4)=4,$ $\rho(1)= 2,$ $\rho(2)= 3,$ $\rho(3)= 1,$ $\rho(4)= 4.$ Then $\lambda \circ \rho\ne \rho\circ \lambda $ and $W^{\lambda,\rho}_I(G)$ is a non-associative po-groupoid, but $K_{I}^{\lambda,\rho}(G)_{ea}\cong\Gamma(W^{\lambda,\rho}_I(G),u).$ \[le:8.3\] Let $I$ be a non-void set, $\lambda, \rho\colon I \to I$ be bijections, and let there exist a decomposition $\{I_t\colon t \in T\}$ of $I,$ where each $I_t$ is non-void, such that $\lambda\circ\rho(I_t)=\rho\circ \lambda (I_t)$ for each $t \in T$ and, for each $t \in T$ there are $s_1,s_2 \in T$ such that $\lambda(I_t)=I_{s_1}$ and $\rho(I_t)=I_{s_2}.$ Then, for all integers $m,n \in \mathbb Z,$ we have $\lambda^m\circ \rho^n(I_t)=\rho^n\circ \lambda^m(I_t)$ for each $t \in T.$ \(i) First we show that $\rho^{n+1}\circ \lambda(I_t)=\lambda\circ \rho^{n+1}(I_t)$ for each integer $n \ge 1$ and any $t \in T.$ Indeed, $\rho^2\circ \lambda(I_t)= \rho(\rho\circ \lambda(I_t))=\rho(\lambda\circ \rho(I_t))=\rho\circ\lambda(\rho(I_t))=\lambda\circ \rho(\rho(I_t))=\lambda\circ \rho^2(I_t).$ Now by induction we assume that $\rho^{i+1}\circ\lambda(I_s) = \lambda\circ \rho^{i+1}(I_s)$ holds for each $1\le i\le n$ and each $s \in T.$ Then $\rho^{n+1}\circ \lambda(I_t)= \rho^n(\rho\circ \lambda (I_t))=\rho^n(\lambda \circ \rho(I_t))= \rho^n\circ \lambda(\rho(I_t)) = \lambda\circ \rho^n(\rho(I_t))=\lambda\circ \rho^{n+1}(I_t).$ \(ii) In the same way we have $\rho\circ \lambda^{m+1}(I_t)=\lambda^{m+1}\circ \rho(I_t)$ for each integer $m\ge 1$ and $t \in T.$ \(iii) Using the analogous reasoning, we can show that $\lambda^m\circ \rho^n(I_t)=\rho^n\circ \lambda^m(I_t)$ for all integers $m,n\ge 1$ and each $t \in T.$ \(iv) Using (iii), we have $I_t=\rho^{n}\circ \lambda^{m}(\lambda^{-m} \circ \rho^{-n}(I_t))= \lambda^m \circ \rho^n(\lambda^{-m} \circ \rho^{-n}(I_t))$ which yields $\rho^{-n}\circ \lambda^{-m}(I_t)= \lambda^{-m}\circ\rho^{-n}(I_t)$ for each $m,n\ge 0$ and each $t\in T.$ \(v) By (iii), we have $\lambda^m\circ \rho^n(\lambda^{-m}(I_t))= \rho^n\circ \lambda^m(\lambda^{-m}(I_t))=\rho(I_t)$ which gives $\rho^n\circ \lambda^{-m}(I_t)=\lambda^{-m}\rho^n(I_t)$ for all $m,n \ge 0$ and each $t \in T.$ In the same way, we can prove $\rho^{-n}\circ \lambda^{m}(I_t)=\lambda^{m}\rho^{-n}(I_t)$ for $m,n \ge 0,$ $t\in T.$ Summarizing (i)-(v), we have proved the statement in question. We note that in Example \[ex:8.2\], we have $I=\{1,2,3\}\cup \{4\}$ and $\lambda(\{1,2,3\})=\{1,2,3\},$ $\rho(\{1,2,3\})=\{1,2,3\}$, $\lambda(\{4\})=\{4\}=\rho(\{4\})$ and the conditions of Lemma \[le:8.3\] are satisfied. \[ex:8.4\] Let $I=\{1,2,3,4\},$ $\lambda(1)=1,$ $\lambda(2)=3,$ $\lambda(3)= 2,$ $\lambda(4)=4,$ $\rho(1)= 2,$ $\rho(2)= 1,$ $\rho(3)= 4,$ $\rho(4)= 3.$ If we set $I_1=\{1,4\},$ $I_2=\{2,3\},$ then $\lambda(I_1)=I_1,$ $\lambda(I_2)=I_2,$ $\rho(I_1)=I_2$ and $\rho(I_1).$ The conditions of Lemma [\[le:8.3\]]{} are satisfied. \[ex:8.5\] Let $I=\{1,2,3,4\},$ $\lambda(1)=2,$ $\lambda(2)=3,$ $\lambda(3)= 1,$ $\lambda(4)=4,$ $\rho(1)= 1,$ $\rho(2)= 3,$ $\rho(3)= 4,$ $\rho(4)= 2.$ If we set $I_1=\{1,4\},$ $I_2=\{2,3\},$ then $\lambda\circ\rho(I_1)= I_2=\rho\circ\lambda(I_1)$ and $\lambda\circ\rho(I_2)= I_1=\rho\circ\lambda(I_2).$ But $\lambda(I_1)=\{2,4\},$ $\lambda(I_2)=\{1,3\},$ $\rho(I_1)=\{1,2\}$ and $\rho(I_2)=\{3,4\}.$ Therefore, the decomposition $\{I_1,I_2\}$ does not satisfy the conditions of Lemma [\[le:8.3\]]{}, only the decomposition $\{I\}$ does. \[pr:8.6\] Let $I,\lambda,\rho$ and the decomposition $\{I_t\colon t \in T\}$ of $I$ satisfy the conditions of Lemma [\[le:8.3\]]{} and let $G$ be a po-group. Let $H=\{(m,x_i)\in W^{\lambda,\rho}_I(G)\colon$ for each $i,j \in I_t,\ x_i=x_j, \ t\in T\}.$ Then $H$ is a po-group which is a po-subloop of $W^{\lambda,\rho}_I(G)$, and if $u:=(1,(e)) \in H,$ then $(H,u)$ is a unital po-group, and $\Gamma(H,u)$ is a subalgebra of the pseudo effect algebra $\Gamma(W^{\lambda,\rho}_I(G),u).$ The conditions of the proposition entail that $H$ is a subloop of the loop $W^{\lambda,\rho}_I(G)$ because the conditions imply by Lemma \[le:8.3\] $x_{\lambda^m\circ\rho^n(i)}=x_{\rho^n\circ \lambda^m(i)}$ for all $m,n \in \mathbb Z,$ $i \in I_t,$ $t \in T,$ that is, associativity of the product holds in $H.$ Consequently, $\Gamma(H,u)$ is a pseudo effect algebra. \[re:8.7\] [We note that if $\phi$ is the embedding from the proof of Theorem [\[th:8.1\]]{} of the kite pseudo effect algebra $K_{I}^{\lambda,\rho}(G)_{ea}$ onto the pseudo effect algebra $\Gamma(W^{\lambda,\rho}_I(G),u),$ then $\phi^{-1}(\Gamma(H,u))$ is a pseudo effect subalgebra of the kite pseudo effect algebra $K_{I}^{\lambda,\rho}(G)_{ea}.$]{} We note, in Example \[ex:8.5\], the only decomposition of $I$ satisfying Lemma \[le:8.3\] is the family $\{I\}.$ Then the corresponding group $H$ from Proposition \[pr:8.6\] consists only of the elements of the form $(m,(g)),$ where $(m,(g)):=(m,g_i)$ with $g_i = g,$ $i \in I$ $(g \in G),$ therefore, $\Gamma(H,u)\cong \Gamma(\mathbb Z{\,\overrightarrow{\times}\,}G,(1,0)).$ The same is true concerning the group $H$ from Proposition \[pr:8.6\] for any case of $I$ and the decomposition $\{I\}.$ If we use the decomposition of $I$ consisting only of the singletons of $I,$ $\{\{i\}\colon i \in I\},$ then Lemma \[le:8.3\] holds for this decomposition iff $\lambda \circ \rho=\rho\circ \lambda,$ and for the po-group $H$ from Proposition \[pr:8.6\], we have $H=W^{\lambda,\rho}_I(G), $ and the po-loop $W^{\lambda,\rho}_I(G)$ is associative, that is, it is a po-group. Such a case can happen, e.g., if $\rho = \lambda^m$ for some integer $m \in \mathbb Z.$ But this is not a unique case as the following example shows also another possibility. \[ex:3.8\] Let $I=\{1,2,3,4\},$ $\lambda(1)=2,$ $\lambda(2)=1,$ $\lambda(3)=4,$ $\lambda(4)=3,$ and $\rho(1)= 4,$ $\rho(2)= 3,$ $\rho(3)= 2,$ $\rho(4)= 1.$ Then $\lambda\circ\rho =\rho\circ \lambda,$ $\lambda^2=id_I=\rho^2,$ but $\rho \ne \lambda^m$ for any $m \in \mathbb Z.$ In addition, if $H$ is an arbitrary subloop of the po-loop $W^{\lambda,\rho}_I(G)$ such that $u=(1,(e)) \in H,$ then $\Gamma(H,u)$ is a pseudo effect subalgebra of the pseudo effect algebra $\Gamma(W^{\lambda,\rho}_I(G),u),$ and $\phi^{-1}(\Gamma(H,u))$ is also a pseudo effect subalgebra of the kite pseudo effect algebra $K_{I}^{\lambda,\rho}(G)_{ea}.$ From theory of pseudo effect algebras we know, [@DvVe2 Thm 7.2], that for every pseudo effect algebra $E$ with RDP$_1,$ there exists a unique (up to isomorphism) unital po-group $(H,u)$ with RDP$_1$ such that $E \cong \Gamma(H,u).$ From [@DvuK Thm 4.1], it follows that if a po-group $G$ satisfies RDP$_1,$ then so does the kite pseudo effect algebra $K_{I}^{\lambda,\rho}(G)_{ea},$ and consequently, there is a unital po-group $(H,u)$ with RDP$_1$ such that $K_{I}^{\lambda,\rho}(G)_{ea}=\Gamma(H,u).$ Theorem \[th:8.1\] guarantees the existence of a unital po-loop $(H,u)$ such that $K_{I}^{\lambda,\rho}(G)_{ea}\cong \Gamma(H,u),$ and by Examples \[ex:8.4\]–\[ex:8.5\], we have seen that the po-loop $W^{\lambda,\rho}_I(G)$ is not associative. \[prob:3.9\] [(1)]{} In [@DvuK], there was formulated an open problem to describe a unital po-group $(H,u)$ such that $K_{I}^{\lambda,\rho}(G)_{ea}\cong \Gamma(H,u).$ This is still not answered. [(2)]{} In addition, if there are two unital po-loops $(H_1,u_1)$ and $(H_2,u_2)$ such that $\Gamma(H_1,u_1)\cong K_{I}^{\lambda,\rho}(G)_{ea}\cong \Gamma(H_2,u_2),$ when are these po-loops isomorphic to each other as po-loops ? [DvVe2]{} S. Burris, H.P. Sankappanavar, [*“A Course in Universal Algebra"*]{}, The Millennium Edition, 2000. A. Dvurečenskij, [*Pseudo MV-algebras are intervals in $\ell$-groups*]{}, J. Austral. Math. Soc. [**72**]{} (2002), 427–445. A. Dvurečenskij, [*Ideals of pseudo-effect algebras and their applications*]{}, Tatra Mt. Math. Publ. [**27**]{} (2003), 45–65. A. Dvurečenskij, [*Kite pseudo effect algebras,*]{} http://arxiv.org/abs/1306.0304 A. Dvurečenskij, T. Kowalski, [*Kites and pseudo BL-algebras*]{}, Algebra Univer. (to appear). http://arxiv.org/submit/510377 A. Dvurečenskij, T. Vetterlein, [*Pseudoeffect algebras. I. Basic properties*]{}, Inter. J. Theor. Phys. [**40**]{} (2001), 685–701. A. Dvurečenskij, T. Vetterlein, [*Pseudoeffect algebras. II. Group representation*]{}, Inter. J. Theor. Phys. [**40**]{} (2001), 703–726. A. Dvurečenskij, T. Vetterlein, [*Congruences and states on pseudo-effect algebras*]{}, Found. Phys. Letters [**14**]{} (2001), 425–446. A. Dvurečenskij, T. Vetterlein, [*Non-commutative algebras and quantum structures*]{}, Inter. J. Theor. Phys. [**43**]{} (2004), 1599–1612. D.J. Foulis, M.K. Bennett, [*Effect algebras and unsharp quantum logics*]{}, Found. Phys. [**24**]{} (1994), 1325–1346. L. Fuchs, [*“Partially Ordered Algebraic Systems"*]{}, Pergamon Press, Oxford, London, NY, Paris, 1963. G. Georgescu, A. Iorgulescu, [*Pseudo-MV algebras*]{}, Multi Valued Logic [**6**]{} (2001), 95–135. A.M.W. Glass [*“Partially Ordered Groups"*]{}, World Scientific, Singapore, 1999. P. Jipsen and F. Montagna, *On the structure of generalized BL-algebras*, Algebra Universalis [**55**]{} (2006), 226–237. J. Rachunek, [*A non-commutative generalization of MV-algebras*]{}, Czechoslovak Math. J. [**52**]{} (2002), 255–273. K. Ravindran, [*On a structure theory of effect algebras*]{}, PhD thesis, Kansas State Univ., Manhattan, Kansas, 1996. [^1]: Keywords: Pseudo MV-algebra, pseudo effect algebra $\ell$-group, po-group, strong unit, kite pseudo effect algebra, subdirect product, po-loop. AMS classification: 03G12 81P15, 03B50 The paper has been supported by Slovak Research and Development Agency under the contract APVV-0178-11, the grant VEGA No. 2/0059/12 SAV, and by CZ.1.07/2.3.00/20.0051.

--- abstract: 'Presented in this work are some results relative to sequences found in the logistic equation bifurcation diagram, which is the unimodal quadratic map prototype. All of the different saddle-node bifurcation cascades, associated to every last appearance $p$-periodic orbit ($p=3,\,4,\,5,\dots$), can also be generated from the very Feigenbaum cascade. In this way it is evidenced the relationship between both cascades. The orbits of every saddle-node bifurcation cascade, mentioned above, are located in different chaotic bands, and this determines a sequence of orbits converging to every band-merging Misiurewicz point. In turn, these accumulation points form a sequence whose accumulation point is the Myrberg-Feigenbaum point. It is also proven that the first appearance orbits in the $n$-chaotic band converge to the same point as the last appearance orbits of the $(n+1)$-chaotic band. The symbolic sequences of band-merging Misiurewicz points are computed for any window.' author: - 'Jesús San Martín${}^{a,b}$, Daniel Rodríguez-Pérez${}^{b}$' title: Conjugation of cascades --- $^{a}$ Departamento de Matemática Aplicada, E.U.I.T.I, Universidad Politécnica de Madrid. Ronda de Valencia 3, 28012 Madrid Spain $^{b}$ Departamento de Física Matemática y Fluidos, U.N.E.D. Senda del Rey 9, 28040 Madrid Spain Corresponding author: jsm@dfmf.uned.es Introduction ============ Nonlinear dynamical systems exhibit a rich variety of behaviors. Bifurcations, chaos, patterns, phase transitions are among the most investigated phenomena, and they have led to many discussions within scientific literature. The former mentioned phenomena appears in a lot of dynamical systems, and this is an indicative of the existence of underlaying common structures whose origin claims to be uncovered. To be able to undertake the study of a dynamical system, diverse tools have been developed, which make the study of such problems easier, by means of simplifications, classifications or through the identification of universality classes among transitions from one behavior to another. Near a hyperbolic fixed point, the well known Hartman-Gro[ß]{}man theorem states that the local phase portrait is topologically conjugate to the phase portrait of the linearized system. However, when the fixed point is a non-hyperbolic one, the center manifold must be considered. The Center Manifold theorem allows one to generalize the ideas of the Hartman-Gro[ß]{}man theorem. It is deduced that the flow, restricted to the center manifold, determines the topological behavior close to a non-hyperbolic fixed point. In general, when the flow is restricted to the center manifold the problem is simplified quite a bit, yet the restricted flow itself can be very complicated. The normal form theorem is the next mathematical tool that must be used: a smooth vector field is substituted by a polynomial vector field. Another technique very similar, in certain way, to the tool stated above consists in constructing a Poincaré section. After that, the Poincaré map is created, that is, a map from the Poincaré section in itself. In dissipative systems this map contracts the Poincaré section surface after each iteration, and the initial problem is thus transformed into another problem with an attractor of lower dimensionality. Fortunately, one-dimensional maps provide a good approximation to many dynamical systems. Furthermore one-dimensional maps can be classified into universality classes. That is why the bifurcation route to chaos, discovered by Feigenbaum [@Feigenbaum78; @Feigenbaum79] and characterized by its universal scaling is so important. And for the same reason the logistic equation$$x_{n+1}=f(x_{n})=\mu x_{n}(1-x_{n})\quad\mu\in[0,4],\; x_{n}\in[0,1]\label{eq:logistic}$$ is so intensively studied. The different behaviors of this map can be extended to any quadratic one-dimensional map, because they are topologically conjugated [@Milnor88]. Therefore, any dynamical system ruled by a quadratic polynomial behaves as if it were ruled by the logistic equation.   The goal of this paper is to show in the logistic equation the interesting connection between the Feigenbaum cascade and the saddle-node cascade [@SanMartin07]. Also this connection affects the global structures of windows and Misiurewicz points [@Misiurewicz91] where chaotic bands mix. Given the multitude of dynamical systems that exhibit Feigenbaum’s universality, it is very probable that they also show the new universal behaviors described in this paper. Periodic windows, apart from being an irreplaceable tool in system dynamics, also play other roles and appear in different problems of nonlinear dynamics. They can be used, by applying the Poincaré map, to spot the windows of a dynamical systems which are ruled by ordinary differential equations [@Churukian96]. Chaos control techniques use periodic windows too and they use unstable periodic orbits inside them [@Xu96; @Bishop98]. Periodic windows are also useful to study chaos-induced diffusion [@Blackburn98] which is affected by the presence of proximate periodic windows. Therefore any contribution aimed at the identification of the laws ruling periodic windows will be useful in these related fields.   This paper is organized as follows: In section \[sec:CofC\], it is proven that there is an analytic relation among saddle-node bifurcation cascades [@SanMartin07] and Feigenbaum cascade [@Feigenbaum78; @Feigenbaum79]. The former is derived from the latter. We will refer to this relation by *conjugation of cascades*. The section \[sec:CofC\] is preceded by section \[sec:Definitions\] where later used definitions and theorems are introduced. In the following section \[sec:1st-and-last\], the previous result is used to prove that saddle-node bifurcation cascades associated to last appearance orbits in the $(n+1)$-chaotic band and the ones associated to first appearance orbits in the $n$-chaotic band converge to the same Misiurewicz point where the $(n+1)$-chaotic and $n$-chaotic bands mix. Section \[sec:generalization\] generalizes the former results, which were only valid in the canonical window, to any periodic window. In particular, we will also generalize the rule to get the saddle-node bifurcation cascade in any window. As a spin-off, a geometrical interpretation of the Derrida, Gervois and Pomeau (DGP) formula [@Derrida79] is achieved. In Section \[sec:disc-and-conclusions\], some potential applications of the theorems to experimental research are indicated. \[sec:Definitions\]Definitions ============================== The following notation, definitions and theorems will be referred to in what follows. Whenever they are based on or copied from a previous work, the original reference is given. [Definition 1.]{} : $S_{n}$ will denote a $n$-periodic symbolic pattern. The symbols are $R$ or $L$, depending on wether the iterates $x_{i}$ of a $n$-periodic orbit of the logistic map (\[eq:logistic\]) are $x_{i}>\frac{1}{2}$ or $x_{i}<\frac{1}{2}$ ($i=1,2,\,3,\dots n$), respectively [@MetropolisSteinStein73]. In particular, to determine in an unique way the periodic sequence $S_{n}$, we will always take as the first letter in the sequence that corresponding to the position $x_{i}$ nearest to $1/2$, the rest of the symbols being determined by the succesive iterations of this first value [@SanMartin07]. [Definition 2.]{} : The pattern $S_{n}$ has even “$R$-parity” if it has an even number of symbols $R$ in it. Otherwise $S_{n}$ “$R$-parity” will be termed odd [@MetropolisSteinStein73].\ We will use $R$-parity whenever we work with logistic-like maps (upwards convex). Should we were working with the real Mandelbrot map, for instance (which is downwards convex), all $R$’s and $L$’s in what follows would appear interchanged, and we would use $L$-parity instead. [Definition 3.]{} : The pattern $\overline{S_{n}}$ obtained exchanging $R$ and $L$ in $S_{n}$ will be called the conjugated sequence of $S_{n}$. In the case of a supercycle pattern $CS_{n-1}$ ($C$ corresponds to $x_{1}=1/2$), its conjugate will only change the $R$ and $L$ in the pattern, whereas the initial $C$ remains in place. [Definition 4.]{} : $(S_{n})^{p}=\underbrace{S_{n}S_{n}\dots S_{n}}_{p\mbox{ times}}$, is the pattern of $n\cdot p$ symbols constructed by $p$ repetitions of the $n$-symbols pattern $S_{n}$. As a consequence, we may write $\left[(S_{n})^{p}\right]^{q}=(S_{n})^{p\cdot q}$. We will adopt an analogous notation for the case of supercycles. [Definition 5.]{} : $CS_{n-1}$ is a $n$-symbols pattern formed by replacing the first letter in sequence $S_{n}$ by the symbol $C$. This orbit corresponds to a supercycle periodic orbit (starting at $x_{1}=\frac{1}{2}$). [Definition 6.]{} : $(CS_{n})^{p}\vert S_{p}$ is the pattern resulting from the substitution of the $p$ succesive $C$-letters in the pattern $(CS_{n})^{p}=\underbrace{CS_{n}CS_{n}\dots CS_{n}}_{p\mbox{ times}}$ by the $p$ succesive symbols in $S_{p}$. [Definition 7.]{} : $CP_{n,q}$ represents the pattern of the $n$-th pitchfork bifurcation supercycle of the $q$-periodic orbit. For $q=1$ we will write simply $CP_{n}$. [Definition 8.]{} : The first F-harmonic of $CS_{n}$, denoted by $H_{F}^{(1)}(CS_{n})$, is formed appending $CS_{n}$ to itself and changing the second $C$ to $R$ ($L$) if the $R$-parity of $CS_{n}$ is even (odd). The second F-harmonic, $H_{F}^{(2)}(CS_{n})$, is formed appending $CS_{n}$ to $H_{F}^{(1)}(CS_{n})$ and changing the new $C$ to $R$ ($L$) is the $R$-parity of $H_{F}^{(1)}(CS_{n})$ is even (odd). The successive F-harmonics are constructed in the same way. [@Pastor96] The F-harmonics (or Fourier harmonics) were introduced by Romera, Pastor and Montoya in [@Pastor96] and have been widely employed by these authors in the study of dynamical systems [@Romera96; @Pastor02; @Pastor03]. It must be emphasized that, while those harmonics introduced by Metropolis, Stein and Stein [@MetropolisSteinStein73] give us the patterns corresponding to the Feigenbaum period doubling cascade, the F-harmonics are used to compute the patterns of the last appearance orbits (see Definition 10 below) in the current window. This property will be amply used in our proofs; they will be, together with the saddle-node bifurcation cascades, the main tools used in this paper. [Definition 9.]{} : A saddle-node bifurcation cascade is a sequence of saddle-node bifurcations in which the number of fixed points showing this kind of bifurcation is duplicated [@SanMartin07]. The successive elements of the sequence are given by an equation identical to the one that Feigenbaum found for a period doubling cascade [@Feigenbaum78; @Feigenbaum79]. This implies that both bifurcation cascades (Feigenbaum’s and saddle-node) scale in the same way. [Definition 10.]{} : We will consider a $q$-periodic window to be a first appearance window within the $p$-periodic one if it corresponds to values of parameter $\mu$ of expression (\[eq:logistic\]) smaller than any other $q$-periodic window inside the given $p$-periodic one. Similarly, we will define a last appearance $q$-periodic orbit (within the $p$-periodic one) as that having the larger $\mu$ parameter among the period-$q$ windows (of the $p$-periodic one)**.**\ A periodic window of period $q\cdot p$ will be called the $q$-periodic window inside the given $p$-periodic one, to stress the similarity of the structures in period-$p$ window with those in the canonical window. [Definition 11.]{} : Within the chaotic region of the logistic equation bifurcation diagram (to the right of the Myrberg-Feigenbaum point), there exist chaotic bands of $\mu$ values where the iterates of expression (\[eq:logistic\]) tend to be grouped in $2^{n}$ intervals (separated by $2^{n}-1$ “empty” intervals); we will refer to one of these bands as the $n$-th chaotic band. Following the same naming convention as in Definition 10, we will speak of the first and last appearance $q\cdot2^{n}$-periodic orbits inside a chaotic band, as those corresponding to the smaller and greater parameter $\mu$ values, respectively, within that $n$-chaotic band. In our proofs we will often use saddle-node cascades inside the canonical window. They can be worked out applying the following theorem, which is a particular case of that shown in [@SanMartin07]. [Theorem 1]{} : (*Saddle-Node bifurcation cascade* *in the canonical window*). *The sequence of the $p\cdot2^{n}$-periodic saddle-node orbit of a saddle-node bifurcation cascade starting from a $p$-periodic orbit in the canonical window is obtained with the following process:* 1. *Write the sequence of the orbit of the supercycle of $f^{2^{n}}$, that is $CP_{n}$ (see Definition 7).* 2. *Write consecutively $p$ times the sequence obtained in the former point “i”, getting a sequence like$$\underbrace{CP_{n}\dots CP_{n}}_{p\mbox{ times}}$$* 3. *Write the sequence of period-$p$ saddle-node orbit, that is, the sequence of the saddle-node orbit of $f^{p}$. The first point in the sequence must be the nearest saddle-node point to $C$.* 4. *If $n$ is odd then conjugate the letters obtained in point “iii” by means of $L\leftrightarrow R$. Bear in mind that $n\in{\cal Z}^{+}$.* 5. *Replace the $i$-th letter $C$ ($i=1,2,\dots p$) of the sequence obtained in “ii” by the $i$-th letter of the sequence obtained in “iv”.* \[sec:CofC\]Conjugation of cascades. ==================================== [Theorem 2.]{} : (*Conjugation of cascades* *in the canonical window*) *Let $S_{p\cdot2^{n}}$ be the pattern of the $p\cdot2^{n}$-periodic orbit of a saddle-node bifurcation cascade in the canonical window. If $CS_{p\cdot2^{n}-1}$ denotes the supercycle associated to $S_{p\cdot2^{n}}$ (see Definition 5) and $CP_{n}$ denotes the supercycle of the $n$-th pitchfork bifurcation (see Definition 7) then $$CS_{p\cdot2^{n}-1}=H_{F}^{(p-1)}(CP_{n})$$ That is, the supercycle of the $n$-th element of the cascade can be computed from the saddle-node bifurcation cascade (using Theorem 1) or from the Feigenbaum cascade (using F-harmonics, see Definition 8).* **Proof.** To prove this statement, we are going to compute the pattern of one of these $2^{n}p$-periodic orbits in two ways: 1) applying the **Theorem 1** *(Saddle-Node bifurcation cascade in the canonical window),* this way we get the $n$-th term of the saddle-node bifurcation cascade, and 2) computing the $(p-1)$-th F-harmonic to the $2^{n}$-periodic orbit generated after the $n$-th pitchfork bifurcation of the $1$-periodic orbit. In our proof we must consider separately the cases of even and odd $n$. **Let $n$ be even:** 1. Applying Theorem 1: Theorem 1 involves the following steps: - determine the last apperarance $p$-periodic orbit pattern, that is $CRL^{p-2}$ [@Post91]. Given that $n$ is even, this pattern will not be conjugated (i.e. $R$’s and $L$’s will not be swapped). - repeat $p$ times the sequence $CP_{n}$ which gives$$\underbrace{CP_{n}CP_{n}\dots CP_{n}}_{p\mbox{ times}}\label{eq:th1patt1}$$ - substitute the succesive $C$ symbols in the pattern (\[eq:th1patt1\]) by the succesive symbols in the pattern $CRL^{p-2}$, which gives the pattern sought for:$$CP_{n}RP_{n}(LP_{n})^{p-2}\label{eq:th1patt2}$$ where $p=3,\,4,\,5,\dots$ (there is no cascade for $p=2$). 2. Applying F-harmonics: The construction of the F-harmonics follows the next steps: - append $CP_{n}$ to the pattern $CP_{n}$, giving$$CP_{n}CP_{n}$$ - substitute the second $C$ by $R$ (or $L$) if the $R$-parity of $CP_{n}$ is even (or odd). Given that $n$ is assumed to be even, the $R$-parity of $CP_{n}$ is also even (see Note 1 in Appendix); as a consequence the first F-harmonic becomes$$H_{F}^{(1)}(CP_{n})=CP_{n}RP_{n}$$ - generate the $n$-th F-harmonic appending $CP_{n}$ to the $(n-1)$-th F-harmonic, and changing the second $C$ of the obtained sequence by $R$ (or $L$) if the $R$-parity of $H_{F}^{(n-1)}(CP_{n})$ is even (or odd). It is easy to see that $H_{F}^{(1)}(CP_{n})$ $R$-parity is odd. As a consequence, on appending the expression $CP_{n}$ to $H_{F}^{(1)}(CP_{n})$ in order to obtain the $H_{F}^{(2)}(CP_{n})$, the $C$ will be changed to $L$, thus conserving $R$-parity. The same will happen for the next F-harmonics thus getting$$H_{F}^{(p-1)}(CP_{n})=CP_{n}RP_{n}(LP_{n})^{p-2}\label{eq:th1patt3}$$ where $p=2,\,3,\,4,\dots$ (although in the cascade there is no $p=2$).   **Let $n$ be odd:** 1. Applying Theorem 1: In this case of $n$ odd, Theorem 1 would be equally applied but pattern $CRL^{p-2}$ must be conjugated to $CLR^{p-2}$. Thus the final pattern will be$$CP_{n}LP_{n}(RP_{n})^{p-2}\label{eq:th1patt4}$$ 2. Applying F-harmonics: Being $n$ odd, the $R$-parity of $CP_{n}$ is also odd (see Note 1 in Appendix), thus modifying the first F-harmonic to$$H_{F}^{(1)}(CP_{n})=CP_{n}LP_{n}$$ having even $R$-parity. As a consequence, on appending $CP_{n}$ to $H_{F}^{(1)}(CP_{n})$ to get to $H_{F}^{(2)}(CP_{n})$ the second $C$ will be replaced by $R$ thus conserving the $R$-parity. The result is$$H_{F}^{(p-1)}(CP_{n})=CP_{n}LP_{n}(RP_{n})^{p-2}$$ with $p=2,\,3,\,4,\,5,\dots$   In summary, the theorem is proven because both Theorem 1 and the F-harmonics give the same results $$\begin{array}{ll} CP_{n}RP_{n}(LP_{n})^{p-2} & \, p=3,4,5,\dots\mbox{ for }n\mbox{ even}\\ CP_{n}LP_{n}(RP_{n})^{p-2} & \, p=3,4,5,\dots\mbox{ for }n\mbox{ odd}\end{array}\label{eq:th1canonic}$$ The terms with $p=2$ in the F-harmonic have been disregarded, given that they correspond to a pitchfork bifurcation which, thus, cannot account for a saddle-node one.$\blacktriangleleft$ Examples. {#examples. .unnumbered} --------- As an example, we will show how to obtain the first saddle-node bifurcation orbits of the last appearance period-$5$ orbit in the canonical window, whose pattern is $CRL^{3}$. It will be enough to examine the cases $n=1,2$ to show how to perform the calculation in both cases when the $P_{n}$ orbit has to be conjugated or not. For $n=1$, that is odd $n$: - Using Theorem 1 we compute: - the corresponding $CP_{1}$ in the canonical window is simply $CR$ - the pattern $CRL^{3}$ will be conjugated because $n$ is odd, resulting: $CRL^{3}\rightarrow CLR^{3}$ - repeat $CR$ $p=5$ times: $CRCRCRCRCR$ - replace $C$’s in the latter sequence by $CLR^{3}$, resulting: $CRLR^{7}$ - Using Theorem 2 and F-harmonics: - $CP_{1}=CR$ - its first F-harmonic is $CRCR\rightarrow CRLR$ ($C$ changes to $L$ because $R$-parity of $CR$ is odd) - its second F-harmonic is $CRLRCR\rightarrow CRLR^{3}$ ($C$ changes to $R$ because $R$-parity of $CRLR$ is even) - and its fourth F-harmonic is $CRLR^{7}$ Both sequences coincide, as it was expected. For $n=2$, that is even $n$: - Using Theorem 1 we compute: - $CP_{2}=CRLR$ - the pattern $CRL^{3}$ will not be conjugated because $n$ is even - repeat $CRLR$ $p=5$ times: $CRLRCRLRCRLRCRLRCRLR$ - replace $C$’s in the latter sequence by $CRL^{3}$, resulting: $CRLR^{3}(LR)^{7}$ - Using Theorem 2 and F-harmonics: - $CP_{2}=CRLR$ - its first F-harmonic is $CRLRCRLR\rightarrow CRLR^{3}LR$ ($C$ changes to $R$ because $R$-parity of $CRLR$ is even) - and its fourth F-harmonic is $CRLR^{3}(LR)^{7}$ Both sequences also coincide. \[sec:1st-and-last\]Convergence of first and last appearance orbits. ==================================================================== [Theorem 3.]{} : *In the canonical window, the last appearance orbits within the $(n+1)$-chaotic band have the same accumulation point as the first appearance orbits within the $n$-chaotic band.* **Proof.** To prove the theorem, we need to compute the patterns of the first and last appearance orbits within an arbitrary chaotic band. These patterns are given by the saddle-node bifurcation cascades associated, respectively, to the first and last orbits in the $1$-chaotic band. The saddle-node cascades will be computed by means of Theorem 1. Given that this theorem treats the cases of even and odd $n$ differently, we will study these two cases separately. **Let $n$ be even:** 1. Computation of the first appearance orbit patterns within the $n$-chaotic band. As we have already indicated, we compute the requested pattern applying Theorem 1 following the next steps: - give the pattern of the first appearance superstable $p$-periodic orbit within the $1$-chaotic band. This pattern is $CRLR^{p-3}$, with $p=3,5,7,\dots$ - repeat $p$ times the pattern $CP_{n}$ (see Definition 7) as in the following$$\underbrace{CP_{n}CP_{n}\dots CP_{n}}_{p\mbox{ times}}\label{eq:th2patt1}$$ Given that we assume $n$ to be even, the symbols in the pattern $CRLR^{p-3}$ are not conjugated, and succesive $C$’s in expression (\[eq:th2patt1\]) are simply replaced by the successive $C$’s in the pattern $CRLR^{p-3}$. This substitution results in the sought for pattern$$CP_{n}RP_{n}LP_{n}(RP_{n})^{p-3}\label{eq:th2patt2}$$ where $p=3,\,5,\,7,\dots$ As expected, the pattern (\[eq:th2patt2\]) has $p\cdot2^{n}$ symbols. 2. Computation of the last appearance orbit patterns within the $(n+1)$-chaotic band. To apply Theorem 1 we accomplish the following steps: - give the pattern of the last appearance superstable $p$-periodic orbit within the $1$-chaotic band, that is [@Post91]$$CRL^{p-2}$$ with $p=3,\,4,\,5,\dots$ - repeat $p$ times the pattern $CP_{n+1}$. Given that $n$ is even, $CP_{n+1}=CP_{n}RP_{n}$ (see Note 1 in Appendix), this becomes$$\underbrace{CP_{n}RP_{n}CP_{n}RP_{n}\dots CP_{n}RP_{n}}_{p\mbox{ times}}\label{eq:th2patt3}$$ - given that $n+1$ is odd the pattern $CRL^{p-2}$ will be conjugated, resulting in $CLR^{p-2}$. Next, succesive $C$’s in expression (\[eq:th2patt3\]) are replaced by successive symbols in $CLR^{p-2}$. This substitution gives the sought for pattern, that is$$CP_{n}RP_{n}LP_{n}(RP_{n})^{2p-3}$$ where $p=3,\,4,\,5,\dots$   **Let $n$ be odd:** 1. Computation of the first appearance orbit patterns within the $n$-chaotic orbit. We take an identical approach to “i”, but given that $n$ is odd, the pattern $CRLR^{p-3}$ must be conjugated to $CLRL^{p-3}$. The consequence of this conjugation is that the following pattern is generated$$CP_{n}LP_{n}RP_{n}(LP_{n})^{p-3}$$ with $p=3,\,5,\,7,\dots$ 2. Computation of the last appearance orbit patterns within the $(n+1)$-chaotic orbit. We take an identical approach to “ii”, but given that $(n+1)$ is even, the pattern $CRL^{p-2}$ must not be conjugated. On the other hand, given that $n$ is odd, we have $CP_{n+1}=CP_{n}LP_{n}$ (see Note 1 in Appendix). As a consequence, the pattern $$CP_{n}LP_{n}RP_{n}(LP_{n})^{2p-3}$$ is obtained, with $p=3,\,4,\,5,\dots$   In summary we get: - the first appearance orbit patterns within the $n$-chaotic band are: $$\begin{array}{ll} CP_{n}RP_{n}LP_{n}(RP_{n})^{p-3} & \, p=3,5,7,\dots\mbox{ for }n\mbox{ even}\\ CP_{n}LP_{n}RP_{n}(LP_{n})^{p-3} & \, p=3,5,7,\dots\mbox{ for }n\mbox{ odd}\end{array}\label{eq:bandas-canonica_1}$$ - the last appearance orbit patterns within the $(n+1)$-chaotic band are:$$\begin{array}{ll} CP_{n}RP_{n}LP_{n}(RP_{n})^{2p-3} & \, p=3,4,5,\dots\mbox{ for }n\mbox{ even}\\ CP_{n}LP_{n}RP_{n}(LP_{n})^{2p-3} & \, p=3,4,5,\dots\mbox{ for }n\mbox{ odd}\end{array}\label{eq:bandas-canonica_2}$$ Myrberg’s formula [@Myrberg63] shows that the parameters corresponding to these orbits have the same accumulation point, and this proves the theorem. $\blacktriangleleft$ ![\[fig:FandL-app-orbits\]First and last appearance orbits approaching the band merging Misiurewicz point.](figs/FandL-app-orbits){width="100.00000%"} Consequences of Theorem 3 (Misiurewicz point cascade) ----------------------------------------------------- The limit $p\rightarrow\infty$ of expressions (\[eq:bandas-canonica\_1\]) and (\[eq:bandas-canonica\_2\]) corresponds to those points where the $n$ and $n+1$ bands merge. These are Misiurewicz points having a preperiod (within square brackets in the following expression) followed by a period that corresponds to an unstable periodic orbit:$$\begin{array}{rl} {}[CP_{n}RP_{n}LP_{n}]\; RP_{n} & \quad\mbox{ for }n\mbox{ even}\\ {}[CP_{n}LP_{n}RP_{n}]\; LP_{n} & \quad\mbox{ for }n\mbox{ odd}\end{array}\label{eq:misiurewicz-A}$$ These expressions were previously obtained in [@Romera96] using a different approach. The value of the parameter $\mu$ in expression (\[eq:logistic\]), where these points are localized, is the smallest solution of the equation $${\cal P}_{2^{n}+1}(\mu)={\cal P}_{2^{n-1}}(\mu)$$ being ${\cal P}_{n}(\mu)$ the dynamical polynomials defined in [@Myrberg63; @Zheng84], which can be obtained recursively (for equation (\[eq:logistic\])) as$${\cal P}_{0}=1/2,\quad{\cal P}_{n+1}=\mu{\cal P}_{n}(1-{\cal P}_{n})$$ Given that patterns (\[eq:misiurewicz-A\]) have been obtained as the limit of saddle-node orbit sequences elements (each of these elements belongs to a different saddle-node bifurcation cascade) the following corollary can be stated: [Corollary to Theorem 3.]{} : *Band merging points, whose patterns are given in (\[eq:misiurewicz-A\]) as the limit orbits of saddle-node orbit sequences, form a Misiurewicz point cascade. Furthermore, this Misiurewicz point cascade converges to the Myrberg-Feigenbaum point, driven by the behavior of the saddle-node bifurcation cascades, as is described in [@SanMartin07].* \[sec:generalization\]Generalization to any $q$-periodic window =============================================================== The selfsimilarity shown by the behaviors of the logistic equation is well known. It is revealed by the existence of periodic windows that mimic the different structures of the canonic window. Inside every subwindow the same bifurcation diagram structure exists, with its periodic and chaotic regions, intermittencies, and so on. In particular, every window shows in its beginning a periodic behavior, followed by a pitchfork bifurcation cascade; similarly every window has saddle-node bifurcation cascades, which reflect the ones existing in the canonical window. This circumstance is a clear suggestion to look for the relationship between the orbits of the pitchfork bifurcation cascade and those of the saddle-node bifurcation cascade, both belonging to a given $q$-periodic window. In this way we generalize the results obtained above, for the canonical window, to any arbitrary window. Nonetheless, although the results for the canonical window will become a particular case of the more general theorems, they will be used in the proofs of the general theorems. The Bifurcation Rigidity theorem [@Hunt99] allows us to approximately compute the parameter values for which the different orbits in a $p$-periodic window, mimicking those in the canonical one, take place. This is achieved through a linear mapping of the windows, the deeper the subwindow is located the more exact the result is. Nevertheless, this useful tool does not allow us to determine the patterns of the orbits and thus, we cannot establish a relationship among them. To obtain the patterns of the orbits, we need to compute that of the $p\cdot2^{n}$-periodic orbits in a saddle-node bifurcation cascade, located inside a $q$-periodic window. A similar problem (that of finding the pattern of a $q$-periodic orbit inside the $p\cdot2^{n}$-periodic window) was already addressed in [@SanMartin07] and is thus known to us. Merely exchanging the names of the windows involved, brings us the requested pattern. This result, using the compact notation introduced in section \[sec:Definitions\], is stated as follows. [Theorem 4.]{} : *Let $S_{p\cdot2^{n}}$ ($n=1,2,3\dots$) be the patterns of the orbits in a saddle-node bifurcation cascade located inside the canonical window. Let $S_{q}$ be the pattern of the $q$-periodic saddle-node orbit which marks the origin of the $q$-periodic window, and let $CS_{q-1}$ be the pattern of its associated supercycle. The pattern $S_{q\cdot p\cdot2^{n}}$ of the $p\cdot2^{n}$-periodic orbit in the saddle-node bifurcation cascade inside the $q$-periodic window considered is $$\left((CS_{q-1})^{p\cdot2^{n}}\vert S_{p\cdot2^{n}}\right)\qquad\mbox{for even }R\mbox{-parity of the }q\mbox{-periodic supercycle}$$ $$\left((CS_{q-1})^{p\cdot2^{n}}\vert\overline{S_{p\cdot2^{n}}}\right)\qquad\mbox{for odd }R\mbox{-parity of the }q\mbox{-periodic supercycle}$$* **Proof.** The proof is achieved by just exchanging $f^{q}\leftrightarrow f^{p\cdot2^{n}}$ in the original proof (see [@SanMartin07]). With this substitution, expression $$\left((CS_{q-1})^{p\cdot2^{n}}\vert S_{p\cdot2^{n}}\right)$$ is obtained, which is valid for $f^{q}$ having a maximum at $x=1/2$. This corresponds to an even $R$-parity of the $q$-periodic orbit (see Lemma 1 in the appendix). Should $f^{q}$ have a minimum at $x=1/2$, that is, for a $q$-periodic orbit having odd $R$-parity (see Lemma 2 in the appendix), it is enough to conjugate $S_{p\cdot2^{n}}\rightarrow\overline{S_{p\cdot2^{n}}}$ to get the correct result, that is$$\left((CS_{q-1})^{p\cdot2^{n}}\vert\overline{S_{p\cdot2^{n}}}\right)$$ $\blacktriangleleft$ As a consequence of replacing $f^{q}\leftrightarrow f^{p\cdot2^{n}}$, we observe that the orbit of $f^{p\cdot2^{n}}$ gets reproduced around every extremum of $f^{q}$ lying near the line $x_{i+1}=x_{i}$. This is shown in figure \[fig:3in5\], where the shape of the $f^{3}$ orbit (upper panel in figure \[fig:3in5\]) can be identified in those extremes of $f^{5}$ (right panel in figure \[fig:3in5\]) near the diagonal $x_{i+1}=x_{i}$. In figure \[fig:3x2in5\] the same situation is shown, this time for the supercycle of $f^{5\cdot3\cdot2}$ (saddle-node bifurcation of that of $f^{5\cdot3}$); in this case it is $f^{3\cdot2}$ that gets reproduced in each maximum of $f^{5}$. ![\[fig:3in5\]Graph of $f^{5\cdot3}$ (the $f^{3}$ periodic orbit inside the period-$5$ window). It can be seen how in one of the extrema of $f^{5}$ (highlighted in the right panel plot) the shape of $f^{3}$ is reproduced.](figs/3in5){width="70.00000%"} ![\[fig:3x2in5\]Graph of $f^{5\cdot3\cdot2}$ (the $f^{3\cdot2}$ saddle-node bifurcation of the $f^{3}$ periodic orbit inside the period-$5$ window, which is located in the $1$-chaotic band of the period-$5$ window). It can be seen how in one of the extrema of $f^{5}$ (drawn in the right panel and encircled by the ellipse) the shape of $f^{3\cdot2}$ is reproduced.](figs/3x2in5){width="60.00000%"} Let us emphasize that, what is being obtained is the pattern of a $p\cdot2^{n}$-periodic saddle-node orbit located inside a $q$-periodic window. Given that this $q$-periodic window has its origin at a $q$-periodic saddle-node orbit, the resulting pattern will be that of a $q\cdot p\cdot2^{n}$-periodic saddle-node orbit. Moreover, the geometrical construction used in the proof, allows us to show the geometrical meaning of the DGP formula, as can be seen in what follows. The substitution of the first symbol in the pattern $S_{q\cdot p\cdot2^{n}}$ by a $C$ is enough to obtain its associated supercycle, that we denote by $CS_{q\cdot p\cdot2^{n}-1}$ (see Definition 5), whose geometrical interpretation was given above. Given that this same supercycle can be also constructed applying DGP rule to compute the product of the orbits $CS_{p\cdot2^{n}-1}$ and $CS_{q-1}$, what is obtained is a geometrical interpretation of the DGP rule. We thus conclude (see figure \[fig:p2n-in-e\] for a graphical sketch) that: 1. The composition of the supercycles $CS_{p\cdot2^{n}-1}$ and $CS_{q-1}$ , according to DGP rule, represents the supercycle associated to the $p\cdot2^{n}$-periodic orbit, inside the $q$-periodic window. That is so because, in the geometric proof, we have forced the shape of a $p\cdot2^{n}$-periodic saddle-node orbit to appear around the extrema of a saddle-node $q$-periodic orbit. This is shown in figures \[fig:3in5\] and \[fig:3x2in5\]. 2. However, we could have forced the reverse situation, that is, to reproduce the shape of the $q$-periodic orbit around the extrema of a $p\cdot2^{n}$-periodic orbit, arising after $n$ pitchfork bifurcations of a $p$-periodic saddle-node orbit. In that case, we would be focusing on the $n$-th pitchfork bifurcation of a $p$-periodic orbit, localized inside the $q$-periodic window. This second case was already known and is not a conclusion of our geometrical proof. ![\[fig:p2n-in-e\]Example of the application of Theorem 5, within the period-$3$ window to find the last appearance orbits in the $1$- and $2$-chaotic bands, applying F-harmonics to the orbits in the Feigenbaum cascade. Also shown, is how the Feigenbaum cascade is computed from Metropolis-Stein-Stein harmonics [@MetropolisSteinStein73], and the saddle-node bifurcation cascade is obtained using Theorem 4.](figs/p2n-in-e){width="80.00000%"} Theorem 4 allows us to state Theorem 2 and Theorem 3 generalized to subwindows inside the canonical one. The proofs are given in the next two subsections. \[sub:CoC-general\]Conjugation of cascades in an arbitrary window. ------------------------------------------------------------------ [Theorem 5]{} : (*Conjugation of cascades in an arbitrary window*). *Let $S_{q\cdot p\cdot2^{n}}$ be the pattern of the $p\cdot2^{n}$-periodic orbit in the saddle-node bifurcation cascade inside the $q$-periodic window, such that $p$ is a last appearance orbit. If $CS_{q\cdot p\cdot2^{n}-1}$ denotes the supercycle associated to $S_{q\cdot p\cdot2^{n}}$ (see Definition 5) and $CP_{n,q}$ denotes the $n$-th pitchfork bifurcation supercycle of the $q$-periodic orbit (see Definition 7) then$$CS_{q\cdot p\cdot2^{n}-1}=H_{F}^{(p-1)}(CP_{n,q})$$* What the theorem says is that, inside an arbitrary $q$-periodic window, superstable orbit patterns of a saddle-node bifurcation cascade, which is associated to a last appearance orbit, coincide with orbit patterns generated by appying F-harmonics to Feigenbaum cascade supercycles. **Proof.** Let us consider a $q$-periodic window, whose origin is in a $q$-periodic saddle-node orbit. This $q$-periodic window needs not to be a primary window, any window located inside any other will do. Let us also consider the last appearance $p$-periodic orbit inside the chosen $q$-periodic window, where $p=3,\,4,\,5,\dots$. This $p$-periodic orbit has a saddle-node bifurcation cascade associated to it, whose orbits have periods $p\cdot2^{n}$, $n=0,\,1,\,2,\dots$, and are all located inside the $q$-periodic window (that is, they are really $q\cdot p\cdot2^{n}$-periodic orbits). We want to show that the $p\cdot2^{n}$-periodic saddle-node orbit supercycle inside the $q$-periodic window can be obtained equivalently by applying Theorem 4 or computing the $(p-1)$-th F-harmonic of the $n$-th pitchfork bifurcation of the $q$-periodic orbit.   **Let the $R$-parity of the $q$-periodic supercycle pattern be even.** 1. Applying Theorem 4. This theorem (see section \[sec:generalization\]) says that the pattern $CS_{q\cdot p\cdot2^{n}-1}$ is given by$$\left((CS_{q-1})^{p\cdot2^{n}}\vert CS_{p\cdot2^{n}-1}\right)\mbox{ for even }R\mbox{-parity of the }q\mbox{-periodic orbit}$$ On the other hand, the pattern $CS_{p\cdot2^{n}-1}$ is given by expression (\[eq:th1canonic\]). 1. Let $n$ be odd. Then $CS_{p\cdot2^{n}-1}=CP_{n}LP_{n}(RP_{n})^{p-2}$ according to expression (\[eq:th1canonic\]). Then we perform the following computation$$\begin{array}{rl} \left((CS_{q-1})^{p\cdot2^{n}}\vert CS_{p\cdot2^{n}-1}\right)= & \left((CS_{q-1})^{p\cdot2^{n}}\vert CP_{n}LP_{n}(RP_{n})^{p-2}\right)\\ = & \left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)\left((CS_{q-1})^{2^{n}}\vert LP_{n}\right)\\ & \left((CS_{q-1})^{2^{n}}\vert RP_{n}\right)^{p-2}\end{array}$$ 2. Let $n$ be even. Then $CS_{p\cdot2^{n}-1}=CP_{n}RP_{n}(LP_{n})^{p-2}$, with $p=3,\,4,\,5,\dots$ according to expression (\[eq:th1canonic\]). In this case the computation goes as follows$$\begin{array}{rl} \left((CS_{q-1})^{p\cdot2^{n}}\vert CS_{p\cdot2^{n}-1}\right)= & \left((CS_{q-1})^{p\cdot2^{n}}\vert CP_{n}RP_{n}(LP_{n})^{p-2}\right)\\ = & \left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)\left((CS_{q-1})^{2^{n}}\vert RP_{n}\right)\\ & \left((CS_{q-1})^{2^{n}}\vert LP_{n}\right)^{p-2}\end{array}$$ 2. Applying F-harmonics. Let us now apply F-harmonics to the $n$-th pitchfork bifurcation of the $q$-periodic orbit. The $n$-th pitchfork bifurcation of $CS_{q-1}$ , that is $CP_{n,q}$, is given by $$\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)$$ and has the same $R$-parity as $CP_{n}$, according to Lemma 2 in the Appendix. 1. Let $n$ be odd, then the $R$-parity of $CP_{n}$ is odd and, hence, the $R$-parity of $\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)$ is also odd (for $n\geq1$). The F-harmonic is computed as$$\begin{array}{l} H_{F}^{(1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]=\\ =\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\end{array}$$ where the $C$’s in the last bracket have been changed to $L$’s because of the assumed odd $R$-parity. As a consequence, $H_{F}^{(1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]$ has even $R$-parity. To compute the second F-harmonic $H_{F}^{(2)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]$, those $C$’s will be replaced by $R$ thus keeping the same $R$-parity. Hence, the result for the $(p-1)$-th F-harmonic is $$\begin{array}{l} H_{F}^{(p-1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]=\\ =\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]^{p-2}\end{array}$$ 2. Let $n$ be even, thus the $R$-parity of $CP_{n}$ is even as well and the $R$-parity of $\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)$ is also even. The first F-harmonic of $\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)$ is$$H_{F}^{(1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]=\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]$$ It is easy to see that $H_{F}^{(1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]$ has odd $R$-parity. As a consequence, on appending the expression $\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]$ to $H_{F}^{(1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]$, in order to obtain the second F-harmonic, the $C$ will be switched to $L$, keeping the same $R$-parity. The same will happen for the next F-harmonics, thus giving$$\begin{array}{rl} H_{F}^{(p-1)}\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]= & \left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\\ & \left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]^{p-2}\end{array}$$ We have thus proven that, for even as well as for odd $n$, the same results are achieved both ways. **Let the $R$-parity of the $q$-periodic supercycle pattern be odd.** In this case we will have to compute $$\left((CS_{q-1})^{p\cdot2^{n}}\vert\overline{CS_{p\cdot2^{n}-1}}\right)$$ using Theorem 4. On the other hand, the $n$-th pitchfork bifurcation will be $$\left((CS_{q-1})^{2^{n}}\vert\overline{CP_{n}}\right)$$ and it has the same $R$-parity as $\overline{CP_{n}}$ according to Lemma 2 in the Appendix. Furthermore $\overline{CP_{n}}$ has odd (even) $R$-parity when $n$ is even (odd) as indicated in Note 1. Therefore, when F-harmonics are calculated, the same results will be obtained.$\blacktriangleleft$ \[sub:1st-and-last-general\]Convergence of first and last appearance orbits in an arbitrary window. --------------------------------------------------------------------------------------------------- [Theorem 6:]{} : *Inside any $q$-periodic window, the last apperance orbits within the $(n+1)$-chaotic band have the same accumulation point as the first appearance orbits within the $n$-chaotic band.* **Proof.** Once again we study separately the cases of even and odd $R$-parity: **Let the $R$-parity of the $q$-periodic supercycle pattern be even**. We want to compute the pattern of the supercycles of the $p\cdot2^{n}$ first appearance and $p\cdot2^{n+1}$ last appearance saddle-node orbits within the $q$-window. These orbits are located in the $n$- and $(n+1)$-chaotic bands of that window, respectively. In order to do so, we will apply Theorem 4, and hence we need to know the patterns of the $p\cdot2^{n}$ first appearance and $p\cdot2^{n+1}$ last appearance saddle-node orbits within the canonical window, which are located in the $n$- and $(n+1)$-chaotic bands, respectively. 1. Let $n$ be even. The pattern of the supercycle associated to the first appearance $p\cdot2^{n}$ saddle-node orbit, in the canonical window, is (see \[eq:bandas-canonica\_1\]) $$CS_{p\cdot2^{n}-1}=CP_{n}RP_{n}LP_{n}(RP_{n})^{p-3}\label{eq:th2Gpatt1}$$ where $p=3,\,5,\,7,\,\dots$. Also, the pattern of the supercycle of the last appearance $p\cdot2^{n+1}$ saddle-node orbit, in the canonical window, is given by (see \[eq:bandas-canonica\_2\])$$CS_{p\cdot2^{n+1}-1}=CP_{n}RP_{n}LP_{n}(RP_{n})^{2\cdot p-3}\label{eq:th2Gpatt2}$$ - The pattern of the supercycle associated to the $p\cdot2^{n}$ first appearance saddle-node orbit, within the $q$-window, is computed by applying Theorem 4 as follows: 1. Repeat $p\cdot2^{n}$ times the pattern $CS_{q-1}$ $$(CS_{q-1})^{p\cdot2^{n}}$$ 2. Conjugate it according to (\[eq:th2Gpatt1\])$$\begin{array}{r} \left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\\ \left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]^{p-3}\end{array}\label{eq:th2Gpatt3}$$ - The pattern of the supercycle associated to the $p\cdot2^{n+1}$ last appearance saddle-node orbit, within the $q$-window, by applying Theorem 4 is computed as: 1. Repeat $p\cdot2^{n+1}$ times the pattern $CS_{q-1}$ $$(CS_{q-1})^{p\cdot2^{n+1}}$$ 2. Conjugate it according to (\[eq:th2Gpatt2\])$$\begin{array}{r} \left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\\ \left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]^{2p-3}\end{array}\label{eq:th2Gpatt4}$$ For $p\rightarrow\infty$, the patterns (\[eq:th2Gpatt3\]) and (\[eq:th2Gpatt4\]) converge to the same Misiurewicz point with preperiod$$\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\label{eq:misiurewicz-even-pre}$$ and period$$\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\label{eq:misiurewicz-even-per}$$ 2. Let $n$ be odd. The pattern of the supercycle associated to the first appearance $p\cdot2^{n}$ saddle-node orbit in the canonical window is (see \[eq:bandas-canonica\_1\]) $$CS_{p\cdot2^{n}-1}=CP_{n}LP_{n}RP_{n}(LP_{n})^{p-3}\label{eq:th2Gpatt5}$$ where $p=3,\,5,\,7,\,\dots$. Also, the pattern of the supercycle associated to the last appearance $p\cdot2^{n+1}$ saddle-node orbit in the canonical window is given by (see \[eq:bandas-canonica\_2\])$$CS_{p\cdot2^{n+1}-1}=CP_{n}LP_{n}RP_{n}(LP_{n})^{2p-3}\label{eq:th2Gpatt6}$$ - Pattern of the supercycle associated to the $p\cdot2^{n}$ first appearance saddle-node orbit, within the $q$-window, by applying Theorem 4 is computed as: After repeating $p\cdot2^{n}$ times the pattern $CS_{q-1}$ and conjugating it according to (\[eq:th2Gpatt5\]) we obtain$$\begin{array}{r} \left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\\ \left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]^{p-3}\end{array}\label{eq:th2Gpatt7}$$ - Pattern of the supercycle associated to the $p\cdot2^{n+1}$ last appearance saddle-node orbit, within the $q$-window, by applying Theorem 4 is computed as: After repeating $p\cdot2^{n+1}$ times the pattern $CS_{q-1}$ and conjugating it according to (\[eq:th2Gpatt6\]) we obtain$$\begin{array}{r} \left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\\ \left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]^{2p-3}\end{array}\label{eq:th2Gpatt8}$$ For $p\rightarrow\infty$, the patterns (\[eq:th2Gpatt7\]) and (\[eq:th2Gpatt8\]) converge to the same Misiurewicz point with preperiod$$\left[(CS_{q-1})^{2^{n}}\vert CP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\left[(CS_{q-1})^{2^{n}}\vert RP_{n}\right]\label{eq:misiurewicz-odd-pre}$$ and period$$\left[(CS_{q-1})^{2^{n}}\vert LP_{n}\right]\label{eq:misiurewicz-odd-per}$$ $\blacktriangleleft$ **Let the $R$-parity of the $q$-periodic supercycle pattern be odd**. Given that the $R$-parity of $CS_{q-1}$ is odd, the conjugated patterns $\overline{CS_{p\cdot2^{n}-1}}$ and $\overline{CS_{p\cdot2^{n+1}-1}}$ will be used on applying Theorem 4. Hence, the results obtained will be the same. Corollary to Theorem 3 is now rewritten, valid for any periodic window in the bifurcation diagram. [Corollary to Theorem 6.]{} : *Band merging points in a $q$-periodic window, whose patterns are given in (\[eq:misiurewicz-even-pre\])-(\[eq:misiurewicz-even-per\]) and (\[eq:misiurewicz-odd-pre\])-(\[eq:misiurewicz-odd-per\]) as the limit orbits of saddle-node orbit sequences, form a Misiurewicz point cascade. Furthermore, this Misiurewicz point cascade converges to the Myrberg-Feigenbaum point of that window, driven by the behavior of the saddle-node bifurcation cascades, described in [@SanMartin07].* It is important to notice that the symbolic sequences of the Misiurewicz, given by (\[eq:misiurewicz-even-pre\]-\[eq:misiurewicz-even-per\]) and (\[eq:misiurewicz-odd-pre\]-\[eq:misiurewicz-odd-per\]), show the band merging points in any window, that is, not only those (already known) corresponding to the canonic window, but also those located inside every subwindow. \[sec:disc-and-conclusions\]Discussion and conclusions ====================================================== It has been shown, in the preceding sections, how the patterns of those periodic orbits of the Feigenbaum cascade determine the patterns of the orbits of the saddle-node bifurcation cascades. The accumulation points corresponding to these latter patterns are the band merging points, that are also Misiurewicz points. The sequence of these Misiurewicz points converges to the Myrberg-Feigenbaum point, and determines the chaotic band structure of the bifurcation diagram. This chain of relations shows how the periodic part of a window determines its chaotic part: the hyperbolic components of the set, determine the non-hyperbolic ones. The relationship between the Feigenbaum cascade and the saddle-node bifurcation cascade could be expected, because both cascades share the same scaling law found by Feigenbaum [@Feigenbaum78]. On the other hand, this scaling is the direct consequence of the renormalization equation. This equation has only one fixed point in the functional space, whose stable manifold is one-dimensional. Thus, Feigenbaum’s cascade and saddle-node cascade can but pertain to the same scaling universality [@deMelo93]. The problem of localizing the chaotic band merging points, as well as the convergence of that sequence of points, is a problem largely addressed numerically by many authors as, for instance, by Hao and Zhang [@Hao82] in their “Bruselator oscillator” model. Both problems can be approached using the Corollary to Theorem 6, stated above, and the saddle-node cascade convergency, described in [@SanMartin07]. From the symbolic sequences of a pitchfork orbit in the Feigenbaum’s cascade, the symbolic sequences of the supercycles of the saddle-node orbits can be computed. After which, Myrberg’s formula [@Myrberg63] can be used to compute the value of the parameter giving rise to those orbits. This circumstance can be very useful to the experimenter, because once the Feigenbaum cascade has been observed, the corresponding saddle-node bifurcations can be spotted from it. Given that many dynamical systems, in particular lasers, present period doubling cascades, it easy for the experimenter to explicitly localize the saddle-node bifurcation cascades from the experimentally obtained return-map [@Pisarchik00] . As a result, it is also possible to spot the intermittency cascades associated with that saddle-node bifurcation cascade. We must take into account that without an exact knowledge of the parameter values, it is practically impossible to localize these cascades, because of the geometrical stretching of those regions under saddle-node bifurcations. We want to point out other possible uses from an experimental point of view. A relevant fact for the experimenter is the possibility to assess the localization of the Misiurewicz points which separate chaotic bands within any window. In this way, parameter values can be bound to the particular region of interest where the sought phenomena take place. There is yet another interesting point. In all the results just obtained, orbit sequences have been considered. Each of them had a symbolic sequence associated to it. These symbolic sequences on their own, can be used to assess the underlying universal scaling laws [@LopezRuiz06]. This can be reflected, for instance, in the fact that the previously signaled Misiurewicz points scale according to the same Feigenbaum scaling scheme. All that has been shown in this paper is immediately extendible to unimodal maps with extrema of the form $x^{2n}$ under very general considerations. Appendix {#appendix .unnumbered} ======== [Lema 1]{} : *Let $f(x;\,\mu)$ be an unimodal $C^{2}$ class map of the interval $[a,\, b]$ into itself, such that its critical point $x_{c}\in(a,\, b)$ is a maximum (minimum). If for $\mu=\mu_{0}$, there exists a $q$-periodic supercycle having a pattern $CS_{q-1}$, then $f^{q}(x_{c};\,\mu_{0})$ has a maximum (minimum) when the $R$-parity of $CS_{q-1}$ is even, while it has a minimum (maximum) in case of odd $R$-parity of $CS_{q-1}$.* [Proof:]{} : Let us consider a value of $\mu$ in a small neighborhood of $\mu_{0}$, such that $f^{q}(x;\,\mu)$ has a fixed point at some $x=x_{c}+\varepsilon$, for some arbitrarily small $\varepsilon>0$. At $x=x_{c}$, $f^{q}(x_{c};\,\mu)$ has a critical point, given that $f(x_{c};\,\mu)$ has a critical point, for $x=x_{c}$. To determine whether it is a maximum or a minimum, it is enough to see whether, in a neighborhood of $x_{c}+\varepsilon$, the iterated function $f^{q}(x_{c}+\varepsilon;\,\mu)$ is increasing (corresponding to $f$ having a minimum at $x_{c}$) or decreasing (corresponding to a maximum). That is, whether $\left[f^{q}\right]^{\prime}(x_{c}+\varepsilon;\,\mu)$ is positive or negative. $$\left[f^{q}\right]^{\prime}(x_{c}+\varepsilon;\,\mu)=\prod_{i=1}^{q}f^{\prime}(x_{i})$$ where the $x_{i}$ represent each of the $q$ iterates of the function $f$ period, starting at $x_{1}=x_{c}+\varepsilon$. It turns out that$$\mbox{sign}\left[f^{q}\right]^{\prime}(x_{c}+\varepsilon;\,\mu)=\prod_{i=1}^{q}\mbox{sign }f^{\prime}(x_{i};\,\mu)$$ Let us assume $f$ to have a maximum at $x_{c}$, that is, to be an increasing function in $(a,\, x_{c})$ (i.e. with positive derivative) and a decreasing function in $(x_{c},\, b)$ (i.e. with negative derivative). When $f^{q}$ has a minimum at $x_{c}$ (i.e. $\mbox{sign}\left[f^{q}\right]^{\prime}(x_{c}+\varepsilon;\,\mu)$ is positive) the number of $x_{i}$ laying in $(x_{c},\, b)$ will be even, therefore, the number of $R$-symbols in the pattern of the orbit will be even. Conversely, when $f^{q}$ has a maximum at $x_{c}$, the number of $R$-symbols in the pattern will be odd. Considering $x_{1}=x_{c}+\varepsilon\rightarrow x_{c}$, we get back the superstable orbit, $CS_{q-1}$. This represents the deletion of the first $R$ of the pattern, corresponding to $x_{1}$. Based on the continuous variation of the function with $\mu$, we find that for an even $R$-parity of the supercycle pattern, a maximum will be present at $x=x_{c}$, while for an odd $R$-parity, there will be a minimum. $\blacktriangleleft$ [Note 1:]{} : We need to know the $R$-parity of $CP_{n}$ in order to compute F-harmonics of the pitchfork orbits in the proof. It is clear that for the period-$1$ orbit supercycle, whose sequence is $C$, its first pitchfork has the sequence $CR$ and odd $R$-parity; the orbit originated by the second pitchfork has sequence $CR[C]R\rightarrow CRLR$, as given by the Metropolis-Stein-Stein harmonic [@MetropolisSteinStein73], having even $R$-parity; the iteration of these harmonics gives odd $R$-parities for those $CP_{n}$ with $n$ odd, while those $CP_{n}$ with even $n$, have even $R$-parity. As a consequence of this, it is easy to see that $\overline{CP_{n}}$ has odd (even) $R$-parity whenever $n$ is even (odd). [Lemma 2]{} : *The $n$-th pitchfork bifurcation of $CS_{q-1}$, that is, $CP_{n,q}$ is given by$$\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)\mbox{ for even }R\mbox{-parity of }CS_{q-1}$$ $$\left((CS_{q-1})^{2^{n}}\vert\overline{CP_{n}}\right)\mbox{ for odd }R\mbox{-parity of }CS_{q-1}$$ Furthermore, $CP_{n,q}$ has the same $R$-parity as $CP_{n}$ (respectively, $\overline{CP_{n}}$) for even (respectively, odd) $R$-parity of $CS_{q-1}$.* [Proof.]{} :   **Let the $R$-parity of $CS_{q-1}$ be even.** When the MSS composition rule [@MetropolisSteinStein73] is applied, the sequence $S_{q-1}$ does not play any role, because it does not change the $R$-parity of the new pattern. Only the $C$’s formerly changed have a role. Therefore we work as if with the $n$-th pitchfork bifurcation of the period-$1$ orbit. **Let the $R$-parity of $CS_{q-1}$ be odd.** The proof is similar to the former point, but the sequence $S_{q-1}$ changes the $R$-parity of the new pattern when the composition rule is applied. Therefore we work as if with the conjugated of the $n$-th pitchfork bifurcation of the period-$1$ orbit. Whatever the $R$-parity of $S_{q-1}$ is, the $(CS_{q-1})^{2^{n}}$ sequence always has an even number of $S_{q-1}$ sequences, therefore this even number of $S_{q-1}$ gives an even $R$-parity. So, the $R$-parity of $\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)$ comes from the $C$’s inserted from $CP_{n}$, that is, the $R$-parity of $\left((CS_{q-1})^{2^{n}}\vert CP_{n}\right)$ is that of $CP_{n}$. In the same way it is proven that $\left((CS_{q-1})^{2^{n}}\vert\overline{CP_{n}}\right)$ has the same $R$-parity as $\overline{CP_{n}}$. [10]{} M. J. Feigenbaum. 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--- abstract: | *The notion of vertex sparsification (in particular cut-sparsification) is introduced in [@M], where it was shown that for any graph $G = (V, E)$ and a subset of $k$ terminals $K \subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ *on just the terminal set* so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. Then approximation algorithms can be run directly on $H$ as a proxy for running on $G$, yielding approximation guarantees independent of the size of the graph. In this work, we consider how well cuts in the sparsifier $H$ can approximate the minimum cuts in $G$, and whether algorithms that use such reductions need to incur a multiplicative penalty in the approximation guarantee depending on the quality of the sparsifier.* *We give the first super-constant lower bounds for how well a cut-sparsifier $H$ can simultaneously approximate all minimum cuts in $G$. We prove a lower bound of $\Omega(\log^{1/4} k)$ – this is polynomially-related to the known upper bound of $O(\log k/\log \log k)$. This is an exponential improvement on the $\Omega(\log \log k)$ bound given in [@LM] which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers.* *Despite this negative result, we show that for many natural problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the $0$-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy. Using this, we obtain optimal $O(\log k)$-competitive Steiner oblivious routing schemes, which generalize the results in [@R]. We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most $O(\log k)$ times the integrality gap restricted to trees. This result helps to explain the ubiquity of the $O(\log k)$ guarantees for such problems. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results – this was previously open. Our algorithm makes novel use of Earth-mover constraints.* author: - 'Moses Charikar [^1]' - 'Tom Leighton [^2]' - 'Shi Li [^3]' - 'Ankur Moitra [^4]' bibliography: - 'latex4.bib' nocite: '[@JLS]' title: Vertex Sparsifiers and Abstract Rounding Algorithms --- =10000 =10000 Introduction ============ Background ---------- The notion of vertex sparsification (in particular cut-sparsification) is introduced in [@M]: Given a graph $G = (V, E)$ and a subset of terminals $K \subset V$, the goal is to construct a graph $H = (K, E_H)$ *on just the terminal set* so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. If for all cuts $(A, K - A)$, the the value of the cut in $H$ is at least the value of the corresponding minimum cut in $G$ and is at most $\alpha$ times this value, then we call $H$ a cut-sparsifier of quality $\alpha$. The motivation for considering such questions is in obtaining approximation algorithms with guarantees that are independent of the size of the graph. For many graph partitioning and multicommodity flow questions, the value of the optimum solution can be approximated given just the values of the minimum cut separating $A$ from $K -A$ in $G$ (for *every* $A \subset K$). As a result the value of the optimum solution is approximately preserved, when mapping the optimization problem to $H$. So approximation algorithms can be run on $H$ as a proxy for running directly on $G$, and because the size (number of nodes) of $H$ is $|K|$, any approximation algorithm that achieves a $poly(\log |V|)$-approximation guarantee in general will achieve a $poly(\log |K|)$ approximation guarantee when run on $H$ (provided that the quality $\alpha$ is also $poly(\log |K|)$). Feasible solutions in $H$ can also be mapped back to feasible solutions in $G$ for many of these problems, so polynomial time constructions for good cut-sparsifiers yield black box techniques for designing approximation algorithms with guarantees $poly(\log |K|)$ (and independent of the size of the graph). In addition to being useful for designing approximation algorithms with improved guarantees, the notion of cut-sparsification is also a natural generalization of many methods in combinatorial optimization that attempt to preserve certain cuts in $G$ (as opposed to all minimum cuts) in a smaller graph $H$ - for example Gomory-Hu Trees, and Mader’s Theorem. Here we consider a number of questions related to cut-sparsification: 1. Is there a super-constant lower bound on the quality of cut-sparsifiers? Do the best (or even near-best) cut-sparsifiers necessarily result from (a distribution on) contractions? 2. Do we really need to pay a price (in the approximation guarantee) when applying vertex sparsification to an optimization problem? 3. Can we construct (in polynomial time) cut-sparsifiers with quality as good as the current best *existential* results? We resolve all of these questions in this paper. In the preceding subsections, we will describe what is currently known about each of these questions, our results, and our techniques. [^5] Super-Constant Lower Bounds and Separations ------------------------------------------- In [@M], it is proven that in general there are always cut-sparsifiers $H$ of quality at most $O(\log k / \log \log k)$. In fact, if $G$ excludes any fixed minor then this bound improves to $O(1)$. Yet prior to this work, no super-constant lower bound was known for the quality of cut-sparsifiers in general. We prove  \[thm:lower\] There is an infinite family of graphs that admits no cut-sparsifiers of quality better than $\Omega(\log^{1/4} k)$. Some results are known in more general settings. In particular, one could require that the graph $H$ not only approximately preserve minimum cuts but also approximately preserve the congestion of all multicommodity flows (with demands endpoints restricted to be in the terminal set). This notion of vertex-sparsification is referred to as flow-sparsification (see [@LM]) and admits a similar definition of quality. [@LM] gives a lower bound of $\Omega(\log \log k)$ for the quality of flow-sparsifiers. However, this does not apply to cut sparsifiers and in fact, for the example given in [@LM], there is an $O(1)$-quality cut-sparsifier! Additionally, there are examples in which cuts can be preserved within a constant factor, yet flows cannot: Benczur and Karger [@BK] proved that given any graph on $n$ nodes, there is a sparse (weighted) graph $G'$ that approximate all cuts in $G$ within a multiplicative $(1 + \epsilon)$ factor, but one provably cannot preserve the congestion of all multicommodity flows within a factor better than $\Omega(\frac{\log n}{\log \log n})$ on a sparse graph (consider the complete graph $K_n$). So here the limits of sparsification are much different for cuts than for flows. In this paper, we give a super-constant lower bound on the quality of cut-sparsifiers in general and in fact this implies a stronger lower bound than is given in [@LM]. Our bound is polynomially related to the current best upper-bound, which is $O(\log k / \log \log k)$. We note that the current best upper bound is actually a reduction from the upper bound on the integrality gap of a particular LP relaxation for the $0$-extension problem [@CKR], [@FHRT]. The integrality gap of this LP relaxation is known to be $\Omega(\sqrt{\log k})$. Yet, the best lower bound we are able to obtain here is $\Omega(\log^{1/4} k)$. This leads us to our next question: Do integrality gaps for the $0$-extension LP immediately imply lower bounds for cut-sparsification? This question, as we will see, is essentially equivalent to the question of whether or not the best cut-sparsifiers necessarily come from a distribution on contractions. Lower bounds on the quality of cut-sparsifiers (in this paper) and flow-sparsifiers ([@LM]) are substantially more complicated than integrality gap examples for the $0$-extension LP relaxation. If the best cut-sparsifiers or flow-sparsifiers were actually always generated from some distribution on contractions in the original graph via strong duality (see Section $3$), any integrality gap would immediately imply a lower bound for cut-sparsificatin or flow-sparsification. But as we demonstrate here, this is not the case:  \[thm:sep\] There is an infinite family of graphs so that the ratio of the best quality cut-sparsifier to the best quality cut-sparsifier that can be achieved through a distribution on contractions is $o(1) = O(\frac{\log\log\log\log k}{\log^2 \log \log k})$ We also note that in order to prove this result we establish a somewhat surprising connection between cut-sparsification and the harmonic analysis of Boolean functions. The particular cut-sparsifier that we construct in order to prove this result is inspired by the noise stability operator, and as a result, we can use tools from harmonic analysis (Bourgain’s Junta Theorem [@Bou] and the Hypercontractive Inequality [@Bon], [@Bec]) to analyze the quality of the cut-sparsifier. Casting this question of bounding the quality as a question in harmonic analysis allows us to reason about many cuts simultaneously without worrying about the messy details of the combinatorics. Abstract Integrality Gaps and Rounding Algorithms ------------------------------------------------- As described earlier, running an approximation algorithm on the sparsifier $H = (K, E_H)$ as a proxy for the graph $G = (V, E)$ pays an additional price in the approximation guarantee that corresponds to how well $H$ approximates $G$. Here we consider the question of whether this loss can be avoided. As a motivating example, consider the problem of Steiner oblivious routing [@M]. Previous techniques for constructing Steiner oblivious routing schemes [@M], [@LM] first construct a flow-sparsifier $H$ for $G$, construct an oblivious routing scheme in $H$ and then map this back to a Steiner oblivious routing scheme in $G$. Any such approach must pay a price in the competitive ratio, and cannot achieve an $O(\log k)$-competitive guarantee because (for example) expanders do not admit constant factor flow-sparsifiers [@LM]. So black box reductions pay a price in the competitive ratio, yet here we present a technique for *combining* the flow-sparsification techniques in [@LM] and the oblivious routing constructions in [@R] into a single step, and we prove that there are $O(\log k)$-competitive Steiner oblivious routing schemes, which is optimal. This result is a corollary of a more general idea: The constructions of flow-sparsifiers given in [@LM] (which is an extension of the techniques in [@M]) can be regarded as a dual to the rounding algorithm in [@FHRT] for the $0$-extension problem. What we observe here is: Suppose we are given a rounding algorithm that is used to round the fractional solution of some relaxation to an integral solution for some optimization problem. If this rounding algorithm also works for the relaxation for the $0$-extension problem given in [@K] (and also used in [@CKR], [@FHRT]), then we can use the techniques in [@M], [@LM] to obtain *stronger* flow-sparsifiers which are not only good quality flow-sparsifiers, but also for which the optimization problem is easy. So in this way we do not need to pay an additional price in the approximation guarantee in order to replace the dependence on $n$ with a dependence on $k$. With these ideas in mind, what we observe is that the rounding algorithm in [@FRT] wh ich embed s metric spaces into distributions on dominating tree-metrics, can also be used to round the $0$-extension relaxation. This allows us to construct flow-sparsifiers that have $O(\log k)$-quality, and also can be explicitly written as a convex combination of $0$-extensions that are tree-like. On trees, oblivious routing is easy, and so this gives us a way to simultaneously construct good flow-sparsifiers and good oblivious routing schemes on the sparsifier in one step! Of course, the rounding algorithm in [@FRT] for embedding metric spaces into distributions on dominating tree-metrics is a *very* common first step in rounding fractional relaxations of graph partitioning, graph layout and clustering problems. So for all problems that use this embedding as the main step, we are able to replace the dependence on $n$ with dependence on $k$, and we do not introduce any additional poly-logarithmic factors as in previous work! One can also interpret our result as giving a generalization of the hierarchical decompositions given in [@R] for approximating the cuts in a graph $G$ on trees. We state our results more formally, below, and we refer to such a statement as an [Abstract Integrality Gap]{}.  \[def:gpp\] We call a fractional packing problem $P$ a graph packing problem if the goal of the dual covering problem $D$ is to minimize the ratio of the total units of distance $\times$ capacity allocated in the graph divided by some monotone increasing function of the distances between terminals. This definition is quite general, and captures maximum concurrent flow, maximum multiflow, and multicast routing as special cases, in addition to many other common optimization problems. The integraldual $ID$ problems are generalized sparsest cut, multicut and requirement cut respectively.  \[thm:gpp\] For any graph packing problem $P$, the maximum ratio of the integral dual to the fractional primal is at most $O(\log k)$ times the maximum ratio restricted to trees. For a packing problem that fits into this class, this theorem allows us to reduce bounding the integrality gap in general graphs to bounding the integrality gap on trees, which is often substantially easier than for general graphs (i.e. for the example problems given above). We believe that this result helps to explain the intrinsic robustness of fractional packing problems into undirected graphs, in particular the ubiquity of the $O(\log k)$ bound for the flow-cut gap for a wide range of multicommodity flow problems. We also give a polynomial time algorithm to reduce any graph packing problem $P$ to a corresponding problem on a tree: Again, let $K$ be the set of terminals. Let $OPT(P, G)$ be the optimal value of the fractional graph packing problem $P$ on the graph $G$.  \[thm:agpp\] There is a polynomial time algorithm to construct a distribution $\mu$ on (a polynomial number of) trees on the terminal set $K$, s.t. $$E_{T \leftarrow \mu}[OPT(P, T)] \leq O(\log k) OPT(P, G)$$ and such that any valid integral dual of cost $C$ (for any tree $T$ in the support of $\mu$) can be immediately transformed into a valid integral dual in $G$ of cost at most $C$. As a corollary, given an approximation algorithm that achieves an approximation ratio of $C$ for the integral dual to a graph packing problem on trees, we obtain an approximation algorithm with a guarantee of $O(C \log k)$ for general graphs. We will refer to this last result as an [Abstract Rounding Algorithm]{}. We also give a polynomial time construction of $O(\log k/ \log \log k)$ quality flow-sparsifiers (and consequently cut-sparsifiers as well), which were previously only known to exist, but finding a polynomial time construction was still open. We accomplish this by performing a lifting (inspired by Earth-mover constraints) on an appropriate linear program. This lifting allows us to implicitly enforce a constraint that previously was difficult to enforce, and required an approximate separation oracle rather than an exact separation oracle. We give the details in section  \[sec:alift\]. Maximum Concurrent Flow ======================= An instance of the maximum concurrent flow problem consists of an undirected graph $G = (V, E)$, a capacity function $c: E \rightarrow \Re^+ $ that assigns a non-negative capacity to each edge, and a set of demands $\{ (s_i, t_i, f_i)\}$ where $s_i, t_i \in V$ and $f_i$ is a non-negative demand. We denote $K = \cup_i \{s_i, t_i\}$. The maximum concurrent flow question asks, given such an instance, what is the largest fraction of the demand that can be simultaneously satisfied? This problem can be formulated as a polynomial-sized linear program, and hence can be solved in polynomial time. However, a more natural formulation of the maximum concurrent flow problem can be written using an exponential number of variables. For any $a, b \in V$ let $P_{a, b}$ be the set of all (simple) paths from $a$ to $b$ in $G$. Then the maximum concurrent flow problem and the corresponding dual can be written as : $$\begin{array}{rlrl} \max & \lambda \hspace{5.0pc} & \hspace{4.0pc} \min & \sum_{e} d(e) c(e)\\ \mbox{s.t.} & & \hspace{4.0pc} \mbox{s.t.} & \\ & \sum_{P \in P_{s_i, t_i}} x(P) \geq \lambda f_i & \hspace{4.0pc} & \forall_{P \in P_{s_i, t_i}} \sum_{e \in P} d(e) \geq D(s_i, t_i) \\ & \sum_{P \owns e} x(P) \leq c(e) & \hspace{4.0pc} & \sum_i D(s_i, t_i) f_i \geq 1\\ & x(P) \geq 0 & \hspace{4.0pc} & d(e) \geq 0, D(s_i, t_i) \geq 0 \end{array}$$ For a maximum concurrent flow problem, let $\lambda^*$ denote the optimum. Let $|K| = k$. Then for a given set of demands $\{s_i, t_i, f_i\}$, we associate a vector $\vec{f} \in \Re^{k \choose 2}$ in which each coordinate corresponds to a pair $(x, y) \in {K \choose 2}$ and the value $\vec{f}_{x, y}$ is defined as the demand $f_i$ for the terminal pair $s_i = x, t_i = y$. We denote $cong_G(\vec{f}) = \frac{1}{\lambda^*}$ Or equivalently $cong_G(\vec{f})$ is the minimum $C$ s.t. $\vec{f}$ can be routed in $G$ and the total flow on any edge is at most $C$ times the capacity of the edge. Throughout we will use the notation that graphs $G_1, G_2$ (on the same node set) are “summed” by taking the union of their edge set (and allowing parallel edges). Cut Sparsifiers --------------- Suppose we are given an undirected, capacitated graph $G = (V, E)$ and a set $K \subset V$ of terminals of size $k$. Let $h: 2^V \rightarrow \Re^+$ denote the cut function of $G$: $h(A) = \sum_{(u, v) \in E \mbox{ s.t. } u \in A, v \in V -A} c( u, v)$. We define the function $h_K : 2^K \rightarrow \Re^+$ which we refer to as the terminal cut function on $K$: $h_K(U) = \min_{A \subset V \mbox{ s.t. } A \cap K = U} h(A)$. $G'$ is a [*cut-sparsifier*]{} for the graph $G = (V, E)$ and the terminal set $K$ if $G'$ is a graph on just the terminal set $K$ (i.e. $G' = (K, E')$) and if the cut function $h' : 2^K \rightarrow \Re^+$ of $G'$ satisfies (for all $U \subset K$) $$h_K(U) \leq h'(U)$$ We can define a notion of quality for any particular cut-sparsifier: The [*quality*]{} of a cut-sparsifier $G'$ is defined as $$max_{U \subset K} \frac{h'(U) }{h_K(U)}$$ We will abuse notation and define $\frac{0}{0} = 1$ so that when $U$ is disconnected from $K - U$ in $G$ or if $U = \emptyset$ or $U = K$, the ratio of the two cut functions is $1$ and we ignore these cases when computing the worst-case ratio and consequently the quality of a cut-sparsifier. $0$-Extensions -------------- $f: V \rightarrow K$ is a $0$-extension if for all $a \in K$, $f(a) = a$. So a $0$-extension $f$ is a clustering of the nodes in $V$ into sets, with the property that each set contains exactly one terminal. Given a graph $G = (V, E)$ and a set $K \subset V$, and $0$-extension $f$, $G_f = (K, E_f)$ is a capacitated graph in which for all $a, b \in K$, the capacity $c_f(a,b)$ of edge $(a, b) \in E_f$ is $$\sum_{(u,v) \in E \mbox{ s.t. } f(u) = a, f(v) = b} c(u, v)$$ Lower Bounds for Cut Sparsifiers {#sec:lb} ================================ Consider the following construction for a graph $G$. Let $Y$ be the hypercube of size $2^d$ for $d = \log k$. Then for every node $y_s \in Y$ (i.e. $s \in \{0, 1\}^d$), we add a terminal $z_s$ and connect the terminal $z_s$ to $y_s$ using an edge of capacity $\sqrt{d}$. All the edges in the hypercube are given capacity $1$. We’ll use this instance to show 2 lower bounds, one for 0-extension cut sparsifiers and the other for arbitrary cut sparisifers. Lower bound for Cut Sparsifiers from 0-extensions ------------------------------------------------- In this subseciton, we give an $\Omega(\sqrt{d})$ integrality gap for the semi-metric relaxation of the $0$-extension problem on this graph, even when the semi-metric (actually on all of $V$) is $\ell_1$. Such a bound is actually implicit in the work of [@JLS] too. Also , we show a strong duality between the worst case integrality gap for the semi-metric relaxation (when the semi-metric on $V$ must be $\ell_1$) and the quality of the best cut-sparsifer that can result from contractions. This gives an $\Omega(\sqrt{\log k})$ lower bound on how well a distribution on $0$-extensions can approximate the minimum cuts in $G$. Also, given the graph $G = (V, E)$ a set $K \subset V$ of terminals, and a semi-metric $D$ on $K$ we define the $0$-extension problem as: The **0-Extension Problem** is defined as $$\min_{\mbox{0-Extensions} f} \sum_{(u,v) \in E} c(u,v) D(f(a), f(b))$$ We denote $OPT(G, K, D)$ as the value of this optimum. Let $\Delta_U$ denote the cut-metric in which $\Delta_U (u,v) = 1_{|U \cap \{u, v\} |=1}$. Also, given an partition $\cP$ of $V$, we will refer to $\Delta_{\cP}$ as the partition metric (induced by $\cP$) which is $1$ if $u$ and $v$ are contained in different subsets of the partition $\cP$, and is $0$ otherwise. $$\begin{array}{rl} \min & \sum_{(u, v) \in E} c(u, v) \delta(u, v) \\ \mbox{s.t.} & \\ & \delta \mbox{ is a semi-metric on } V\\ & \forall_{t, t' \in K} \delta(t, t') = D(t, t'). \end{array}$$ We refer to this linear program as the **Semi-Metric Relaxation**. For a particular instance $(G, K, D)$ of the $0$-extension problem, we denote the optimal solution to this linear program as $OPT_{sm}(G, K, D)$.  \[thm:fhrt\] [@FHRT] $$OPT_{sm}(G, K, D) \leq OPT \leq O(\frac{\log k}{\log \log k}) OPT_{sm}(G, K, D)$$ If we are given a semi-metric $D$ which is $\ell_1$, we can additionally define a stronger (exponentially) sized linear program. $$\begin{array}{rl} \min & \sum_{U} \delta(U) h(U) \\ \mbox{s.t.} & \\ & \forall_{t, t' \in K} \sum_{U}\delta(U) \Delta_U(t, t') = D(t, t'). \end{array}$$ We will refer to this linear program as the **Cut-Cut Relaxation**. For a particular instance $(G, K, D)$ of the $0$-extension problem, we denote the optimal solution to this linear program as $OPT_{cc}(G, K, D)$. The value of this linear program is that an upper bound on the integrality gap of this linear program (for a particular graph $G$ and a set of terminals $K$) gives an upper bound on the quality of cut-sparsifiers. In fact, a stronger statement is true, and the quality of the best cut-sparsifier that can be achieved through contractions will be exactly equal to the maximum integrality gap of this linear program. The upper bound is given in [@M] -and here we exhibit a strong duality: The *[Contraction Quality]{} of $G, K$ is defined to be the minimum $\alpha$ such that there is a distribution on $0$-extensions $\gamma$ and $H = \sum_f \gamma(f) G_f$ is a $\alpha$ quality cut-sparsifier.* Let $\nu$ be the maximum integrality gap of the Cut-Cut Relaxation for a particular graph $G = (V, E)$, a particular set $K \subset V$ of terminals, over all $\ell_1$ semi-metrics $D$ on $K$. Then the Contraction Quality of $G, K$ is exactly $\nu$. Let $\alpha$ be the Contraction Quality of $G, K$. Then implicitly in [@M], $\alpha \leq \nu$. Suppose $\gamma$ is a distribution on $0$-extensions s.t. $H = \sum_f \gamma(f) G_f$ is a $\alpha$-quality cut sparsifier. Given any $\ell_1$ semi-metric $D$ on $K$, we can solve the Cut-Cut Linear Program given above. Notice that cut $(U, V-U)$ that is assigned positive weight in an optimal solution must be the minimum cut separating $U \cap K = A$ from $K - A = (V - U) \cap K$ in $G$. If not, we could replace this cut $(U, V-U)$ with the minimum cut separating $A$ from $K - A$ without affecting the feasibility and simultaneously reducing the cost of the solution. So for all $U$ for which $\delta(U) > 0$, $h(U) = h_K(U \cap K)$. Consider then the cost of the semi-metric $D$ against the cut-sparsifier $H$ which is defined to be $\sum_{(a, b)} c_H(a,b) D(a,b) = \sum_f \gamma(f) \sum_{(a,b)} c_f(a,b) D(a,b)$ which is just the average cost of $D$ against $G_f$ where $f$ is sampled from the distribution $\gamma$. The Cut-Cut Linear Program gives a decomposition of $D$ into a weighted sum of cut-metrics - i.e. $D(a,b) = \sum_{U} \delta(U) \Delta_U(a,b)$. Also, the cost of $D$ against $H$ is linear in $D$ so this implies that $$\sum_{(a, b)} c_H(a,b) D(a,b) = \sum_{(a,b)} \sum_{U} c_H(a,b) \delta(U) \Delta_U(a,b) = \sum_{(a,b)} c_H(a,b) \delta(U) h'(U \cap K)$$ In the last line, we use $\sum_{(a,b)} c_H(a,b) \Delta_U(a,b) = h'(U \cap K)$. Then $$\sum_{(a, b)} c_H(a,b) D(a,b) \leq \sum_{U} \delta(U) \alpha h_K(U \cap K) = \alpha OPT_{cc}(G, K, D)$$ In the inequality, we have used the fact that $H$ is an $\alpha$-quality cut-sparsifier, and in the last line we have used that $\delta(U) > 0$ implies that $h(U) = h_K(U \cap K)$. This completes the proof because the average cost of $D$ against $G_f$ where $f$ is sampled from $\gamma$ is at most $\alpha OPT_{cc}(G, K, D)$, so there must be some $f$ s.t. the cost against $D$ is at most $\alpha OPT_{cc}(G, K, D)$. We will use this strong duality between the Cut-Cut Relaxation and the Contraction Quality to show that for the graph $G$ given above, no distribution on $0$-extensions gives better than an $\Omega(\sqrt{\log k})$ quality cut-sparsifier, and all we need to accomplish this is to demonstrate an integrality gap on the example for the Cut-Cut Relaxation. Let’s repeat the construction of $G$ here. Let $Y$ be the hypercube of size $2^d$ for $d = \log k$. Then for every node $y_s \in Y$ (i.e. $s \in \{0, 1\}^d$), we add a terminal $z_s$ and connect the terminal $z_s$ to $y_s$ using an edge of capacity $\sqrt{d}$. All the edges in the hypercube are given capacity $1$. Then consider the distance assignment to the edges: Each edge connecting a terminal to a node in the hypercube - i.e. an edge of the form $(z_s, y_s)$ is assigned distance $\sqrt{d}$ and every other edge in the graph is assigned distance $1$. Then let $\sigma$ be the shortest path metric on $V$ given these edge distances. $\sigma$ is an $\ell_1$ semi-metric on $V$, and in fact there is a weighted combination of cuts s.t. $\sigma(u,v) = \sum_{U} \delta(U) \Delta_U(u,v)$ and $\sum_U \delta(U) h(U) = O(k d)$ We can take $\delta(U) = 1$ for any cut $(U, V-U)$ s.t. $U = \{z_s \cup y_s | s_i = 1\}$ - i.e. $U$ is the axis-cut corresponding to the $i^{th}$ bit. We also take $\delta(U) = \sqrt{d}$ for each $U = \{z_s\}$. This set of weights will achieve $\sigma(u,v) = \sum_{U} \delta(U) \Delta_U(u,v)$, and also there are $d$ axis cuts each of which has capacity $h(U) = \frac{k}{2}$ and there are $k$ singleton cuts of weight $\sqrt{d}$ and capacity $\sqrt{d}$ so the total cost is $O(k d)$. Yet if we take $D$ equal to the restriction of $\sigma$ on $K$, then $OPT(G, K, D) = \Omega(k d^{3/2})$: $OPT(G, K, D) = \Omega(k d^{3/2})$ Consider any $0$-extension $f$. And we can define the weight of any terminal $a$ as $weight_f(a) = |f^{-1}(a) | = |\{v | f(v) = a\}|$. Then $\sum_a weight_f(a) = n$ because each node in $V$ is assigned to some terminal. We can define a terminal as heavy with respect to $f$ if $weight_f(a) \geq \sqrt{k}$ and light otherwise. Obviously, $\sum_a weight_f(a) = \sum_{a \mbox{ s.t. } a \mbox{ is light}} weight_f(a) + \sum_{a \mbox{ s.t. } a \mbox{ is heavy}} weight_f(a)$ so the sum of the sizes of either all heavy terminals or of all light terminals is at least $\frac{n}{2} = \Omega(k)$. Suppose that $ \sum_{a \mbox{ s.t. } a \mbox{ is light}} weight_f(a) = \Omega(k)$. For any pair of terminals $a, b$, $D(a,b) \geq \sqrt{d}$. Also for any light terminal $a$, $f^{-1}(a) - \{a\}$ is a subset of the Hypercube of at most $\sqrt{k}$ nodes, and the small-set expansion of the Hypercube implies that the number of edges out of this set is at least $\Omega(weight_f(a) \log k) = \Omega(weight_f(a) d)$. Each such edge pays at least $\sqrt{d}$ cost, because $D(a,b) \geq \sqrt{d}$ for all pairs of terminals. So this implies that the total cost of the $0$-extension $f$ is at least $\sum_{a \mbox{ s.t. } a \mbox{ is light}} \Omega(weight_f(a) d^{3/2})$. Suppose that $ \sum_{a \mbox{ s.t. } a \mbox{ is heavy}} weight_f(a) = \Omega(k)$. Consider any heavy terminal $z_t$, and consider any $y_s \in f^{-1}(z_t)$ and $t \neq s$. Then the edge $(y_s, z_s)$ is capacity $\sqrt{d}$ and pays a total distance of $D(z_t, z_s) \geq \sigma(y_t, y_s)$. Consider any set $U$ of $\sqrt{k}$ nodes in the Hypercube. If we attempt to pack these nodes so as to minimize $\sum_{y_s \in U} \sigma(y_s, y_t)$ for some fixed node $y_t$, then the packing that minimizes the quantity is an appropriately sized Hamming ball centered at $y_t$. In a Hamming ball centered at the node $y_t$ of at least $\sqrt{k}$ total nodes, the average distance from $y_t$ is $\Omega(\log k) = \Omega(d)$, and so this implies that $\sum_{y_s \in f^{-1}(z_t)} D(z_t, z_s) \geq \sum_{y_s \in f^{-1}(z_t)} D(y_t, y_s) \geq \Omega(weight_f(z_t) d)$. Each such edge has capacity $\sqrt{d}$ so the total cost of the $0$-extension $f$ is at least $\sum_{a \mbox{ s.t. } a \mbox{ is heavy}} \ Omega(weight_f(a) d^{3/2})$ And of course using our strong duality result, this integrality gap implies that any cut-sparsifier that results from a distribution on $0$-extensions has quality at least $\Omega(\sqrt{\log k})$, and this matches the current best lower bound on the integrality gap of the Semi-Metric Relaxation for $0$-extension, so in principle this could be the best lower bound we could hope for (if the integrality gap of the Semi-Metric Relaxation is in fact $O(\sqrt{\log k})$ then there are always cut-sparsifiers that results from a distribution on $0$-extensions that are quality at most $O(\sqrt{\log k})$). Lower bounds for Arbitrary Cut sparsifiers ------------------------------------------ We will in fact use the above example to give a lower bound on the quality of *any* cut-sparisifer. We will show that for the above graph, no cut-sparsifier achieves quality better than $\Omega(\log^{1/4} k)$, and this gives an exponential improvement over the previous lower bound on the quality of flow-sparsifiers (which is even a stronger requirement for sparsifiers, and hence a weaker lower bound). The particular example $G$ that we gave above has many symmetries, and we can use these symmetries to justify considering only symmetric cut-sparsifiers. The fact that these cut-sparsifiers can be assumed without loss of generality to have nice symmetry properties, translates to that any such cut-sparsifier $H$ is characterized by a much smaller set of variables rather than one variable for every pair of terminals. In fact, we will be able to reduce the number of variables from $k \choose 2$ to $\log k$. This in turn will allow us to consider a much smaller family of cuts in $G$ in order to derive that the system is infeasible. In fact, we will only consider sub-cube cuts (cuts in which $U = \{z_s \cup y_s | s = [0, 0, 0, .... 0, *, *, ...,*]\}$) and the Hamming ball $U = \{ z_s \cup y_s | d(y_s, y_0) \leq \frac{d}{2}\}$. The operation $J_s$ for some $s \in \{0, 1\}^d$ which is defined as $J_s(y_t) = y_{t + s \mod 2}$ and $J_s(z_t) = z_{t + s \mod 2}$. Also let $J_s(U) = \cup_{u \in U} J_s(u)$. For any permutation $\pi: [d] \rightarrow [d]$, $\pi(s) = [s_{\pi(1)}, s_{\pi(2)}, ... s_{\pi(d)}]$. Then the operation $J_\pi$ for any permutation $\pi$ is defined at $J_\pi(y_t) = y_{\pi(t)}$ and $T_{\pi}(z_t) = z_{\pi(t)}$. Also let $J_\pi(U) = \cup_{u \in U} T_{\pi}(u)$. For any subset $U \subset V$ and any $s \in \{0, 1\}^d$, $h(U) = h(J_s(U))$. For any subset $U \subset V$ and any permutation $\pi: [d] \rightarrow [d]$, $h(U) = h(J_\pi(U))$. Both of these operations are automorphisms of the weighted graph $G$ and also send the set $K$ to $K$.  \[lemma:auto\] If there is a cut-sparsifier $H$ for $G$ which has quality $\alpha$, then there is a cut-sparsifier $H'$ which has quality at most $\alpha$ and is invariant under the automorphisms of the weighted graph $G$ that send $K$ to $K$. Given the cut-sparsifier $H$, we can apply an automorphism $J$ to $G$, and because $h(U) = h(J(U))$, this implies that $h_K(A) = \min_{U \mbox{ s.t. } U \cap K = A} h(U) = \min_{U \mbox{ s.t. } U \cap K = A} h(J(U))$. Also $J(U \cap K) = J(U) \cap J(K) = J(U) \cap K$ so we can re-write this last line as $$\min_{U \mbox{ s.t. } U \cap K = A} h(J(U)) = \min_{U' \mbox{ s.t. } J(U') \cap K = J(A)} h(J(U'))$$ And if we set $U' = J^{-1}(U)$ then this last line becomes equivalent to $$\min_{U' \mbox{ s.t. } J(U') \cap K = J(A)} h(J(U'))= \min_{U \mbox{ s.t. } U \cap K = J(A)} h(U)= h_K(J(A))$$ So the result is that $h_K(A) = h_K(J(A))$ and this implies that if we do not re-label $H$ according to $J$, but we do re-label $G$, then for any subset $A$, we are checking whether the minimum cut in $G$ re-labeled according to $J$, that separates $A$ from $K -A$ is close to the cut in $H$ that separates $A$ from $K-A$. The minimum cut in the re-labeled $G$ that separates $A$ from $K -A$, is just the minimum cut in $G$ that separates $J^{-1}(A)$ from $K - J^{-1}(A)$ (because the set $J^{-1}(A)$ is the set that is mapped to $A$ under $J$). So $H$ is an $\alpha$-quality cut-sparsifier for the re-labeled $G$ iff for all $A$: $$h_K(A) = h_K(J^{-1}(A)) \leq h'(A) \leq \alpha h_K(J^{-1}(A)) = \alpha h_K(A)$$ which is of course true because $H$ is an $\alpha$-quality cut-sparsifier for $G$. So alternatively, we could have applied the automorphism $J^{-1}$ to $H$ and not re-labeled $G$, and this resulting graph $H_{J^{-1}}$ would also be an $\alpha$-quality cut-sparsifier for $G$. Also, since the set of $\alpha$-quality cut-sparsifiers is convex (it is defined by a system of inequalities), we can find a cut-sparsifier $H'$ that has quality at most $\alpha$ and is a fixed point of the group of automorphisms, and hence invariant under the automorphisms of $G$ as desired. \[cor:auto\] If $\alpha$ is the best quality cut-sparsifier for the above graph $G$, then there is an $\alpha$ quality cut-sparsifier $H$ in which the capacity between two terminals $z_s$ and $z_t$ is only dependent on the Hamming distance $Hamm(s, t)$. Given any quadruple $z_s, z_t$ and $z_{s'}, z_{t'}$ s.t. $Hamm(s, t) = Hamm(s', t')$, there is a concatenation of operations from $J_s$, $J_\pi$ that sends $s$ to $s'$ and $t$ to $t'$. This concatenation of operations $J$ is in the group of automorphisms that send $K$ to $K$, and hence we can assume that $H$ is invariant under this operation which implies that $c_H(s,t) = c_H(s',t')$. One can regard any cut-sparsifier (not just ones that result from contractions) as a set of $k \choose 2$ variables, one for the capacity of each edge in $H$. Then the constraints that $H$ be an $\alpha$-quality cut-sparsifier are just a system of inequalities, one for each subset $A \subset K$ that enforces that the cut in $H$ is at least as large as the minimum cut in $G$ (i.e. $h'(A) \geq h_K(A)$) and one enforcing that the cut is not too large (i.e. $h'(A) \leq \alpha h_K(A)$). Then in general, one can derive lower bounds on the quality of cut-sparsifiers by showing that if $\alpha$ is not large enough, then this system of inequalities is infeasible meaning that there is not cut-sparsifier achieving quality $\alpha$. Unlike the above argument, this form of a lower bound is much stronger and does not assume anything about how the cut-sparsifier is generated. [Theorem]{}[thm:lower]{} For $\alpha = \Omega(\log^{1/4} k)$, there is no cut-sparsifier $H$ for $G$ which has quality at most $\alpha$. Assume that there is a cut-sparsifier $H'$ of quality at most $\alpha$. Then using the above corollary, there is a cut-sparsifier $H$ of quality at most $\alpha$ in which the weight from $a$ to $b$ is only a function of $Hamm(a,b)$. Then for each $i \in [d]$, we can define a variable $w_i$ as the total weight of edges incident to any terminal of length $i$. I.e. $w_i = \sum_{b \mbox{ s.t. } Hamm(a,b) = i} c_H(a,b)$. For simplicity, here we will assume that all cuts in the sparsifier $H$ are at most the cost of the corresponding minimum cut in $G$ and at least $\frac{1}{\alpha}$ times the corresponding minimum cut. This of course is an identical set of constraints that we get from dividing the standard definition that we use in this paper for $\alpha$-quality cut-sparsifiers by $\alpha$. We need to derive a contradiction from the system of inequalities that characterize the set of $\alpha$-quality cut sparsifiers for $G$. As we noted, we will consider only the sub-cube cuts (cuts in which $U = \{z_s \cup y_s | s = [0, 0, 0, .... 0, *, *, ... *]\}$) and the Hamming ball $U = \{ z_s \cup y_s | d(y_s, y_0) \leq \frac{d}{2}\}$, which we refer to as the Majority Cut. Consider the Majority Cut: There are $\Theta(k)$ terminals on each side of the cut, and most terminals have Hamming weight close to $\frac{d}{2}$. In fact, we can sort the terminals by Hamming weight and each weight level around Hamming weight $\frac{d}{2}$ has roughly a $\Theta(\frac{1}{\sqrt{d}})$ fraction of the terminals. Any terminal of Hamming weight $\frac{d}{2} - \sqrt{i}$ has roughly a constant fraction of their weight $w_i$ crossing the cut in $H$, because choosing a random terminal Hamming distance $i$ from any such terminal corresponds to flipping $i$ coordinates at random, and throughout this process there are almost an equal number of $1$s and $0$s so this process is well-approximated by a random walk starting at $\sqrt{i}$ on the integers, which equally likely moves forwards and backwards at each step for $i$ total steps, and asking the probability that the walk ends at a negative integer. In particular, for any terminal of Hamming weight $\frac{d}{2} - t$, the fraction of the weight $w_i$ that crosses the Majority Cut is $O(exp\{-\frac{t^2}{i})$. So the total weight of length $i$ edges (i.e. edges connecting two terminals at Hamming distance $i$) cut by the Majority Cut is $O(w_i |\{z_s | Hamm(s, 0) \geq \frac{d}{2} - \sqrt{i}\}|) = O(w_i \sqrt{i / d})k$ because each weight close to the boundary of the Majority cut contains roughly a $\Theta(\frac{1}{\sqrt{d}})$ fraction of the terminals. So the total weight of edges crossing the Majority Cut in $H$ is $O(k \sum_{i = 1}^d w_i \sqrt{i / d})$ And the total weight crossing the minimum cut in $G$ separating $A = \{ z_s | d(y_s, y_0) \leq \frac{d}{2}\}$ from $K - A$ is $\Theta(k \sqrt{d})$. And because the cuts in $H$ are at least $\frac{1}{\alpha}$ times the corresponding minimum cut in $G$, this implies $\sum_{i = 1}^d w_i \sqrt{i / d} \geq \Omega(\frac{\sqrt{d}}{\alpha})$ Next, we consider the set of sub-cube cuts. For $j \in [d]$, let $A_j = \{z_s | s_1 = 0, s_2 = 0, .. s_j = 0\}$. Then the minimum cut in $G$ separating $A_j$ from $K - A_j$ is $\Theta(|A_j| \min(j, \sqrt{d}))$, because each node in the Hypercube which has the first $j$ coordinates as zero has $j$ edges out of the sub-cube, and when $j > \sqrt{d}$, we would instead choose cutting each terminal $z_s \in A_j$ from the graph directly by cutting the edge $(y_s, z_s)$. Also, for any terminal in $A_j$, the fraction of length $i$ edges that cross the cut is approximately $1 - (1 - \frac{j}{d})^i = \Theta(\min(\frac{ij}{d}, 1))$. So the constraints that each cut in $H$ be at most the corresponding minimum cut in $G$ give the inequalities $ \sum_{i = 1}^d \min(\frac{i j}{d}, 1) w_i \leq O(\min(j, \sqrt{d}))$ We refer to the above constraint as $B_j$. Multiply each $B_j$ constraint by $\frac{1}{j^{3/2}}$ and adding up the constraints yields a linear combination of the variables $w_i$ on the left-hand side. The coefficient of any $w_i$ is $$\sum_{j=1}^{d-1} \frac{\min(\frac{i j}{d}, 1)}{j^{3/2}} \geq \sum_{j=1}^{d/i} \frac{\frac{i j}{d}}{j^{3/2}}$$ And using the Integration Rule this is $\Omega(\sqrt{\frac{i}{d}})$. This implies that the coefficients of the constraint $B$ resulting from adding up $\frac{1}{j^{3/2}}$ times each $B_j$ for each $w_i$ are at least as a constant times the coefficient of $w_i$ in the Majority Cut Inequality. So we get $$\sum_{j=1}^{d-1} \frac{1}{j^{3/2}}\min(j, \sqrt{d}) \geq \Omega \Big (\sum_{j=1}^{d-1} \frac{1}{j^{3/2}} \sum_{i = 1}^d \min(\frac{i j}{d}, 1) w_i \Big) \geq \Omega \Big(\sum_{i = 1}^d w_i \sqrt{\frac{i}{d}} \Big) \geq \Omega \Big(\frac{\sqrt{d}}{\alpha} \Big)$$ And we can evaluate the constant $\sum_{j=1}^{d-1} j^{-3/2} \min(j, \sqrt{d}) =\sum_{j=1}^{\sqrt{d}} j^{-1/2} + \sqrt{d} \sum_{j=\sqrt{d}+1}^{d-1} j^{-3/2}$ using the Integration Rule, this evaluates to $O(d^{1/4})$. This implies $O(d^{1/4}) \geq \frac{\sqrt{d}}{\alpha}$ and in particular this implies $\alpha \geq \Omega(d^{1/4})$. So the quality of the best cut-sparsifier for $H$ is at least $\Omega(\log^{1/4} k)$. We note that this is the first super-constant lower bound on the quality of cut-sparsifiers. Recent work gives a super-constant lower bound on the quality of flow-sparsifiers in an infinite family of expander-like graphs. However, for this family there are constant-quality cut-sparsifiers. In fact, lower bounds for cut-sparsifiers imply lower bounds for flow-sparsifiers, so we are able to improve the lower bound of $\Omega(\log \log k)$ in the previous work for flow-sparsifiers by an exponential factor to $\Omega(\log^{1/4} k)$, and this is the first lower bound that is tight to within a polynomial factor of the current best upper bound of $O(\frac{\log k}{\log \log k})$. This bound is not as good as the lower bound we obtained earlier in the restricted case in which the cut-sparsifier is generated as a convex combination of $0$-extension graphs $G_f$. As we will demonstrate, there are actually cut-sparsifiers that achieve quality $o(\sqrt{\log k})$ for $G$, and so in general restricting to convex combinations of $0$-extensions is sub-optimal, and we leave open the possibility that the ideas in this improved bound may result in better constructions of cut (or flow)-sparsifiers that are able to beat the current best upper bound on the integrality gap of the $0$-extension linear program. Noise Sensitive Cut-Sparsifiers =============================== In Appendix \[sec:aha\], we give a brief introduction to the harmonic analysis of Boolean functions, along with formal statements that we will use in the proof of our main theorem in this section. A Candidate Cut-Sparsifier -------------------------- Here we give a cut-sparsifier $H$ which will achieve quality $o(\sqrt{\log k})$ for the graph $G$ given in Section \[sec:lb\], which is asymptotically better than the best cut-sparsifier that can be generated from contractions. As we noted, we can assume that the weight assigned between a pair of terminals in $H$, $c_H(a,b)$ is only a function of the Hamming distance from $a$ to $b$. In $G$, the minimum cut separating any singleton terminal $\{z_s\}$ from $K - \{z_s\}$ is just the cut that deletes the edge $(z_s, y_s)$. So the capacity of this cut is $\sqrt{d}$. We want a good cut-sparsifier to approximately preserve this cut, so the total capacity incident to any terminal in $H$ will also be $\sqrt{d}$ - i.e. $c'(\{z_s\}) = \sqrt{d}$. We distribute this capacity among the other terminals as follows: We sample $t \sim_\rho s$, and allocate an infinitesimal fraction of the total weight $\sqrt{d}$ to the edge $(z_s, z_t)$. Equivalently, the capacity of the edge connecting $z_s$ and $z_t$ is just $Pr_{u \sim_\rho t}[u = s] \sqrt{d}$. We choose $\rho = 1 - \frac{1}{\sqrt{d}}$. This choice of $\rho$ corresponds to flipping each bit in $t$ with probability $\Theta(\frac{1}{\sqrt{d}})$ when generating $u$ from $t$. We prove that the graph $H$ has cuts at most the corresponding minimum-cut in $G$. This cut-sparsifier $H$ has cuts at most the corresponding minimum-cut in $G$. In fact, a stronger statement is true: $\vec{H}$ can be routed as a flow in $G$ with congestion $O(1)$. Consider the following explicit routing scheme for $\vec{H}$: Route the $\sqrt{d}$ total flow in $\vec{H}$ out of $z_s$ to the node $y_s$ in $G$. Now we need to route these flows through the Hypercube in a way that does not incur too much congestion on any edge. Our routing scheme for routing the edge from $z_s$ to $z_t$ in $\vec{H}$ from $y_s$ to $y_t$ will be symmetric with respect to the edges in the Hypercube: choose a random permutation of the bits $\pi : [d] \rightarrow [d]$, and given $u \sim_\rho t$, fix each bit in the order defined by $\pi$. So consider $i_1 = \pi(1)$. If $t_{i_1} \neq u_{i_1}$, and the flow is currently at the node $x$, then flip the $i_1^{th}$ bit of $x$, and continue for $i_2 = \pi(2)$, $i_3, ... i_d = \pi(d)$. Each permutation $\pi$ defines a routing scheme, and we can average over all permutations $\pi$ and this results in a routing scheme that routes $\vec{H}$ in $G$. This routing scheme is symmetric with respect to the automorphisms $J_s$ and $J_\pi$ of $G$ defined above. The congestion on any edge in the Hypercube incurred by this routing scheme is the same. The above routing scheme will achieve congestion at most $O(1)$ for routing $\vec{H}$ in $G$. Since the congestion of any edge in the Hypercube under this routing scheme is the same, we can calculate the worst case congestion on any edge by calculating the average congestion. Using a symmetry argument, we can consider any fixed terminal $z_s$ and calculate the expected increase in average congestion when sampling a random permutation $\pi: [d] \rightarrow [d]$ and routing all the edges out of $z_s$ in $H$ using $\pi$. This expected value will be $k$ times the average congestion, and hence the worst-case congestion of routing $\vec{H}$ in $G$ according to the above routing scheme. As we noted above, we can define $H$ equivalently as arising from the random process of sampling $u \sim_\rho t$, and routing an infinitesimal fraction of the $\sqrt{d}$ total capacity out of $z_t$ to $z_u$, and repeating until all of the $\sqrt{d}$ capacity is allocated. We can then calculate the the expected increase in average congestion (under a random permutation $\pi$) caused by routing the edges out of $z_s$ as the expected increase in average congestion divided by the total fraction of the $\sqrt{d}$ capacity allocated when we choose the target $u$ from $u \sim_\rho t$. In particular, if we allocated a $\Delta $ fraction of the $\sqrt{d}$ capacity, the expected increase in total congestion is just the total capacity that we route multiplied by the length of the path. Of course, the length of this path is just the number of bits in which $u$ and $t$ differ, which in expectation is $\Theta(\sqrt{d})$ by our choice of $\rho$. So in this procedure, we allocate $\Delta \sqrt{d}$ total capacity, and the expected increase in total congestion is the total capacity routed $\Delta \sqrt{d}$ times the expected path length $\Theta(\sqrt{d})$. We repeat this procedure $\frac{1}{\Delta}$ times, and so the expected increase in total congestion caused by routing the edges out of $z_t$ in $G$ is $\Theta(d)$. If we perform this procedure for each terminal, the resulting total congestion is $\Theta(kd)$, and because there are $\frac{kd}{2}$ edges in the Hypercube, the average congestion is $\Theta(1)$ which implies that the worst-case congestion on any edge in the Hypercube is also $O(1)$, as desired. Also, the congestion on any edge $(z_s, y_s)$ is $1$ because there is a total of $\sqrt{d}$ capacity out of $z_s$ in $H$, and this is the only flow routed on this edge, which has capacity $\sqrt{d}$ in $G$ by construction. So the worst-case congestion on any edge in the above routing scheme is $O(1)$. \[cor:cutlow\] For any $A \subset K$, $h'(A) \leq O(1) h_K(A)$. Consider any set $A \subset K$. Let $U$ be the minimum cut in $G$ separating $A$ from $K - A$. Then the total flow routed from $A$ to $K - A$ in $\vec{H}$ is just $h'(A)$, and if this flow can be routed in $G$ with congestion $O(1)$, this implies that the total capacity crossing the cut from $U$ to $V-U$ is at least $\Omega(1) h'(A)$. And of course the total capacity crossing the cut from $U$ to $V-U$ is just $h_K(A)$ by the definition of $U$, which implies the corollary. So we know that the cuts in $H$ are never too much larger than the corresponding minimum cut in $G$, and all that remains to show that the quality of $H$ is $o(\sqrt{\log k})$ is to show that the cuts in $H$ are never too small. We conjecture that the quality of $H$ is actually $\Theta(\log^{1/4} k)$, and this seems natural since the quality of $H$ just restricted to the Majority Cut and the sub-cube cuts is actually $\Theta(\log^{1/4} k)$, and often the Boolean functions corresponding to these cuts serve as extremal examples in the harmonic analysis of Boolean functions. In fact, our lower bound on the quality of any cut-sparsifier for $G$ is based only on analyzing these cuts so in a sense, our lower bound is tight given the choice of cuts in $G$ that we used to derive infeasibility in the system of equalities characterizing $\alpha$-quality cut-sparsifiers. A Fourier Theoretic Characterization of Cuts in $H$ --------------------------------------------------- Here we give a simple formula for the size of a cut in $H$, given the Fourier representation of the cut. So here we consider cuts $A \subset K$ to be Boolean functions of the form $f_A: \{-1, +1\}^d \rightarrow \{-1, +1\}$ s.t. $f_A(s) = +1$ iff $z_s \in A$. $h'(A) = k \frac{\sqrt{d}}{2} \frac{1 - NS_{\rho}[f_A(x)]}{2}$ We can again use the infinitesimal characterization for $H$, in which we choose $u \sim_\rho t$ and allocate $\Delta$ units of capacity from $z_s $ to $z_t$ and repeat until all $\sqrt{d}$ units of capacity are spent. If we instead choose $z_s$ uniformly at random, and then choose $u \sim_\rho t$ and allocate $\Delta$ units of capacity from $z_s $ to $z_t$, and repeat this procedure until all $k \frac{\sqrt{d}}{2}$ units of capacity are spent, then at each step the expected contribution to the cut is exactly $\Delta \frac{1 - NS_{\rho}[f_A(x)]}{2}$ because $\frac{1 - NS_{\rho}[f_A(x)]}{2}$ is exactly the probability that if we choose $t$ uniformly at random, and $u \sim_\rho t$ that $f_A(u) \neq f_A(t)$ which means that this edge contributes to the cut. We repeat this procedure $\frac{k \sqrt{d}}{2 \Delta}$ times, so this implies the lemma. \[lemma:fcut\] $h'(A) = \Theta \Big (k \sum_S \hat{f}_S^2 \min(|S|, \sqrt{d}) \Big )$ Using the setting $\rho = 1 - \frac{1}{\sqrt{d}}$, we can compute $h'(A)$ using the above lemma: $$h'(A) = k \frac{\sqrt{d}}{4} (1 - NS_{\rho}[f_A(x)])$$ And using Parseval’s Theorem, $\sum_S \hat{f}_S^2 = ||f||_2 = 1$, so we can replace $1$ with $\sum_S \hat{f}_S^2$ in the above equation and this implies $$h'(A) = k \frac{\sqrt{d}}{4} \sum_S \hat{f}_S^2 (1 - (1 - \frac{1}{\sqrt{d}})^{|S|})$$ Consider the term $(1 - (1 - \frac{1}{\sqrt{d}})^{|S|})$. For $|S| \leq \sqrt{d}$, this term is $\Theta(\frac{|S|}{\sqrt{d}})$, and if $|S| \geq \sqrt{d}$, this term is $\Theta(1)$. So this implies $$h'(A) = \Theta \Big (k \sum_S \hat{f}_S^2 \min(|S|, \sqrt{d}) \Big )$$ Small Set Expansion of $H$ -------------------------- The edge-isoperimetric constant of the Hypercube is $1$, but on subsets of the cube that are imbalanced, the Hypercube expands more than this. For a given set $A \subset \{-1, +1\}^{[d]}$, we define $bal(A) = \frac{1}{k} \min(|A|, k - |A|)$ as the balance of the set $A$. Given any set $A \subset \{-1, +1\}^{[d]}$ of balance $b = bal(A)$, the number of edges crossing the cut $(A, \{-1, +1\}^{[d]} - A)$ in the Hypercube is $\Omega(b k \log \frac{1}{b})$. So the Hypercube expands better on small sets, and we will prove a similar small set expansion result for the cut-sparsifier $H$. In fact, for any set $A \subset K$ (which we will associated with a subset of $\{-1, +1\}^{[d]} $ and abuse notation), $h'(A) \geq bal(A) k \Omega(\min(\log \frac{1}{bal(A)}, \sqrt{d}))$. We will prove this result using the Hypercontractive Inequality. \[lemma:imb\] $h'(A) \geq bal(A) k \Omega(\min(\log \frac{1}{bal(A)}, \sqrt{d}))$ Assume that $|A| \leq |\{-1, +1\}^{[d]} - A|$ without loss of generality. Throughout just this proof, we will use the notation that $f_A : \{-1, +1\}^d \rightarrow \{0, 1\}$ and $f_A(s) = 1$ iff $s \in A$. Also we will denote $b = bal(A)$. Let $\gamma << 1$ be chose later. Then we will invoke the Hypercontractive inequality with $q =2$, $p = 2 - \gamma$, and $\rho = \sqrt{\frac{p-1}{q-1}} = \sqrt{1 - \gamma}$. Then $$||f||_p = E_x[f(x)^p]^{1/p} = b^{1/p} \approx b^{1/2 (1 + \gamma/2)}$$ Also $||T_{\rho}(f(x))||_q = ||T_{\rho}(f(x))||_2 = \sqrt{\sum_S \rho^{2 |S|} \hat{f}_S^2}$. So the Hypercontractive Inequality implies $$\sum_S \rho^{2 |S|} \hat{f}_S^2 \leq b^{1 + \gamma/2} = b e^{-\frac{\gamma}{2} \ln\frac{1}{b}}$$ And $\rho^{2 |S|} = (1 - \gamma)^{|S|}$. Using Parseval’s Theorem, $\sum_S \hat{f}_S^2 = ||f||_2^2 = b$, and so we can re-write the above inequality as $$b - b e^{\frac{-\gamma}{2} \ln\frac{1}{b}} \leq b - \sum_S (1 - \gamma)^{ |S|} \hat{f}_S^2 \leq b^{1 - \gamma/2} = \sum_S \hat{f}_S^2 (1 - (1 - \gamma)^{ |S|}) = \Theta \Big (\sum_S \hat{f}_S^2 \gamma \min(|S|, \frac{1}{\gamma}) \Big )$$ This implies $$\frac{b}{\gamma} (1 - e^{-\frac{\gamma}{2} \ln\frac{1}{b}}) \leq \Theta \Big (\sum_S \hat{f}_S^2 \min(|S|, \frac{1}{\gamma}) \Big )$$ And as long as $\frac{1}{\gamma} \leq \sqrt{d}$, $$\sum_S \hat{f}_S^2 \min(|S|, \frac{1}{\gamma}) \leq \frac{1}{k}O(h'(A)) = O \Big ( \sum_S \hat{f}_S^2 \min(|S|, \sqrt{d}) \Big )$$ If $\frac{\gamma}{2} \ln\frac{1}{b} \leq 1$, then $e^{-\frac{\gamma}{2} \ln\frac{1}{b}} = 1 - \Omega(\frac{\gamma}{2} \ln\frac{1}{b})$ which implies $$\frac{b}{\gamma} (1 - e^{-\frac{\gamma}{2} \ln\frac{1}{b}}) \geq \Omega( b \ln \frac{1}{b})$$ However if $\ln \frac{1}{b} = \Omega(\sqrt{d})$, then we cannot choose $\gamma$ to be small enough (we must choose $\frac{1}{\gamma} \leq \sqrt{d}$) in order to make $\frac{\gamma}{2} \ln\frac{1}{b}$ small. So the only remaining case is when $\ln \frac{1}{b} = \Omega(\sqrt{d})$. Then notice that the quantity $(1 - e^{-\frac{\gamma}{2} \ln\frac{1}{b}})$ is increasing with decreasing $b$. So we can lower bound this term by substituting $b = e^{-\Theta(\sqrt{d})}$. If we choose $\gamma = \frac{1}{\sqrt{d}}$ then this implies $$\frac{1}{\gamma}(1 - e^{-\frac{\gamma}{2} \ln\frac{1}{b}}) = \Omega(\sqrt{d})$$ And this in turn implies that $$\frac{b}{\gamma}(1 - e^{-\frac{\gamma}{2} \ln\frac{1}{b}}) = \Omega(b \sqrt{d})$$ which yields $h'(A) \geq \Omega(bk \sqrt{d})$. So in either case, $h'(A)$ is lower bounded by either $\Omega(bk \sqrt{d}) $ or $\Omega(bk \ln \frac{1}{b})$, as desired. Interpolating Between Cuts via Bourgain’s Junta Theorem ------------------------------------------------------- In this section, we show that the quality of the cut-sparsifier $H$ is $o(\sqrt{\log k})$, thus beating how well the best distribution on $0$-extensions can approximate cuts in $G$ by a super-constant factor. We will first give an outline of how we intend to combine Bourgain’s Junta Theorem, and the small set expansion of $H$ in order to yield this result. In a previous section, we gave a Fourier theoretic characterization of the cut function of $H$. We will consider an arbitrary cut $A \subset K$ and assume for simplicity that $|A| \leq |K-A|$. If the Boolean function $f_A$ that corresponds to this cut has significant mass at the tail of the spectrum, this will imply (by our Fourier theoretic characterization of the cut function) that the capacity of the corresponding cut in $H$ is $\omega(k)$. Every cut in $G$ has capacity at most $O(k \sqrt{d})$ because we can just cut every edge $(z_s, y_s)$ for each terminal $z_s \in A$, and each such edge has capacity $\sqrt{d}$. Then in this case, the ratio of the minimum cut in $G$ to the corresponding cut in $H$ is $o(\sqrt{d})$. But if the tail of the Fourier spectrum of $f_A$ is not significant, and applying Bourgain’s Junta Theorem implies that the function $f_A$ is close to a junta. Any junta will have a small cut in $G$ (we can take axis cuts corresponding to each variable in the junta) and so for any function that is different from a junta on a vanishing fraction of the inputs, we will be able to construct a cut in $G$ (not necessarily minimum) that has capacity $o(k \sqrt{d})$. On all balanced cuts (i.e. $|A| = \Theta(k)$), the capacity of the cut in $H$ will be $\Omega(k)$, so again in this case the ratio of the minimum cut in $G$ to the corresponding cut in $H$ is $o(\sqrt{d})$. So the only remaining case is when $|A| = o(k)$, and from the small set expansion of $H$ the capacity of the cut in $H$ is $\omega(|A|)$ because the cut is imbalanced. Yet the minimum cut in $G$ is again at most $|A| \sqrt{d}$, so in this case as well the ratio of the minimum cut in $G$ to the corresponding cut in $H$ is $o(\sqrt{d})$. [Theorem]{}[thm:sep]{} There is an infinite family of graphs for which the quality of the best cut-sparsifier is $\Omega(\frac{\log^2 \log \log k}{\log\log\log\log k})$ better than the best that a distribution on $0$-extensions can achieve. We repeat Bourgain’s Junta Theorem: \[Bourgain\] [@Bou], [@KN] Let $f \{-1, +1\}^d \rightarrow \{-1, +1\}$ be a Boolean function. Then fix any $\epsilon, \delta \in (0, 1/10)$. Suppose that $$\sum_S (1-\epsilon)^{|S|} \hat{f}_S^2 \geq 1 - \delta$$ then for every $\beta > 0$, $f$ is a $$\Big (2^{c \sqrt{\log 1/\delta \log \log 1/\epsilon}}\Big ( \frac{\delta}{\sqrt{\epsilon}} + 4^{1/\epsilon} \sqrt{\beta}\Big), \frac{1}{\epsilon \beta} \Big ) \mbox{-junta}$$ We will choose: $\frac{1}{\epsilon} = \frac{1}{32} \log d$ $\frac{1}{\beta} = d^{1/4}$ $\frac{1}{\delta'} = \log^{2/3} d$ And also let $b = bal(A) = \frac{|A|}{k}$, and remember for simplicity we have assumed that $|A| \leq |K-A|$, so $b \leq \frac{1}{2}$. $\delta = \delta'b$ If $\sum_S (1-\epsilon)^{|S|} \hat{f}_S^2 \leq 1 - \delta$ then this implies $\sum_S \hat{f}_S^2 \min(|S|, \sqrt{d}) \geq \Omega \Big (\frac{\delta}{\epsilon} \Big ) = \Omega(b\log^{1/3} d)$ The condition $\sum_S (1-\epsilon)^{|S|} \hat{f}_S^2 \geq 1 - \delta$ implies $\delta \leq 1 - \sum_S (1-\epsilon)^{|S|} \hat{f}_S^2 = O(\sum_S \hat{f}_S^2 \min(|S| \epsilon, 1)) $ and rearranging terms this implies $$\frac{\delta}{\epsilon} \leq O(\sum_S \hat{f}_S^2 \min(|S|, \frac{1}{\epsilon})) = O(\sum_S \hat{f}_S^2 \min(|S|, \sqrt{d}))$$ where the last line follows because $\frac{1}{\epsilon} = O(\log d) \leq O(\sqrt{d})$. So combining this lemma and Lemma \[lemma:fcut\]: if the conditions of Bourgain’s Junta Theorem are not met, then the capacity of the cut in the sparsifier is $\Omega(kb \log^{1/3} d)$. And of course, the capacity of the minimum cut in $G$ is at most $kb \sqrt{d}$, because for each $z_s \in A$ we could separate $A$ from $K -A $ by cutting the edge $(z_s, y_s)$, each of which has capacity $\sqrt{d}$. If the conditions of Bourgain’s Junta Theorem are not met, then the ratio of the minimum cut in $G$ separating $A$ from $K -A$ to the corresponding cut in $H$ is at most $O(\frac{\sqrt{d}}{\log^{1/3}d})$. But what if the conditions of Bourgain’s Junta Theorem are met? We can check what Bourgain’s Junta Theorem implies for the given choice of parameters. We first consider the case when $b$ is not too small. In particular, for our choice of parameters the following 3 inequalities hold: $$\begin{aligned} 2^{c \sqrt{\log 1/\delta \log \log 1/\epsilon}} &\leq& \log^{1/24} d \label{inequ:b-is-small-1}\\ \frac{\delta}{\sqrt{\epsilon}} &\geq& 4^{1/\epsilon}\sqrt{\beta} \label{inequ:b-is-small-2}\\ \sqrt{d}b\log^{-1/8}d &\geq& d^{1/4}\log d \label{inequ:b-is-small-3}\end{aligned}$$ If (\[inequ:b-is-small-2\]) is true, $\left( \frac{\delta}{\sqrt{\epsilon}} + 4^{1/\epsilon} \sqrt{\beta}\right) = O\left(b\log^{-1/6} d\right)$ If (\[inequ:b-is-small-1\]) and (\[inequ:b-is-small-2\]) are true, $2^{c \sqrt{\log 1/\delta \log \log 1/\epsilon}}\Big ( \frac{\delta}{\sqrt{\epsilon}} + 4^{1/\epsilon} \sqrt{\beta}\Big) = O\left(b\log^{-1/8}d\right)$ So when we apply Bourgain’s Junta Theorem, if the conditions are met (for our given choice of parameters), we get that $f_A$ is an $\left(O\left(b\log^{-1/8}d\right), O(d^{1/4} \log d)\right)$-junta. If $f_A$ is a $(\nu, j)$-junta, then $h_K(A) \leq k \nu \sqrt{d} + j \frac{k}{2}$ Let $g$ be a $j$-junta s.t. $Pr_x[f_A(x) \neq g(x)] \leq \nu$. Then we can disconnect the set of nodes on the Hypercube where $g$ takes a value $+1$ from the set of nodes where $g$ takes a value $-1$ by performing an axis cut for each variable that $g$ depends on. Each such axis cut, cuts $\frac{k}{2}$ edges in the Hypercube, so the total cost of cutting these edges is $j \frac{k}{2}$ and then we can alternatively cut the edge $(z_s, y_s)$ for any $s$ s.t. $f_A(s) \neq g(s)$, and this will be a cut separating $A$ from $K -A$ and these extra edges cut are each capacity $\sqrt{d}$ and we cut at most $\nu k$ of these edges in total. So if $f_A$ is an$\left(O\left(b\log^{-1/8}d\right), O(d^{1/4} \log d)\right)$-junta and (\[inequ:b-is-small-3\]) holds, then $h_K(A) \leq O\left(\frac{kb \sqrt{d}}{\log^{1/8}d }\right)$. Suppose the conditions of Bourgain’s Junta Theorem are met, and (\[inequ:b-is-small-1\])(\[inequ:b-is-small-2\]) and (\[inequ:b-is-small-3\]) are true, then the ratio of the minimum cut in $G$ separating $A$ from $K -A$ to the corresponding cut in $H$ is at most $O(\frac{\sqrt{d}}{\log^{1/8}d})$. Lemma \[lemma:imb\] also implies that the edge expansion of $H$ is $\Omega(1)$, so given a cut $|A|$, $h'(A) \geq \Omega (|A|) = \Omega(kb)$. Yet under the conditions of this case, the capacity of the cut in $G$ is $O\left(\frac{kb\sqrt{d}}{\log^{1/8}k }\right)$ and this implies the statement. So, the only remaining case is when the conditions of Bourgain’s Junta Theorem are met at least 1 of the 3 conditions is not true. Yet we can apply Lemma  \[lemma:imb\] directly to get that in this case $h'(A) = \omega(|A|)$ and of course $h_K(A) \leq |A|\sqrt{d}$. Suppose the conditions of Bourgain’s Junta Theorem are met, and at least 1 of the 3 inequalities is not true, then the ratio of the minimum cut in $G$ separating $A$ from $K-A$ to the corresponding cut in $H$ is at most $O(\frac{\sqrt{d}\log\log\log d}{\log^2\log d})$. If (\[inequ:b-is-small-1\]) is false, $\log(1/\delta') + \log(1/b) = \log(1/\delta) > \frac{(\log \log^{1/30}d/c)^2}{\log\log 1/\epsilon}=\Omega\left(\frac{\log^2\log d}{\log\log\log d}\right)$. Since $1 / \delta' = O(\log\log d)$, it must be the case that $\log(1/b) = \Omega\left(\frac{\log^2\log d}{\log\log\log d}\right)$. If (\[inequ:b-is-small-2\]) is false, $b < \frac{4^{1/\epsilon}\sqrt\beta\sqrt\epsilon}{\delta'}=O(d^{-1/8}\log^{1/6}d)$, and $\log (1/b) = \Omega(\log d)$. If (\[inequ:b-is-small-3\]) is false, $b < d^{-1/4}\log^{9/8}d$ and $\log (1/b) = \Omega(\log d)$. The minimum of the 3 bounds is the first one. So, $\log(1/b) = \Omega\left(\frac{\log^2\log d}{\log\log\log d}\right)$ if at least 1 of the 3 conditions is false. Applying Lemma \[lemma:imb\], we get that $h'(A) \geq \Omega(|A|\log\frac1b)=\Omega(|A|\frac{\log^2\log d}{\log\log\log d})$. And yet $h_K(A) \leq |A|\sqrt{d}$, and this implies the statement. Combining the cases, this implies that the quality of $H$ is $O(\frac{\sqrt{d}\log\log\log d}{\log^2 \log d})$. The quality of $H$ as a cut-sparsifier for $G$ is $O(d^{1/4})$ [Theorem]{}[thm:sep]{} There is an infinite family of graphs for which the quality of the best cut-sparsifier is $\Omega(\frac{\log^2 \log \log k}{\log\log\log\log k})$ better than the best that a distribution on $0$-extensions can achieve. Improved Constructions via Lifting {#sec:alift} ================================== In this section we give a polynomial time construction for a flow-sparsifier that achieves quality at most the quality of the best flow-sparsifier that can be realized as a distribution over $0$-extensions. Thus we give a construction for flow-sparsifiers (and thus also cut-sparsifier) that achieve quality $O(\frac{\log k}{\log \log k})$. Given that the current best upper bounds on the quality of both flow and cut-sparsifiers are achieved as a distribution over $0$-extensions, the constructive result we present here matches the best known *existential* bounds on the quality of cut or flow-sparsifiers. All previous constructions [@M], [@LM] need to sacrifice some super-constant factor in order to actually construct cut or flow-sparsifiers. We achieve this using a linear program that can be interpreted as a lifting of previous linear programs used in constructive results. Our technique, we believe, is of independent interest: we perform a lifting on an appropriate linear program. This lifting allows us to implicitly enforce a constraint automatically that previously was difficult to enforce, and required an approximate separation oracle rather than an exact separation oracle. There are known ways for implicitly enforcing this constraint using an exponential number of variables, but surprisingly we are able to implicitly enforce this constraint using only polynomially many variables, after just a single lifting operation. The lifting operation that we perform is inspired by Earth-mover relaxations, and makes it a rare example of when an algorithm is actually able to use the Earth-mover constraints, as opposed to the usual use of such constraints in obtaining hardness from integrality gaps. \[thm:lift\] Given a flow sparsifier instance $\mathcal{H} = (G, k)$, there is a polynomial (in $n$ and $k$) time algorithm that outputs a flow sparsifier $H$ of quality $\alpha \leq \alpha'(\mathcal{H})$, where $\alpha'(\mathcal{H})$ is the quality of the best flow sparsifier that can be realized as a distributions over $0$-extensions. We show that the following LP can give a flow-sparsifier with the desired properties: $$\begin{aligned} \min & \alpha \\ \mbox{s.t} & \\ & \begin{array}{l} \begin{array}{rcl} cong_G(\vec{w}) &\leq& \alpha \\ w_{i,j} &=& \displaystyle\sum_{u,v \in V, u\neq v} c(u,v)x^{u,v}_{i,j} \qquad \forall i, j \in K\\ \end{array} \\ \left \{ \begin{array}{llll} x^{u,v}_{i,j} &=& x^{v,u}_{j,i} & \qquad \forall u, v\in V,u\neq v, i,j\in K\\ \displaystyle\sum_{j\in K}x^{u,v}_{i,j} &=& x^u_i & \qquad \forall u, v\in V, u\neq v, i \in K\\ \displaystyle\sum_{i\in K}x^u_i &=& 1 & \qquad \forall u \in V\\ x^i_i &=& 1 &\qquad \forall i \in K\\ x^{u,v}_{i,j} &\geq& 0 &\qquad \forall u, v \in V, u\neq v, i,j \in K\\ \end{array} \right \} \mbox{\large \textbf{Earth-mover Constraints}}\\ \end{array}\end{aligned}$$ The value of the LP is $\alpha \leq \alpha'(\mathcal{H})$. Let $\mathcal{F}$ be the best distribution of 0-extensions. We explicitly give a satisfying assignment for all the variables : $$\begin{aligned} \alpha &=& \alpha'(\mathcal{H})\\ x^u_i &=& \Pr_{f \sim \mathcal{F}}\left[f(u) = i\right]\\ x^{u,v}_{i,j} &=& \Pr_{f \sim \mathcal{F}}\left[f(u) = i \wedge f(v) = j\right]\\ w_{i,j} &=& \sum_{u,v \in V, u\neq v} c(u,v)x^{u,v}_{i,j} \end{aligned}$$ It’s easy to see that the graph $H$ formed by ${\left\{w_{i,j}\right\}}$ is exactly the same as the flow sparsifier obtained from $\mathcal{F}$. So $H$ can be routed in $G$ with conjestion at most $\alpha'$. One can also verify that all the other constraints are satisfied. Thus, the value of the LP is at most $\alpha'(\mathcal{H})$. There are qualitatively two types of constraints that are associated with good flow-sparsifiers $H$: All flows routable in $H$ with congestion at most $1$ must be routable in $G$ with congestion at most $\alpha$. Actually, there is a notion of a hardest flow feasible in $H$ to route in $G$: the flow that saturates all edges in $H$ (i.e. $\vec{H}$). So the constraint that all flows routable in $H$ with congestion at most $1$ be also routable in $G$ with congestion at most $\alpha$ can be enforced by ensuring that $\vec{H}$ can be routed in $G$ with congestion at most $\alpha$. This constraint can be written using an infinite number of linear constraints on $H$ associated with the dual to a maximum concurrent flow problem, and in fact an oracle for the maximum concurrent flow problem can serve as a separation oracle for these constraints. The second set of constraints associated with good flow-sparsifiers are that all flows routable in $G$ with congestion at most $1$ can also be routed in $H$ with congestion at most $1$. This constraint can also be written as an infinite number of linear constraints on $H$, but no polynomial time separation oracle is known for these constraints. Instead, previous work relied on using oblivious routing guarantees to get an approximate separation oracle for this problem. Intuitively, the constraint that all flows routable in $G$ can be routed in $H$ can be enforced in a number of ways. The strategy outlined in the preceding paragraph attempts to incorporate these constraints into the linear program. Alternatively, one could enforce that $H$ be realized as a distribution over $0$-extensions $G_f$. This would automatically enforce that all flows routable in $G$ would also be routable in $H$. However, this would require a variable for each $0$-extension $G_f$, and there would be exponentially many such variables. Yet the above linear programming formulation is a hybrid between these two approaches. In previous linear programming formulations, the sparsifier $H$ was not required to be explicitly generated from $G$, hence the need to enforce that it actually be a flow-sparsifer. When there is a variable for each $0$-extension, then $H$ is forced to be generated from $G$ and this constraint is implicitly satisfied. Yet just enforcing the Earth-mover constraints, as above, actually forces $H$ to have enough structure inherited from $G$ that $H$ is automatically a flow-sparsifier! This is the reason that we are able to get improved constructive results. To re-iterate, a simple lifting (corresponding to the Earth-mover constraints) does actually impose enough structure on $H$, that we can implicitly impose the constraint that $H$ be a flow-sparsifier without using exponentially many variables for each $0$-extension $G_f$! ${\left\{w_{i,j}:i,j \in K, i<j \right\}}$ is a flow sparsifier of quality $\alpha$. Let $H$ be the capacitated graph on $K$ formed by ${\left\{w_{i,j}\right\}}$. The LP system guarantees that $H$ can be routed in $G$ with conjestion at most $\alpha$, and thus we only need to show the other direction: every multi-commodity flow in $G$ with end points in $K$ can be routed in $H$ with conjestion at most 1. Consider a multi-commoditiy flow ${\left\{f_{i,j}:i, j \in K, i < j\right\}}$ that can be routed in $G$. By the LP duality, we have $$\sum_{u < v}c(u,v)\delta(u,v) \geq \sum_{i < j}f_{i,j}\delta(i,j)$$ for every metric $\delta$ over $V$. Let $\delta'$ be any metric over $K$, then $$\begin{aligned} \sum_{i<j}\delta'(i,j)w_{i,j} = \sum_{i<j}\delta'(i,j)\sum_{u\neq v}c(u,v)x^{u,v}_{i,j}=\sum_{u<v}c(u,v)\sum_{i\neq j}x^{u,v}_{i,j}\delta'(i,j)\geq \sum_{u < v}c(u,v)EMD_{\delta'}(x^u, x^v).\end{aligned}$$ Define $\delta(u,v) = EMD_{\delta'}(x^u, x^v)$. Clearly, $\delta$ is a metric over $V$ and $\delta(i,j) = \delta'(i,j)$ for every $i, j \in K$. We have $$\begin{aligned} \sum_{i<j}\delta'(i,j)w_{i,j} \geq \sum_{u < v}c(u,v)\delta(u,v)\geq \sum_{i<j}f_{i,j}\delta(i,j) = \sum_{i<j}f_{i,j}\delta'(i,j)\end{aligned}$$ We have proved that $\sum_{i<j}\delta'(i,j)w_{i,j}\geq \sum_{i<j}f_{i,j}\delta'(i,j)$ for every metric $\delta'$ over $K$. By the LP duality, $f$ can be routed in $H$ with conjestion 1. The LP can be solved in polynomial (in $n$ and $k$) time. The LP contains polynomial number of variables and hence it is sufficient to give a separation oracle between a given point and the polytope defined by the LP. All constraints except whether or not $cong_G(\vec{H}) \leq \alpha$ can be directly checked, and for this remaining constraint the exact separation oracle is given by solving a maximum concurrent flow problem. Abstract Integrality Gaps and Rounding Algorithms ================================================= In this section, we give a generalization of the hierarchical decompositions constructed in [@R]. This immediately yields an $O(\log k)$-competitive Steiner oblivious routing scheme, which is optimal. Also, from our hierarchical decompositions we can recover the $O(\log k)$ bound on the flow-cut gap for maximum concurrent flows given in [@LLR] and [@AR]. Additionally, we can also give an $O(\log k)$ flow-cut gap for the maximum multiflow problem, which was originally given in [@GVY]. This even yields an $O(\log k)$ flow-cut gap for the relaxation for the requirement cut problem, which is given in [@NR]. In fact, we will be able to give an abstract framework to which the results in this section apply (and yield $O(\log k)$ flow-cut gaps for), and in this sense we are able to help explain the intrinsic robustness of the worst-case ratio between integral cover compared to fractional packing problems in graphs. Philosophically, this section aims to answer the question: Do we really need to pay a price in the approximation guarantee for reducing to a graph on size $k$? In fact, as we will see, there is often a way to combine both the reduction to a graph on size $k$ and the rounding needed to actually obtain a flow-cut gap on the reduced graph, into one step! This is exactly the observation that leads to our improved approximation guarantee for Steiner oblivious routing. $0$-Decomposition ----------------- We extend the notion of $0$-extensions, which we previously defined, to a notion of $0$-decompositions. Intuitively, we would like to combine the notion of a $0$-extension with that of a decomposition tree. Again, given a $0$-extension $f$, we will denote $G_f$ as the graph on $K$ that results from contracting all sets of nodes mapped to any single terminal. Then we will use $c_f$ to denote the capacity function of this graph. Given a tree $T$ on $K$, and a $0$-extension $f$, we can generate a $0$-decomposition $G_{f, T} = (K, E_{f, T})$ as follows: The only edges present in $G_{f, T}$ will be those in $T$, and for any edge $(a, b) \in E(T)$, let $T_a, T_b$ be the subtrees containing $a, b$ respectively that result from deleting $(a, b)$ from $T$. Then $c_{f, T}(a,b)$ (i.e. the capacity assigned to $(a, b)$ in $G_{f, T}$ is: $c_{f, T}(a,b) = \sum_{u, v \in K \mbox{ and } u \in T_a, v \in T_b} c_f(u, v)$. Let $\Lambda$ denote the set of $0$-extensions, and let $\Pi$ denote the set of trees on $K$. For any distribution $\gamma$ on $\Lambda \times \Pi$, and for any demand $\vec{d} \in \Re^{K \choose 2}$, $cong_H(\vec{d}) \leq cong_G(\vec{d})$ where $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$ Clearly for all $f, T$, $\gamma(f, T) \vec{d}$ is feasible in $\gamma(f, T) G_f$ (because contracting edges only makes routing flow easier), and so because $G_{f, T}$ is a hierarchical decomposition tree for $G_f$, then it follows that $\gamma(f, T) \vec{d}$ is also feasible in $G_{f, T}$. Given any distribution $\gamma$ on $\Lambda \times \Pi$, let $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$. Then $\sup_{\vec{d} \in \Re^{K \choose 2}} \frac{cong_G(\vec{d})}{cong_H(\vec{d})} = cong_G(\vec{H})$ \[thm:zdconst\] There is a polynomial time algorithm to construct a distribution $\gamma$ on $\Lambda \times \Pi$ such that $cong_G(\vec{H}) = O(\log k)$ where $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$. We want to show that there is a distribution $\gamma$ on $\Lambda \times \Pi$ such that $cong_G(\vec{H}) = O(\log k)$. This will yield a generalization of [@R]. So as in [@M], we set up a zero-sum game in which the first player chooses $f, T$ and plays $G_{f, T}$. The second player then chooses some metric space $d: K \times K \rightarrow \Re^+$ s.t. there is some extension of $d$ to a metric space on $V$ s.t. $\sum_{(u,v) \in E} d(u, v) c(u, v) \leq 1$. Then the first player loses $\sum_{(a, b)} c_{f, T}(a, b) d(a, b)$, which we will refer to as the cost of the metric space $d$ against $G_{f, T}$. It follows immediately from [@M] or [@LM] that a bound of $O(\log k)$ on the game value will imply our desired structural result. The game value $\nu$ is $O(\log k)$ We consider an arbitrary strategy $\lambda$ for the second player, which is a distribution on metric spaces $d$ that can be realized in $G$ with distance $\times$ capacity units at most $1$. In fact, if we take the average metric space $\Delta = \sum_{d} \lambda(d) d$, then this metric space can also be realized in $G$ with at most $1$ unit of distance $\times$ capacity. So we can bound the game value by showing that for all metric spaces $\Delta$ that can be realized with distance $\times$ capacity units at most $1$, there is a $0$-decomposition $G_{f, T}$ for which the cost against $\Delta$ is at most $O(\log k)$. We can prove this by a randomized rounding procedure that is almost the same as the rounding procedure in [@FRT]: Scaling up the metric space, we can assume that all distances in the extension of $\Delta$ to a metric space on $V$ have distance at least $1$, and we assume $2^{\delta}$ is an upper bound on the diameter of the metric space. Then we need to first choose a $0$-extension $f$ for which the cost against $\Delta$ is $O(\log k)$ times the cost of realizing $\Delta$ in $G$. We do this as follows: Choose a random permutation $\pi(1), \pi(2), ..., \pi(k)$ of $K$ Choose $\beta$ uniformly at random from $[1, 2]$ $D_\delta \leftarrow \{V\}, i \leftarrow \delta -1$ $\beta_i \leftarrow 2^{i - 1} \beta$ Create a new cluster of all unassigned vertices in $S$ closer than $\beta_i$ to $\pi(\ell)$ $i \leftarrow i - 1$ Then, exactly as in [@FRT], we can construct a decomposition tree from the rounding procedure. The root node is $V$ corresponding to the partition $D_\delta$, and the children of this node are all sets in the partition $D_{\delta - 1}$. Each successive $D_i$ is a refinement of $D_{i + 1}$, so each set of $D_i$ is made to be a child of the corresponding set in $D_{i + 1}$ that contains it. At each level $i$ of this tree, the distance to the layer above is $2^{i}$, and one can verify that this tree-metric associated with the decomposition tree dominates the original metric space $\Delta$ restricted to the set $K$. Note that the tree metric does not dominate $\Delta$ on $V$, because there are some nodes which are mapped to the same leaf node in this tree, and correspondingly have distance $0$. If we consider any edge $(u, v)$, we can bound the expected distance in this tree metric from the leaf node containing $u$ to the leaf-node containing $v$. In fact, this expected distance is only a function of the metric space $\Delta$ restricted to $K \cup \{u, v\}$. Accordingly, for any $(u, v)$, we can regard the metric space that generates the tree-metric as a metric space on just $k + 2$ points. Formally, the rounding procedure in [@FRT] is: Choose a random permutation $\pi(1), \pi(2), ..., \pi(n)$ of $V$ Choose $\beta$ uniformly at random from $[1, 2]$ $D_\delta \leftarrow \{V\}, i \leftarrow \delta -1$ $\beta_i \leftarrow 2^{i - 1} \beta$ Create a new cluster of all unassigned vertices in $S$ closer than $\beta_i$ to $\pi(\ell)$ $i \leftarrow i - 1$ Formally, [@FRT] proves a stronger statement than just that the expected distance (according to the tree-metric generated from the above rounding procedure) is $O(\log n)$ times the original distance. We will say that $u, v$ are split at level $i$ if these two nodes are contained in different sets of $D_i$. Let $X_i$ be the indicator variable for this event. Then the distance in the tree-metric $\Delta_T$ generated from the above rounding procedure is $\Delta_T(u, v) = \sum_i 2^{i + 1} X_i$. In fact, [@FRT] proves the stronger statement that this is true even if $u, v$ are not in the metric space (i.e. $u, v \notin V$) but are always grouped in the cluster which they would be if they were in fact in the set $V$ (provided of course that they can be grouped in such a cluster). More formally, we set $V' = V \cup \{u, v\}$ and if the step “Create a new cluster of all unassigned vertices in $S$ closer than $\beta_i$ to $\pi(\ell)$” is replaced with “Create a new cluster of all unassigned vertices in V’ in $S$ closer than $\beta_i$ to $\pi(\ell)$”, then [@FRT] actually proves in this more general context that $$\sum_{i} 2^{i + 1} X_i \leq O(\log n) \Delta(u, v)$$ When we input the metric space $\Delta$ restricted to $K$ into the above rounding procedure (but at each clustering stage we consider all of $V$) then we get exactly our rounding procedure. So then the main theorem in [@FRT] (or rather our restatement of it) is (If $\Delta_T$ is the tree-metric generated from the above rounding procedure) [@FRT] For all $u, v$, $E[\Delta_T(u, v)] \leq O(\log k) \Delta(u, v)$. So at the end of the rounding procedure, we have a tree in which each leaf correspond to a subset of $V$ that contains at most $1$ terminal. We are given a tree-metric $\Delta_T$ on $V$ associated with the output of the algorithm, and this tree-metric has the property that $\sum_{(u, v) \in E} c(u, v) \Delta_T(u, v) \leq O(\log k)$. We would like to construct a tree $T'$ from $T$ which has only leafs which contain exactly one terminal. We first state a simple claim that will be instructive in order to do this:  \[claim:tree\] Given a tree metric $\Delta_T$ on a tree $T$ on $K$, $cost(G_{f, T}, \Delta_T) = cost(G_f, \Delta_T)$. The graph $G_{f, T}$ can be obtained from $G_f$ by iteratively re-routing some edge $(a, b) \in E_f$ along the path connecting $a$ and $b$ in $T$ and adding $c_f(a, b)$ capacity to each edge on this path, and finally deleting the edge $(a, b)$. The original cost of this edge is $c(a, b) \Delta_T(a, b)$, and if $a = p_1, p_2, ..., p_r = b$ is the path connecting $a$ and $b$ in $T$, the cost after performing this operation is $c(a, b) \sum_{i = 1}^{r-1} \Delta_T(p_i, p_{i+1}) = c(a, b) \Delta_T(a, b)$ because $\Delta_T$ is a tree-metric. We can think of each edge $(u, v)$ as being routed between the deepest nodes in the tree that contain $u$ and $v$ respectively, and the edge pays $c(u, v)$ times the distance according to the tree-metric on this path. Then we can perform the following procedure: each time we find a node in the tree which has only leaf nodes as children and none of these leaf nodes contains a terminal, we can delete these leaf nodes. This cannot increase the cost of the edges against the tree-metric because every edge (which we regard as routed in the tree) is routed on the same, or a shorter path. After this procedure is done, every leaf node that doesn’t contain a terminal contains a parent $p$ that has a terminal node $a$. Suppose that the deepest node in the tree that contains $a$ is $c$ We can take this leaf node, and delete it, and place all nodes in the tree-node $c$. This procedure only affects the cost of edges with one endpoint in the leaf node that we deleted, and at most doubles th e cost paid by the edge because distances in the tree are geometrically decreasing. So if we iteratively perform the above steps, the total cost after performing these operations is at most $4$ times the original cost. And it is easy to see that this results in a natural $0$-extension in which each node $u$ is mapped to the terminal corresponding to the deepest node that $u$ is contained in. Each edge pays a cost proportional to a tree-metric distance between the endpoints of the edge. So we know that $cost(G_f, \Delta_T) = O(\log k)$ because the cost increased by at most a factor of $4$ from iteratively performing the above steps. Yet using the above Claim, we get a $0$-extension $f$ and a tree $T$ such that $cost(G_{f, T}, \Delta_T) = O(\log k)$ and because $\Delta_T$ dominates $\Delta$ when restricted to $K$, this implies that $cost(G_{f, T}, \Delta) \leq cost(G_{f, T}, \Delta_T) = O(\log k)$ and this implies the bound on the game value. In turn, using the arguments in [@LM], implies:  \[thm:zdexist\] There is a distribution $\gamma$ on $\Lambda \times \Pi$ such that $cong_G(\vec{H}) = O(\log k)$ where $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$. Also, using the arguments in [@R] (because each $G_{f, T}$ is a tree and hence has a unique routing scheme), this gives us an $O(\log k)$-competitive Steiner oblivious routing scheme: Given $G = (V, E)$ and $K \subset V$, there is a set of unit flows for all $a, b \in K$ that sends a unit flow from $a$ to $b$, such that given any demand restricted to $K$, $\vec{d}$, the congestion incurred by this oblivious routing scheme is $O(\log k)$ times the minimum congestion routing of $\vec{d}$. Actually, the above theorem can be made constructive directly using the techniques in [@R], which build on [@Y]. We will not repeat the proof, instead we note the only minor difference in the proof. Let $\cR$ denote the set of pairs $(G_{f, T}, g)$ where $G_{f, T}$ is a $0$-decomposition of $G$, and $g$ is a function from edges in $G_{f, T}$ to paths in $g$ so that an edge $(a, b)$ in $G_{f, T}$ is mapped to a path connecting $a$ and $b$ in $G$. Given a metric space $\delta$ on $V$, we can define the notion of the cost of a $(G_{f, T}, g)$ against $\delta$: $$cost((G_{f, T}, g), \delta) = \sum_{(a, b) \in E(G_{f, T})} \sum_{(u, v) \in g(a,b)} c_{f,T}(a, b) \delta(u, v) = \sum_{(a, b) \in E(G_{f, T})} c_{f, T}(a, b) \delta(g(a, b))$$  \[cor:zdexist\] For any metric $\delta$ on $V$, there is some $(G_{f, T}, g) \in \cR$ such that: $$cost((G_{f, T}, g), \delta) \leq O(\log k) \sum_{(u, v)} c(u,v) \delta(u, v)$$ We can apply Theorem \[thm:zdexist\] which implies that there is a distribution $\mu$ on $\cR$ s.t. for all edges $e \in E$, $$E_{(G_{f, T}, g) \leftarrow \mu}[\sum_{(a, b) \in E(G_{f, T}) \mbox{ s.t. } e \ni g(a, b)} c_{f,T}(a,b)] \leq O(\log k) c(e)$$ because we can take the optimal routing of $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$ in $G$, which requires congestion at most $O(\log k)$ and if we compute a path decomposition of the routing schemes of each $G_{f, T}$ in the support of $\gamma$, we can use these to express the routing scheme as a convex combination of pairs from $\cR$. So we can use an identical proof as in [@R] to actually construct a distribution $\gamma$ on $0$-decompositions s.t. for $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$ we have $cong_G(\vec{H}) = O(\log k)$. All we need to modify is the actual packing problem. In [@R], the goal of the packing problem is to pack a convex combination of decomposition trees into the graph $G$ s.t. the expected relative load on any edge is at most $O(\log n)$. Here our goal is to pack a convex combination of $0$-decompositions into $G$. So instead of writing a packing problem over decomposition trees, we write a packing problem over pairs $(G_{f, T}, g) \in \cR$ and the goal is to find a convex combination of these pairs s.t. the relative load on any edge is $O(\log k)$. [@R] find a polynomial time algorithm by relating the change (when a decomposition tree is added to the convex combination) of the worst-case relative load (actually a convex function that dominates this maximum) to the cost of a decomposition tree against a metric. Analogously, as long as we can always (for any metric space $\delta$ on $V$) find a pair $(G_{f, T}, g)$ as in Corollary \[cor:zdexist\] an identical proof as in [@R] will give us a constructive version of Theorem \[thm:zdexist\]. And we can do this by again using the Theorem due to [@FHRT] (which we restated above in a more convenient notation for our purposes). This will give us a $0$-decomposition $G_{f, T}$ for which $\sum_{(a, b)} c_{f, T}(a,b) \delta(a, b) \leq O(\log k) \sum_{(u, v)} c(u,v) \delta(u, v)$ and we still need to choose a routing of $G_{f, T}$ in $G$. We can do this in a easy way: for each edge $(a, b)$ in $G_{f, T}$, just choose the shortest path according to $\delta$ connecting $a$ and $b$ in $G$. The length of this path will be $\delta(a, b)$, and so we have that $cost((G_{f, T}, g), \delta) \leq O(\log k) \sum_{(u, v)} c(u,v) \delta(u, v)$ as desired. Then using the proof in [@R] in our context, this immediately yields Theorem \[thm:zdconst\] Applications ------------ Also, as we noted, this gives us an alternate proof of the main results in [@LLR], [@AR] and [@GVY]. We first give an abstract framework into which these problems all fit: [Definition]{}[def:gpp]{} We call a fractional packing problem $P$ a graph packing problem if the goal of the dual covering problem $D$ is to minimize the ratio of the total units of distance $\times$ capacity allocated in the graph divided by some monotone increasing function of the distances between terminals. Let $ID$ denote the integral dual graph covering problem. To make this definition seem more natural, we demonstrate that a number of well-studied problems fit into this framework. [@LR], [@LLR], [@AR] P: maximum concurrent flow; ID: generalized sparsest cut Here we are given some demand vector $\vec{f} \in \Re^{K \choose 2}$, and the goal is to maximize the value $r$ such that $r \vec{f}$ is feasible in $G$. Then the dual to this problem corresponds to minimizing the total distance $\times$ capacity units, divided by $\sum_{(a, b)} \vec{f}_{a, b} d(a, b)$, where $d$ is the induced semi-metric on $K$. The function in the denominator is clearly a monotone increasing function of the distances between pairs of terminals, and hence is an example of what we call a graph packing problem. The generalized sparsest cut problem corresponds to the “integral” constraint on the dual, that the distance function be a cut metric. [@GVY] P: maximum multiflow; ID: multicut Here we are given some pairs of terminals $T \subset { K \choose 2}$, and the goal is to find a flow $\vec{f}$ that can be routed in $G$ that maximizes $\sum_{(a, b) \in T } \vec{f}_{a, b}$. The dual to this problem corresponds to minimizing the total distance $\times$ capacity units divided by $\min_{(a, b) \in T} \{d(a, b) \} $, again where where $d$ is the induced semi-metric on $K$. Also the function in the denominator is again a monotone increasing function of the distances between pairs of terminals, and hence is another an example of what we call a graph packing problem. The multicut problem corresponds to the “integral” constraint on the dual that the distance function be a partition metric. ID: Steiner multi-cut ID: Steiner minimum-bisection [@NR] P: multicast routing; ID: requirement cut This is another partitioning problem, and the input is again a set of subsets $\{R_i\}_i$. Each subset $R_i$ is also given at requirement $r_i$, and the goal is to minimize the total capacity removed from $G$, in order to ensure that each subset $R_i$ is contained in at least $r_i$ different components. Similarly to the Steiner multi-cut problem, the standard relaxation for this problem is to minimize the total amount of distance $\times$ capacity units allocated in $G$, s.t. for each $i$ the minimum spanning tree $T_i$ (on the induced metric on $K$) on every subset $R_i$ has total distance at least $r_i$. Let $\Pi_i$ be the set of spanning trees on the subset $R_i$. Then we can again cast this relaxation in the above framework because the goal is to minimize the total distance $\times$ capacity units divided by $\min_i \{ \frac{ \min_{T \in \Pi_i} \sum_{(a, b) \in T} d(a, b)}{r_i} \} $. The dual to this fractional covering problem is actually a common encoding of multicast routing problems, and so these problems as well are examples of graph packing problems. Here the requirement cut problem corresponds to the “integral” constraint that the distance function be a partition metric. In fact, one could imagine many other examples of interesting problems that fit into this framework. One can regard maximum multiflow as an *unrooted* problem of packing an edge fractionally into a graph $G$, and the maximum concurrent flow problem is a rooted graph packing problem where we are given a fixed graph on the terminals (corresponding to the demand) and the goal is to pack as many copies as we can into $G$ (i.e. maximizing throughput). The dual to the Steiner multi-cut is more interesting, and is actually a combination of rooted and unrooted problems where we are given subset $R_i$ of terminals, and the goal is to maximize the total spanning trees over the sets $R_i$ that we pack into $G$. This is a combination of a unrooted (each spanning tree on any set $R_i$ counts the same) and a rooted problem (once we fix the $R_i$, we need a spanning tree on these terminals). Then any other flow-problem that is combinatorially restricted can also be seen to fit into this framework. As an application of our theorem in the previous section, we demonstrate that all graph packing problems can be reduced to graph packing problems on trees at the loss of an $O(\log k)$. So whenever we are given a bound on the ratio of the integral covering problem to the fractional packing problem on trees of say $C$, this immediately translates to an $O(C \log k)$ bound in general graphs. So in some sense, these embeddings into distributions on $0$-decompositions helps explain the intrinsic robustness of graph packing problems, and why the integrality gap always seems to be $O(\log k)$. In fact, since we can actually construct these distributions on $0$-decompositions, we obtain an [Abstract Rounding Algorithm]{} that works for general graph packing problems. [Theorem]{}[thm:agpp]{} There is a polynomial time algorithm to construct a distribution $\mu$ on (a polynomial number of) trees on the terminal set $K$, s.t. $$E_{T \leftarrow \mu}[OPT(P, T)] \leq O(\log k) OPT(P, G)$$ and such that any valid integral dual of cost $C$ (for any tree $T$ in the support of $\mu$) can be immediately transformed into a valid integral dual in $G$ of cost at most $C$. We first demonstrate that the operations we need to construct a $0$-decomposition only make the dual to a graph packing problem more difficult: Let $\nu(G, K)$ be the optimal value of a dual to a graph packing problem on $G = (V, E)$, $K \subset V$. Replacing any edge $(u, v)$ of capacity $c(u, v)$ with a path $u = p_1, p_2, ..., p_r = v$, deleting the edge $(u, v)$ and adding $c(u, v)$ units of capacity along the path does not decrease the optimal value of the dual. We can scale the distance function of the optimal dual so that the monotone increasing function of the distances between terminals is exactly $1$. Then the value of the dual is exactly the total capacity $\times$ distance units allocated. If we maintain the same metric space on the vertex set $V$, then the monotone increasing function of terminal distances is still exactly $1$ after replacing the edge $(u, v)$ by the path $u = p_1, p_2, ..., p_r = v$. However this replacement does change the cost (in terms of the total distance $\times$ capacity units). Deleting the edge reduces the cost by $c(u, v) d(u, v)$, and augmenting along the path increases the cost by $c(u, v) \sum_{i = 1}^{r-1} d(p_i, p_{i + 1})$ which, using the triangle inequality, is at least $c(u, v) d(u, v)$. Suppose we join two nodes $u, v$ (s.t. not both of $u, v$ are terminals) into a new node $u'$, and replace each edge into $u$ or $v$ with a corresponding edge of the same capacity into $u'$. Then the optimal value of the dual does not decrease. We can equivalently regard this operation as placing an edge of infinite capacity connecting $u$ and $v$, and this operation clearly does not change the set of distance functions for which the monotone increasing function of the terminal distances is at least $1$. And so this operation can only increase the cost of the optimal dual solution. We can obtain any $0$-decomposition $G_{f, T}$ from some combination of these operations. So we get that for any $f, T$: $\nu(G_{f, T}, K) \geq \nu(G, K)$ Let $\gamma$ be the distribution on $\Lambda \times \Pi$ s.t. $H = \sum_{f \in \Lambda, T \in \Pi} \gamma(f, T) G_{f, T}$ and $cong_{G}(\vec{H}) \leq O(\log k)$. $E_{(f, T) \leftarrow \gamma}[\nu(G_{f, T}, K)] \leq O(\log k) \nu(G, K)$. We know that there is a metric $d$ on $V$ s.t. $\sum_{(u, v)} c(u, v) d(u, v) = \nu(G, K)$ and that the monotone increasing function of $d$ (restricted to $K$) is at least $1$. We also know that there is a simultaneous routing of each $\gamma(f, T) G_{f, T}$ in $G$ so that the congestion on any edge in $G$ is $O(\log k)$. Then consider the routing of one such $\gamma(f, T) G_{f, T}$ in this simultaneous routing. Each edge $(a, b) \in E_{f, T}$ is routed to some distribution on paths connecting $a$ and $b$ in $G$. In total $\gamma(f, T) c_f(a, b)$ flow is routed on some distribution on paths, and consider a path $p$ that carries $C(p)$ total flow from $a$ to $b$ in the routing of $\gamma(f, T) G_{f, T}$. If the total distance along this path is $d(p)$, we increment the distance $d_{f, T}$ on the edge $(a, b)$ in $G_{f, T}$ by $\frac{d(p) C(p)}{\gamma(f, T) c_{f, T}(a, b)}$, and we do this for all such paths. We do this also for each $(a, b)$ in $G_{f, T}$. If $d_{f, T}$ is the resulting semi-metric on $G_{f, T}$, then this distance function dominates $d$ restricted to $K$, because the distance that we allocate to the edge $(a, b)$ in $G_{f, T}$ is a convex combination of the distances along paths connecting $a$ and $b$ in $G$, each of which is at least $d(a, b)$. So if we perform the above distance allocation for each $G_{f, T}$, then each resulting $d_{f, T}, G_{f, T}$ pair satisfies the condition that the monotone increasing function of terminal distances ($d_{f, T}$) is at least $1$. But how much distance $\times$ capacity units have we allocated in expectation? $$E_{(f, T) \leftarrow \gamma}[\nu(G_{f, T}, K)] \leq \sum_{f, T} \gamma(f, T) \sum_{(a, b) \in E_{f, T}} c_{f, T}(a, b) d_{f, T}(a, b)$$ We can re-write $$\sum_{f, T} \gamma(f, T) \sum_{(a, b) \in E_{f, T}} c_{f, T}(a, b) d_{f, T}(a, b) = \sum_{(a, b) \in E} flow_{\vec{H}}(e) d(a, b) \leq cong_{G}(\vec{H}) \sum_{(a, b) \in E} c(a, b) d(a, b) \leq O(\log k) \nu(G, K)$$ And this implies: [Theorem]{}[thm:gpp]{} For any graph packing problem $P$, the maximum ratio of the integral dual to the fractional primal is at most $O(\log k)$ times the maximum ratio restricted to trees. And since we can actually construct such a distribution on $0$-decompositions in polynomial time, using Theorem \[thm:zdconst\], this actually gives us an [Abstract Rounding Algorithm]{}: We can just construct such a distribution on $0$-decompositions, sample one at random, apply a rounding algorithm to the tree to obtain a integral dual on the $0$-decomposition $G_{f, T}$ within $O(\log k)C$ times the value of the primal packing problem on $G$. This integral dual on the $0$-decomposition $G_{f, T}$ can then be easily mapped back to an integral dual on $G$ at no additional cost precisely because we can set the distance in $G$ of any edge $(a, b)$ to be the tree-distance according to the integral dual on $G_{f, T}$ between $a$ and $b$. Using Claim \[claim:tree\], this implies that the cost of the dual in $G_f$ is equal the cost of the dual in $G_{f, T}$. And we can choose an integral dual $\delta'$ in $G$ in which for all $u, v$, $\delta'(u, v) = \delta(f(u), f(v))$ and the cost of this dual $\delta'$ on $G$ is exactly the cost of $G_f$ on $\delta$. And so we have an integral dual solution in $G$ of cost at most $O(\log k) C$ times the cost of the fractional primal packing value in $G$, where $C$ is the maximum integrality gap of the graph packing problem restricted to trees. This yields our [Abstract Rounding Algorithm]{}: [Theorem]{}[thm:agpp]{} There is a polynomial time algorithm to construct a distribution $\mu$ on (a polynomial number of) trees on the terminal set $K$, s.t. $$E_{T \leftarrow \mu}[OPT(P, T)] \leq O(\log k) OPT(P, G)$$ and such that any valid integral dual of cost $C$ (for any tree $T$ in the support of $\mu$) can be immediately transformed into a valid integral dual in $G$ of cost at most $C$. If there is a $C$-approximation algorithm for a graph partitioning problem restricted to trees, then there is an $O(C\log k)$ approximation algorithm for the graph partitioning problem in general graphs. So, there is a natural, generic algorithm associated with this theorem : Decompose $G$ into an $O(\log k)$-oblivious distribution $\mu$ of 0-decompostion trees; Randomly select a tree $G_{f,\tau}$ from the distribution $\mu$; Solve the problem on the tree $G_{f, \tau}$, let $\delta$ be the metric the algorithm output; Return $(\delta, f)$. For example, this gives a generic algorithm that achieves an $O(\log k)$ guarantee for *both* generalized sparsest cut and multicut. The previous techniques for rounding a fractional solution to generalized sparsest cut [@LLR], [@AR] rely on metric embedding results, and the techniques for rounding fractional solutions to multicut [@GVY] rely on purely combinatorial, region-growing arguments. Yet, through this theorem, we can give a unified rounding algorithm that achieves an $O(\log k)$ guarantee for both of these problems, and more generally for graph packing problems (whenever the integrality gap restricted to trees is a constant). Acknowledgments =============== We would like to thank Swastik Kopparty, Ryan O’Donnell and Yuval Rabani for many helpful discussions. Harmonic Analysis {#sec:aha} ================= We consider the group $F_2^d = \{-1, +1\}^d$ equipped with the group operation $s \circ t = [s_1 * t_1, s_2 * t_2, ... s_d * t_d] \in F_2^d$. Any subset $S \subset [d]$ defines a character $\chi_S(x) = \prod_{i \in S} x_i : F_2^d \rightarrow \{-1, +1\}$. See [@O] for an introduction to the harmonic analysis of Boolean functions. Then any function $f: \{-1, +1\}^d \rightarrow \Re$ can be written as: $$f(x) = \sum_S \hat{f}_S \chi_S(x)$$ For any $S, T \subset [d]$ s.t. $S \neq T$, $E_x [\chi_S(x) \chi_T(x)] = 0$ For any $p > 0$, we will denote the $p$-norm of $f$ as $||f||_p = \Big( E_x[f(x)^p] \Big)^{1/p}$. Then \[Parseval\] $$\sum_S \hat{f}_S^2 = E_x[f(x)]^2] = ||f||_2^2$$ Given $-1 \leq \rho \leq 1$, Let $y \sim_\rho x$ denote choosing $y$ depending on $x$ s.t. for each coordinate $i$, $E[y_i x_i] = \rho$. Given $-1 \leq \rho \leq 1$, the operator $T_\rho$ maps functions on the Boolean cube to functions on the Boolean cube, and for $f: \{-1, +1\}^d \rightarrow \Re$, $T_\rho(f(x)) = E_{y \sim_\rho x} [f(y)]$. $T_\rho(\chi_S(x)) = \chi_S(x) \rho^{|S|}$ In fact, because $T_\rho$ is a linear operator on functions, we can use the Fourier representation of a function $f$ to easily write the effect of applying the operator $T_\rho$ to the function $f$: $T_\rho(f(x)) = \sum_S \rho^{|S|} \hat{f}_S \chi_S(x)$ The *Noise Stability* of a function $f$ is $NS_{\rho}(f) = E_{x, y \sim_{\rho} x}[f(x) f(y)]$ $NS_\rho(f) = \sum_S \rho^{|S|} \hat{f}_S^2$ \[Hypercontractivity\] [@Bon] [@Bec] For any $q \geq p \geq 1$, for any $\rho \leq \sqrt{\frac{p-1}{q-1}}$ $$||T_\rho f||_q \leq ||f||_p$$ A statement of this theorem is given in [@O] and [@DFKO] for example. A function $g: \{-1, +1\}^d \rightarrow \Re$ is a $j$-junta if there is a set $S \subset [d]$ s.t. $|S| \leq j$ and $g$ depends only on variables in $S$ - i.e. for any $x, y \in F_2^d$ s.t. $\forall_{i \in S} x_i = y_i$ we have $g(x) = g(y)$. We will call a function $f$ an $(\epsilon, j)$-junta if there is a function $g: \{-1, +1\}^d \rightarrow \Re$ that is a $j$-junta and $Pr_x[f(x) \neq g(x)] \leq \epsilon$. We will use a quantitative version of Bourgain’s Junta Theorem [@Bou] that is given by Khot and Naor in [@KN]: \[Bourgain\] [@Bou], [@KN] Let $f \{-1, +1\}^d \rightarrow \{-1, +1\}$ be a Boolean function. Then fix any $\epsilon, \delta \in (0, 1/10)$. Suppose that $$\sum_S (1-\epsilon)^{|S|} \hat{f}_S^2 \geq 1 - \delta$$ then for every $\beta > 0$, $f$ is a $$\Big (2^{c \sqrt{\log 1/\delta \log \log 1/\epsilon}}\Big ( \frac{\delta}{\sqrt{\epsilon}} + 4^{1/\epsilon} \sqrt{\beta}\Big), \frac{1}{\epsilon \beta} \Big ) \mbox{-junta}$$ This theorem is often described as mysterious, or deep, and has lead to some breakthrough results in theoretical computer science [@KN], [@KV] and is also quite subtle. For example, this theorem crucially relies on the property that $f$ is a Boolean function, and in more general cases only much weaker bounds are known [@DFKO]. [^1]: moses@cs.princeton.edu, Center for Computational Intractability, Department of Computer Science, Princeton University, supported by NSF awards MSPA-MCS 0528414, CCF 0832797, and AF 0916218 [^2]: ftl@math.mit.edu, Mathematics Department, MIT and Akamai Technologies, Inc. [^3]: shili@cs.princeton.edu, Center for Computational Intractability, Department of Computer Science, Princeton University, supported by NSF awards MSPA-MCS 0528414, CCF 0832797, and AF 0916218. [^4]: moitra@mit.edu, Computer Science and Artificial Intelligence Laboratory, MIT, This research was supported in part by a Fannie and John Hertz Foundation Fellowship. Part of this work was done while the author was visiting Princeton University. [^5]: Recently, it has come to our attention that, independent of and concurrent to our work, Makarychev and Makarychev, and independently, Englert, Gupta, Krauthgamer, Raecke, Talgam and Talwar obtained results similar to some in this paper.

--- abstract: 'We address the problem of anomaly detection in videos. The goal is to identify unusual behaviours automatically by learning exclusively from normal videos. Most existing approaches are usually data-hungry and have limited generalization abilities. They usually need to be trained on a large number of videos from a target scene to achieve good results in that scene. In this paper, we propose a novel few-shot scene-adaptive anomaly detection problem to address the limitations of previous approaches. Our goal is to learn to detect anomalies in a previously unseen scene with only a few frames. A reliable solution for this new problem will have huge potential in real-world applications since it is expensive to collect a massive amount of data for each target scene. We propose a meta-learning based approach for solving this new problem; extensive experimental results demonstrate the effectiveness of our proposed method. All codes are released in .' author: - Yiwei Lu - Frank Yu - Mahesh Kumar Krishna Reddy - Yang Wang bibliography: - 'egbib.bib' title: 'Few-Shot Scene-Adaptive Anomaly Detection' ---

--- author: - 'Anton Molina$^\dag$' - 'Pranav Vyas$^\dag$' - 'Nikita Khlystov$^\dag$' - Shailabh Kumar - Anesta Kothari - Dave Deriso - Zhiru Liu - Samhita Banavar - Eliott Flaum - 'Manu Prakash\*' bibliography: - 'references.bib' title: 'Project 1000 x 1000: Centrifugal melt spinning for distributed manufacturing of N95 filtering facepiece respirators' --- Introduction {#introduction .unnumbered} ============ The COVID-19 pandemic caused by the SARS-CoV-2 coronavirus has resulted in widespread shortages of personal protective equipment, especially N95 filtering facepiece respirators (FFRs), which are critical to the safety of patients, caretakers, and healthcare workers exposed to high volumes of aerosolized viral pathogens. The unprecedented demand for N95 FFRs has rapidly depleted existing supply chains, causing medical centers in major cities to initiate efforts to decontaminate N95 FFRs for reuse. Resource limited regions that normally have very limited access to N95 FFRs have limited choice but to utilize ineffective substitutes such as T-shirts and tissues, which places their already-limited number of healthcare workers at great risk of infection. While existing N95 supply chains have struggled to meet the surge in demand, bad actors have begun flooding the market with counterfeit FFRs that are incapable of providing appropriate respiratory protection [^1]. For these reasons, there is an urgent need to augment the existing supply chain of valid N95 FFRs through distributed, rapidly accessible manufacturing methods combined with quality control and local testing and validation techniques. The National Institute for Occupational Safety and Health (NIOSH) require filters with an N95 rating to remove at least 95% of particles $\ge 0.3$ in size [@national1996niosh]. Commercial N95 filters typically consist of three to four layers of non-woven fibrous material that trap aerosolized viral particles within its fiber matrix using a combination of inertial and electrostatic forces. The fiber matrix is typically composed of a blend of nano- and micrometer diameter polymer-based fibers. Previous studies[@kim2007direct; @kim2010application; @bonilla2012direct] using electrostatic field meters suggest[^2] that these fibers hold a baseline electrostatic potential of up to at a tip-sample distance of 75 . The N95 filter material is produced on an industrial scale in a process called “meltblowing” [@huang2017; @huang2019], where high velocity air streams are blown through nozzles that coaxially extrude molten polymer. Thermoplastics such as polypropylene and poly-4-methyl-1-pentene are commonly used in this process because of their low water retention and desirable melt-flow properties [@drabek2019]. Electrostatic charge contributes as much as 95% of the filtration efficiency [@tsai2020] and is typically accomplished using corona charging [@klaase1984]. However, expanding production to new industrial meltblowing facilities is a major effort that includes precision manufacturing of large-scale, specialized extrusion dies as well as design of an extensive fabrication workflow, requiring months of construction before operation is possible [@reifenhauser2020]. Existing N95 FFR manufacturing methods by meltblowing therefore are incompatible with the urgent need for increased supply. In this study, we investigate centrifugal melt spinning (CMS) as a small-scale, distributed manufacturing approach for N95 FFR production. With very simple equipment and operational methods, we hypothesize that CMS can be rapidly deployed to address the increased demand in N95 FFRs. Laboratory-scale CMS-based methods are estimated to have 50-fold greater throughput than equivalent electrospinning methods, with production rates of up to 60 per orifice [@rogalski2017]. The physical footprint of CMS operation is significantly reduced compared to that of industrial meltblowing operations, allowing multiple CMS production lines to be run in parallel at smaller, local settings. CMS-based methods have been previously used for the fabrication of poly(lactic acid), poly(ethylene oxide), and polypropylene nano- and microfibers [@raghavan2013; @parker2010]. Here, we apply centrifugal melt spinning to address the ongoing N95 FFR shortage and construct a readily distributable, low-cost and modular laboratory-scale fiber production apparatus. We report characterization of fiber morphology and filtration performance of CMS-produced polypropylene fiber material and investigate electrostatic charging through the application of an external electric field in two different ways. We perform preliminary optimization of parameters relevant to N95 FFR manufacturing using CMS and propose how the process may be scaled up through distributed manufacturing. Results and Discussion {#results-and-discussion .unnumbered} ====================== ![ **Production of Filtration Media Using e-CMS.** **A** A commercial cotton candy machine is connected to a van de Graaff generator to produce a high voltage on the collection drum. **B** The raw material is collected as a loosely packed bale **right, top**. The raw material is subjected to compression **left** to produce samples of controlled grammage. Our highest performing samples gave a filtration efficiency equivalent to N95 **right, bottom**. **C** the collection tub surrounds a spinneret that is filled with polypropylene resin and heated so that the resin flows through orifices along the perimeter of the spinneret. **D** The high-throughput nature of this process means bulk quantities of this material can be rapidly produced. []{data-label="fig:samplematerial"}](fig_1.PNG){width="0.75\linewidth"} Process Design {#process-design .unnumbered} -------------- Centrifugal melt electrospinning of non-woven polypropylene fibers requires high temperatures (165), high rotational speeds (3,000 - 12,000 RPM) for producing fine fibers, and a means to impart electrostatic charge on the nascent fibers. Commercial cotton candy machines, normally used for centrifugal spinning of sugar-based fibers, are a convenient apparatus for producing sufficiently high temperatures, although many designs do not have sufficiently fast rotational speeds. Larger industrial-grade machines, such as the one used in this report, achieve higher rotational speeds, enabling production of fibers suitable for use as a filtration material (**Figure \[fig:samplematerial\]**). The material studied in this report was produced using a cotton candy machine modified only by replacing the standard aluminum mesh with a solid aluminum ring. The ring has several small apertures (600 ) along its perimeter, thereby allowing more effective distribution of heat and better control over fiber morphology. The choice of polymer resin is an important consideration for producing fibers. For example, previous work has shown a strong dependence of fiber diameter on melt flow rate, a measure closely related to molecular weight and viscosity [@raghavan2013]. In this study, we used three different types of polypropylene resin: (1) high molecular weight isotactic (high-MW), (2) low molecular weight isotactic (low-MW), and (3) amorphous. We observed fiber formation for all three resins. The resulting fibers are collected as a loose bale, similar to how cotton candy appears. It is necessary to increase the density of the material before its filtration efficiency can be evaluated. ![ **Optical microscopy analysis.** (Top) confocal images of high (left) and low (right) MW fibers. Histogram (bottom) showing fiber diameters for the samples above. Average fiber diameter was calculated from N=50 unique fibers. []{data-label="fig:Lightmicro"}](fig_histogram_uncharged.jpeg){width="0.75\linewidth"} ![ **Electron microscopy analysis.** SEM image taken for a N95 FFR (filtering facepiece respirator) material on the left; as compared to a high molecular weight CMS material on the right. []{data-label="fig:SEM"}](fig_sem.jpeg){width="0.6\linewidth"} Material Characterization {#material-characterization .unnumbered} ------------------------- We hypothesized that the polymer properties of the polypropylene feedstock would significantly determine the material properties of filtration media produced by our CMS process. We found that material produced using isotactic polypropylene of low molecular weight (average $M_w$ $\sim$12,000, average $M_n$ $\sim$5,000) yielded fibers that were mechanically brittle and more prone to disintegration as compared to higher molecular weight isotactic polypropylene. Amorphous polypropylene produced dense material that displayed high cohesiveness and adhesiveness, meaning it was not compatible for application as FFR material. Confocal imaging of fibers produced from isotactic polypropylene revealed an average fiber diameter of around 7-8  (**Figure \[fig:Lightmicro\]**), with a greater spread in fiber diameter in the case of high molecular weight polypropylene. After compaction, isotactic polypropylene samples exhibited grammages of about 620-695 , significantly greater than commercial N95 FFR material (141 ). Compacted samples produced by CMS were also significantly thicker than commercial N95 FFR material, yielding samples 1.2-1.8  in thickness as compared to about 0.7 . Material density after compaction was also about twice that of commercial N95 FFR material (0.44 vs. 0.20 ) (**Table \[table:1\]**). Scanning electron microscopy (SEM) was used to compare a commercially available N95 filtration material with CMS-produced and compacted high molecular weight material (**Figure \[fig:SEM\]**). The images indicate that the compaction results in the formation of a well-packed fiber matrix, suitable for further testing. Filtration media rated for N95-grade performance requires removal of at least 95% of particles of average diameter 0.3 . To verify the applicability of our CMS material in the context of FFRs, we performed filtration testing using a custom-built setup involving a handheld particle counter and a capsule constructed from threaded PVC piping to hold circular samples excised from bulk, compacted CMS material. Using incense smoke as a source of particles primarily in the 0.3  diameter range, we found that material produced by our CMS process enabled filtration efficiencies that exceeded 95% on average for three independently compacted samples. We found that N95-grade performance was achieved regardless of the molecular weight of polypropylene used (**Figure \[fig:filtefficiency\], right**). Samples excised from commercial N95 FFR material yielded an average filtration efficiency exceeding 97%, as expected. By contrast, samples excised from T-shirt fabric material (suggested for homemade masks by the CDC) gave a filtration efficiency of 45%, similar to previous reports [@mueller2018]. Airflow pressure drop across our CMS material samples was found to significantly greater than that of commercial N95 FFR samples (20 vs. 3.2 ) (**Figures \[fig:filtefficiency\], left,** and **\[fig:PDall\]**). This suggests that high filtration efficiency in the case of CMS material was achieved at the expense of breathability relative to commercially manufactured filtration material. The significantly higher grammage of compacted CMS samples relative to N95 FFR material likely gives rise to this reduced breathability and could be addressed by reducing the mass of material used for compaction. Moreover, given that increased electrostatic charging of non-woven fibrous material enables higher filtration efficiency for a given material density [@tsai2020], introducing electrostatic charging could improve breathability of our CMS material while maintaining high filtration efficiency. ------------- --------------------- --------------------- --------------------- Sample Low MW CMS High MW CMS N95 FFR \[0.5ex\] A 753.1 *(0.440)* 522.0 *(0.446)* 115.5 *(0.165)* B 629.0 *(0.422)* 564.8 *(0.471)* 145.5 *(0.202)* C 701.7 *(0.465)* 770.2 *(0.412)* 162.6 *(0.246)* Grammage 694.61$\pm$62.35 619.01$\pm$132.67 141.20$\pm$23.82 *Density* *0.442*$\pm$*0.021* *0.443*$\pm$*0.030* *0.204*$\pm$*0.041* ------------- --------------------- --------------------- --------------------- : Grammages (in ) and densities (in  ) of compacted samples produced by centrifugal melt spinning of three different types of polypropylene feedstock[]{data-label="table:1"} ![ **Filtration efficiency and pressure drop testing of filtration material produced using CMS.** Particle filtration efficiency of filtration materials were tested using incense smoke as a source of particles and a handheld particle counter (Lighthouse 3016). Circular samples of commercial N95 FFR (Kimberly-Clark), compacted CMS material produced from polypropylene feedstock of two different molecular weights, and T-shirt fabric were excised and characterized in a custom-built filtration testing apparatus. Filtration efficiency was calculated as the ratio of detected particles (0.30-0.49 ) with and without filter. T-shirt fabric material consisted of 50% cotton, 25% polyester, and 25% rayon. Pressure drop testing was performed at a constant flow rate of 4.00  using an in-line pressure sensor (Honeywell) and the same excised material samples. Breathability is reported as the inverse of pressure drop across material and is taken relative to that measured for commercial N95 FFR material. []{data-label="fig:filtefficiency"}](FEPDfig_v4.png){width="0.7\linewidth"} ------------------------------ ---------- ---------- ---------- --------- Material Sample 1 Sample 2 Sample 3 Average \[0.5ex\] Commercial N95 FFR 97.76 98.04 96.22 97.34 High MW CMS 94.94 95.52 98.10 96.19 Low MW CMS 96.57 96.63 93.33 95.51 Woven T-shirt 56.07 23.13 56.17 45.12 ------------------------------ ---------- ---------- ---------- --------- : Filtration efficiency values of filtration material produced using CMS in comparison to other materials for particles in the range of 0.30-0.49 .[]{data-label="table:2"} Effects of Electrostatic Charging {#effects-of-electrostatic-charging .unnumbered} --------------------------------- Although not necessary, electrostatic charging contributes significantly to filtration efficiency, especially for particles of diameter , a range potentially relevant to SARS-CoV-2 transmission [@Kowalski1999; @van_doremalen_aerosol_2020]. We first attempted to introduce electrostatic charge on filtration material after production by CMS using corona charging[@Kao2004; @tsai1993], coupling to a van de Graaff generator, as well as triboelectric charge transfer using polystyrene material. A surface voltmeter was used to measure the voltage close to the insulator surfaces. Surface charge density on the fibers was then estimated based on the measured voltage readings (details in the methods section). Charge density on a commercial N95 FFR material was estimated to be around -2000 . Compacted filtration material produced using high MW polypropylene have exhibited charge densities close to -1000 . After charging using the above three methods, estimated surface charge densities on the CMS-produced fibers have been calculated to be as high as -9500 . However, our surface charge measurement method does not accurately reveal the homogeneity or stability of the transferred charges, and therefore careful interpretation of the measured charge density values is advised. Further measurements which can help analyze the uniformity of charges are necessary to inform how to use electrostatic charging for improved filtration efficiency. We also attempted charging of fibers during the production process (e-CMS) by applying an external electric field between the heated CCM spinneret (containing molten polypropylene) and collection drum. Charging of fibers would occur by trapping induced dipoles in the fibers in the aligned state after solidification [@Kao2004]. Preliminary results indicated that this method of charging did not noticeably influence fiber morphology or improve filtration efficiency (**Figures \[fig:charged\] and \[fig:FEall\]**). Parallel efforts by another group using a CMS process have similarly shown that kilovolt-strength electric potentials are not necessary (OIST) [@bandi2020]. Further development of the CMS process to accommodate and better understand potentially beneficial effects of charging during fiber production remains important. Future Work and Process Scaling {#future-work-and-process-scaling .unnumbered} ------------------------------- We are continuing to develop this approach to increase both output and quality at a lower cost. This involves not only continued tool building but also community engagement. We are currently building a device from the group up using readily available components can reduce cost by avoiding the unnecessary, specialized components associated with a cotton candy machine. The motivation for this is twofold. First, a purpose built device will offer greater control over experimental parameters. For example, a simple spindle rotor offers greater control over and access to higher rotational speeds. This has the potential to reduce fiber diameter, offering improvements in filtration performance, and increasing throughput. Additionally, a modular device design will allow for much more flexibility in testing different approaches for charging the material during production. This is especially valuable since it allows for community based development, where improvements to the process can be made in a distributed way. Second, the use of readily available components can reduce cost by up to 50%. The current unit cost of our prototype is $\sim\$1,000$. This cost is based on a standard cotton candy machine (\$650), a custom machined cylinder (\$50), and a van de Graaff Generator (\$300). A 50% reduction in unit cost increases the economic scalability of this approach. The two main objectives of this work is to develop an FFR manufacturing method in an affordable and also high-throughput manner. During our prototyping phase, we iterated over several unique CMS designs and have been able to consistently produce at least 1-2  of material across a range of processing parameters. Given that a typical N95 FFR contains of filtration material, we can gain some perspective on the extent to which the present proposal can make a meaningful impact. A single CMS can produce enough filtration material for 1000 FFRs in a day. Therefore, a small scale facility consisting of 6 CMS devices operated for 12 hours by a small group of people to produce enough material for 5000 FFRs per day. This is sufficient to supply the daily demand of a large medical facility [@ppe2016]. Alternatively, 1000 CMS devices operating as a distributed network provides a rapidly configurable and resilient manufacturing capacity equivalent to a single, industrial-scale facility. We believe that the approach described here has the potential to make a significant contribution towards addressing the current FFR shortage. Our work represents a starting point for others to construct and develop their own affordable modular processes for producing high quality filtration material. Methods {#methods .unnumbered} ======= Materials {#materials .unnumbered} --------- Amorphous polypropylene, isotactic low molecular weight (average $M_w$ $\sim$12,000, average $M_n$ $\sim$5,000) polypropylene, and isotactic high molecular weight (average $M_w$ $\sim$250,000, average $M_n$ $\sim$67,000) were purchased from Sigma-Aldrich (St. Louis, MO). Commercial N95 FFR material was obtained from a Kimberly-Clark 62126 Particulate Filter Respirator and Surgical Mask (Kimberly-Clark Professional, Roswell, GA). Preparation of filtration material {#preparation-of-filtration-material .unnumbered} ---------------------------------- Nano- and microfibers were prepared using a modified cotton candy machine (Spin Magic 5, Paragon, USA). Polypropylene resin was placed directly into the preheated spinneret while in motion. Fibrous material was collected on the machine collection drum and compacted against a heated metal plate (130, 30 , 4.23 ) using a cylindrical pipe and plunger. Three independent replicates were compacted for each sample tested, sourcing from the same batch of material produced for each of the three types of polypropylene. Circular samples (17.25  diam.) for filtration testing were excised from this compacted material. As shown in Figure \[fig:testing\] (B), the circular samples are placed into a test filter assembly. A test filter consists of a disc of sample material (1  thick) held between two laser-cut Plexiglas mesh screens (1.6  thick, 17.25  diam.) (see Figure \[fig:testing\]). The filter assembly is inserted between two threaded PVC pipe connectors that are press-fit onto the testing jig. The edges of the filter assembly are wrapped with a paraffin wax seal (Parafilm, Bemis Inc, USA) to secure the three layers together and prevent air from leaking around the filter within the PVC pipe. Electron microscopy {#electron-microscopy .unnumbered} ------------------- A Hitachi TM-1000 tabletop SEM was used to obtain the micrographs. The fiber samples were attached to the stage using conductive silver paste. No sputter coat was added to the materials. Electrostatic charging {#electrostatic-charging .unnumbered} ---------------------- Corona charging was done by placing samples on top of a grounded aluminum plate and applying a steady ion current through point electrodes that are connected to a high DC voltage source (Model PMT2000, Advanced Research Instruments Corp.). The distance between the point electrodes and the sample was about 5 . The voltage of the emitters was set to 4800 . Charging beyond 30 minutes brought no additional increase in surface charge. E-CMS was performed by connecting the collection drum with a Van De Graaff generator, resulting in a steady-state voltage of approximately -18 . Triboelectric charging was performed by rubbing the sample directly against a range of materials. In particular, metal and cardboard were found capable of imparting negative charges on polypropylene samples, while polystyrene sheets had the opposite effect. Charge measurements {#charge-measurements .unnumbered} ------------------- A surface DC voltmeter (SVM2, Alphalab Inc., USA) was used to measure the voltage of the fiber material, 2.54  away from the fiber mesh surface. The surface charge density is estimated using manufacturer-provided[@svm2] equations $$\begin{array}{ll} \frac{Q}{A} &= \alpha V f(f-1) \\ f &= \sqrt{1+\frac{D^2}{4L^2}} , \end{array}$$ where $Q$ is the surface charge (), $A$ is the area () of fiber mesh, $V$ is the measured voltage (), $D$ is the diameter () of the fiber mesh, $L$ is the distance () of the mesh surface away from voltmeter sensor, and $\alpha = 3.6\times 10^{-14} $ is device-specific parameter provided by the manufacturer. Filtration testing {#filtration-testing .unnumbered} ------------------ The filter efficiency testing is done using a custom experimental setup that includes a handheld particle counter (Model 3016 IAQ, LightHouse, USA), 100  incense (Nag Champa, Satya Sai Baba, India), and connectors (universal cuff adaptor, teleflex multi-adaptor). Whereas a typical testing setup uses an all-in-one filter tester, *e.g.* an 8130A automated filter tester (TSI Automated, USA) that supports a flow rate up to 110 , our system was run at an airflow rate of 2.83 . The incense produces particles of various sizes, including those in the range picked up by the detector (0.30-10 ), and primarily in the 0.30-0.49  range. To calculate the filtration efficiency, the ratio of unfiltered particles detected to the number of particles detected without filter is subtracted from unity. Pressure drop testing {#pressure-drop-testing .unnumbered} --------------------- ![ **A. Filtration testing setup** 1: Incense stick 2: Test filter assembly. 3: Lighthouse 3016 handheld particle counter **B. Test filter assembly** Compressed sample is placed between two acrylic mesh screens, sealed on the sides with paraffin tape and held in place using the pipe screw setup. **C. Pressure drop testing** 1: Flow control valve 2: Airflow measurement sensor 3: Test filter assembly 4: Pressure sensor and micro-controller. []{data-label="fig:testing"}](fig_testing.jpeg){width="\linewidth"} The same sample-containing capsule used for filtration testing was also used for pressure drop measurements. Compressed air flow was delivered at a constant rate of 4.00 , similar to that experienced during human respiration accounting for the smaller sample cross-sectional area compared to a full FFR. The airflow rate is measured using a Mass Flow Meter SFM3300 (Sensirion AG, Switzerland) and the pressure drop is measured using a Honeywell Trustability Series pressure sensor (Model HSCDANN005PGSA3, Honeywell International Inc., USA). Sensor data is acquired using an Arduino Mega microcontroller development board (Arduino AG, Italy). Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors are greatful to Dr. George Herring, Hongquan Li, Prof. Anna Paradowska of the University of Sydney, and Tyler Orr for helpful discussion relating to project design, parts machining and material characterization. The authors thank Hongquan Li for assistance in pressure drop experiments and Prof. Fabian Pease of the Electrical Engineering Department at Stanford University for assistance in SEM characterization. The authors also deeply thank Edward Mazenc, Yuri Lensky, Daniel Ranard, and Abby Kate Grosskopf for contributing in the process design process. The authors would like to thank financial support from UCSF COVID-19 Response Fund, Schmidt Futures, Moore Foundation, CZ BioHub, NSF CCC grant (DBI 1548297) and HHMI-Gates Faculty Award. Supplementary Information {#supplementary-information .unnumbered} ========================= Distributed manufacturing {#distributed-manufacturing .unnumbered} ------------------------- ![ **Centralized vs. Distributed Manufacturing** []{data-label="fig:distributed"}](fig_dist.PNG){width="0.6\linewidth"} Results of charging on fiber morphology {#results-of-charging-on-fiber-morphology .unnumbered} --------------------------------------- ![ **Optical microscopy analysis.** (Top) confocal images of high (left) and low (right) MW fibers subject to a high voltage applied to the collection drum. Histogram (bottom) showing fiber diameters for the samples above. Average fiber diameter was calculated from N=50 unique fibers. []{data-label="fig:charged"}](fig_histogram_charged.jpeg){width="0.6\linewidth"} Full filtration efficiency and pressure drop measurement results {#full-filtration-efficiency-and-pressure-drop-measurement-results .unnumbered} ---------------------------------------------------------------- ![ **Filtration efficiency of all filtration samples tested.** []{data-label="fig:FEall"}](FEall.png){width="1.0\linewidth"} ![ **Pressure drop of all filtration samples tested.** []{data-label="fig:PDall"}](PDall.png){width="1.0\linewidth"} ![ **Surface charge measurement.** A commercial voltmeter (alphalabs inc.) was used to measure the surface voltage of sample materials. The surface charges were then calculated based on the measured voltages. Samples were kept 2.54  away from the sensor for the measurements. []{data-label="fig:chargemeter"}](chargemeter.jpeg){width="0.75\linewidth"} [^1]: At the time of writing, the CDC is actively maintaining a growing [ list of counterfeit N95 products](https://www.cdc.gov/niosh/npptl/usernotices/counterfeitResp.html). [^2]: There’s limited publicly-available data on the electrostatic charges of commercial polymers, and the measurement process itself is quite involved.

--- abstract: 'Collaborative machine learning and related techniques such as federated learning allow multiple participants, each with his own training dataset, to build a joint model by training locally and periodically exchanging model updates. We demonstrate that these updates leak *unintended* information about participants’ training data and develop passive and active inference attacks to exploit this leakage. First, we show that an adversarial participant can infer the presence of exact data pointsfor example, specific locationsin others’ training data (i.e., *membership* inference). Then, we show how this adversary can infer *properties* that hold only for a subset of the training data and are independent of the properties that the joint model aims to capture. For example, he can infer when a specific person first appears in the photos used to train a binary gender classifier. We evaluate our attacks on a variety of tasks, datasets, and learning configurations, analyze their limitations, and discuss possible defenses.' author: - | Luca Melis$^\dag$\ UCL\ luca.melis.14@alumni.ucl.ac.uk - | Congzheng Song$^\dag$\ Cornell University\ cs2296@cornell.edu - | Emiliano De Cristofaro\ UCL & Alan Turing Institute\ e.decristofaro@ucl.ac.uk - | Vitaly Shmatikov\ Cornell Tech\ shmat@cs.cornell.edu bibliography: - 'citation.bib' title: '**Exploiting Unintended Feature Leakage in Collaborative Learning$^*$**' --- Introduction ============ Collaborative machine learning (ML) has recently emerged as an alternative to conventional ML methodologies where all training data is pooled and the model is trained on this joint pool. It allows two or more participants, each with his own training dataset, to construct a joint model. Each participant trains a local model on his own data and periodically exchanges model parameters, updates to these parameters, or partially constructed models with the other participants. Several architectures have been proposed for distributed, collaborative, and federated learning [@dean2012large; @chilimbi2014project; @xing2015petuum; @moritz2015sparknet; @lin2017deep; @zinkevich2010parallelized]: with and without a central server, with different ways of aggregating models, etc. The main goal is to improve the training speed and reduce overheads, but protecting privacy of the participants’ training data is also an important motivation for several recently proposed techniques [@mcmahan2016communication; @shokri2015privacy]. Because the training data never leave the participants’ machines, collaborative learning may be a good match for the scenarios where this data is sensitive (e.g., health-care records, private images, personally identifiable information, etc.). Compelling applications include training of predictive keyboards on character sequences that users type on their smartphones [@mcmahan2016communication], or using data from multiple hospitals to develop predictive models for patient survival [@jochems2] and side effects of medical treatments [@jochems1]. Collaborative training, however, [*does*]{} disclose information via model updates that are based on the training data. The key question we investigate in this paper is: **what can be inferred about a participant’s training dataset from the model updates** revealed during collaborative model training? Of course, the purpose of ML is to discover new information about the data. Any useful ML model reveals something about the population from which the training data was drawn. For example, in addition to accurately classifying its inputs, a classifier model may reveal the features that characterize a given class or help construct data points that belong to this class. In this paper, we focus on inferring “unintended” features, i.e., properties that hold for certain subsets of the training data, but not generically for all class members. The basic privacy violation in this setting is *membership inference*: given an exact data point, determine if it was used to train the model. Prior work described passive and active membership inference attacks against ML models [@shokri2017membership; @hayes2017logan], but collaborative learning presents interesting new avenues for such inferences. For example, we show that an adversarial participant can infer whether a specific location profile was used to train a gender classifier on the FourSquare location dataset [@yang2015participatory] with 0.99 precision and perfect recall. We then investigate passive and active *property inference* attacks that allow an adversarial participant in collaborative learning to infer properties of other participants’ training data that are not true of the class as a whole, or even *independent* of the features that characterize the classes of the joint model. We also study variations such as inferring when a property appears and disappears in the data during trainingfor example, identifying when a certain person first appears in the photos used to train a generic gender classifier. For a variety of datasets and ML tasks, we demonstrate successful inference attacks against two-party and multi-party collaborative learning based on [@shokri2015privacy] and multi-party federated learning based on [@mcmahan2016communication]. For example, when the model is trained on the LFW dataset [@huang2007labeled] to recognize gender or race, we infer whether people in the training photos wear glassesa property that is [*uncorrelated*]{} with the main task. By contrast, prior property inference attacks [@ateniese2015hacking; @hitaj2017deep] infer only properties that characterize an entire class. We discuss this critical distinction in detail in Section \[sec:privML\]. Our key observation, concretely illustrated by our experiments, is that modern deep-learning models come up with separate internal representations of all kinds of features, some of which are independent of the task being learned. These “unintended” features leak information about participants’ training data. We also demonstrate that an *active* adversary can use multi-task learning to trick the joint model into learning a better internal separation of the features that are of interest to him and thus extract even more information. Some of our inference attacks have direct privacy implications. For example, when training a binary gender classifier on the FaceScrub [@ng2014data] dataset, we infer with high accuracy (0.9 AUC score) that a certain person appears in a single training batch even if half of the photos in the batch depict other people. When training a generic sentiment analysis model on Yelp healthcare-related reviews, we infer the specialty of the doctor being reviewed with perfect accuracy. On another set of Yelp reviews, we identify the author even if their reviews account for less than a third of the batch. We also measure the performance of our attacks vis-à-vis the number of participants (see Section \[sec:experimentsM\]). On the image-classification tasks, AUC degrades once the number of participants exceeds a dozen or so. On sentiment-analysis tasks with Yelp reviews, AUC of author identification remains high for many authors even with 30 participants. Federated learning with model averaging [@mcmahan2016communication] does not reveal individual gradient updates, greatly reducing the information available to the adversary. We demonstrate successful attacks even in this setting, e.g., inferring that photos of a certain person appear in the training data. Finally, we evaluate possible defensessharing fewer gradients, reducing the dimensionality of the input space, dropoutand find that they do not effectively thwart our attacks. We also attempt to use participant-level different privacy [@mcmahan2017learning], which, however, is geared to work with thousands of users, and the joint model fails to converge in our setting. Background {#sec:background} ========== Machine learning (ML) --------------------- An ML model is a function $f_{\theta}:{\mathcal{X}}\mapsto{\mathcal{Y}}$ parameterized by a set of *parameters* ${\theta}$, where ${\mathcal{X}}$ denotes the input (or feature) space, and ${\mathcal{Y}}$ the output space. In this paper, we focus on the supervised learning of classification tasks. The *training data* is a set of data points labeled with their correct classes. We work with models that take as input images or text (i.e., sequences of words) and output a class label. To find the optimal set of parameters that fits the training data, the training algorithm optimizes the *objective (loss) function*, which penalizes the model when it outputs a wrong label on a data point. We use $L(x, y;\theta)$ to denote the loss computed on a data point $(x, y)$ given the model parameters $\theta$, and $L(b;\theta)$ to denote the average loss computed on a batch $b$ of data points. [[***Stochastic Gradient Descent (SGD).***]{}]{} There are many methods to optimize the objective function. Stochastic gradient descent (SGD) and its variants are commonly used to train artificial neural networks, but our inference methodology is not specific to SGD. SGD is an iterative method where at each step the optimizer receives a small batch of the training data and updates the model parameters $\theta$ according to the direction of the negative gradient of the objective function with respect to $\theta$ and scaled by the learning rate $\eta$. Training finishes when the model has converged to a local minimum, where the gradient is close to zero. The trained model is tested using held-out data, which was not used during training. A standard metric is *test accuracy*, i.e., the percentage of held-out data points that are classified correctly. [[***Hyperparameters.***]{}]{} Most modern ML algorithms have a set of tunable hyperparameters, distinct from the model parameters. They control the number of training iterations, the ratio of the regularization term in the loss function (its purpose is to prevent overfitting, i.e., a modeling error that occurs when a function is too closely fitted to a limited set of data points), the size of the training batches, etc. [[***Deep learning (DL).***]{}]{} A family of ML models known as deep learning recently became very popular for many ML tasks, especially related to computer vision and image recognition [@schmidhuber2015deep; @lecun2015deep]. DL models are made of layers of non-linear mappings from input to intermediate hidden states and then to output. Each connection between layers has a floating-point weight matrix as parameters. These weights are updated during training. The topology of the connections between layers is task-dependent and important for the accuracy of the model. Collaborative learning {#sec:collabml} ---------------------- Training a deep neural network on a large dataset can be time- and resource-consuming. A common scaling approach is to partition the training dataset, concurrently train separate models on each subset, and exchange parameters via a parameter server [@chilimbi2014project; @dean2012large]. During training, each local model pulls the parameters from this server, calculates the updates based on its current batch of training data, then pushes these updates back to the server, which updates the global parameters. Collaborative learning may also involve participants who want to hide their training data from each other. We review two architectures for privacy-preserving collaborative learning based on, respectively, [@shokri2015privacy] and [@mcmahan2016communication]. **Server executes:** Initialize ${\theta}_0$ $g_t^k\gets$**ClientUpdate**(${\theta}_{t-1}$) ${\theta}_t\gets {\theta}_{t-1} - \eta \sum_k g_t^k$ **ClientUpdate**(${\theta}$): Select batch $b$ from client’s data **return** local gradients $\nabla L(b;{\theta})$ \[alg:ps\] [[***Collaborative learning with synchronized gradient updates.***]{}]{} Algorithm \[alg:ps\] shows collaborative learning with synchronized gradient updates [@shokri2015privacy]. In every iteration of training, each participant downloads the global model from the parameter server, locally computes gradient updates based on one batch of his training data, and sends the updates to the server. The server waits for the gradient updates from all participants and then applies the aggregated updates to the global model using stochastic gradient descent (SGD). In [@shokri2015privacy], each client may share only a fraction of his gradients. We evaluate if this mitigates our attacks in Section \[sec:grad\_frac\]. Furthermore, [@shokri2015privacy] suggests differential privacy to protect gradient updates. We do not include differential privacy in our experiments. By definition, record-level differential privacy bounds the success of membership inference, but does not prevent property inference that applies to a group of training records. Participant-level differential privacy, on the other hand, bounds the success of all attacks considered in this paper, but we are not aware of any participant-level differential privacy mechanism that enables collaborative learning of an accurate model with a small number of participants. We discuss this further in Section \[ssec:dp\_defense\]. **Server executes:** Initialize ${\theta}_0$ $m \gets max(C \cdot K, 1)$ $S_t \gets \text{(random set of m clients)}$ ${\theta}_t^k\gets$**ClientUpdate**(${\theta}_{t-1}$) ${\theta}_t\gets\sum_{k} \frac{n^k}{n} {\theta}_t^k$ **ClientUpdate**(${\theta}$): ${\theta}\gets {\theta}- \eta \nabla L(b;{\theta})$ **return** local model ${\theta}$ \[alg:fl\] [[***Federated learning with model averaging.***]{}]{} Algorithm \[alg:fl\] shows the federated learning algorithm [@mcmahan2016communication]. We set $C$, the fraction of the participants who update the model in each round, to $1$ (i.e., the server takes updates from all participants), to simplify our experiments and because we ignore the efficiency of the learning protocol when analyzing the leakage. In each round, the $k$-th participant locally takes several steps of SGD on the current model using his entire training dataset of size $n^k$ (i.e., the globally visible updates are based not on batches but on participants’ entire datasets). In Algorithm \[alg:fl\], $n$ is the total size of the training data, i.e., the sum of all $n^k$. Each participant submits the resulting model to the server, which computes a weighted average. The server evaluates the resulting joint model on a held-out dataset and stops training when performance stops improving. The convergence rate of both collaborative learning approaches heavily depends on the learning task and the hyperparameters (e.g., number of participants and batch size). Reasoning about Privacy in Machine\ Learning {#sec:privML} =================================== If a machine learning (ML) model is useful, it must reveal information about the data on which it was trained [@dworknaor]. To argue that the training process and/or the resulting model violate “privacy,” it is not enough to show that the adversary learns something new about the training inputs. At the very least, the adversary must learn *more* about the training inputs than about other members of their respective classes. To position our contributions in the context of related work (surveyed in ) and motivate the need to study unintended feature leakage, we discuss several types of adversarial inference previously considered in the research literature. Inferring class representatives {#badprior} ------------------------------- Given black-box access to a classifier model, *model inversion* attacks [@fredrikson2015model] infer features that characterize each class, making it possible to construct representatives of these classes. In the special caseand only in this special casewhere all class members are similar, the results of model inversion are similar to the training data. For example, in a facial recognition model where each class corresponds to a single individual, all class members depict the same person. Therefore, the outputs of model inversion are visually similar to any image of that person, including the training photos. If the class members are *not* all visually similar, the results of model inversion do not look like the training data [@shokri2017membership]. If the adversary actively participates in training the model (as in the collaborative and federated learning scenarios considered in this paper), he can use GANs [@goodfellow2014generative] to construct class representatives, as done by Hitaj et al. [@hitaj2017deep]. Only in the special case where all class members are similar, GAN-constructed representatives are similar to the training data. For example, all handwritten images of the digit ‘9’ are visually similar. Therefore, a GAN-constructed image for the ‘9’ class looks similar to *any* image of digit 9, including the training images. In a facial recognition model, too, all class members depict the same person. Hence, a GAN-constructed face looks similar to any image of that person, including the training photos. Note that *neither technique reconstructs actual training inputs.* In fact, there is no evidence that GANs, as used in [@hitaj2017deep], can even distinguish between a training input and a random member of the same class. Data points produced by model inversion and GANs are similar to the training inputs only if all class members are similar, as is the case for MNIST (the dataset of handwritten digits used in [@hitaj2017deep]) and facial recognition. This simply shows that ML works as it should. A trained classifier reveals the input features characteristic of each class, thus enabling the adversary to sample from the class population. For instance, Figure \[ganimage\] shows GAN-constructed images for the gender classification task on the LFW dataset, which we use in our experiments (see ). These images show a generic female face, but there is no way to tell from them whether an image of a *specific* female was used in training or not. ![Samples from a GAN attack on a gender classification model where the class is “female.”\[ganimage\]](gan/gan_300 "fig:"){width="11.00000%"} ![Samples from a GAN attack on a gender classification model where the class is “female.”\[ganimage\]](gan/gan_400 "fig:"){width="11.00000%"} ![Samples from a GAN attack on a gender classification model where the class is “female.”\[ganimage\]](gan/gan_500 "fig:"){width="11.00000%"} ![Samples from a GAN attack on a gender classification model where the class is “female.”\[ganimage\]](gan/gan_600 "fig:"){width="11.00000%"} Finally, the active attack in [@hitaj2017deep] works by overfitting the joint model’s representation of a class to a single participant’s training data. This assumes that the entire training corpus for a given class belongs to that participant. We are not aware of any deployment scenario for collaborative learning where this is the case. By contrast, we focus on a more realistic scenario where the training data for each class are distributed across multiple participants, although there may be significant differences between their datasets. Inferring membership in training data {#subsec:membership-def} ------------------------------------- The (arguably) simplest privacy breach is, given a model and an exact data point, inferring whether this point was used to train the model or not. Membership inference attacks against aggregate statistics are well-known [@homer2008resolving; @pyrgelis2017knock; @dwork2015robust], and recent work demonstrated black-box membership inference against ML models [@shokri2017membership; @demyst2018; @long2018understanding; @hayes2017logan], as discussed in . The ability of an adversary to infer the presence of a specific data point in a training dataset constitutes an immediate privacy threat if the dataset is in itself sensitive. For example, if a model was trained on the records of patients with a certain disease, learning that an individual’s record was among them directly affects his or her privacy. Membership inference can also help demonstrate inappropriate uses of data (e.g., using health-care records to train ML models for unauthorized purposes [@bbc]), enforce individual rights such as the “right to be forgotten,” and/or detect violations of data-protection regulations such as the GDPR [@GDPR]. Collaborative learning presents interesting new avenues for such inferences. Inferring properties of training data ------------------------------------- In collaborative and federated learning, participants’ training data may not be identically distributed. Federated learning is explicitly designed to take advantage of the fact that participants may have private training data that are different from the publicly available data for the same class [@mcmahan2016communication]. Prior work [@fredrikson2015model; @hitaj2017deep; @ateniese2015hacking] aimed to infer properties that characterize an entire class: for example, given a face recognition model where one of the classes is Bob, infer what Bob looks like (e.g., Bob wears glasses). It is not clear that hiding this information in a good classifier is possible or desirable. By contrast, we aim to infer *properties that are true of a subset of the training inputs but not of the class as a whole*. For instance, when Bob’s photos are used to train a gender classifier, we infer that Alice appears in some of the photos. We especially focus on the properties that are *independent* of the class’s characteristic features. In contrast to the face recognition example, where “Bob wears glasses” is a characteristic feature of an entire class, in our gender classifier study we infer whether people in Bob’s photos wear glasseseven though wearing glasses has no correlation with gender. There is no legitimate reason for a model to leak this information; it is purely an artifact of the learning process. A participant’s contribution to each iteration of collaborative learning is based on a batch of his training data. We infer *single-batch properties*, i.e., detect that the data in a given batch has the property but other batches do not. We also infer *when a property appears in the training data*. This has serious privacy implications. For instance, we can infer when a certain person starts appearing in a participant’s photos or when the participant starts visiting a certain type of doctors. Finally, we infer properties that characterize a participant’s entire dataset (but not the entire class), e.g., authorship of the texts used to train a sentiment-analysis model. {width="85.00000%"} Inference Attacks {#sec:inference} ================= Threat model {#sec:threatmodel} ------------ We assume that $K$ participants (where $K\geq2$) jointly train an ML model using one of the collaborative learning algorithms described in Section \[sec:collabml\]. One of the participants is the *adversary*. His goal is to infer information about the training data of another, *target participant* by analyzing periodic updates to the joint model during training. Multi-party ($K>2$) collaborative learning also involves honest participants who are neither the adversary, nor the target. In the multi-party case, the identities of the participants may not be known to the adversary. Even if the identities are known but the models are aggregated, the adversary may infer something about the training data but not trace it to a specific participant; we discuss this further in Section \[sec:attribution\]. The updates that adversary observes and uses for inference depend on both $K$ and how collaborative training is done. As inputs to his inference algorithms, the adversary uses the model updates revealed in each round of collaborative training. For synchronized SGD [@shokri2015privacy] with $K=2$, the adversary observes gradient updates computed on a single batch of the target’s data. If $K>2$, he observes an aggregation of gradient updates from all other participants (each computed on a single batch of the respective participant’s data). For federated learning with model averaging [@mcmahan2016communication], the observed updates are the result of two-step aggregation: (1) every participant aggregates the gradients computed on each local batch, and (2) the server aggregates the updates from all participants. For property inference, the adversary needs auxiliary training data correctly labeled with the property he wants to infer (e.g., faces labeled with ages if the goal is to infer ages). For active property inference (Section \[subsec:active-att\]), these auxiliary data points must also be labeled for the main task (e.g., faces labeled with identities for a facial recognition model). Overview of the attacks ----------------------- provides a high-level overview of our inference attacks. At each iteration $t$ of training, the adversary downloads the current joint model, calculates gradient updates as prescribed by the collaborative learning algorithm, and sends his own updates to the server. The adversary saves the snapshot of the joint model parameters $\theta_t$. The difference between the consecutive snapshots $\Delta\theta_t = \theta_t - \theta_{t-1} = \sum_{k}\Delta\theta^k_t$ is equal to the aggregated updates from all participants, hence $\Delta\theta_t - \Delta\theta^\text{adv}_t$ are the aggregated updates from all participants other than the adversary. [[***Leakage from the embedding layer.***]{}]{} \[sec:embedleak\] All deep learning models operating on non-numeric data where the input space is discrete and sparse (e.g., natural-language text or locations) first use an embedding layer to transform inputs into a lower-dimensional vector representation. For convenience, we use *word* to denote discrete tokens, i.e., actual words or specific locations. Let vocabulary $V$ be the set of all words. Each word in the training data is mapped to a word-embedding vector via an embedding matrix $W_\text{emb}\in\mathbb{R}^{|V|\times d}$, where $|V|$ is the size of the vocabulary and $d$ is the dimensionality of the embedding. During training, the embedding matrix is treated as a parameter of the model and optimized collaboratively. The gradient of the embedding layer is sparse with respect to the input words: given a batch of text, the embedding is updated only with the words that appear in the batch. The gradients of the other words are zeros. This difference directly reveals which words occur in the training batches used by the honest participants during collaborative learning. [[***Leakage from the gradients.***]{}]{} In deep learning models, gradients are computed by back-propagating the loss through the entire network from the last to the first layer. Gradients of a given layer are computed using this layer’s features and the error from the layer above. In the case of sequential fully connected layers $h_l, h_{l+1}$ ($h_{l+1} = W_l\cdot h_l$, where $W_l$ is the weight matrix), the gradient of error $E$ with respect to $W_l$ is computed as $\frac{\partial E}{\partial W_l}=\frac{\partial E}{\partial h_{l+1}} \cdot h_l$. The gradients of $W_l$ are inner products of the error from the layer above and the features $h_{l}$. Similarly, for a convolutional layer, the gradients of the weights are convolutions of the error from the layer above and the features $h_{l}$. Observations of gradient updates can thus be used to infer feature values, which are in turn based on the participants’ private training data. Membership inference {#subsec:membership-att} -------------------- As explained above, the non-zero gradients of the embedding layer reveal which words appear in a batch. This helps infer whether a given text or location appears in the training dataset or not. Let $V_t$ be the words included in the updates $\Delta\theta_t$. During training, the attacker collects a vocabulary sequence $[V_1, \dots, V_T]$. Given a text record $r$, with words $V_r$, he can test if $V_r\subseteq V_t$, for some $t$ in the vocabulary sequence. If $r$ is in target’s dataset, then $V_r$ will be included in at least one vocabulary from the sequence. The adversary can use this to decide whether $r$ was a member or not. Passive property inference {#subsec:passive-att} -------------------------- We assume that the adversary has auxiliary data consisting of the data points that have the property of interest ($D_\text{prop}^\text{adv}$) and data points that do not have the property ($D_\text{nonprop}^\text{adv}$). These data points need to be sampled from the same class as the target participant’s data, but otherwise can be unrelated. The intuition behind our attack is that the adversary can leverage the snapshots of the global model to generate aggregated updates based on the data with the property and updates based on the data without the property. This produces labeled examples, which enable the adversary to train a binary *batch property classifier* that determines if the observed updates are based on the data with or without the property. This attack is *passive*, i.e., the adversary observes the updates and performs inference without changing anything in the local or global collaborative training procedure. [[***Batch property classifier.***]{}]{} Algorithm \[alg:train\_infer\] shows how to build a batch property classifier during collaborative training. Given a model snapshot $\theta_t$, calculate gradients $g_\text{prop}$ based on a batch with the property ${b_\text{prop}}^\text{adv} \subset D_\text{prop}^\text{adv}$ and $g_\text{nonprop}$ based on a batch without the property ${b_\text{nonprop}}^\text{adv} \subset D_\text{nonprop}^\text{adv}$. Once enough labeled gradients have been collected, train a binary classifier $f_\text{prop}$. For the property inference attacks that exploit the embedding-layer gradients (e.g., the attack on the Yelp dataset in Section \[yelpattack\]), we use a logistic regression classifier. For all other property inference attacks, we experimented with logistic regression, gradient boosting, and random forests. Random forests with $50$ trees performed the best. The input features in this case correspond to the observed gradient updates. The number of the features is thus equal to the model’s parameters, which can be very large for a realistic model. To downsample the features representation, we apply the max pooling operator [@goodfellow2016deep] on the observed gradient updates. More specifically, max pooling performs a max filter to non-overlapping subregions of the initial features representation, thus reducing the computational cost of the attack. **Inputs:** Attacker’s auxiliary data $D_\text{prop}^\text{adv}, D_\text{nonprop}^\text{adv}$ **Outputs:** Batch property classifier $f_\text{prop}$ $G_\text{prop}\gets\emptyset$ $G_\text{nonprop}\gets\emptyset$ Receive $\theta_t$ from server Run **ClientUpdate**(${\theta}_t$) Sample ${b_\text{prop}}^\text{adv} \subset D_\text{prop}^\text{adv}, {b_\text{nonprop}}^\text{adv} \subset D_\text{nonprop}^\text{adv}$ Calculate $g_\text{prop}=\nabla L({b_\text{prop}}^\text{adv}; \theta_t), g_\text{nonprop}=\nabla L({b_\text{nonprop}}^\text{adv}; \theta_t)$ $G_\text{prop}\gets G_\text{prop}\cup \{g_\text{prop}\}$ $G_\text{nonprop}\gets G_\text{nonprop}\cup \{g_\text{nonprop}\}$ Label $G_\text{prop}$ as positive and $G_\text{nonprop}$ as negative Train a binary classifier $f_\text{prop}$ given $G_\text{prop}, G_\text{nonprop}$ \[alg:train\_infer\] [[***Inference algorithm.***]{}]{} As collaborative training progresses, the adversary observes gradient updates $g_\text{obs} = \Delta\theta_t - \Delta\theta^\text{adv}_t$. For single-batch inference, the adversary simply feeds the observed gradient updates to the batch property classifier $f_\text{prop}$. This attack can be extended from single-batch properties to the target’s entire training dataset. The batch property classifier $f_\text{prop}$ outputs a score in \[0,1\], indicating the probability that a batch has the property. The adversary can use the average score across all iterations to decide whether the target’s entire dataset has the property in question. Active property inference {#subsec:active-att} ------------------------- An active adversary can perform a more powerful attack by using *multi-task learning*. The adversary extends his local copy of the collaboratively trained model with an augmented property classifier connected to the last layer. He trains this model to simultaneously perform well on the main task and recognize batch properties. On the training data where each record has a main label $y$ and a property label $p$, the model’s joint loss is calculated as: $$\begin{aligned} L_\text{mt} = \alpha\cdot L(x, y;\theta) + (1 - \alpha)\cdot L(x, p; \theta) $$ During collaborative training, the adversary uploads the updates $\nabla_\theta L_\text{mt}$ based on this joint loss, causing the joint model to learn separable representations for the data with and without the property. As a result, the gradients will be separable too (e.g., see  in ), enabling the adversary to tell if the training data has the property. This adversary is still “honest-but-curious” in the cryptographic parlance. He faithfully follows the collaborative learning protocol and does not submit any malformed messages. The only difference with the passive attack is that this adversary performs additional *local* computations and submits the resulting values into the collaborative learning protocol. Note that the “honest-but-curious” model does not constrain the parties’ input values, only their messages. Datasets and model architectures ================================ The datasets, collaborative learning tasks, and adversarial inference tasks used in our experiments are reported in . Our choices of hyperparameters are based on the standard models from the ML literature. [[***Labeled Faces In the Wild (LFW).***]{}]{} LFW [@huang2007labeled] contains 13,233 62x47 RGB face images for 5,749 individuals with labels such as gender, race, age, hair color, and eyewear. [[***FaceScrub.***]{}]{} FaceScrub [@ng2014data] contains 76,541 50x50 RGB images for 530 individuals with the gender label: 52.5% are labeled as male, the rest as female. For our experiments, we selected a subset of 100 individuals with the most images, for a total of 18,809 images. On both LFW and FaceScrub, the collaborative models are convolutional neural networks (CNN) with three spatial convolution layers with 32, 64, and 128 filters, kernel size set to $(3,3)$, and max pooling layers with pooling size set to 2, followed by two fully connected layers of size 256 and 2. We use rectified linear unit (ReLU) as the activation function for all layers. Batch size is 32 (except in the experiments where we vary it), SGD learning rate is 0.01. [[***People in Photo Album (PIPA).***]{}]{} PIPA [@zhang2015beyond] contains over 60,000 photos of 2,000 individuals collected from public Flickr photo albums. Each image includes one or more people and is labeled with the number of people and their gender, age, and race. For our experiments, we selected a subset of 18,000 images with three or fewer people and scaled the raw images to 128x128. The collaborative model for PIPA is a VGG-style [@simonyan2014very] 10-layer CNN with two convolution blocks consisting of one convolutional layer and max pooling, followed by three convolution blocks consisting of two convolutional layers and max pooling, followed by two fully connected layers. Batch size is 32, SGD learning rate is 0.01. [[***Yelp-health.***]{}]{} We extracted health care-related reviews from the Yelp dataset[^1] of 5 million reviews of businesses tagged with numeric ratings (1-5) and attributes such as business type and location. Our subset contains 17,938 reviews for 10 types of medical specialists. [[***Yelp-author.***]{}]{} We also extracted a Yelp subset with the reviews of the top 10 most prolific reviewers, 16,207 in total. On both Yelp datasets, the model is a recurrent neural network with a word-embedding layer of dimension 100. Words in a review are mapped to a sequence of word-embedding vectors, which is fed to a gated recurrent unit (GRU [@cho2014learning]) layer that maps it to a sequence of hidden vectors. We add a fully connected classification layer to the last hidden vector of the sequence. SGD learning rate is 0.05. [[***FourSquare.***]{}]{} In [@yang2015nationtelescope; @yang2015participatory], Yang et al. collected a global dataset of FourSquare location “check-ins” (userID, time, location, activity) from April 2012 to September 2013. For our experiments, we selected a subset of 15,548 users who checked in at least 10 different locations in New York City and for whom we know their gender [@yang2016privcheck]. This yields 528,878 check-ins. The model is a gender classifier, a task previously studied by Pang et al. [@pang2016deepcity] on similar datasets. [[***CLiPS Stylometry Investigation (CSI) Corpus.***]{}]{} This annually expanded dataset [@verhoeven2014clips] contains student-written essays and reviews. We obtained 1,412 reviews, equally split between Truthful/Deceptive or Positive/Negative and labeled with attributes of the author (gender, age, sexual orientation, region of origin, personality profile) and the document (timestamp, genre, topic, veracity, sentiment). 80% of the reviews are written by females, 66% by authors from Antwerpen, the rest from other parts of Belgium and the Netherlands. On both the FourSquare and CSI datasets, the model, which is based on [@kim2014convolutional], first uses an embedding layer to turn non-negative integers (locations indices and word tokens) into dense vectors of dimension 320, then applies three spatial convolutional layers with 100 filters and variable kernel windows of size $(3,320)$, $(4,320)$ and $(5,320)$ and max pooling layers with pooling size set to $(l-3, 1)$, $(l-4,1)$, and $(l-5,1)$ where $l$ is the fixed length to which input sequences are padded. The hyperparameter $l$ is 300 on CSI and 100 on FourSquare. After this, the model has two fully connected layers of size 128 and 2 for FourSquare and one fully connected layer of size 2 for CSI. We use RELU as the activation function. Batch size is 100 for FourSquare, 12 for CSI. SGD learning rate is 0.01. Two-Party Experiments {#sec:experiments2} ===================== All experiments were performed on a workstation running Ubuntu Server 16.04 LTS equipped with a 3.4GHz CPU i7-6800K, 32GB RAM, and an NVIDIA TitanX GPU card. We use MxNet [@chen2015mxnet] and Lasagne [@lasagne] to implement deep neural networks and Scikit-learn [@scikit-learn] for conventional ML models. The source code is available upon request. Training our inference models takes less than 60 seconds on average and does not require a GPU. We use AUC scores to evaluate the performance of both the collaborative model and our property inference attacks. For membership inference, we report only precision because our decision rule from Section \[subsec:membership-att\] is binary and does not produce a probability score. Membership inference {#sec:two-member} -------------------- The adversary first builds a Bag of Words (BoW) representation for the input whose membership in the target’s training data he aims to infer. We denote this as the test BoW. During training, as explained in Section \[subsec:membership-att\], the non-zero gradients of the embedding layer reveal which “words” are present in each batch of the target’s data, enabling the adversary to build a batch BoW. If the test BoW is a subset of the batch BoW, the adversary infers that the input of interest occurs in the batch. We evaluate membership inference on the Yelp-health and FourSquare datasets with the vocabulary of 5,000 most frequent words and 30,000 most popular locations, respectively. We split the data evenly between the target and the adversary and train a collaborative model for 3,000 iterations. Table \[tab:location\_mia\] shows the precision of membership inference for different batch sizes. As batch size increases, the adversary observes more words in each batch BoW and the attack produces more false positives. Recall is always perfect (i.e., no false negatives) because any true test BoW must be contained in at least one of the batch BoWs observed by the adversary. Single-batch property inference {#ssec:2p_batch} ------------------------------- We call a training batch ${b_\text{nonprop}}$ if none of the inputs in it have the property, ${b_\text{prop}}$ otherwise. The adversary aims to identify which of the batches are ${b_\text{prop}}$. We split the training data evenly between the target and the adversary and assume that the same fraction of inputs in both subsets have the property. During training, $\frac{1}{m}$ of the target’s batches include only inputs with the property ($m=2$ in the following). [0.255]{} {width="100.00000%"} [0.255]{} {width="100.00000%"} [0.255]{} {width="100.00000%"} [0.255]{} {width="100.00000%"} [0.36]{} {width="100.00000%"} [0.36]{} {width="100.00000%"} [[***LFW.***]{}]{} Table \[tbl:lfw\_prop\] reports the results of single-batch property inference on the LFW dataset. We chose properties that are *uncorrelated* with the main classification label that the collaborative model is trying to learn. The attack has perfect AUC when the main task is gender classification and the inference task is “race:black” (the Pearson correlation between these labels is -0.005). The attack also achieves almost perfect AUC when the main task is “race: black” and the inference task is “eyewear: sunglasses.” It also performs well on several other properties, including “eyewear: glasses” when the main task is “race: Asian.” These results demonstrate that gradients observed during training leak more than the characteristic features of each class. In fact, **collaborative learning leaks properties of the training data that are uncorrelated with class membership**. To understand why, we plot the t-SNE projection [@maaten2008visualizing] of the features from different layers of the joint model in . Observe that the feature vectors are grouped by property in the lower layers pool1, pool2 and pool3, and by class label in the higher layer. Intuitively, the model did not just learn to separate inputs by class. The lower layers of the model also learned to separate inputs by various properties that are uncorrelated with the model’s designated task. Our inference attack exploits this unintended extra functionality. [[***Yelp-health.***]{}]{} \[yelpattack\] On this dataset, we use review-score classification as the main task and the specialty of the doctor being reviewed as the property inference task. Obviously, the latter is more sensitive from the privacy perspective. We use 3,000 most frequent words in the corpus as the vocabulary and train for 3,000 iterations. Using BoWs from the embedding-layer gradients, the attack achieves almost perfect AUC. Table \[tbl:yelp\_words\] shows the words that have the highest predictive power in our logistic regression. [[***Fractional properties.***]{}]{} We now attempt to infer that *some* of the inputs in a batch have the property. For these experiments, we use FaceScrub’s top 5 face IDs and Yelp-author (the latter with the 3,000 most frequent words as the vocabulary). The model is trained for 3,000 iterations. As before, $1/2$ of the target’s batches include inputs with the property, but here we vary the fraction of inputs with the property within each such batch among 0.1, 0.3, 0.5, 0.7, and 0.9. reports the results. On FaceScrub for IDs 0, 1, and 3, AUC scores are above 0.8 even if only 50% of the batch contain that face, i.e., the adversary can successfully infer that photos of a particular person appear in a batch even though (a) the model is trained for generic gender classification, and (b) half of the photos in the batch are of other people. If the fraction is higher, AUC approaches 1. On Yelp-author, AUC scores are above 0.95 for all identities even when the fraction is 0.3, i.e., the attack successfully infers the authors of reviews even though (a) the model is trained for generic sentiment analysis, and (b) more than two thirds of the reviews in the batch are from other authors. [0.43]{} {width="100.00000%"} [0.43]{} {width="100.00000%"} Inferring when a property occurs {#sec:emerge} -------------------------------- Continuous training, when new training data is added to the process as it becomes available, presents interesting opportunities for inference attacks. If the occurrences of a property in the training data can be linked to events outside the training process, privacy violation is exacerbated. For example, suppose a model leaks that a certain third person started appearing in another participant’s training data immediately after that participant uploaded his photos from a trip. [[***PIPA.***]{}]{} Images in the PIPA dataset have between 1 to 3 faces. We train the collaborative model to detect if there is a young adult in the image; the adversary’s inference task is to determine if people in the image are of the same gender. The latter property is a stepping stone to inferring social relationships and thus sensitive. We train the model for 2,500 iterations and let the batches with the “same gender” property appear in iterations 500 to 1500. shows, for each iteration, the probability output by the adversary’s classifier that the batch in that iteration has the property. The appearance and disappearance of the property in the training data are clearly visible in the plot. [[***FaceScrub.***]{}]{} For the gender classification model on FaceScrub, the adversary’s objective is to infer whether and when a certain person appears in the other participant’s photos. The joint model is trained for 2,500 iterations. We arrange the target’s training data so that two specific identities appear during certain iterations: ID 0 in iterations 0 to 500 and 1500 to 2000, ID 1 in iterations 500 to 1000 and 2000 to 2500. The rest of the batches are mixtures of other identities. The adversary trains three property classifiers, for ID 0, ID 1, and also for ID 2 which does not appear in the target’s dataset. reports the scores of all three classifiers. ID 0 and 1 receive the highest scores in the iterations where they appear, whereas ID 2, which never appears in the training data, receives very low scores in all iterations. These experiments show that our attacks can successfully infer *dynamic properties* of the training dataset as collaborative learning progresses. Inference against well-generalized models ----------------------------------------- To show that our attacks work with (1) relatively few observed model updates and (2) against well-generalized models, we experiment with the CSI corpus. reports the accuracy of inferring the author’s gender. The attack reaches $0.98$ AUC after only 2 epochs and improves as the training progresses and the adversary collects more updates. also shows that the model is not overfitted. Its test accuracy on the main sentiment-analysis task is high and improves with the number of the epochs. [0.32]{} {width="95.00000%"} [0.32]{} {width="95.00000%"} [0.32]{} {width="95.00000%"} Active property inference {#sec:active} ------------------------- To show the additional power of the active attack from , we use FaceScrub. The main task is gender classification, the adversary’s task is to infer the presence of ID 4 in the training data. We assume that this ID occurs in a single batch, where it constitutes 50% of the photos. We evaluate the attack with different choices of $\alpha$, which controls the balance between the main-task loss and the property-classification loss in the adversary’s objective function. shows that AUC increases as we increase $\alpha$. and show the t-SNE projection of the final fully connected layer, with $\alpha=0$ and $\alpha=0.7$, respectively. Observe that the data with the property (blue points) is grouped tighter when $\alpha=0.7$ than in the model trained under a passive attack ($\alpha=0$). This illustrates that *as a result of the active attack, the joint model learns a better separation for data with and without the property*. Multi-Party Experiments {#sec:experimentsM} ======================= In the multi-party setting, we only consider passive property inference attacks. We vary the number of participants between 4 and 30 to match the deployment scenarios and applications proposed for collaborative learning, e.g., hospitals or biomedical research institutions training on private medical data [@jochems1; @jochems2]. This is similar to prior work [@hitaj2017deep], which was evaluated on MNIST with 2 participants and face recognition on the AT&T dataset with 41 participants. [0.36]{} {width="100.00000%"}   [0.36]{} {width="100.00000%"} Synchronized SGD {#sec:multi-sgd} ---------------- \[aggregate-attacks\] As the number of honest participants in collaborative learning increases, the adversary’s task becomes harder because the observed gradient updates are aggregated across multiple participants. Furthermore, the inferred information may not directly reveal the identity of the participant to whom the data belongs (see Section \[sec:attribution\]). In the following experiments, we split the training data evenly across all participants, but so that only the target and the adversary have the data with the property. The joint model is trained with the same hyperparameters as in the two-party case. Similar to , the adversary’s goal is to identify which aggregated gradient updates are based on batches ${b_\text{prop}}$ with the property. [[***LFW.***]{}]{} We experiment with (1) gender classification as the main task and “race: black” as the inference task, and (2) smile classification as the main task and “eyewear: sunglasses” as the inference task. shows that the attack still achieves reasonably high performance, with AUC score around 0.8, when the number of participants is 12. Performance then degrades for both tasks. [[***Yelp-author.***]{}]{} The inference task is again author identification. In the multi-party case, the gradients of the embedding layer leak the batch BoWs of all honest participants, not just the target. reports the results. For some authors, AUC scores do not degrade significantly even with many participants. This is likely due to some unique combinations of words used by these authors, which identify them even in multi-party settings. [0.2655]{} {width="95.00000%"} [0.2655]{} {width="95.00000%"} [0.2655]{} {width="95.00000%"} [0.2655]{} {width="95.00000%"} Model averaging {#sec:multi-avg} --------------- In every round $t$ of federated learning with model averaging (see Algorithm \[alg:fl\]), the adversary observes $\theta_t-\theta_{t-1} = \sum_{k}\frac{n^k}{n}\theta_t^k - \sum_{k}\frac{n^k}{n}\theta_{t-1}^k = \sum_{k}\frac{n^k}{n} (\theta_t^k - \theta_{t-1}^k)$, where $\theta_t^k - \theta_{t-1}^k$ are the aggregated gradients computed on the $k$-th participant’s local dataset. In our experiments, we split the training data evenly among honest participants but ensure that in the target participant’s subset, $\hat{p}$% of the inputs have the property, while none of the other honest participants’ data have it. During each epoch of local training, every honest participant splits his local training data into 10 batches and performs one round of training. We assume that the adversary has the same number of inputs with the property as the target. As before, when the adversary trains his binary classifier, he needs to locally “emulate” the collaborative training process, i.e., sample data from his local dataset, compute aggregated updates, and learn to distinguish between the aggregates based on the data without the property and aggregates where one of the underlying updates was based on the data with the property. We perform 8 trials where a subset of the training data has the property and 8 control trials where there are no training inputs with the property. [[***Inferring presence of a face.***]{}]{} We use FaceScrub and select two face IDs (1 and 3) whose presence we want to infer. In the “property” case, $\hat{p}=80\%$, i.e., 80% of one honest participant’s training data consist of the photos that depict the person in question. In the control case, $\hat{p}=0\%$, i.e., the photos of this person do not occur in the training data. shows the scores assigned by the adversary’s classifier to the aggregated updates with 3 and 5 total participants. When the face in question is present in the training dataset, the scores are much higher than when it is absent. Success of the attack depends on the property being inferred, distribution of the data across participants, and other factors. For example, the classifiers for Face IDs 2 and 4, which were trained in the same fashion as the classifiers for Face IDs 1 and 3, failed to infer the presence of the corresponding faces in the training data. [[***Inferring when a face occurs.***]{}]{} In this experiment, we aim to infer when a participant whose local data has a certain property joined collaborative training. We first let the adversary and the rest of the honest participants train the joint model for 250 rounds. The target participant then joins the training at round $t=250$ with the local data that consists of photos depicting ID 1. reports the results of the experiment: the adversary’s AUC scores are around 0 when face ID 1 is not present and then increase almost to 1.0 right after the target participant joins the training. ![Inferring that a participant whose local data has the property of interest has joined the training. $K=2$ for rounds 0 to 250, $K=3$ for rounds 250 to 500.[]{data-label="fig:ma_it"}](figs/fbfl_it){width="40.00000%"} Defenses {#sec:defenses} ======== Sharing fewer gradients {#sec:grad_frac} ----------------------- As suggested in [@shokri2015privacy], participants in collaborative learning could share only a fraction of their gradients during each update. This reduces communication overhead and, potentially, leakage, since the adversary observes fewer gradients. To evaluate this defense, we measure the performance of single-batch inference against a sentiment classifier collaboratively trained on the CSI Corpus by two parties who exchange only a fraction of their gradients. shows the resulting AUC scores: when inferring the region of the texts’ authors, our attack still achieves 0.84 AUC when only 10% of the updates are shared during each iteration, compared to 0.93 AUC when all updates are shared. Dimensionality reduction ------------------------ ![Uniqueness of user profiles with respect to the number of top locations.[]{data-label="fig:uniqueness"}](figs/checkins_uniqueness.png){width="35.00000%"} As discussed in Section \[sec:embedleak\], if the input space of the model is sparse and inputs must be embedded into a lower-dimensional space, non-zero gradient updates in the embedding layer reveal which inputs are present in the training batch. One plausible defense is to only use inputs that occur many times in the training data. This does not work in general, e.g., shows that restricting inputs to the top locations in the FourSquare dataset eliminates most of the training data. A smarter defense is to restrict the model so that it only uses “words” from a pre-defined vocabulary of common words. For example, Google’s federated learning for predictive keyboards uses a fixed vocabulary of 5,000 words [@mcmahan2016communication]. In , we report the accuracy of our membership inference attack and the accuracy of the joint model on its main taskgender classification for the FourSquare dataset, sentiment analysis for the CSI Corpusfor different sizes of the common vocabulary (locations and words, respectively). This approach partially mitigates our attacks but also has a significant negative impact on the quality of the collaboratively trained models. Dropout ------- Another possible defense is to employ *dropout* [@srivastava2014dropout], a popular regularization technique used to mitigate overfitting in neural networks. Dropout randomly deactivates activations between neurons, with probability $p_{drop} \in [0,1]$. Random deactivations may weaken our attacks because the adversary observes fewer gradients corresponding to the active neurons. To evaluate this approach, we add dropout after the max pool layers in the joint model. reports the accuracy of inferring the region of the reviews in the CSI Corpus, for different values of $p_{drop}$. Increasing the randomness of dropout makes our attacks *stronger* while slightly decreasing the accuracy of the joint model. Dropout stochastically removes features at every collaborative training step, thus yielding more *informative* features (similar to feature bagging [@ho1995random; @chang2017dropout]) and increasing variance between participants’ updates. Participant-level differential privacy {#ssec:dp_defense} -------------------------------------- As discussed in Section \[sec:collabml\], record-level $\varepsilon$-differential privacy, by definition, bounds the success of membership inference but does not prevent property inference. Any application of differential privacy entails application-specific tradeoffs between privacy of the training data and accuracy of the resulting model. The participants must also somehow choose the parameters (e.g., $\varepsilon$) that control this tradeoff. In theory, participant-level differential privacy bounds the success of inference attacks described in this paper. We implemented the participant-level differentially private federated learning algorithm by McMahan et al. [@mcmahan2017learning] and attempted to train a gender classifier on LFW, but the model did not converge for any number of participants (we tried at most 30). This is due to the magnitude of noise needed to achieve differential privacy with the moments accountant bound [@abadi2016deep], which is inversely proportional to the number of users (the model in [@mcmahan2017learning] was trained on *thousands* of users). Another participant-level differential privacy mechanism, presented in [@geyer2017differentially], also requires a very large number of participants. Moreover, these two mechanisms have been used, respectively, for language modeling [@mcmahan2017learning] and handwritten digit recognition [@geyer2017differentially]. Adapting them to the specific models and tasks considered in this paper may not be straightforward. Following [@mcmahan2017learning; @geyer2017differentially], we believe that participant-level differential privacy provide reasonable accuracy only in settings involving at least thousands of participants. We believe that further work is needed to investigate whether participant-level differential privacy can be adapted to prevent our inference attacks *and* obtain high-quality models in settings that do not involve thousands of users. Limitations of the attacks ========================== Auxiliary data -------------- Our property inference attacks assume that the adversary has auxiliary training data correctly labeled with the property he wants to infer. For generic properties, such data is easy to find. For example, the auxiliary data for inferring the number and genders of people can be any large dataset of images with males and females, single and in groups, where each image is labeled with the number of people in it and their genders. Similarly, the auxiliary data for inferring the medical specialty of doctors can consist of any texts that include words characteristic of different specialties (see Table \[tbl:yelp\_words\]). More targeted inference attacks require specialized auxiliary data that may not be available to the adversary. For example, to infer that photos of a certain person occurs in another participant’s dataset, the adversary needs (possibly different) photos of that person to train on. To infer the authorship of training texts, the adversary needs a sufficiently large sample of texts known to be written by a particular author. Number of participants ---------------------- In our experiments, the number of participants in collaborative training is relatively small (ranging from 2 to 30), while some federated-learning applications involve thousands or millions of users [@mcmahan2016communication; @mcmahan2017learning]. As discussed in Section \[sec:multi-sgd\], performance of our attacks drops significantly as the number of participants increases. Undetectable properties ----------------------- It may not be possible to infer some properties from model updates. For example, our attack did not detect the presence of some face identities in the multi-party model averaging experiments (Section \[sec:multi-avg\]). If for whatever reason the model does not internally separate the features associated with the target property, inference will fail. Attribution of inferred properties {#sec:attribution} ---------------------------------- In the two-party scenarios considered in Section \[sec:experiments2\], attribution of the inferred properties is trivial because there is only one honest participant. In the multi-party scenarios considered in Section \[sec:experimentsM\], model updates are aggregated. Therefore, even if the adversary successfully infers the presence of inputs with a certain property in the training data, he may not be able to attribute these inputs to a specific participant. Furthermore, he may not be able to tell if all inputs with the property belong to one participant or are distributed across multiple participants. In general, attribution requires auxiliary information specific to the leakage. For example, consider face identification. In some applications of collaborative learning, the identities of all participants are known because they need to communicate with each other. If collaborative learning leaks that a particular person appears in the training images, auxiliary information about the participants (e.g., their social networks) can reveal which of them knows the person in question. Similarly, if collaborative learning leaks the authorship of the training texts, auxiliary information can help infer which participant is likely to train on texts written by this author. Another example of attribution based on auxiliary information is described in Section \[sec:emerge\]. If photos of a certain person first appear in the training data after a new participant has joined collaborative training, the adversary may attribute these photos to the new participant. Note that leakage of medical conditions, locations, images of individuals, or texts written by known authors is a privacy breach even if it cannot be traced to a specific participant or multiple participants. Leaking that a certain person appears in the photos or just the number of people in the photos reveals intimate relationships between people. Locations can reveal people’s addresses, religion, sexual orientation, and relationships with other people. Related Work {#sec:related} ============ [[***Privacy-preserving distributed learning.***]{}]{} Transfer learning in combination with differentially private (DP) techniques tailored for deep learning [@abadi2016deep] has been used in [@papernot2016semi; @papernot2018scalable]. These techniques privately train a “student” model by transferring, through noisy aggregation, the knowledge of an ensemble of “teachers” trained on the disjoint subsets of training data. These are centralized, record-level DP mechanisms with a trusted aggregator and do not apply to federated or collaborative learning. In particular, [@papernot2016semi; @papernot2018scalable] assume that the adversary cannot see the individual models, only the final model trained by the trusted aggregator. Moreover, record-level DP by definition does not prevent property inference. Finally, their effectiveness has been demonstrated only on a few specific tasks (MNIST, SVHN, OCR), which are substantially different from the tasks considered in this paper. Shokri and Shmatikov [@shokri2015privacy] propose making gradient updates differentially private to protect the training data. Their approach requires extremely large values of the $\varepsilon$ parameter (and consequently little privacy protection) to produce an accurate joint model. More recently, participant-level differentially private federated learning methods [@mcmahan2017learning; @geyer2017differentially] showed how to protect participants’ training data by adding Gaussian noise to local updates. As discussed in , these approaches require a large number of users (on the order of thousands) for the training to converge and achieve an acceptable trade-off between privacy and model performance. Furthermore, the results in [@mcmahan2017learning] are reported for a specific language model and use *AccuracyTop1* as the proxy, not the actual accuracy of the non-private model. Pathak et al. [@pathak2010multiparty] present a differentially private global classifier hosted by a trusted third-party and based on locally trained classifiers held by separate, mutually distrusting parties. Hamm et al. [@hamm2016learning] use knowledge transfer to combine a collection of models trained on individual devices into a single model, with differential privacy guarantees. Secure multi-party computation (MPC) has also been used to build privacy-preserving neural networks in a distributed fashion. For example, SecureML [@mohassel2017secureml] starts with the data owners (clients) distributing their private training inputs among two non-colluding servers during the setup phase; the two servers then use MPC to train a global model on the clients’ encrypted joint data. Bonawitz et al. [@bonawitz2017practical] use secure multi-party aggregation techniques, tailored for federated learning, to let participants encrypt their updates so that the central parameter server only recovers the sum of the updates. In Section \[sec:multi-avg\], we showed that inference attacks can be successful even if the adversary only observes aggregated updates. [[***Membership inference.***]{}]{} Prior work demonstrated the feasibility of membership inference from aggregate statistics, e.g., in the context of genomic studies [@homer2008resolving; @backes2016membership], location time-series [@pyrgelis2017knock], or noisy statistics in general [@dwork2015robust]. Membership inference against black-box ML models has also been studied extensively in recent work. Shokri et al. [@shokri2017membership] demonstrate membership inference against black-box supervised models, exploiting the differences in the models’ outputs on training and non-training inputs. Hayes et al. [@hayes2017logan] focus on generative models in machine-learning-as-a-service applications and train GANs [@goodfellow2014generative] to detect overfitting and recognize training inputs. Long et al. [@long2018understanding] and Yeom et al. [@yeom2017unintended] study the relationship between overfitting and information leakage. Truex et al. [@demyst2018] extend [@shokri2017membership] to a more general setting and show how membership inference attacks are data-driven and largely transferable. They also show that an adversary who participates in collaborative learning, with access to individual model updates from all honest participants, can boost the performance of membership inference vs. a centralized model. Nasr et al. [@nasr2018machine] design a privacy mechanism to adversarially train *centralized* machine learning models with provable protections against membership inference. [[***Other attacks on machine learning models.***]{}]{} Several techniques infer class features and/or construct class representatives if the adversary has black-box [@fredrikson2014privacy; @fredrikson2015model] or white-box [@ateniese2015hacking] access to a classifier model. As discussed in detail in Section \[sec:privML\], these techniques infer features that characterize an entire class and not specifically the training data, except in the cases of pathological overfitting where the training sample constitutes the entire membership of the class. Hitaj et al. [@hitaj2017deep] show that a participant in collaborative deep learning can use GANs to construct class representatives. Their technique was evaluated only on models where all members of the same class are visually similar (handwritten digits and faces). As discussed in Section \[badprior\], there is no evidence that it produces actual training images or can distinguish a training image and another image from the same class. The informal property violated by the attacks of [@fredrikson2014privacy; @fredrikson2015model; @ateniese2015hacking; @hitaj2017deep] is: “a classifier should prevent users from generating an input that belongs to a particular class or even learning what such an input looks like.” It is not clear to us why this property is desirable, or whether it is even achievable. Aono et al. [@aono2017privacy] show that, in the collaborative deep learning protocol of [@shokri2015privacy], an honest-but-curious server can partially recover participants’ training inputs from their gradient updates under the (greatly simplified) assumption that the batch consists of a single input. Furthermore, the technique is evaluated only on MNIST where all class members are visually similar. It is not clear if it can distinguish a training image and another image from the same MNIST class. Song et al. [@song2017machine] engineer an ML model that memorizes the training data, which can then be extracted with black-box access to the model. Carlini et al. [@carlini2018secret] show that deep learning-based generative sequence models trained on text data can unintentionally memorize training inputs, which can then be extracted with black-box access. They do this for sequences of digits artificially introduced into the text, which are not affected by the relative word frequencies in the language model. Training data that is explicitly incorporated or otherwise memorized in the model can also be leaked by model stealing attacks [@tramer2016stealing; @wang2018stealing; @joon2018towards]. Concurrently with this work, Ganju et al. [@ganju2018property] developed property inference attacks against fully connected, relatively shallow neural networks. They focus on the post-training, white-box release of models trained on sensitive data, as opposed to collaborative training. In contrast to our attacks, the properties inferred in [@ganju2018property] may be correlated with the main task. Evaluation is limited to simple datasets and tasks such as MNIST, U.S. Census tabular data, and hardware performance counters with short features. Conclusion {#sec:conclusion} ========== In this paper, we proposed and evaluated several inference attacks against collaborative learning. These attacks enable a malicious participant to infer not only *membership*, i.e., the presence of exact data points in other participants’ training data, but also *properties* that characterize subsets of the training data and are independent of the properties that the joint model aims to capture. Deep learning models appear to internally recognize many features of the data that are uncorrelated with the tasks they are being trained for. Consequently, model updates during collaborative learning leak information about these “unintended” features to adversarial participants. Active attacks are potentially very powerful in this setting because they enable the adversary to trick the joint model into learning features of the adversary’s choosing without a significant impact on the model’s performance on its main task. Our results suggest that leakage of unintended features exposes collaborative learning to powerful inference attacks. We also showed that defenses such as selective gradient sharing, reducing dimensionality, and dropout are not effective. This should motivate future work on better defenses. For instance, techniques that learn only the features relevant to a given task [@edwards2015censoring; @osia2017hybrid; @osia2018deep] can potentially serve as the basis for “least-privilege” collaboratively trained models. Further, it may be possible to detect active attacks that manipulate the model into learning extra features. Finally, it remains an open question if participant-level differential privacy mechanisms can produce accurate models when collaborative learning involves relatively few participants. [[***Acknowledgments.***]{}]{} This research was supported in part by the NSF grants 1611770 and 1704296, the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program, the Alan Turing Institute under the EPSRC grant EP/N510129/1, and a grant by Nokia Bell Labs. [^1]: <https://www.yelp.com/dataset>

--- abstract: 'We compare a perturbative QCD-based jet-energy loss model to the measured data of the pion nuclear modification factor and the high-$p_T$ elliptic flow at RHIC and LHC energies. This jet-energy loss model (BBMG) is currently coupled to state-of-the-art hydrodynamic descriptions. We report on a model extension to medium backgrounds generated by the parton cascade BAMPS. In addition, we study the impact of realistic medium transverse flow fields and a jet-medium coupling which includes the effects of the jet energy, the temperature of the bulk medium, and non-equilibrium effects close to the phase transition. By contrasting the two different background models, we point out that the description of the high-$p_T$ elliptic flow for a non-fluctuating medium requires to include such a jet-medium coupling and the transverse flow fields. While the results for both medium backgrounds show a remarkable similarity, there is an impact of the background medium and the background flow on the high-$p_T$ elliptic flow.' author: - 'Barbara Betz$^{1}$, Florian Senzel$^{1}$, Carsten Greiner$^{1}$, and Miklos Gyulassy$^{2,3}$' title: 'The impact of the medium and the jet-medium coupling on jet measurements at RHIC and LHC' --- Introduction ============ One of the open challenges in heavy-ion physics is to gain a precise understanding of the jet-medium dynamics, the jet-medium interactions, and the jet-energy loss formalism. In this letter, we study the impact of the medium and the details of the jet-medium coupling on the pion nuclear modification factor ($R_{AA}$) and the high-$p_T$ elliptic flow ($v_2$) measured at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) [@data1; @data2; @data3; @data4]. We find that both the background medium and the details of the jet-medium coupling play an important role for the simultaneous description of the $R_{AA}$ and the high-$p_T$ $v_2$. This simultaneous description reveals the so-called high-$p_T$ $v_2$-problem [@Betz:2014cza; @Betz:2012qq]: For various theoretical models [@data1; @Xu; @Molnar], the high-$p_T$ elliptic flow below $p_T \sim 20$ GeV is about a factor of two below the measured data [@data1; @data2; @data3; @data4]. This effect has been discussed in literature [@Betz:2014cza; @Betz:2012qq; @Xu; @Molnar]. Recently, it has been shown by CUJET3.0 [@Xu:2014tda] that a jet-medium coupling $\kappa=\kappa(E^2,T)$ can solve the high-$p_T$ $v_2$-problem for non-fluctuating initial conditions. This jet-medium coupling depends on the energy of the jet $E$, the temperature of the medium $T$, and non-equilibrium effects around the phase transition of $T_c \sim 160$ MeV. In this letter, we contrast results obtained for a background medium determined via the viscous hydrodynamic approach VISH2+1 [@VISH2+1] with the parton cascade BAMPS [@BAMPS; @Uphoff:2014cba] and study the impact of the jet-medium coupling $\kappa=\kappa(E^2,T)$ derived by CUJET3.0 [@Xu:2014tda]. We show that both medium backgrounds lead to surprisingly similar results. However, the background flow fields need to be included in both scenarios to enhance the high-$p_T$ elliptic flow which is otherwise too small. Applying the jet-medium coupling $\kappa=\kappa(E^2,T)$ finally leads to a reproduction of the high-$p_T$ $v_2$-data within measured error bars. {width="14.5cm"} With this, we contrast two completely different background models and demonstrate the importance of the background flow fields and the jet-medium coupling for the correct description of the measured jet observables. The Jet-Energy Loss Model BBMG ============================== The jet-energy loss model used in this letter (for convenience referred to as BBMG model) is based on the generic ansatz [@Betz:2014cza; @Betz:2012qq] $$\begin{aligned} \frac{dE}{d\tau}= -\kappa\, E^a(\tau) \, \tau^{z} \, e^{c=(2+z-a)/4} \, \zeta_q \, \Gamma_f , \label{Eq1}\end{aligned}$$ with the jet-energy $E$, the path-length $\tau$, and the energy density of the background medium $e$. In case of a radiative perturbative QCD (pQCD) energy-loss description used here, the explicit form of Eq. (\[Eq1\]) is [@Betz:2014cza; @Betz:2012qq] $$\begin{aligned} \frac{dE}{d\tau}= -\kappa\, E^0(\tau) \, \tau^{1} \, e^{3/4} \, \zeta_q \, \Gamma_f . \label{Eq2}\end{aligned}$$ Jet-energy loss fluctuations are included via the distribution $f_q(\zeta_q)= \frac{(1 + q)}{(q+2)^{1+q}} (q + 2- \zeta_q)^q $ which allows for an easy interpolation between non-fluctuating ($\zeta_{q=-1}=1$) distributions and those ones increasingly skewed towards small $\zeta_{q>-1} < 1$. Unless mentioned otherwise, the jet-energy loss fluctions are included with $q=0$ [@Betz:2014cza]. The background flow fields are incorporated via the flow factor $\Gamma_f=\gamma_f [1 - v_f \cos(\phi_{\rm jet} - \phi_{\rm flow})]$ with the background flow velocities $v_f$ given by VISH2+1 [@VISH2+1] or BAMPS [@BAMPS] and the $\gamma$-factor $\gamma_f = 1/\sqrt{1-v_f^2}$ [@Liu:2006he; @Baier:2006pt; @Renk:2005ta; @Armesto:2004vz]. $\phi_{\rm jet}$ is the jet angle w.r.t. the reaction plane and $\phi_{\rm flow}=\phi_{\rm flow}(\vec{x},t)$ is the corresponding local azimuthal angle of the background flow fields. Initially, the jets are distributed according to a transverse initial profile given by the bulk flow fields of VISH2+1 and BAMPS [@VISH2+1; @BAMPS]. Besides the two background media, we also contrast a jet-medium coupling $\kappa$ that depends only on the collision energy $\kappa=\kappa(\sqrt{s_{NN}})$ with the CUJET3.0 jet-medium coupling $\kappa=\kappa(E^2,T)$ which depends on the jet energy, the local temperature, and includes possible non-perturbative effects around the phase transition as in Ref. [@Xu:2014tda]. The DGLV [@DGLV] jet-medium coupling was generalized in Ref. [@Xu:2014tda] to be of the form $$\begin{aligned} \kappa(E^2,T) &=& \alpha_S^2(E^2)\chi_T\left(f_E^2+f_E^2 f_M^2 \mu^2/E^2\right)\nonumber\\ && - (1-\chi_T)(f_M^2 + f_E^2 f_M^2\mu^2/E^2) \,. \label{Eq3}\end{aligned}$$ Please note that Eq. (\[Eq3\]) is qualitatively similar to CUJET3.0 as the running coupling constant there a function of momentum transfer $\alpha(Q^2)$ while we assume that $Q^2=E^2$. The above expression includes - a running coupling effect via $\alpha_S(E^2)=1/[c+9/4\pi\,\log(E^2/T_c^2)]$ with $c=1.05$, - the Polyakov-loop suppression of the color-electric scattering [@Hidaka:2008dr] via $\chi_T = c_q L + c_g L^2$ with the pre-factors $c_q = (10.5 N_f)/(10.5 N_f + 16)$ for quarks and $c_g = 16/(10.5 N_f + 16)$ for gluons. Here, we consider $N_f = 3$. $L(T) = [\frac{1}{2} + \frac{1}{2} \rm{Tanh}[7.69(T - 0.0726)]]^{10}$ is a fit to lattice QCD [@Bazavov:2009zn; @Borsanyi:2010bp], as in Ref. [@Xu:2014tda]. - and a model of near-critical $T_c$ enhancement of scattering due to emergent magnetic monopoles. The functions $f_E(T)$ and $f_M(T)$ are also fits to lattice QCD [@Nakamura:2003pu]. The electric and magnetic screening masses are given by $\mu_{E,M}(T) = f_{E,M}(T)\mu(T)$ with the Debye screening mass $\mu^2(T) = \sqrt{4\pi\alpha_s(\mu^2)}T\sqrt{1+N_f/6}$. The functions $f_{E,M}(T)$ can be re-written to $f_E(T) = \sqrt{\chi_T}$ and $f_M(T)=0.3 \mu(T)/(T\sqrt{1+N_f/6})$ [@Xu:2014tda]. This jet-medium coupling decreases with the temperature of the background medium and thus shows an effective running with collision energy. {width="14.5cm"} Results and Discussion ====================== Fig. \[Fig01\] shows the pion nuclear modification factor ($R_{AA}$) for central (left panel) and mid-central (middle panel) collisions at RHIC (black) and LHC (red) as well as the high-$p_T$ pion elliptic flow ($v_2$) for mid-central events (right panel). The measured data [@data1; @data2; @data3; @data4] are compared to the pQCD-based energy loss of Eq. (\[Eq2\]) excluding the flow fields $\Gamma_f$, $dE/d\tau=\kappa(\sqrt{s_{NN}}) E^0 \tau^1 e^{3/4} \zeta_{0}$, for the hydrodynamic backgrounds of VISH2+1 [@VISH2+1] (solid lines) and the parton cascade BAMPS [@BAMPS] (dashed-dotted lines). Jet-energy loss fluctuations are considered via $\zeta_0$. The jet-medium coupling in Fig. \[Fig01\] depends on the collision energy \[$\kappa=\kappa(\sqrt{s_{NN}})$\]. Within the present error bars, both the central and the mid-central pion nuclear modification factor can be described using this pQCD ansatz without the background flow fields. However, the high-$p_T$ $v_2$-problem [@Betz:2014cza; @Betz:2012qq] becomes obvious. The right panel of Fig. \[Fig01\] shows that the high-$p_T$ elliptic flow is below the measured data. Including the background flow fields via $\Gamma_f$ in Fig. \[Fig02\] leads to a significant increase of the high-$p_T$ elliptic flow while the pion nuclear modification factor is only affected marginally. Fig. \[Fig02\] reveals the strong influence of the background flow fields on the high-$p_T$ elliptic flow. Besides this, Figs. \[Fig01\] and \[Fig02\] demonstrate a surprising similarity between the results based on a medium described by viscous hydrodynamics [@VISH2+1] and the parton cascade BAMPS [@BAMPS]. This similarity cannot be expected a priori as the two background media are quite different: While the hydrodynamic description of VISH2+1 assumes an equilibrated system, the parton cascade BAMPS also includes non-equilibrium effects in the bulk medium evolution. However, since those effects are small, a temperature can be defined after a very short initial time $t_0$. In this work, we use $t_0=0.3$ fm at RHIC and $t_0=0.2$ fm at LHC energies. In a third step, we include the jet-medium coupling $\kappa=\kappa(E^2,T)$ given by Eq.(\[Eq3\]) [@Xu:2014tda] in our jet-energy loss approach. The result is shown in Fig. \[Fig03\], again for the hydrodynamic background VISH2+1 (solid lines) and a medium determined via the parton cascade BAMPS (dashed-dotted lines). As in Figs. \[Fig01\] and \[Fig02\], the pion nuclear modification factor is well described both at RHIC and LHC. However, the high-$p_T$ elliptic flow increases significantly below $p_T \sim 20$ GeV, especially for the BAMPS background which already includes non-equilibrium effects through microscopic, non-equilibrium transport calculations [@BAMPS; @Uphoff:2014cba]. Fig. \[Fig03\] demonstrates that the jet-medium coupling $\kappa=\kappa(E^2,T)$ suggested by CUJET3.0 [@Xu:2014tda] can solve the high-$p_T$ $v_2$-problem. However, the background medium considered does play an important role for the description of the high-$p_T$ elliptic flow. Please note that the initial conditions studied here are non-fluctuating, i.e. neglect event-by-event short-scale inhomogeneities. The effect of event-by-event fluctuations will be studied elsewhere [@Betz:2016]. {width="14.5cm"} To further investigate the influence of the parton cascade medium, we varied the jet-energy loss fluctuations. The results are shown in Fig. \[Fig04\]. As one can see, the slope of both the $R_{AA}$ and the high-$p_T$ $v_2$ at LHC energies changes when considering non-fluctuating ($\zeta_{q=-1}=1$) jet-energy loss distributions. To be more precise, the results get closer to the slope of the measured data. This is in contrast to results previously obtained with the pQCD-based jet-energy loss ansatz [@Betz:2014cza] for the VISH2+1 background [@VISH2+1]. In Ref. [@Betz:2014cza] the slope of both the $R_{AA}$ and the high-$p_T$ $v_2$ did not change significantly when changing the jet-energy loss fluctuations. In particular, the results of Ref. [@Betz:2014cza] for the elliptic flow at $p_T>20$ GeV coincided for various fluctuation distributions. This result strengthens the observation that the background medium considered plays an important role for the correct description of the jet observables. However, while the nuclear modification factor is less influenced by the background medium, the impact of the background medium and background flow on the high-$p_T$ elliptic flow is quite significant. Conclusions =========== We compared the measured data on the nuclear modification factor and the high-$p_T$ elliptic flow at RHIC and LHC energies to results obtained by the pQCD-based jet-energy loss model BBMG. We contrasted results obtained via a hydrodynamic background (VISH2+1) [@VISH2+1] with results based on the parton cascade BAMPS [@BAMPS; @Uphoff:2014cba]. We showed that the results for both medium backgrounds exhibit a remarkable similarity, especially for the pion nuclear modification factor. We demonstrated that the background medium and background flow strongly influence the high-$p_T$ $v_2$. We found that for event-averaged or non-fluctuating initial conditions, studied here, the simultaneous description of the pion nuclear modification factor and high-$p_T$ elliptic flow requires to consider both the background flow fields and a jet-medium coupling that depends on the energy of the jet, the temperature of the medium, and non-equilibrium effects around the phase transition [@Xu:2014tda]. {width="14.5cm"} Acknowledgement =============== We thank J. Noronha, J. Noronha-Hostler, and J. Xu for helpful discussions as well as U. Heinz and C. Shen for making their hydrodynamic field grids available. This work was supported through the Bundesministerium für Bildung und Forschung, the Helmholtz International Centre for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse, the US-DOE Nuclear Science Grant No. DE-AC02-05CH11231 within the framework of the JET Topical Collaboration, the US-DOE Nuclear Science Grant No.DE-FG02-93ER40764, and IPP/CCNU, Wuhan. Numerical computations have been performed at the Center for Scientific Computing (CSC). [99]{} A. Adare [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. C [**87**]{}, 034911 (2013); Phys. Rev. Lett.  [**105**]{}, 142301 (2010). B. Abelev [*et al.*]{} \[ALICE Collaboration\], Phys. Lett. B [**720**]{}, 52 (2013); Phys. Lett. B [**719**]{}, 18 (2013). S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Eur. Phys. J. C [**72**]{}, 1945 (2012); Phys. Rev. Lett.  [**109**]{}, 022301 (2012). 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--- abstract: 'We present the discovery, from archival [*Chandra*]{} and [*XMM-Newton*]{} data, of X-ray eclipses in two ultraluminous X-ray sources (ULXs), located in the same region of the galaxy M51: CXOM51 J132940.0$+$471237 (ULX-1, for simplicity) and CXOM51 J132939.5$+$471244 (ULX-2). Three eclipses were detected for ULX-1, two for ULX-2. The presence of eclipses puts strong constraints on the viewing angle, suggesting that both ULXs are seen almost edge-on and are certainly not beamed towards us. Despite the similar viewing angles and luminosities ($L_{\rm X} \approx 2 \times 10^{39}$ erg s$^{-1}$ in the 0.3–8 keV band for both sources), their X-ray properties are different. ULX-1 has a soft spectrum, well fitted by Comptonization emission from a medium with electron temperature $kT_e \approx 1$ keV. ULX-2 is harder, well fitted by a slim disk with $kT_{\rm in} \approx 1.5$–1.8 keV and normalization consistent with a $\sim$10$M_{\odot}$ black hole. ULX-1 has a significant contribution from multi-temperature thermal plasma emission ($L_{\rm X,mekal} \approx 2 \times 10^{38}$ erg s$^{-1}$); about 10% of this emission remains visible during the eclipses, proving that the emitting gas comes from a region slightly more extended than the size of the donor star. From the sequence and duration of the [*Chandra*]{} observations in and out of eclipse, we constrain the binary period of ULX-1 to be either $\approx$6.3 days, or $\approx$12.5–13 days. If the donor star fills its Roche lobe (a plausible assumption for ULXs), both cases require an evolved donor; most likely a blue supergiant, given the young age of the stellar population in that galactic environment.' author: - 'R. Urquhart and R. Soria' title: 'Two eclipsing ultraluminous X-ray sources in M51' --- \[firstpage\] Introduction {#intro} ============ Ultraluminous X-ray sources (ULXs) are the high-luminosity end of the X-ray binary population, with X-ray luminosities $> 10^{39}$ erg s$^{-1}$, which is the approximate peak luminosity of Galactic stellar-mass black holes (BHs). The most likely explanation for the vast majority of ULXs is that they are stellar-mass BHs [or neutron stars; @2014Natur.514..202B] accreting well above the critical accretion rate and their luminosity is a few times the classical Eddington limit of spherical accretion. Another possibility is that ULXs are powered by accreting BHs up to $\approx 80 M_{\odot}$ [@2010ApJ...714.1217B], several times more massive than typical Galactic stellar-mass BHs ($M \sim 5$–$15 M_{\odot}$: @2012ApJ...757...36K). In addition, ULXs may appear more luminous because their X-ray emission is partly collimated along our line of sight. This may happen at super-critical accretion rates, because of the predicted formation of a dense radiatively-driven disk outflow and a lower-density polar funnel, along which more photons can escape [@2011ApJ...736....2O; @2014ApJ...796..106J; @2015MNRAS.453.3213S]. Finally, some of the brightest ULXs may contain a population of intermediate-mass BHs [$10^2\,{{\rm M}_{\odot}}\leq M \leq10^4\,{{\rm M}_{\odot}}$; @2009Natur.460...73F; @2016ApJ...817...88Z]. It is difficult to determine the relative contribution of those three factors (mass, accretion rate and viewing angle), and therefore also determine the true isotropic luminosity and accretion rate of ULXs, without at least a direct constraint on their viewing angle. There is already indirect evidence that ULXs are not strongly beamed. Modelling of the optical light curve from the irradiated donor star in NGC7793-P13 showed [@2014Natur.514..198M] that the source is viewed at an angle $>$20$^{\circ}$ and more likely much higher; thus, in that case, super-Eddington accretion is the reason for the high luminosity, not a heavier BH or a down-the-funnel view. Studies of large ($\ga$100 pc) photo-ionized and/or shock-ionized plasma bubbles around ULXs [@2002astro.ph..2488P; @2008AIPC.1010..303P] provide other clues about viewing angles: if fast-accreting BHs appeared as ULXs only for a narrow range of face-on inclinations, we would see many more of those large ionized bubbles without a ULX inside; moreover, the true X-ray luminosity of a beamed source (much lower than the apparent luminosity) would not be enough to explain the strong He emission observed from some of the photo-ionized ULX bubbles [@2006IAUS..230..293P]. Both the fact that most ULX bubbles do contain a bright, central X-ray source, and the fact that (in photo-ionized bubbles) the apparent X-ray photon flux from the central source is consistent with the He photon flux from the bubble, suggest that, statistically, ULXs are seen over a broad range of viewing angles. X-ray spectroscopic studies can also be used to qualitatively constrain ULX viewing angles: it was suggested [@2013MNRAS.435.1758S] that ULXs seen at lower inclination (down the polar funnel) have harder X-ray spectra while those seen at higher inclination (through the disk wind) have softer spectra with a lower-energy downturn, due to a higher degree of Compton scattering in the wind. This interpretation is consistent with the presence of absorption and emission features (interpreted as signatures of the outflow) in the X-ray spectra of ULXs with softer spectra [@2014MNRAS.438L..51M; @2015MNRAS.454.3134M]. It is also in agreement with a higher degree of short-term variability (interpreted as the imprint of a clumpy wind) in sources with softer spectra [@2015MNRAS.447.3243M]. Apart from those indirect or statistical arguments, until recently there was no bright extragalactic stellar-mass BH for which the viewing angle could be directly pinned down. We have now discovered two such sources, both located in the same spiral arm of the spiral galaxy M51; in fact, surprisingly, they appear projected in the sky within only $\approx$ $350\,{\text{pc}}$ of each other (see Figure \[chand\_img\]). Both sources have X-ray luminosities $\ga$10$^{39}$ erg s$^{-1}$, and crucially, they both show sharp X-ray drops and rebrightenings, which we interpret as eclipses by their donor stars, occulting the inner region of the disk. The presence of eclipses places a lower limit on the inclination angle ($i \gtrsim 75^{\circ}$) as we must be viewing the X-ray sources near edge-on. In this paper, we present the eclipse discovery and the main X-ray timing and spectral properties of the two sources. We will also briefly discuss more general implications and opportunities provided by the detection of eclipses, for our modelling of these systems. In a companion paper (Soria et al., in prep.) we will present a study of the optical counterparts and other interesting, newly-discovered properties of those same two ULXs, which show optical and radio evidence of jets and outflows. Targets of our study {#obs_sec} ==================== M51, also known as the Whirlpool Galaxy, is an interacting face-on spiral at a distance of $8.0 \pm 0.6$ Mpc [@M51dist]. The two eclipsing sources discussed in this paper are those catalogued as CXOM51 J132940.0$+$471237 (henceforth, ULX-1) and CXOM51 J132939.5$+$471244 (henceforth, ULX-2) in [@2004ApJ...601..735T]. We re-estimated their positions using all the [*Chandra*]{} data available to-date, and obtained RA (J2000) $=13^{\rm{h}} 29^{\rm{m}} 39^{\rm s}.94$, Dec. (J2000) $= +47^{\circ} 12' 36''.6$ for ULX-1, and RA (J2000) $= 13^{\rm{h}} 29^{\rm{m}} 39^{\rm s}44$, Dec. (J2000) $= +47^{\circ} 12' 43''.3$ for ULX-2. Both positions are subject to the standard uncertainty in the absolute astrometry of [*Chandra*]{} pointings, $\approx$0”.6 at the 90% confidence level[^1]. A more precise determination of their positions is left to a follow-up study (Soria et al., in prep.) of their optical and radio counterparts. ULX-1 and ULX-2 were first discovered as a single unresolved source by the [*Einstein Observatory*]{} [@1985ApJ...298..259P]. This was followed up with observations with [*ROSAT*]{} (source C in ; source R7 in @1995ApJ...438..663M). The higher spatial resolution of [*Chandra*]{}’s Advanced CCD Imaging Spectrometer (ACIS) finally led to the two sources being resolved [source 6 and source 5 in @2004ApJ...601..735T]. ULX-1 was found to be a relatively soft source, with very few counts above $2\,{\text{keV}}$. ULX-2 was found to be variable, decreasing in luminosity by a factor of $\approx2.5$ between observations [@2004ApJ...601..735T]. Further spectral studies of the two sources, based on a 2003 [*XMM-Newton*]{} observation, were carried out by [@2005ApJ...635..198D]. With a much larger database of [*Chandra*]{} and [*XMM-Newton*]{} observation available since then, we have now studied the two sources in more detail, and found more intriguing properties. ![Top panel: [*Chandra*]{}/ACIS-S adaptively smoothed 190-ks image of M51 during ObsID 13814, showing the location of ULX-1 and ULX-2 with respect to the nuclear region. Red represents the 0.3–1 keV band, green the 1–2 keV band, and blue the 2–7 keV band. Bottom panel: as in the top panel, but only for the portion of ObsID 13814 during which ULX-1 is in eclipse (70 ks). []{data-label="chand_img"}](full.eps "fig:"){width="49.00000%"}\ ![Top panel: [*Chandra*]{}/ACIS-S adaptively smoothed 190-ks image of M51 during ObsID 13814, showing the location of ULX-1 and ULX-2 with respect to the nuclear region. Red represents the 0.3–1 keV band, green the 1–2 keV band, and blue the 2–7 keV band. Bottom panel: as in the top panel, but only for the portion of ObsID 13814 during which ULX-1 is in eclipse (70 ks). []{data-label="chand_img"}](eclipse.eps "fig:"){width="49.00000%"} [lccr]{} & & & ------------------------------------------------------------------------ \ &&& ------------------------------------------------------------------------ \ &   354 &  14.9 & 2000-03-21 ------------------------------------------------------------------------ \ &  1622 &  26.8 & 2001-06-23\ &  3932 &  48.0 & 2003-08-07\ & 13813 & 179.2 & 2012-09-09\ & 13812 & 157.5 & 2012-09-12\ & 15496 &  41.0 & 2012-09-19\ & 13814 & 189.9 & 2012-09-20\ & 13815 &  67.2 & 2012-09-23\ & 13816 &  73.1 & 2012-09-26\ & 15553 &  37.6 & 2012-10-10 ------------------------------------------------------------------------ \ & 0112840201 & 20.9 & 2003-01-15 ------------------------------------------------------------------------ \ & 0212480801 & 49.2 & 2005-07-01\ & 0212480901 &  closed & 2005-07-01\ & 0303420101 & 54.1 & 2006-05-20\ & 0303420301 &  closed & 2006-05-20\ & 0303420201 & 36.8 & 2006-05-24\ & 0303420401 &  closed & 2006-05-24\ & 0677980701 & 13.3 & 2011-06-07\ & 0677980801 & 13.3[^2] & 2011-06-11 ------------------------------------------------------------------------ \ \[obs\_table\] [ccccccc]{} & & & & & & ------------------------------------------------------------------------ \ &&&&&& ------------------------------------------------------------------------ \   354 &  0 &  15 &51715.34–51715.51&&& $18.1\pm1.1$ ------------------------------------------------------------------------ \  1622 & 25 &   2& 52083.78–52084.09& 52083.81–\[52084.09\] & $1.2\pm0.2$ & $10.7\pm1.9$\  3932 &  0 &  48 &52858.61–52859.16 &&& $12.0\pm0.5$\ 13813 & 40 & 139 & 56179.74–56181.82&\[56179.74\]–56180.20 & $0.4\pm0.1$ & $14.7\pm0.3$\ 13812 &  0 & 157 & 56182.77–56184.59&&& $12.2\pm0.3$\ 15496 &  0 &  41 &56189.39–56189.86&&& $14.8\pm0.6$\ 13814 & 70 & 120 & 56190.31–56192.50& 56191.64–\[56192.50\] & $0.3\pm0.1$ & $15.6\pm0.4$\ 13815 &  0 &  67 &56193.34–56194.12 &&& $14.0\pm0.5$\ 13816 &  0 &  73 &56196.22–56197.06 &&& $12.0\pm0.4$\ 15553 &  0 &  38 &56210.03–56210.47&&& $10.2\pm0.5$ ------------------------------------------------------------------------ \ \[ec1\_table\] [ccccccc]{} & & & & & & ------------------------------------------------------------------------ \ &&&&& ------------------------------------------------------------------------ \   354 &  0 &  15 &51715.34–51715.51&&& $18.0\pm1.1$ ------------------------------------------------------------------------ \  1622 &  0 &  27 & 52083.78–52084.09& && $7.6\pm0.5$\  3932 &  0&  48 &52858.61–52859.16 &&& $9.9\pm0.5$\ 13813 & 48 & 131 & 56179.74–56181.82& \[56179.74\]–56180.30 &$0.6\pm0.1$& $10.5\pm0.3$\ 13812 &  0 & 157 & 56182.77–56184.59&&& $10.7\pm0.3$\ 15496 &  0 &  41 &56189.39–56189.86&&& $13.7\pm0.6$\ 13814 &  0& 190 & 56190.31–56192.50& && $11.5\pm0.2$\ 13815 &  0 &  67 &56193.34–56194.12&&& $3.3\pm0.2$\ 13816 &  0 &  73 &56196.22–56197.06&&& $14.6\pm0.5$\ 15553 &  0 &  38 &56210.03–56210.47&&& $7.9\pm0.5$ ------------------------------------------------------------------------ \ \[ec2\_table\] Data Analysis {#data_sec} ============= M51 was observed by [*Chandra*]{}/ACIS-S fourteen times between 2000 and 2012: two of those observations were too short ($\le$2 ks) to be useful, and another two did not include our sources in the field of view; the other 10 observations are listed in Table \[obs\_table\]. (See @2016..Kuntz for a full catalog and discussion of all the [*Chandra*]{} sources in M51.) We downloaded the [*Chandra*]{} data from the public archives and re-processed them using standard tasks within the Chandra Interactive Analysis of Observations () Version 4.7 software package [@2006SPIE.6270E..1VF]. Any intervals with high particle backgrounds were filtered out. We extracted spectra and light-curves for ULX-1 and ULX-2 using circular regions of $\approx$4 radii and local background regions three times as large as the source regions. For each observation, background-subtracted light curves were created with the task [*dmextract*]{}. Spectra were extracted with [*specextract*]{}, and were then grouped to a minimum of 15 counts per bin, for $\chi^2$ fitting. M51 was also observed by [*XMM-Newton*]{} nine times between 2003 and 2011, although no data were recorded on three occasions due to strong background flaring (Table \[obs\_table\]). We downloaded the [*XMM-Newton*]{} data from NASA’s High Energy Astrophysics Science Archive Research Centre (HEASARC) archive. We used the European Photon Imaging Camera (EPIC) observations and re-processed them using standard tasks in the Science Analysis System ([SAS]{}) version 14.0.0 software package; we filtered out high particle background exposure intervals. Due to the lower spatial resolution of [*XMM-Newton*]{}/EPIC, the ULXs cannot be entirely visually resolved, although the elongated appearance of the EPIC source is consistent with the two separate [*Chandra*]{} sources (as discussed in Section \[eclipses\_sec\]). For each observation we extracted a single background-subtracted light-curve and spectrum for both sources combined, using a circular extraction region of 20 radius, and a local background region that is at least three times larger, does not fall onto any chip gap and is of similar distance to the readout nodes as the source region. Standard flagging criteria `#XMMEA_EP` and `#XMMEA_EM` were used for pn and MOS respectively, along with `FLAG=0`. We also selected patterns 0–4 for pn and 0–12 for MOS. For our timing study, we extracted light-curves with the [SAS]{} tasks [*evselect*]{} and [*epiclccorr*]{}. For our spectral study, we extracted individual pn, MOS1 and MOS2 spectra with standard [*xmmselect*]{} tasks; whenever possible, we combined the pn, MOS1 and MOS2 spectra of each observation with [*epicspeccombine*]{}, to create a weighted-average EPIC spectrum. In some observations, the pn data were not usable because the source falls onto a chip gap; in those cases, we used only the MOS1 and MOS2 data in [*epicspeccombine*]{}. Finally, we grouped the spectra to a minimum of 20 counts per bin so that we could use Gaussian statistics. For both [*Chandra*]{} and [*XMM-Newton*]{} data, spectral fitting was performed with [XSPEC]{} version 12.8.2 [@1996ASPC..101...17A]. Timing analysis was conducted with standard [FTOOLS]{} tasks [@1995ASPC...77..367B], such as [*lcurve*]{}, [*efsearch*]{} and [*statistics*]{}. Imaging analysis was done with HEASARC’s visualization package, and adaptive image smoothing with ’s [*csmooth*]{} routine. ![Top panel: [*Chandra*]{}/ACIS-S background-subtracted light-curve of ULX-1 from observation 1622, split into a soft band (0.3–1.2 keV: red datapoints) and a hard band (1.2–7.0 keV; blue datapoints). It shows a sharp drop in flux about 2 ks into the observation. The data are binned into 1000-s intervals. Bottom panel: as in the top panel, for ULX-2 in the same observation.[]{data-label="1622_lc"}](1622_ULX1_1000.eps "fig:"){width="48.00000%"} ![Top panel: [*Chandra*]{}/ACIS-S background-subtracted light-curve of ULX-1 from observation 1622, split into a soft band (0.3–1.2 keV: red datapoints) and a hard band (1.2–7.0 keV; blue datapoints). It shows a sharp drop in flux about 2 ks into the observation. The data are binned into 1000-s intervals. Bottom panel: as in the top panel, for ULX-2 in the same observation.[]{data-label="1622_lc"}](1622_ULX2_1000.eps "fig:"){width="48.00000%"} {width="48.00000%"} {width="48.00000%"}\ {width="48.00000%"} {width="48.00000%"}\ ![Top panel: [*Chandra*]{}/ACIS-S background-subtracted light-curves of ULX-1 during ObsID 13814 (red for the 0.3–1.2 keV band, blue for the 1.2–7.0 keV band), showing the beginning of an eclipse about 110 ks into the observation. The data are binned into 1000-s intervals. Middle panel: soft (red curve, 0.3–1.2 keV), hard (blue curve, 1.2–7.0 keV) and total (green curve, 0.3–7.0 keV) [*Chandra*]{}/ACIS-S background-subtracted light-curves of ULX-1 during ObsID 13814, zoomed in around the time of eclipse ingress. The data are binned into 1000-s intervals. Bottom panel: as in the top panel, for ULX-2 during the same [*Chandra*]{} observation.[]{data-label="13814_lc"}](13814_ULX1.eps "fig:"){width="48.00000%"} ![Top panel: [*Chandra*]{}/ACIS-S background-subtracted light-curves of ULX-1 during ObsID 13814 (red for the 0.3–1.2 keV band, blue for the 1.2–7.0 keV band), showing the beginning of an eclipse about 110 ks into the observation. The data are binned into 1000-s intervals. Middle panel: soft (red curve, 0.3–1.2 keV), hard (blue curve, 1.2–7.0 keV) and total (green curve, 0.3–7.0 keV) [*Chandra*]{}/ACIS-S background-subtracted light-curves of ULX-1 during ObsID 13814, zoomed in around the time of eclipse ingress. The data are binned into 1000-s intervals. Bottom panel: as in the top panel, for ULX-2 during the same [*Chandra*]{} observation.[]{data-label="13814_lc"}](13814_decay_ULX1.eps "fig:"){width="48.00000%"} ![Top panel: [*Chandra*]{}/ACIS-S background-subtracted light-curves of ULX-1 during ObsID 13814 (red for the 0.3–1.2 keV band, blue for the 1.2–7.0 keV band), showing the beginning of an eclipse about 110 ks into the observation. The data are binned into 1000-s intervals. Middle panel: soft (red curve, 0.3–1.2 keV), hard (blue curve, 1.2–7.0 keV) and total (green curve, 0.3–7.0 keV) [*Chandra*]{}/ACIS-S background-subtracted light-curves of ULX-1 during ObsID 13814, zoomed in around the time of eclipse ingress. The data are binned into 1000-s intervals. Bottom panel: as in the top panel, for ULX-2 during the same [*Chandra*]{} observation.[]{data-label="13814_lc"}](13814_ULX2.eps "fig:"){width="48.00000%"} Results ======= Eclipses {#eclipses_sec} -------- ### ULX-1 eclipses and dips in the [*Chandra*]{} data {#ulx1_ec_sec} From our inspection of the [*Chandra*]{} light-curves, we have discovered 3 epochs (ObsIDs 1622, 13813 and 13814) in which the flux of ULX-1 is strongly reduced for at least part of the observation (Table \[ec1\_table\] and Figures \[1622\_lc\], \[13813\_lc\], \[13814\_lc\]). The transition between the long-term-average flux level and the lower level occurs too quickly ($\Delta t \sim 10^3$ s) to be explained by a state transition in the inflow, or a change in the mass accretion rate. Our identification of the low state in ObsID 1622 as a true stellar eclipse rather than a dip may be debatable, given that the flux drop happens right at the start of the observation; however, the presence of eclipses is very clear in ObsIDs 13813 (2012 September 9) and 13814 (2012 September 20), which show a low-to-high and a high-to-low transition, respectively. We also checked that ULX-1 is not at the edge of the chip, there are no instrumental glitches, and no other source in the field has a count-rate step change at the same time. We conclude that the simplest and most logical explanation is an eclipse of the X-ray emitting region by the donor star. The flux during the eclipse is not exactly zero: by stacking the time intervals during eclipses, we can find a faint but statistically significant residual emission, softer than the emission outside eclipses. We will discuss the spectrum of the residual emission in Section \[spectral\_sec\]. The way ULX-1 enters the eclipse in ObsID 13814 (Figure \[13814\_lc\]) is also interesting. The transition to eclipse in the soft band (0.3–1.2 keV) appears less sharp than the transition in the hard band (1.2–7.0 keV): the soft-band count rate drops to effectively zero in $\approx$4 ks, while the same transition happens in $\la$1 ks for the hard band. This can be explained if the softer X-ray photons come from a more extended region that takes longer to be completely occulted than the effectively point-like central region responsible for the harder X-ray photons ; for example, the softer emission may have contributions from the outer, cooler parts of an outflow. However, we cannot rule out that the discrepancy is simply due to small-number statistics. Finally, we find a deep dip in the [*Chandra*]{} light curve of ULX-1 during ObsID 13812 (Figure \[13812\_lc\]). The count rate drops to zero and then recovers to the pre-dip level, just like during an eclipse. However, the short duration ($\approx$20 ks) and double-dipping substructure of this phase suggest that this occultation is not due to the companion star; we suggest that it is more likely the result of lumps or other inhomogeneities in the thick outer rim of the disk, or is caused by the accretion stream overshooting the point of impact in the outer disk and covering our view of the inner regions . Analogous X-ray dips are seen in several Galactic X-ray binaries and are interpreted as evidence of a high viewing angle. Assuming that the occultation is produced by a geometrically thick structure in Keplerian rotation, we can estimate the angular extent of this feature by scaling the duration of the dipping phase to the binary period of ULX-1. If the period is $\approx$6d (see Section \[ulx1\_bp\_sec\]), the occulting structure spans $\Delta \phi \approx 14^{\circ}$; for a $\approx$13d period, $\Delta \phi \approx 6^{\circ}$. ![Top panel: as in Figure \[13814\_lc\], for the [*Chandra*]{}/ACIS-S observation 13812, showing a dip around 70–90 ks into the observation. All data in this panel and in those below are binned to $1000\,{\text{s}}$. Middle panel: zoomed-in view of the dip in the soft band (red datapoints), hard band (blue datapoints) and total band (green datapoints). Botttom panel: as in Figure \[13814\_lc\], for observation 13812.[]{data-label="13812_lc"}](13812_ULX1.eps "fig:"){width="48.00000%"} ![Top panel: as in Figure \[13814\_lc\], for the [*Chandra*]{}/ACIS-S observation 13812, showing a dip around 70–90 ks into the observation. All data in this panel and in those below are binned to $1000\,{\text{s}}$. Middle panel: zoomed-in view of the dip in the soft band (red datapoints), hard band (blue datapoints) and total band (green datapoints). Botttom panel: as in Figure \[13814\_lc\], for observation 13812.[]{data-label="13812_lc"}](13812_decay.eps "fig:"){width="48.00000%"} ![Top panel: as in Figure \[13814\_lc\], for the [*Chandra*]{}/ACIS-S observation 13812, showing a dip around 70–90 ks into the observation. All data in this panel and in those below are binned to $1000\,{\text{s}}$. Middle panel: zoomed-in view of the dip in the soft band (red datapoints), hard band (blue datapoints) and total band (green datapoints). Botttom panel: as in Figure \[13814\_lc\], for observation 13812.[]{data-label="13812_lc"}](13812_ULX2.eps "fig:"){width="48.00000%"} ### ULX-2 eclipse in the [*Chandra*]{} data In the same set of [*Chandra*]{} observations, we also discovered one eclipse in ULX-2, in observation 13813 (Figure \[13813\_lc\], bottom panel). The abrupt nature of the transition from low to high count rates once again suggests that we are looking at an occultation by the companion star. Remarkably, the egress from the eclipse of ULX-2 happens only $\approx$8ks later than the egress from the ULX-1 eclipse, at MJD 56180.30 and 56180.20, respectively (cf. bottom and top panels of Figure \[13813\_lc\]). The small but significant time difference guarantees that the two count-rate jumps seen in the two ULXs are not instrumental anomalies but real physical events. Moreover, we did extensive checks on other bright X-ray sources in the same ACIS-S3 chip, and found that none of them shows similar jumps around that time; this also rules out instrumental problems. We also examined the light-curves of ULX-2 in all other [*Chandra*]{} observations, including those where eclipses or dips were found in the light-curve of ULX-1 (bottom panels of Figures \[1622\_lc\], \[13814\_lc\], and \[13812\_lc\]). We found no other unambiguous eclipses or deep dips. ULX-2 does show significant intra-observational variability in ObsID 13815. Throughout the 67-ks observation, the source displays a much lower count rate than its average out of eclipse count rate, in both the soft and the hard band (Table 3 and Figure \[13815\_lc\]). The count rate further decreases during that [*Chandra*]{} epoch, until it becomes consistent with a non-detection at the end of the observation. The decrease is slow enough (compared with the eclipse in ObsID 13813, Figure \[13813\_lc\]) to rule out a stellar occultation. We do not have enough evidence or enough counts to test whether this flux decrease is due to intrinsic variability of ULX-2, or to an increased absorption by colder material in the outer disk. As usual, we checked the behaviour of ULX-1 and other bright sources in ObsID 13815 to ascertain that the lower count rate seen from ULX-2 is not an instrumental problem. ### ULX-2 eclipse in the [*XMM-Newton*]{} data We then searched for possible eclipses of either ULX-1 or ULX-2 during the [*XMM-Newton*]{} observations. Due to the poorer spatial resolution of EPIC relative to ACIS-S, ULX-1 and ULX-2 are not completely resolved by [*XMM-Newton*]{}; however, the point spread function in the combined EPIC MOS1+MOS2 images is clearly peanut-shaped, consistent with the position and relative intensity of the two [*Chandra*]{} sources, and the upper source (ULX-2) has significantly harder colors (Figure \[XMM\_image\], top panel). Firstly, we extracted and examined background-subtracted EPIC light-curves for the combined emission of the two unresolved sources, for each [*XMM-Newton*]{} observation. Because the two sources have comparable count rates (Tables 2 and 3), an eclipse in either source would cause the observed count rate to drop by a factor of $\approx$2. This is the scenario we find in observation 0303420101: there is an apparent increase in the observed EPIC-MOS count rate by a factor of $\approx$2, some 22 ks from the start of the observation, which we tentatively interpret as the egress from an eclipse (Figure \[XMM\_lc\], top panel), superposed on short-term intrinsic variability. Unfortunately we cannot use EPIC-pn data for this crucial epoch, because the source falls onto a chip gap. To quantify the step change in the count rate between the first and second part of the observation (green and blue datapoints in Figure \[XMM\_lc\]), we performed a Kolmogorov-Smirnov (KS) test on the two distributions of datapoints, to determine whether they are drawn from different populations. We find a KS statistic of $0.65$ and p-value of $3.8\times10^{-11}$, suggesting that the two sections of the light-curve are indeed statistically different. The average MOS1+MOS2 net count rate in the “eclipse" part of the light-curve is $\approx 0.026 \pm 0.003$ ct s$^{-1}$ (90% confidence limit), while in the “non-eclipse" part it is $\approx 0.046 \pm 0.003$ ct s$^{-1}$. Having ascertained from the X-ray light-curve that ObsID 0303420101 probably includes an eclipse, we extracted MOS1+MOS2 images from the low-rate and high-rate sections of that observation, and confirmed (Figure \[XMM\_image\]) that in the low-rate interval, the emission from ULX-2 is missing. We extracted and inspected the light-curves of every other [*XMM-Newton*]{} observation. No eclipses of ULX-1 and no further eclipses of ULX-2 were detected; however, several of those observations are much shorter than the typical [*Chandra*]{} observations, and the background count rate is much higher in the EPIC cameras. Thus, ruling out the presence of an eclipse as opposed to intrinsic variability is no easy task in some of the [*XMM-Newton*]{} observations. ![[*Chandra*]{}/ACIS-S background-subtracted light-curves for ULX-2 during observation 13815; red datapoints are for the 0.3–1.2 keV band, blue datapoints for the 1.2–7.0 keV band. Data are binned to 300 s. As a comparison, the dashed and dotted lines represent the average count rates for the soft and hard band, respectively, during the previous [*Chandra*]{} observation, ObsID 13814, taken 3 days earlier.[]{data-label="13815_lc"}](13815_ULX2_3000.eps){width="48.00000%"} ![Top panel: stacked [*XMM-Newton*]{} MOS1+MOS2 image for non-eclipse interval of observation 0303420101. Red represents photons in the $0.3$–1 keV band, green is for 1–2 keV and blue is for 2–7 keV. The green ellipses indicate the location of ULX-1 and ULX-2 as determined from the [*Chandra*]{}/ACIS-S images; their point spread functions appear elongated because the ULXs were observed a few arcmin away from the ACIS-S3 aimpoint. The two sources are not clearly resolved by [*XMM-Newton*]{}, but the color difference between the two ends of the peanut-shaped EPIC-MOS source is consistent with the color and spectral differences seen by [*Chandra*]{}. Bottom panel: as in the top panel, but for the ULX-2 eclipse interval of observation 0303420101.[]{data-label="XMM_image"}](XMM-nonec.eps "fig:"){width="47.00000%"}\ ![Top panel: stacked [*XMM-Newton*]{} MOS1+MOS2 image for non-eclipse interval of observation 0303420101. Red represents photons in the $0.3$–1 keV band, green is for 1–2 keV and blue is for 2–7 keV. The green ellipses indicate the location of ULX-1 and ULX-2 as determined from the [*Chandra*]{}/ACIS-S images; their point spread functions appear elongated because the ULXs were observed a few arcmin away from the ACIS-S3 aimpoint. The two sources are not clearly resolved by [*XMM-Newton*]{}, but the color difference between the two ends of the peanut-shaped EPIC-MOS source is consistent with the color and spectral differences seen by [*Chandra*]{}. Bottom panel: as in the top panel, but for the ULX-2 eclipse interval of observation 0303420101.[]{data-label="XMM_image"}](XMM-ec.eps "fig:"){width="47.00000%"} ![Background-subtracted [*XMM-Newton*]{}/EPIC MOS1+MOS2 light-curve for the unresolved ULX source in observation 0303420101. Time intervals affected by background flaring have been removed. The light-curve was extracted in the $0.2$–8 keV band and the datapoints have been binned to 300 s for display purposes. The light-curve is broken into two sections: the first $\approx$22 ks (green datapoints) have a lower count rate and correspond to an eclipse of ULX-2; in the remaining $\approx$20 ks, both ULXs are out of eclipse (blue datapoints). Dotted lines indicate the average count rates for the two sub-intervals.[]{data-label="XMM_lc"}](XMM_ULX2_edit.eps){width="48.00000%"} Constraints on the binary period of ULX-1 {#ulx1_bp_sec} ----------------------------------------- We noted (Table \[ec1\_table\] and Section \[ulx1\_ec\_sec\]) that for ULX-1, two fractions of eclipses are seen $\approx$12 days apart, in ObsID 13813 and ObsID 13814. The egress from the eclipse in ObsID 13813 occurs at MJD 56180.21; the ingress into the eclipse in ObsID 13814 occurs at MJD 56191.64. This enables us to place some constraints on binary period, which must be, $$\begin{gathered} P \approx \frac{(11.43 + {\rm eclipse~duration})}{n} \ \, {\rm days}, \label{bp_eq} \end{gathered}$$ with $n \ge 1$. To refine this constraint, we take into account that the minimum duration of an eclipse is $\approx$90 ks ($\approx$1.0 days), as observed in ObsID 13814. We also know that the maximum duration of an eclipse is $\approx$150 ks ($\approx$1.7 days) as this is the time between the start of the eclipse in ObsID 13814 and the start of the next observation, ObsID 13815, which has no eclipse. Assuming the shortest possible duration of the eclipse implies a binary period of $\approx$12.5$/n$ days. If we use the maximum eclipse time, $\approx$1.7 days, the binary period is $\approx$13.1$/n$ days. We tested a range of eclipse durations and binary periods, to determine which combination of parameters is consistent with the observed sequence of eclipses/non-eclipses in our [*Chandra*]{} observations. Based solely on the minimum duration of an uninterrupted non-eclipse phase ($\approx$160 ks) and the minimum duration of the eclipse ($\approx$90 ks), the minimum acceptable binary period, from Equations , is $P\approx$230 ks $\approx$2.7 d (that is, $n=4$). However, if the binary period were $\approx$3 days, the eclipse found during ObsID 13813 implies that another eclipse should be detected in ObsID 13812. The start of ObsID 13812 is only 2.56 days after the end of the eclipse in ObsID 13813. We do not find an eclipse in ObsID 13812, and this rules out a period of $\approx$3 days. Moreover, an eclipse time $\ga$90 ks over a period of about 3 days would imply that ULX-1 should be in eclipse $\ga$30% of the time. A Roche-lobe filling donor star can eclipse a point-like X-ray source for such a long fraction of the orbit only for mass ratios $q \equiv M_2/M_1 \ga$ a few 100 [Fig. 2 in @1976ApJ...208..512C], which is impossible for any combination of compact objects and normal donor stars. Next, we consider the possibility that $n=3$ in Equation , which corresponds to a period range between $\approx$4.08 and $\approx$4.38 days. In this case, too, we would have seen at least part and more likely all of an eclipse in ObsID 13812, which is not the case: this rules out the $n=3$ case, too. Therefore, the only two acceptable options for the binary period are $n=2$ ($P \sim 6$–6.5 days) or $n=1$ ($P \sim 12$–13 days). We summarize the acceptable region of the period versus eclipse duration parameter space in Figure \[ec\_test\]. We iterated over all possible eclipse durations (1.04–1.70 days, in iteration steps of 0.01 days) and for values of $n = 1,2,3,4$, and compared the predicted occurrences of eclipses with what is detected in the seven [*Chandra*]{} observations between 2012 September 9 to 2012 October 10. Along the line corresponding to each value of $n$, some periods are consistent with the [*Chandra*]{} data (red intervals), others are ruled out (black intervals). In addition, for Roche-lobe-filling donors, eclipse durations $> 20\%$ of the binary period (dark shaded area in Figure \[ec\_test\]) require a mass ratio $q \ga 8$ at an inclination angle of 90$^{\circ}$, or $q \ga 10$ at an inclination of 80$^{\circ}$ [@1976ApJ...208..512C]. This is very implausible if the accretor is a BH, but it is acceptable for a neutron star accreting from an OB star. In the assumption that ULX-1 has a BH primary, the mass-ratio constraint further restricts the viable $n=2$ case to the narrow range $P = 6.23$–6.35 days, with an eclipse duration range of $1.04$–$1.26$ days. If we allow for a neutron star primary, the period can be as long as 6.55 days, corresponding to an eclipse fraction of 26%. Finally, for the $n=1$ case, the predicted fractional time in eclipse goes from $\approx$8% ($P = 12.48$d, eclipse duration $\approx$1.0d) to $\approx$13% ($P = 13.14$d, eclipse duration $\approx$1.7d), with mass ratios $q \sim 0.3$–1, more typical of a BH primary orbiting an OB star. Based on the previous analysis, we compared the predicted eclipse fractions with the total fraction of time ULX-1 was observed in eclipse. Over all [*Chandra*]{} epochs, the system is seen in eclipse for a total of $\approx$135 ks out of $\approx$835 ks, equating to a total eclipsing fraction of $\approx 16\%$. No eclipses of ULX-1 are significantly detected in 177 ks of [*XMM-Newton*]{}/EPIC observations; therefore, the combined eclipse fraction observed by [*Chandra*]{} plus [*XMM-Newton*]{} becomes $\approx$13.3%. This is slightly lower than the predicted time in eclipse in the case of $n=2$ (fractional eclipse duration $\ga 16.7\%$: Figure \[ec\_test\]). Conversely, for the case of $n=1$, the observed time in eclipse is slightly larger than expected (between $\approx$8% and $\approx$13%). We do not regard such discrepancies as particularly significant, because of the limited and uneven sampling of the system; we may have been slightly lucky or slightly unlucky in catching ULX-1 during its eclipses. We also note that dips in the X-ray flux can sometime provide phasing information in binary systems, if they are caused by bulging, denser material located where the accretion stream splashes onto the disk. For example, regular dips at phases $\sim$0.6–0.7 are sometimes seen in low-mass X-ray binaries , and other Roche-lobe overflow systems. We have already mentioned (Section \[ulx1\_ec\_sec\]) that ULX-1 shows a dip in ObsID 13812. A second possible dip can also be seen in the full light-curve (Figure \[full\_lc\]) at the start of the final observation, ObsID 13816. Although only detected in a single 1000-s bin (the first 1000 s of the observation), this drop in flux appears to be intrinsic to ULX-1, as other nearby sources do not show this feature and there are no instrumental problems in those first 1000 s. Intriguingly, both dips appear to be at the same phase with respect to the preceding eclipses ([*i.e.*]{}, $\approx$3.5 days after the eclipse), which strengthens our confidence that the second dip is also real. For a binary period $\approx$6 days, the dips would be at phase $\approx$0.6; for the alternative period range $\approx$12.5–13 days, the dips would be at phase $\approx$0.25–0.30. Finally, two eclipses were found for ULX-2. Unfortunately, the large time interval between the two eclipses seen by [*Chandra*]{} in 2012 September and by [*XMM-Newton*]{} in 2006 May precludes any attempt to constrain the binary period. All we can say is that the total fraction of time spent in eclipse by ULX-2 in our $\approx 1$Ms [*Chandra*]{} plus [*XMM-Newton*]{} dataset is $\approx$7%. Since the minimum eclipse duration is 48 ks ($\approx$0.55 d), we expect the binary period to be $\sim$10 d. ![Test of potential binary periods for ULX-1, based on the spacing between observed eclipses in the [*Chandra*]{} series of observations. We know that $P \approx$(11.43 + eclipse duration)$/n$ days, with $n \ge 1$ and the eclipse duration is between 1.0 days and 1.7 days (horizontal dashed black lines). Each line segment represents a choice of $n$ (from left to right: $n = 4,3,2,1$), and is plotted between the minimum (1.04 days) and maximum (1.70 days) permitted value of the eclipse duration. On each segment, black intervals indicate a combination of period and eclipse duration that is not consistent with the sequence of [*Chandra*]{} observations; instead, red intervals do fit the observed data. The dashed blue line is the region of the parameter space where the eclipse duration is 13.3% of the period, which is the observed eclipsing fraction from all [*Chandra*]{} and [*XMM-Newton*]{} observations. The grey shaded region marks the region of the parameter space where the eclipse duration is greater than 20% of the period, which we consider less likely for empirical reasons (too far from the observed value). For each value of the ratio between eclipse duration and binary period, there is a unique value of the mass ratio $q(\theta)$ (see Section 5.3 for details). For $\theta = 90^{\circ}$, points A, B, C, D correspond to $q=3.6,9.7,0.25,1.2$, respectively; the two points marked with crosses correspond to $q(90^{\circ})=0.5$ and $1.0$. Acceptable solutions in the shaded region require mass ratios $q(\theta) \geq q(90^{\circ}) \ga 10$ (Section \[ulx1\_bp\_sec\]); such high values are ruled out in the case of a BH accretor, but are still possible if ULX-1 is powered by a neutron star.[]{data-label="ec_test"}](eclipse_trials.eps){width="48.00000%"} ![[*Chandra*]{}/ACIS-S background subtracted light-curve for ULX-1 for all epochs in September 2012 (in chronological order: ObsIDs 13813, 13812, 15496, 13814, 13815, 13816). We overlaid two schematic lightcurves corresponding to two alternative periods consistent with the observations: a $6.3$-day period with a $1.3$-day eclipse (dashed blue line), and a $13.1$-day period with a $1.7$-day eclipse (dashed red line, slightly shifted upwards for clarity). In addition to the two eclipses, two shorter dips are also seen. The second dip appears only in the first data point from the final epoch, but is at approximately the same orbital phase as the first dip with respect to their preceding eclipses. The first 2 datapoints of the final epoch are plotted as 1000-s bins (to highlight the short dip), while all other datapoints are binned to 2000 s.[]{data-label="full_lc"}](full_lc_2000.eps){width="48.00000%"} Hardness ratios in and out of eclipses {#col_sec} -------------------------------------- Residual emission is detected at the position of ULX-1 during eclipses (Table 2). This is particularly evident in ObsID 1622, with a residual eclipse count rate $\approx$10% of the average out of eclipse count rate. It is also marginally significant in ObsIDs 13813 and 13814. The reason why the residual emission appears less significant in ObsID 13813 and 13814 than in ObsID 1622 is likely because of the decreased sensitivity of ACIS-S in the soft band between 2001 and 2012 [@2004SPIE.5488..251P]. We stacked the eclipse intervals from all three ULX-1 eclipses and display the resulting 135-ks ACIS-S X-ray-color image in Figure \[chan\_res\_im\] (top panel). The residual emission of ULX-1 is centred at the same coordinates as the out of eclipse emission, and is unresolved, but appears softer (most photons below 1 keV). We also show (Figure \[chan\_res\_im\], bottom panel) the 48-ks ACIS-S image corresponding to the only [*Chandra*]{} eclipse of ULX-2; the signal-to-noise ratio is lower, but there is significant residual emission for ULX-2 in eclipse, as well. ![Top panel: stacked [*Chandra*]{}/ACIS-S image during the ULX-1 eclipse intervals from ObsIDs 1622, 13813 and 13814. Colors are red for 0.3–1 keV, green for 1–2 keV, blue for 2–7 keV. The dashed green ellipses represent the [*Chandra*]{} extraction regions for ULX-1 and ULX-2. Bottom panel: same as the top panel, for the ULX-2 eclipse interval during [*Chandra*]{} ObsID 13813.[]{data-label="chan_res_im"}](rgb_ec_ULX1.eps "fig:"){width="47.00000%"}\ ![Top panel: stacked [*Chandra*]{}/ACIS-S image during the ULX-1 eclipse intervals from ObsIDs 1622, 13813 and 13814. Colors are red for 0.3–1 keV, green for 1–2 keV, blue for 2–7 keV. The dashed green ellipses represent the [*Chandra*]{} extraction regions for ULX-1 and ULX-2. Bottom panel: same as the top panel, for the ULX-2 eclipse interval during [*Chandra*]{} ObsID 13813.[]{data-label="chan_res_im"}](rgb_ec_ULX2.eps "fig:"){width="47.00000%"} To quantify the colors and the color differences in and out of eclipse, we determined the hardness ratio between the net count rates in the 1.2–7 keV band and in the 0.3–1.2 keV band ([*i.e.*]{}, the same bands used in our light-curve plots). It appears (particularly in ObsID 13813) that ULX-1 is softer in eclipse than out of eclipse (Figure \[hard\_ratio\_fig\] and Table \[HR\_tab\]).The difference becomes more significant when we compare the hardness ratio of the stacked eclipse data (Table \[HR\_tab\]) with that of the stacked out of eclipse ones. For ULX-2, we cannot identify significant color differences in and out of eclipse, because of the short duration of the lone detected [*Chandra*]{} eclipse. Our hardness ratio study also clearly shows (Table \[HR\_tab\] and Figure \[hard\_ratio\_fig\]) that ULX-1 is always softer than ULX-2, both in and out of eclipse. Another difference between the two ULXs is their degree of hardness ratio variability from epoch to epoch in the [*Chandra*]{} series. For ULX-1, all the 2012 observations are consistent with the same hardness ratio (Table \[HR\_tab\]). The source appears softer in ObsIDs 354, 1622 and 3932; however, this is misleading because such observations were taken in Cycle 1, Cycle 2 and Cycle 4, respectively, when ACIS-S was more sensitive to soft photons [@2004SPIE.5488..251P]. A rough way to account for this effect is to assume simple power-law models and use the [*Chandra*]{} X-Ray Center online installation of (Version 4.8) to convert the observed count rates into “equivalent" count rates that would have been observed in Cycle 13 (year 2012) when all the other observations took place. A more accurate conversion from observed count rates to Cycle 13-equivalent count rates requires proper spectral modelling in the various epochs. The corrected count rates listed in Table \[HR\_tab\] and plotted in Figure \[hard\_ratio\_fig\], for both ULX-1 and ULX-2, were obtained with the latter method, after we carried out the spectral analysis discussed in Section \[spectral\_sec\]; the best-fitting spectral models were convolved with response and auxiliary response functions of the detector at different epochs, to determine the predicted count rates. Inspection of the corrected count rates confirms that the hardness of ULX-1 is approximately constant; instead, that of ULX-2 is intrinsically variable from epoch to epoch. Based on the observed hardness of the residual eclipse emission of ULX-1, we more plausibly interpret it as thermal-plasma emission, for the purpose of converting count rates into fluxes and luminosities. Assuming a temperature $\sim$0.5 keV, and using again the online tool, we estimate a residual ULX-1 luminosity $L_{\rm X} \sim 10^{37}$ erg s$^{-1}$ in the 0.3–8 keV band. Again, this is only a simple, preliminary estimate. We will present a more accurate estimate of the residual emission based on spectral fitting, and we will discuss its physical origin, after we carry out a full spectral modelling of ULX-1 (Section \[spectral\_mod\_sec\]). We do not have any constraints on plausible models for the residual eclipse emission of ULX-2; however, a selection of thermal-plasma and power-law models also give typical luminosities $L_{\rm X} \sim 10^{37}$ erg s$^{-1}$. What is clear is that it is harder than the residual emission of ULX-1. ![(1.2–7.0)/(0.3–1.2) hardness ratio versus 0.3–7.0 keV count rate for ULX-1 and ULX-2 in eclipse and non-eclipse intervals during the [*Chandra*]{} observations. Red datapoints correspond to ULX-1 in eclipse, green datapoints to ULX-1 out of eclipse, the single cyan datapoint to ULX-2 in eclipse and blue datapoints to ULX-2 out of eclipse. Colors have been corrected for the change in sensitivity of the ACIS-S detector over the years (Section \[col\_sec\], Table \[HR\_tab\])[]{data-label="hard_ratio_fig"}](hardness_ratio.eps){width="47.00000%"} ----------------- ----------------------- ---------------------------- ----------------------- --------------------- Epoch \[2pt\] 354 [$0.27\pm0.04$]{} [$1.71\pm0.23$]{} \[1pt\] \[[$0.49\pm0.07$]{}\] \[[$2.75\pm0.36$]{}\] \[1pt\] 1622 [$0.28\pm0.12$]{} [$0.17\pm0.10$]{} [$1.31\pm0.19$]{} \[1pt\] \[[$0.40\pm0.17$]{}\] \[[$0.31\pm0.18$]{}\] \[[$1.90\pm0.27$]{}\] \[1pt\] 3932 [$0.33\pm0.03$]{} [$1.11\pm0.10$]{} \[1pt\] \[[$0.42\pm0.04$]{}\] \[[$1.43\pm0.13$]{}\] \[1pt\] 13812 [$0.58\pm0.03$]{} [$1.87\pm0.10$]{} \[1pt\] 13813 [$0.58\pm0.03$]{} [$0.07_{-0.07}^{+0.14}$]{} [$2.01\pm0.11$]{} [$1.4\pm0.6$]{} \[1pt\] 13814 [$0.56\pm0.03$]{} [$0.35\pm0.25$]{} [$1.69\pm0.08$]{} \[1pt\] 13815 [$0.56\pm0.04$]{} [$1.32\pm0.18$]{} \[1pt\] 13816 [$0.47\pm0.03$]{} [$1.99\pm0.13$]{} \[1pt\] 15496 [$0.51\pm0.04$]{} [$2.00\pm0.18$]{} \[1pt\] 15553 [$0.51\pm0.06$]{} [$1.80\pm0.22$]{} \[1pt\] stacked [$0.55\pm0.01$]{} [$0.21\pm0.09$]{} [$1.78\pm0.04$]{} [$1.4\pm0.6$]{} \[1pt\] \[[$0.56\pm0.01$]{}\] \[[$0.23\pm0.10$]{}\] \[[$1.86\pm0.06$]{}\] \[[$1.4\pm0.6$]{}\] \[2pt\] ----------------- ----------------------- ---------------------------- ----------------------- --------------------- : Hardness ratios of ULX-1 and ULX-2 for eclipsing and non-eclipsing intervals of the [*Chandra*]{} observations. Values in brackets are rescaled to their [*Chandra*]{} Cycle 13-equivalents. \[HR\_tab\] Spectral properties {#spectral_sec} ------------------- The presence of eclipses implies that both system are viewed at high inclination. Therefore, these two ULXs, although not exceptionally luminous, can help us investigate the relationship between the spectral appearance of ULXs and their viewing angles. During out of eclipse intervals, both ULXs have sufficiently high count rates for multi-component spectral fitting. Here, we present the results of spectral fitting to the three longest [*Chandra*]{} observations: ObsIDs 13812 (158 ks), 13813 (179 ks) and 13814 (190 ks), taken between 2012 September 9–20. For each source, we fitted the three spectra simultaneously, keeping the intrinsic absorption column density and the parameters of any possible thermal-plasma components locked between them but leaving all other model parameters free. The reason for this choice is that we are assuming for simplicity that cold absorption and thermal-plasma emission vary on timescales longer than a few days, while the emission from the inner disk and corona may change rapidly. In addition, we assumed a line-of-sight absorption column $N_{\rm H,0} = 2 \times 10^{20}$ cm$^{-2}$ . [lrr]{} & ------------------------------------------------------------------------ \ & & ------------------------------------------------------------------------ \ & $1.94\, (478.4/246)$&$ 0.99\, (259.3/261) $ ------------------------------------------------------------------------ \ [*diskbb*]{} & $1.61\, (395.8/246)$& $1.01\, (264.3/261)$\ [*diskir*]{} & $1.25\, (297.2/237)$ & $0.96\, (241.8/252)$\ [*diskpbb*]{} & $1.50\, (371.1/243)$ & $0.93\, (238.9/258)$\ [*cutoffpl*]{} & $1.23\, (298.5/243)$& $0.93\, (239.5/258)$\ [*diskbb+powerlaw*]{} & $1.39\, (334.6/240)$ & $0.96\, (245.6/255)$\ [*diskbb+cutoffpl*]{} & $1.11\, (264.1/237)$& $0.93\, (234.8/252)$\ [*diskbb+comptt*]{} & $1.17\, (278.2/237)$& $0.93\, (234.1/252)$\ [*bb+comptt*]{} & $1.17\, (278.0/237)$& $0.93\, (234.1/252)$\ [*diskbb+powerlaw+mk$_1$+mk$_2$*]{} & $1.02\, (241.6/236)$& $0.95\, (239.5/251)$\ [*diskbb+comptt+mk$_1$+mk$_2$*]{} & $1.01\, (234.4/233)$& $0.93\, (230.2/248)$\ [*bb+comptt+mk$_1$+mk$_2$*]{} & $1.01\, (236.3/233)$& $0.94\, (233.0/248)$\ [*diskir+mk$_1$+mk$_2$*]{} & $1.01\, (235.6/233)$ & $0.96\, (237.2/248)$\ [*diskpbb+mk$_1$+mk$_2$*]{} & $1.00\, (238.3/239)$ & $0.92\, (234.3/254)$ ------------------------------------------------------------------------ \ \[tab:my\_label\] ### Spectral models for ULX-1 {#spectral_mod_sec} The first obvious result of our modelling (Table \[tab:my\_label\]) is that the spectrum of ULX-1 is intrinsically curved, not well fitted by a simple power-law ($\chi^2_{\nu} \approx 1.9$) regardless of the value of $N_{\rm H,int}$. Therefore, we tried several other models, roughly belonging to two typical classes: disk-dominated models, in which the disk is responsible for most of the emission above 1 keV and for the high-energy spectral curvature; and Comptonization-dominated models, in which the disk (or other thermal component) provides the seed photon emission below 1 keV, and a cut-off power-law or Comptonization component provides the bulk of the emission above 1 keV. Much of the debate in the literature about the spectral classification of ULXs reduces to the choice between these two interpretations [[*e.g.,*]{} @2009MNRAS.397.1836G; @2011NewAR..55..166F; @2013MNRAS.435.1758S]. Finally, we tested whether the addition of a thermal-plasma emission component improves the fit: the justification for this component is that some ULXs (especially those seen at high viewing angles) may show emission features in the $\sim$1 keV region [@2014MNRAS.438L..51M; @2015MNRAS.454.3134M]. {width="22.00000%"} {width="22.00000%"} {width="22.00000%"}\ {width="22.00000%"} {width="22.00000%"} {width="22.00000%"} We started by fitting single-component disk models: [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ [*diskbb*]{} for a standard disk [@1984PASJ...36..741M; @1986ApJ...308..635M], and [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ [*diskpbb*]{} for a slim disk [@2005ApJ...631.1062K]. They fare relatively better ($\chi^2_{\nu} \approx 1.6$ and $\chi^2_{\nu} \approx 1.5$, respectively) than a power-law model, but they are still not good fits. They also require a surprisingly low peak color temperature, $kT_{\rm in} \approx 0.6$–0.7 keV; this is inconsistent with the disk temperatures expected near or just above the Eddington limit , and would require a heavy stellar-mass BH (as we shall discuss later). Next, we tried adding a power-law component to the disk model: [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ ([*diskpbb*]{} $+$ [*powerlaw*]{}). This is probably the most commonly used model in the literature for the classification of accretion states in stellar mass BHs (despite the interpretation problems caused by the unphysically high contribution of the power-law component at low energies). The quality of the fit improves slightly ($\chi^2_{\nu} \approx 1.4$) but there are still significant systematic residuals. One source of fit residuals is the high-energy downturn. By using instead a [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ ([*diskbb*]{} $+$ [*cutoffpl*]{}) model, we obtain a substantially better fit ($\chi^2_{\nu} \approx 1.1$), with an F-test statistical significance $\approx (1-10^{-12})$. The presence of a high-energy downturn is of course one of the main spectral features of ULXs [@2006MNRAS.368..397S], compared with stellar-mass BHs in sub-Eddington states. However, in this case the best-fitting cut-off energy $E \approx 1$ keV, much lower than the typical $\sim$5-keV high-energy cutoff seen in other ULXs [@2009MNRAS.397.1836G]; this is quantitative evidence that the spectrum of ULX-1 is extremely soft compared with average ULX spectra. Fitting a cutoff power-law alone (without the disk component) gives a $\chi^2_{\nu} \approx 1.2$; from this, we verify that an additional soft thermal component is significant to $>99.99\%$. There is still a third source of fit residuals, at energies around 1 keV, which we will discuss later. The successful models discussed so far are phenomenological approximations of physical models; therefore, we fitted several alternative Comptonization models that produce a soft excess and a high-energy downturn: [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ ([*diskbb*]{} $+$ [*comptt*]{}) [@1994ApJ...434..570T], [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ ([*bb*]{} $+$ [*comptt*]{}), and [*TBabs*]{} $\times$ [*TBabs*]{} $\times$ [*diskir*]{} [@2009MNRAS.392.1106G]. They provide moderately good fits, with $\chi^2_{\nu} \approx 1.2$ (Table \[tab:my\_label\]). In this class of Comptonization models, the disk component is used as the source of seed photons, and the electron temperature sets the location of the high-energy cutoff. For ULX-1, typical seed photon temperatures are $kT_0 \la 0.3$ keV, and the range of electron temperatures in the Comptonizing region is $kT_e \approx 0.8$–1.2 keV, with optical depths $\approx 7$–9. The electron temperature of the Comptonization region is substantially cooler than in most other two-component ULXs (where $kT_e \sim 1.5$–3 keV: @2009MNRAS.397.1836G). This is the physical reason why ULX-1 appears as one of the softest sources in its class, with an unfolded $E\,F_{E}$ spectrum peaking at $\approx$1 keV [c.f. the ULX classification of @2013MNRAS.435.1758S]. The optically-thick thermal continuum component used as seed to the Comptonization models can be equally well modelled with a disk-blackbody or a simple blackbody, given its low temperature at the lowest edge of the ACIS-S sensitivity. Its direct flux contribution to the observed spectrum is small, although difficult to constrain precisely, because of the low number of counts at very soft energies. Individual fits to the three longest observations with a [*diskir*]{} model suggest a direct disk contribution an order of magnitude lower than the Comptonized component. The main reason why none of the smooth continuum models described above are really good fits is the presence of strong residuals (F-test level of significance $>$99.99%) below and around 1 keV. A single-temperature [*mekal*]{} component is not sufficient to eliminate the residuals. Instead, two [*mekal*]{} components with temperatures $kT_1 \approx 0.2$ keV and $T_2 \approx 0.9$ keV significantly improve the fits (Figure \[b1\_model\]), providing $\chi^2_{\nu} \approx 1.01$ for the Comptonization models and $\chi^2_{\nu} \approx 1.00$ for the disk models. In the latter case, the temperature profile index is $p \lesssim 0.6$ (“broadened disk"), favouring the slim disk over the standard disk model. The presence of soft X-ray residuals and the strong continuum curvature are robust and independent of the choice of cold absorption model. We also tried combinations of neutral and ionized absorbers ([*tbabs*]{} $\times$ [*varabs*]{}), but they do not reproduce the strong residual feature at energies $\approx$0.8–1.0 keV. Lower-energy residuals at $\approx$0.5–0.6 keV are relatively less constrained because of the degraded sensitivity of ACIS-S at low energies, rather than because of intrinsic absorption. For all our Comptonization-type and disk-type models, the intrinsic cold absorption $N_{\rm H}$ is $\lesssim$ a few $10^{20}$ cm$^{-2}$ and in most cases, consistent with 0 within the 90% confidence level. Disk models with additional thermal plasma emission produce equally good $\chi^2$ values as Comptonization models with thermal plasma emission (Table \[tab:my\_label\]). Therefore, it is worth examining in more details whether disk models are physically self-consistent, and what their physical interpretation could be. Let us start with a standard disk, to account for the possibility that ULX-1 is a rather massive BH accreting at sub-Eddington rates. We may be tempted to discard this possibility straight away, because the best-fitting temperature profile index $p \lesssim 0.6$ (rather than $p = 0.75$) is generally considered the hallmark of a disk at the Eddington accretion rate, not of a standard disk; however, it was recently shown that broadened disks with $p \approx 0.6$ may also occur in the sub-Eddington regime with accretion rates an order of magnitude lower [@2016..Sutton]. Therefore, we will consider that case here. In standard accretion disk models, with the inner disk fixed at the innermost stable circular orbit, there are two approximate relations between the observable quantities $T_{\rm in}$ and $L$, and the non-directly-observable physical properties $\dot{m}$ (Eddington-scaled mass accretion rate) and $M$ (BH mass): $$\begin{aligned} L & \approx & 1.3 \times 10^{39} \, \dot{m} M_{10}\ \ {\mathrm {erg~s}}^{-1}\label{SS_Lum}\\ kT_{\rm in} & \approx & 1.3 \, (\dot{m}/M_{10})^{1/4}\ \ {\mathrm{keV}}\label{SS_Temp}\end{aligned}$$ , where $M_{10}$ is the BH mass in units of 10 $M_{\odot}$. Equation is simply the luminosity as a fraction of Eddington, in the radiatively efficient regime; Equation follows from the relation $L \approx 4\pi r_{\rm in}^2 \sigma T_{\rm in}^4$. For ULX-1, the best-fitting peak temperature is $kT_{\rm in} \approx (0.7 \pm 0.1)$ keV, and the luminosity $L \approx 2 \times 10^{39}$ erg s$^{-1}$ (as we shall discuss later). From Equations and , this would correspond to a BH mass $M \approx 40$–50 $M_{\odot}$ at accretion rates $\dot{m} \approx 0.3$–0.4. Even if we allow for the possibility of $p < 0.6$ in a sub-Eddington disk, the presence of strong line residuals in the soft X-ray band is another, stronger piece of evidence against the standard disk model; it is instead indicative of Eddington accretion and associated outflows [@2015MNRAS.454.3134M; @2016Natur.533...64P]. For these reasons, we dis-favour the sub-Eddington standard disk model for ULX-1. Next, we test the self-consistency of the slim disk model. In the super-Eddington regime, the disk luminosity is modified as $$L \approx 1.3 \times 10^{39} \, (1+0.6 \ln \dot{m})\, M_{10}\ \ {\mathrm {erg~s}}^{-1}\label{slim_Lum}$$ . A slim disk is no longer truncated exactly at the innermost stable circular orbit: the fitted inner disk radius can be approximated by the empirical scaling $r_{\rm in}(\dot{m}) \approx r_{\rm in}(\dot{m}=1)\,[T_{\rm in}(\dot{m}=1)/T_{\rm in}(\dot{m})]$ [@2000PASJ...52..133W]. As a result, the disk luminosity becomes $L \propto M_{10}^{3/2} T_{\rm in}^2$. Substituting $L$ from Equation , and matching the normalization to that of the sub-Eddington case (Equation ), we obtain: $$kT_{\rm in} \approx 1.3 \, (1+0.6 \ln \dot{m})^{1/2}\, M_{10}^{-1/4}\ \ {\mathrm{keV}}\label{slim_Temp}.$$ A proper treatment of the observed properties of a slim disk requires additional parameters such as the viewing angle and the BH spin [@2008PASJ...60..653V; @2009ApJS..183..171S; @2013MNRAS.436...71V]; however, Equations and are already good enough as a first-order approximation to test the consistency of a slim disk model for ULX-1. From Equation , we find that a slim disk peak temperature $kT_{\rm in} \approx 0.7$ keV requires a BH mass $M \ga 100 M_{\odot}$ (confirmed also by the numerical results of @2008PASJ...60..653V); but such BH would be far below Eddington at the observed luminosity $L \approx 2 \times 10^{39}$ erg s$^{-1}$, contrary to our initial slim disk assumption. Therefore, the slim disk model cannot be self-consistently applied to the spectrum of ULX-1. Based on those physical arguments, we conclude that the best fits in all three epochs are obtained with a Comptonization model, with the addition of multi-temperature optically-thin thermal-plasma emission. From the fits statistics alone, we cannot rule out a broadened disk model, with a heavy stellar-mass BH at sub-Eddington accretion rates; however, the observed presence of strong soft X-ray residuals points to an ultraluminous regime. We list the best-fitting parameters of two equivalent Comptonization models in Tables \[tab\_ulx1\_comptt\], \[tab\_ulx1\_diskir\]; for comparison, we also list the best-fitting parameters of the broadened-disk model (Table \[tab\_ulx1\_diskpbb\]). [llrrr]{} Component & Parameter & ------------------------------------------------------------------------ \ && & & ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & \[0.02\] & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ &$<0.02$ & $<0.02$ & $<0.02$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ mekal &$kT_1$ $({\text{keV}})$ && [$0.17_{-0.04}^{+0.05}$]{} & ------------------------------------------------------------------------ \ &$N_1$[^3] && [$4.1_{-1.9}^{+2.1}$]{}$\times10^{-6}$ & ------------------------------------------------------------------------ \ mekal&$kT_2$ $({\text{keV}})$ && [$0.87_{-0.08}^{+0.09}$]{} & ------------------------------------------------------------------------ \ &$N_2^{\rm ~a}$ && [$4.6_{-0.9}^{+1.0}$]{}$\times10^{-6}$ & ------------------------------------------------------------------------ \ diskbb&$kT_{\rm in}$ $({\text{keV}})$& [$0.32_{-0.08}^{+0.23}$]{} & [$0.19_{-0.03}^{+0.02}$]{}& [$0.18_{-0.01}^{+0.02}$]{} ------------------------------------------------------------------------ \ &$K$[^4] & $<0.6$& $<0.5$& $<2.5$ ------------------------------------------------------------------------ \ comptt & $kT_{0}$ $({\text{keV}})$[^5] & [$0.32_{-0.05}^{+0.23}$]{} & [$0.19_{-0.03}^{+0.02}$]{}& [$0.18_{-0.01}^{+0.02}$]{} ------------------------------------------------------------------------ \ &$kT_{\rm e}$ $({\text{keV}})$ & [$1.1_{-0.3}^{+*}$]{} & [$0.9_{-0.2}^{+2.0}$]{}& [$0.9_{-0.2}^{+0.9}$]{}\ &$\tau$ & [$7.6_{-0.4}^{+0.4}$]{} & [$7.4_{-0.4}^{+0.4}$]{}& [$8.3_{-0.4}^{+0.5}$]{}\ &$N_{\rm c}$ & [$1.4_{-0.3}^{+0.3}$]{}$\times10^{-5}$ & [$8.0_{-0.6}^{+0.5}$]{}$\times10^{-5}$ &[$7.0_{-0.8}^{+0.7}$]{}$\times10^{-5}$ ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$7.32_{-0.41}^{+0.19}$]{}& [$8.59_{-0.39}^{+0.27}$]{} & [$9.26_{-0.35}^{+0.39}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$1.5_{-0.3}^{+0.3}$]{} &[$1.8_{-0.3}^{+0.3}$]{}& [$2.0_{-0.4}^{+0.4}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.1_{-0.4}^{+0.4}$]{} & [$2.2_{-0.5}^{+0.5}$]{} & [$2.6_{-0.5}^{+0.5}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx1\_comptt\] [llrrr]{} Component & Parameter & ------------------------------------------------------------------------ \ && & & ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & \[0.02\] & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & $<0.06$ & $<0.08$& $<0.12$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ mekal &$kT_1$ $({\text{keV}})$ && [$0.17_{-0.03}^{+0.05}$]{} & ------------------------------------------------------------------------ \ &$N_1$ && [$3.8_{-1.4}^{+3.9}$]{}$\times10^{-6}$ & ------------------------------------------------------------------------ \ mekal&$kT_2$ $({\text{keV}})$ && [$0.87_{-0.10}^{+0.08}$]{} & ------------------------------------------------------------------------ \ &$N_2$ && [$4.5_{-0.8}^{+1.1}$]{}$\times10^{-6}$ & ------------------------------------------------------------------------ \ diskir&$kT_{\rm in}$ $({\text{keV}})$& [$0.13_{-0.05}^{+0.40}$]{} & [$0.13_{-0.01}^{+0.09}$]{}& [$0.11_{-0.01}^{+0.21}$]{} ------------------------------------------------------------------------ \ & $\Gamma$ & [$2.85_{-*}^{+0.16}$]{}&[$2.49_{-0.15}^{+0.18}$]{} & [$2.43_{-0.13}^{+0.13}$]{}\ &$kT_{\rm e}$ $({\text{keV}})$& [$1.2_{-0.5}^{+0.5}$]{} & [$0.71_{-0.17}^{+0.11}$]{}& [$0.87_{-0.12}^{+0.17}$]{}\ & L$_{\rm c}$/L$_{\rm d}$ & [$7.2_{-0.4}^{+*}$]{}& $>9.5$ & $>4.4$\ & $f_{\rm in}$ & \[0.1\] & \[0.1\] & \[0.1\]\ & $r_{\rm irr}$ & \[1.2\] & \[1.2\] & \[1.2\]\ & $f_{\rm out}$ & $[1{\times10^{-3}}]$ & $[1{\times10^{-3}}]$ & $[1{\times10^{-3}}]$\ & log($r_{\rm out}$) & \[4\] & \[4\] & \[4\]\ &$K$[^6] & [$1.05_{-0.06}^{+0.05}$]{}& [$1.23_{-0.05}^{+0.06}$]{}& [$3.03_{-0.16}^{+0.15}$]{} ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$7.13_{-0.29}^{+0.28}$]{}& [$8.46_{-0.36}^{+0.28}$]{} & [$9.05_{-0.60}^{+0.18}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$1.5_{-0.3}^{+0.3}$]{} &[$1.8_{-0.4}^{+0.3}$]{}& [$2.0_{-0.3}^{+0.3}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.0_{-0.5}^{+0.5}$]{} & [$2.3_{-0.6}^{+0.5}$]{} & [$2.6_{-0.6}^{+0.5}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx1\_diskir\] ### Continuum and line luminosity of ULX-1 The unabsorbed X-ray luminosity $L_{\rm X}$ is related to the absorption-corrected flux $f_{\rm X}$ by the relation $L_{\rm X} = 2\pi d^2 f_{\rm X}/\cos \theta$, where $d$ is the distance to the source and $\theta$ is the viewing angle, when the emission is from a (sub-Eddington) standard disk surface, and $L_{\rm X} = 4\pi d^2 f_{\rm X}$ for a spherical or point-like emitter. We do not have direct information on the geometry of the emitting region in ULX-1; however, analytical models and numerical simulations of near-Eddington and super-Eddington sources predict mild geometrical beaming, that is, most of the X-ray flux is emitted along the direction perpendicular to the disk plane [@2012ApJ...752...18K; @2014ApJ...796..106J; @2016MNRAS.456.3929S], and the emission should appear fainter and down-scattered in a disk wind when a source is observed at high inclination (as in our case, given the presence of eclipses). Therefore, we choose to use the simplest angle-dependent expression for the luminosity $L_{\rm X} = 2\pi d^2 f_{\rm X}/\cos \theta$. We also identify for simplicity (and in the absence of conflicting evidence) the viewing angle $\theta$ to the plane of the inner disk with the inclination angle of the binary system, which we have constrained to be high from the presence of eclipses; that is, we neglect the possibility of a warped, precessing disk. Instead, we estimate the unabsorbed X-ray luminosity of the thermal plasma components as $L_{\rm X} = 4\pi d^2 f_{\rm X}$, because we assume that the distribution of hot plasma is quasi-spherical above and beyond the disk plane, and its emission is approximately isotropic. With those caveats in mind, we estimate an emitted 0.3–8.0 keV luminosity of the two-temperature thermal-plasma component $L_{\rm X,mekal} \approx 1.3 \times 10^{38}$ erg s$^{-1}$ (Tables \[tab\_ulx1\_comptt\], \[tab\_ulx1\_diskir\], \[tab\_ulx1\_diskpbb\]), essentially all below 2 keV. Assuming $\theta \approx 80^{\circ}$, we then estimate the total ([*i.e.*]{}, thermal plasma plus continuum) 0.3–8.0 keV luminosity as $L_{\rm X} \approx (1.5$–$2.0) \times 10^{39}$ erg s$^{-1}$, during the three longest [*Chandra*]{} observations, regardless of whether the continuum is fitted with a Comptonization model or with a slim disk. To increase the signal-to-noise ratio of the thermal-plasma emission component, we extracted and combined (using [CIAO]{}’s [*specextract*]{} tool) the spectra and responses of all ten [*Chandra*]{} observations (Table 1), for a grand total of $\approx$700 ks out of eclipse. We fitted the resulting spectrum with the same smooth continuum models (Comptonization and slim-disk models) used for the three long spectra: regardless of the choice of continuum, strong systematic residuals appear at energies around 1 keV. As a representative case, we show the residuals corresponding to a [*comptt*]{} fit (Figure \[ulx1\_residuals\], top panel); for this continuum-only model, $\chi^2_{\nu} = 1.54 (217.0/141)$. When two [*mekal*]{} components are added to the continuum (as we did for the spectra of ObsIDs 13812, 13813 and 13814), the goodness-of-fit improves to $\chi^2_{\nu} = 1.10 (150.7/137)$. We tried introducing a third [*mekal*]{} component, and obtained a further improvement to the fit (significant to the 99.5% level), down to $\chi^2_{\nu} = 1.03 (139.5/135)$. We show the 3-[*mekal*]{} model fit and its residuals in Figure \[ulx1\_residuals\] (bottom panel), and list the best-fitting parameters in Table \[tab\_ulx1\_stack\]. The best-fitting [*mekal*]{} temperatures are $kT_1 \approx 0.13$ keV, $kT_2 \approx 0.7$ keV, and $kT_3 \approx 1.7$ keV. (Instead, adding a third [*mekal*]{} component to the best-fitting model for ObsIDs 13812, 13813 and 13814 does not significantly improve that fit.) We estimate an unabsorbed 0.3–8.0 keV luminosity of the three-temperature thermal-plasma component $L_{\rm X,mekal} \approx 2.4 \times 10^{38}$ erg s$^{-1}$. This is moderately higher than the value we estimated for a two-temperature model, because now part of the emission at energies $\gtrsim$2 keV is also attributed to optically-thin thermal plasma. The total (continuum plus thermal plasma) unabsorbed luminosity in the 0.3–8.0 keV band is $L_{\rm X} \approx 1.5 \times 10^{39}$ erg s$^{-1}$, consistent with the luminosity estimated in ObsIDs 13812, 13813 and 13814. Alternatively, we replaced the three [*mekal*]{} components with a [*cemekl*]{}, which is a multi-temperature thermal-plasma model with a power-law distribution of temperatures. The best-fitting [*cemekl*]{} $+$ [*diskbb*]{} $+$ [*comptt*]{} model has $\chi^2_{\nu} = 1.11 (154.1/139)$, maximum temperature $kT_{\rm max} \approx 2.2$ keV, thermal-plasma luminosity $L_{\rm X,cemekl} \approx 1.4 \times 10^{38}$ erg s$^{-1}$, and total luminosity $L_{\rm X} \approx 2 \times 10^{39}$ erg s$^{-1}$. As noted earlier, the role of the disk component in this class of models is to provide the seed photons for the Comptonization component, as well as a soft excess due to the fraction of disk photons that reach us directly. A best-fitting seed temperature $kT_{\rm in} \approx 0.17$ keV and inner-disk size $r_{\rm in} (\cos \theta)^{1/2} \approx 700$ km are consistent with the characteristic temperatures and sizes of the soft thermal components seen in other ULXs ([*e.g.*]{}, @2004ApJ...614L.117M [@2006MNRAS.368..397S; @2009MNRAS.398.1450K]). The direct luminosity contribution of the disk in the 0.3–8 keV band is $\approx (4\pm 1) \times 10^{38}$ erg s$^{-1}$. [llrrr]{} Component & Parameter & ------------------------------------------------------------------------ \ && & & ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & \[0.02\] & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & [$0.05_{-0.05}^{+0.02}$]{} & [$0.06_{-0.04}^{+0.03}$]{} & [$0.04_{-0.04}^{+0.04}$]{} ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ mekal &$kT_1$ $({\text{keV}})$ && [$0.18_{-0.03}^{+0.04}$]{} & ------------------------------------------------------------------------ \ &$N_1$ && [$8.1_{-4.5}^{+7.0}$]{}$\times10^{-6}$ & ------------------------------------------------------------------------ \ mekal&$kT_2$ $({\text{keV}})$ && [$0.84_{-0.08}^{+0.10}$]{} & ------------------------------------------------------------------------ \ &$N_2$ && [$6.7_{-1.0}^{+1.1}$]{}$\times10^{-6}$ & ------------------------------------------------------------------------ \ diskpbb&$kT_{\rm in}$ $({\text{keV}})$& [$0.69_{-0.09}^{+0.11}$]{} & [$0.63_{-0.07}^{+0.08}$]{}& [$0.72_{-0.09}^{+0.12}$]{} ------------------------------------------------------------------------ \ &$p$& $<0.60$ & $<0.57$ & $<0.58$\ &$K$[^7] & [$3.5_{-1.7}^{+7.6}$]{}$\times10^{-3}$& [$7.7_{-3.2}^{+8.8}$]{}$\times10^{-2}$& [$3.8_{-1.7}^{+5.8}$]{}$\times10^{-3}$ ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$7.00_{-0.38}^{+0.36}$]{}& [$8.16_{-0.42}^{+0.40}$]{} & [$9.00_{-0.50}^{+0.48}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$1.8_{-0.4}^{+0.4}$]{} & [$2.4_{-0.4}^{+0.4}$]{} & [$1.5_{-0.3}^{+0.3}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.8_{-0.6}^{+0.6}$]{} & [$3.2_{-0.7}^{+0.7}$]{} & [$2.8_{-0.6}^{+0.6}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx1\_diskpbb\] [llr]{} Component & Parameter & Value ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & \[0.02\] ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & [$0.001_{-0.001}^{+0.007}$]{} ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ mekal &$kT_1$ $({\text{keV}})$ & [$0.13_{-0.01}^{+0.01}$]{} ------------------------------------------------------------------------ \ &$N_1$[^8] & [$5.0_{-1.6}^{+1.7}$]{}$\times10^{-6}$ ------------------------------------------------------------------------ \ mekal&$kT_2$ $({\text{keV}})$ & [$0.73_{-0.05}^{+0.05}$]{} ------------------------------------------------------------------------ \ &$N_2^{\rm ~a}$ & [$3.2_{-0.5}^{+0.5}$]{}$\times10^{-6}$ ------------------------------------------------------------------------ \ mekal&$kT_3$ $({\text{keV}})$ & [$1.74_{-0.12}^{+0.16}$]{} ------------------------------------------------------------------------ \ &$N_3^{\rm ~a}$ & [$1.10_{-0.18}^{+0.18}$]{}$\times10^{-5}$ ------------------------------------------------------------------------ \ diskbb&$kT_{\rm in}$ $({\text{keV}})$& [$0.17_{-0.01}^{+0.02}$]{} ------------------------------------------------------------------------ \ &$K$[^9] & [$0.72_{-0.25}^{+0.25}$]{} ------------------------------------------------------------------------ \ comptt & $kT_{0}$ $({\text{keV}})$[^10]& [$0.17_{-0.01}^{+0.02}$]{} ------------------------------------------------------------------------ \ &$kT_{\rm e}$ $({\text{keV}})$ & [$0.64_{-0.18}^{+0.27}$]{}\ &$\tau$ & [$9.7_{-0.3}^{+0.3}$]{}\ &$N_{\rm c}$ &[$7.8_{-0.7}^{+0.7}$]{}$\times10^{-5}$ ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$7.75_{-0.22}^{+0.22}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$&[$1.4_{-0.3}^{+0.3}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$&[$2.2_{-0.6}^{+0.6}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx1\_stack\] Finally, we inspected the spectral emission in eclipse. We extracted a combined spectrum of the eclipse intervals in ObsIDs 1622, 13813 and 13814. Although we only have $\approx$60 counts, the energy distribution of the counts is similar (Figure \[ULX1\_resid\]) to the thermal-plasma emission component out of eclipse, rather than to the continuum emission. After rebinning the eclipse spectrum to 1 count per bin, we applied the Cash statistics (Cash 1979) to fit the normalization of the same two [*mekal*]{} components previously found in the out-of-eclipse spectra of ObsIDs 13812, 13813 and 13814 (temperatures fixed at $kT_1 \approx 0.2$ keV and $T_2 \approx 0.9$ keV). We find a C-stat value of 49.5 over 54 degrees of freedom for the best-fitting model. The emitted luminosity $\approx 2.4 \times 10^{37}$ erg s$^{-1}$, consistent with our previous simpler estimate based on count rates (Section \[ulx1\_bp\_sec\]). We then let the temperatures free, but did not obtain any significant improvement to the quality of the fit (C-stat value of 49.2 over 52 degrees of freedom). Nor do we improve the fit by adding a third [*mekal*]{} component. ### Spectral models and luminosity of ULX-2 As we did for ULX-1, we started by fitting the spectra of ULX-2 during [*Chandra*]{} ObsIDs 13812, 13813 and 13814 with a simple power-law model (Table \[tab:my\_label\]). The fit is good ($\chi^2_{\nu} \approx 0.99$), but there are residuals consistent with a high-energy downturn. The best-fitting power-law index is $\Gamma = 2.1 \pm 0.1$; however, this value may be an over-estimate if the high-energy steepening is not properly accounted for. Hence, we re-fitted the spectrum with a cutoff power-law ([*TBabs $\times$ TBabs $\times$ cutoffpl)*]{}, and found that the fit is significantly improved: $\chi^2_{\nu} \approx 0.93$, with an F-test significance $\approx$ 99.99% with respect to the unbroken power-law. The power-law index below the cutoff is $\Gamma = 1.1 \pm 0.2$ and the characteristic energy of the cutoff is ($3.0 \pm 0.6$) keV. This is evidence that the spectrum of ULX-2 is significantly curved. Therefore, as we did for ULX-1, we tried a series of models suitable to curved spectra: disk models and Comptonization model. Among disk models, we find that a broadened disk is a significantly better fit (F-test significance $>$99.99%) than a standard disk; a [*TBabs $\times$ TBabs $\times$ diskpbb*]{} model provides $\chi^2_{\nu} \approx 0.93$ (Table \[tab:my\_label\]). The peak disk temperature $kT_{\rm in} \approx$ 1.4–2.0 keV and $p \approx 0.6$, perfectly in line with the expected values for a mildly super-Eddington slim-disk model around a stellar-mass BH. The best-fit parameters can be found in Table \[tab\_ulx2\_diskpbb\]; the model is illustrated in Figure \[b2\_model\]. The [*diskpbb*]{} normalization, $K$, translates into a characteristic inner disk radius $$R_{\rm in} \approx 3.18 K^{1/2} \, d_{10{\rm kpc}} \, (\cos \theta)^{-1/2} \ \ {\rm km},$$ using the conversion factors suitable for slim-disk models [@2008PASJ...60..653V]; $d_{10{\rm kpc}}$ is the distance in units of 10 kpc. A feature of super-critical slim disks is that $R_{\rm in}$ is located slightly inside the innermost stable circular orbit [@2003PASJ...55..959W; @2008PASJ...60..653V]. When this correction is taken into account, the mass $M_{\bullet}$ of a non-rotating BH can be estimated as $M_{\bullet} \approx 1.2 \times R_{\rm in}c^2/(6G) \approx 1.2 R_{\rm in}/(8.9\,{\rm km}) M_{\odot}$. Characteristic radii $R_{\rm in} (\cos \theta)^{1/2} \approx$29–56 km are consistent with all the three long [*Chandra*]{} observations considered here (Table \[tab\_ulx2\_diskpbb\]). For $\theta \approx 80^{\circ}$, this corresponds to characteristic masses $M_{\bullet} \approx 9$–18$M_{\odot}$, consistent with the observed mass distribution of Galactic BHs [@2012ApJ...757...36K]. For a range of viewing angles $70^{\circ} \lesssim \theta \lesssim 85^{\circ}$, the corresponding BH mass range becomes $M_{\bullet} \approx 7$–25$M_{\odot}$. The emitted luminosity in the 0.3–8.0 keV band is $\approx 2 \times 10^{39}$ erg s$^{-1}$ (assuming again a viewing angle $\theta = 80^{\circ}$) and the bolometric disk luminosity is $\approx 3 \times 10^{39}$ erg s$^{-1}$ $\approx$1–3$L_{\rm Edd}$ for the range of BH masses estimated earlier. In this model, ULX-2 would be classified as a broadened-disk ULX in the scheme of @2013MNRAS.435.1758S. Although we favour the slim disk model because of its self-consistency, we cannot rule out the possibility that ULX-2 is fitted by a Comptonization model (Table \[tab\_ulx2\_comptt\]): for example, [*TBabs $\times$ TBabs $\times$ (diskbb $+$ comptt)*]{} yields $\chi^2_{\nu} \approx 0.93$, statistically equivalent to the slim disk model (Table \[tab:my\_label\]), with electron temperatures $kT_e \approx 1$–1.5 keV and optical depth $\tau \approx 9$–13 (slightly hotter and more optically thick than the best-fitting [*comptt*]{} models in ULX-1). Similar values of $\chi^2_{\nu}$ and $kT_e$ are also obtained from other Comptonization models such as [*diskir*]{}. Regardless of the model, the unfolded $E\,F_{E}$ spectrum peaks at $\approx$5 keV, similar to the sources classified as hard ultraluminous by @2013MNRAS.435.1758S. The original definition of the hard ultraluminous regime requires also the presence of a soft excess. In our spectra, it is difficult to constrain the significance of a direct soft emission component (in addition to the Comptonized component or the cutoff power-law) because of the low sensitivity of ACIS-S below 0.5 keV. When we fit the spectrum with a [*diskbb*]{} $+$ [*comptt*]{} model, we find that no more than $\sim$50% of the flux in the 0.3–1.0 keV band is in the direct [*diskbb*]{} component (90% upper limit), but the [*diskbb*]{} normalization is also consistent with 0 within the 90% confidence limit. Regardless of classification semantics, it is clear that ULX-2 has a hard spectrum in the [*Chandra*]{} band, with a high-energy curvature. No significant residuals are found at $\approx$0.8–1 keV in the individual spectra from ObsIDs 13812, 13813 and 13814; however, in at least one observation (ObsID 13812), the spectrum shows two emission features with $>$90% significance at $E \approx 1.3$ keV and $E \approx 1.8$ keV. Similar lines are typically found in thermal plasma emission. They are usually interpreted as emission from a blend of Mg XI lines at 1.33–1.35 keV, and from a Si XIII line at 1.84 keV (with the likely additional contribution of slightly weaker Mg XII lines at 1.75 keV and 1.84 keV). To investigate these and possible other emission features, we extracted a combined [*Chandra*]{} spectrum of ULX-2 from all ten observations, as we did for ULX-1. We fitted the combined spectrum with a [*diskpbb*]{} model, and obtain an excellent fit, $\chi^2_{\nu} = 0.86$ (Table \[tab\_ulx2\_stacked\]). The significance of the two candidate emission features seen in ObsID 13812 fades to $<$90% in the combined spectrum (Figure \[ulx2\_residuals\]). Adding thermal-plasma components to the combined spectrum does not produce any significant improvement. The characteristic disk temperature $kT_{\rm in} \approx 1.6$ keV, radial temperature index $p \approx 0.57$ and inner-disk radius $R_{\rm in} (\cos \theta)^{1/2} \approx 40$ km (Table \[tab\_ulx2\_stacked\]) are consistent with those expected for a super-critical disk, and with the values obtained from the individual fits to ObsIDs 13812, 13813 and 13814. The corresponding range of BH masses is $M_{\bullet} \approx 8$–20$M_{\odot}$, for a non-rotating BH and a viewing angle $\theta = 80^{\circ}$. Finally, we examined the spectrum of ULX-2 in eclipse (Figure \[ULX2\_resid\]). It appears different from what is seen in ULX-1: there is no evidence of a bimodal distribution of counts and it is not possible (from the few counts available) to determine whether the eclipse emission has the same origin as the out of eclipse continuum ([*e.g.*]{}, a small fraction of the direct emission scattered into our line-of-sight by an extended corona), or comes from thermal-plasma at higher temperatures or from bremsstrahlung emission. ![Top panel: datapoints, best-fitting continuum model and spectral residuals for the combined spectrum of all ten [*Chandra*]{}/ACIS-S observations of ULX-1, selecting only non-eclipse time intervals. The model fitted to the combined spectrum is [*TBabs*]{}$\times$[*TBabs*]{}$\times$([*diskbb*]{}+[*comptt*]{}). Significant residuals are seen at photon energies $\sim$1 keV. The datapoints have been binned to a signal-to-noise ratio $\ge9$. Bottom panel: the same spectrum and residuals after the addition of three thermal-plasma emission components ([*mekal*]{} model) at $kT_1 \approx 0.13$ keV, $kT_2 \approx 0.73$ keV and $kT_3 \approx 1.74$ keV, which account well for the residuals.[]{data-label="ulx1_residuals"}](ulx1_nomekal.eps "fig:"){width="34.00000%"}\ ![Top panel: datapoints, best-fitting continuum model and spectral residuals for the combined spectrum of all ten [*Chandra*]{}/ACIS-S observations of ULX-1, selecting only non-eclipse time intervals. The model fitted to the combined spectrum is [*TBabs*]{}$\times$[*TBabs*]{}$\times$([*diskbb*]{}+[*comptt*]{}). Significant residuals are seen at photon energies $\sim$1 keV. The datapoints have been binned to a signal-to-noise ratio $\ge9$. Bottom panel: the same spectrum and residuals after the addition of three thermal-plasma emission components ([*mekal*]{} model) at $kT_1 \approx 0.13$ keV, $kT_2 \approx 0.73$ keV and $kT_3 \approx 1.74$ keV, which account well for the residuals.[]{data-label="ulx1_residuals"}](ulx1_mekal.eps "fig:"){width="34.00000%"}\ ![Combined [*Chandra*]{}/ACIS-S spectrum of ULX-1 during the three eclipses in ObsIDs 1622, 13813 and 13814. The datapoints have been grouped to 1 count per bin. The green curve illustrates the contribution from the best-fitting [*mekal*]{} components (at $T_1\approx 0.18$ keV and $T_2\approx0.86$ keV) during the non-eclipse intervals of the three longest [*Chandra*]{} observations (Table \[tab\_ulx1\_comptt\]). The red curve is the contribution from two [*mekal*]{} components at the same fixed temperatures but with free normalizations, fitted to the eclipse data with the Cash statistics. This plot supports our suggestion that the residual emission during eclipses is due to thermal plasma.[]{data-label="ULX1_resid"}](ec_all.eps){width="34.00000%"} Discussion {#disc_sec} ========== Two eclipsing ULXs in one field: too unlikely? ---------------------------------------------- Luminous stellar-mass BH X-ray binaries or ULXs with X-ray eclipses are very rare sources. SS433 in the Milky Way shows eclipses of its X-ray emission caused by the donor star on a 13.1-d binary period . Unlike M51 ULX-1 and ULX-2, SS433 does not appear as luminous as a ULX because the direct X-ray emission from the inner disk/corona region is already occulted from us. Its donor star periodically eclipses the thermal bremsstrahlung radiation ($L_{\rm X} \sim 10^{36}$ erg s$^{-1}$) from the base of the jet. The first unambiguous eclipsing behaviour in a candidate BH X-ray binary outside the Milky Way was found in IC10 X-1, located in a Local Group dwarf galaxy, with a Wolf-Rayet donor star, a binary period of 1.45 days, and an X-ray luminosity $L_{\rm X} \approx 10^{38}$ erg s$^{-1}$ [@2007ApJ...669L..21P; @2015MNRAS.446.1399L; @2016ApJ...817..154S]. For IC10 X-1, it is still disputed whether the accreting compact object is a BH or a neutron star [@2015MNRAS.452L..31L]. Outside the Local Group, NGC300 X-1 ($L_{\rm X} \approx 5 \times 10^{38}$ erg s$^{-1}$; binary period $\approx$33 hr) shows X-ray dips, consistent with occultation from geometrically thick structures in the outer disk, or absorption in the wind of the donor star, but not with true eclipses [@2015MNRAS.451.4471B]. A strong candidate for a true eclipse is the sharp dip in the [*Swift*]{}/X-Ray Telescope flux recorded once from the ULX P13 in NGC7793, at an orbital phase consistent with the inferior conjunction of its supergiant donor star [@2014Natur.514..198M]; however, there is no further confirmation of that single monitoring datapoint at subsequent epochs. Thus, we argue that the two M51 ULXs discussed in this paper are the first unambiguous eclipsing sources observed at or near the Eddington regime. {width="22.00000%"} {width="22.00000%"} {width="22.00000%"}\ {width="22.00000%"} {width="22.00000%"} {width="22.00000%"} It is rare enough to find two such bright sources projected close to each other in what is not a particularly active starburst region: it is obviously even stranger that both of them show eclipses. Therefore, we tried to assess the statistical significance of this finding. Firstly, we assume that any distance between ULX-1 and ULX-2 not in the plane of M51 is negligible and thus only consider the $\approx350\,{\text{pc}}$ separation. We want to discover the chances of finding two randomly distributed, luminous X-ray binaries in the same galaxy within $350\,{\text{pc}}$ of each other, both having inclination angles $>$80$^{\circ}$. Assuming for example 10 ULXs with $L_{\rm X} \ga 10^{39}$ erg s$^{-1}$ in the same spiral galaxy within a radius of $8\,{\text{k}}{\text{pc}}$ (an over-estimate of the real number of ULXs detected in local-universe galaxies), we used a Monte-Carlo simulation, placing ULXs at random and recording the number of occurrences in which two ULXs were found within a radius of $350\,{\text{pc}}$; for 10 million trials, we find $P_1 \approx 6.6\%$. The probability of finding two nearby ULXs then has to be multiplied by the probability $(P_2)^2$ that they both show eclipses. Assuming no preferential orientation angle, the likelihood of finding a ULX with a viewing angle, for example, $\theta > \theta_{\rm min} = 70^{\circ}$ is $P_2 = \cos 70$. However, if the orientation is too close to $90^{\circ}$, the direct X-ray emission is likely blocked by the outer disk and the source would not appear as a ULX. The thickness of the disk in ULXs is unknown (likely a few degrees), and the minimum angle $\theta_{\rm min}$ that produces eclipses is model-dependent, as a function of the ratio between stellar radius $R_{\ast}$ and binary separation $a$, namely $\cos \theta_{\rm min} \approx R_{\ast}/a$. For plausible distributions of such quantities, $P_2 \la 0.3$ [@2005ApJ...634L..85P]. The final probability becomes $P_1(P_2)^2 \la$ a few $10^{-3}$: we only expect this to happen once every few hundred major galaxies. Spectral properties: broadened disks, Comptonization and thermal plasma ----------------------------------------------------------------------- Luminosity (or more precisely, mass accretion rate) and viewing angle are thought to be the main parameters that determine the observational appearance of ULXs in the X-ray band [[*e.g.*]{}, @2013MNRAS.435.1758S; @2015MNRAS.447.3243M; @2016MNRAS.456.1859U]; disentangling and quantifying their roles is still an unsolved problem. M51 ULX-1 and ULX-2 have approximately the same luminosity and inclination angle (as they both show eclipses); however, they spectral appearance is substantially different. ULX-1 has soft colors in the [*Chandra*]{} band and is well modelled by a soft thermal component (blackbody or disk-blackbody) plus Comptonization; ULX-2 has hard colors and is well modelled by a slim disk with $kT_{\rm in} \approx 1.5$–2.0 keV (hotter than a standard disk). Also, ULX-1 has significant line residuals around 1 keV (consistent with thermal-plasma emission), which are not seen in ULX-2. Thus, we propose that there are other physical parameters that determine the spectral appearance of a ULX in addition to Eddington ratio and viewing angle. We also discovered that ULX-1 has strong radio and optical evidence of a jet (as we will discuss in a separate paper; Soria et al., in prep.) while ULX-2 does not; understanding the relation between outflow structure and spectral appearance remains a key unsolved problem. ![Datapoints, best-fitting continuum model and spectral residuals for the combined spectrum of all ten [*Chandra*]{}/ACIS-S observations of ULX-2, selecting only non-eclipse time intervals. The model fitted to the combined spectrum is [*TBabs*]{}$\times$[*TBabs*]{}$\times$[*diskpbb*]{}. The datapoints have been binned to a signal-to-noise ratio $\ge7$. The combined spectrum of ULX-2 does not show any significant systematic residuals around 1 keV (unlike the spectrum of ULX-1).[]{data-label="ulx2_residuals"}](ulx2_resids.eps){width="34.00000%"} ![[*Chandra*]{}/ACIS-S spectrum of ULX-2 during the eclipse in ObsID 13813. The green curve illustrates the contribution from the best-fitting [*diskpbb*]{} component during the combined non-eclipse observations. The datapoints have been grouped to 1 count per bin. []{data-label="ULX2_resid"}](ULX2_resid.eps){width="34.00000%"} [llrrr]{} Component & Parameter & ------------------------------------------------------------------------ \ && & & ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & \[0.02\] & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & [$0.09_{-0.04}^{+0.04}$]{} & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ diskpbb&$kT_{\rm in}$ $({\text{keV}})$& [$1.5_{-0.2}^{+0.4}$]{} & [$1.7_{-0.4}^{+0.8}$]{}& [$1.8_{-0.4}^{+0.7}$]{} ------------------------------------------------------------------------ \ &$p$& [$0.58_{-0.04}^{+0.07}$]{} & [$0.56_{-0.04}^{+0.06}$]{} & [$0.54_{-0.03}^{+0.04}$]{}\ &$K$[^11] & [$4.5_{-3.2}^{+7.9}$]{}$\times10^{-4}$& [$2.1_{-1.7}^{+5.6}$]{}$\times10^{-4}$& [$1.5_{-1.1}^{+3.3}$]{}$\times10^{-4}$ ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$8.18_{-0.50}^{+0.49}$]{} & [$8.23_{-0.59}^{+0.60}$]{} & [$8.67_{-0.53}^{+0.50}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.2_{-0.4}^{+0.4}$]{} & [$2.3_{-0.4}^{+0.4}$]{}&[$2.5_{-0.4}^{+0.4}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.7_{-0.4}^{+0.4}$]{} & [$3.0_{-0.5}^{+0.5}$]{} & [$3.9_{-0.6}^{+0.6}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx2\_diskpbb\] [llrrr]{} Component & Parameter & ------------------------------------------------------------------------ \ && & & ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & \[0.02\] & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & & $<0.15$ & ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ diskbb&$kT_{\rm in}$ $({\text{keV}})$& [$0.16_{-0.06}^{+0.76}$]{} & [$0.26_{-0.11}^{+0.66}$]{}& [$0.23_{-0.15}^{+0.60}$]{} ------------------------------------------------------------------------ \ &$K$[^12] & $<0.25$& $<0.12$& $<0.40$ ------------------------------------------------------------------------ \ comptt & $kT_{0}$ $({\text{keV}})$[^13] & [$0.16_{-0.06}^{+0.76}$]{} & [$0.26_{-0.11}^{+0.66}$]{}& [$0.23_{-0.15}^{+0.60}$]{} ------------------------------------------------------------------------ \ &$kT_{\rm e}$ $({\text{keV}})$ & [$1.0_{-0.1}^{+0.2}$]{} & [$1.5_{-0.4}^{+*}$]{}& [$1.1_{-0.3}^{+1.0}$]{}\ &$\tau$ & [$13.3_{-2.7}^{+2.6}$]{} & [$9.2_{-7.4}^{+3.3}$]{}& [$11.7_{-3.9}^{+*}$]{}\ &$K_{\rm c}$ & [$4.5_{-0.9}^{+0.8}$]{}$\times10^{-5}$ & [$2.5_{-2.2}^{+1.3}$]{}$\times10^{-5}$ &[$3.4_{-1.7}^{+2.0}$]{}$\times10^{-5}$ ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$8.12_{-0.48}^{+0.47}$]{} & [$8.28_{-0.63}^{+0.62}$]{} & [$8.65_{-0.52}^{+0.53}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$1.9_{-0.3}^{+0.3}$]{} & [$1.9_{-0.3}^{+0.3}$]{}&[$2.1_{-0.3}^{+0.3}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$1.9_{-0.3}^{+0.3}$]{} & [$2.1_{-0.4}^{+0.4}$]{} & [$2.3_{-0.4}^{+0.4}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx2\_comptt\] [llr]{} Component & Parameter & Value ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,0}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$ & \[0.02\] ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ TBabs & $N_{\rm H,int}$ $(10^{22} {\text{c}}{\text{m}}^{-2})$& [$0.08_{-0.03}^{+0.03}$]{} ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ diskpbb&$kT_{\rm in}$ $({\text{keV}})$ & [$1.5_{-0.2}^{+0.2}$]{} ------------------------------------------------------------------------ \ &$p$& [$0.58_{-0.03}^{+0.04}$]{}\ &$K$[^14] & [$3.9_{-2.1}^{+3.9}$]{}$\times10^{-4}$ ------------------------------------------------------------------------ \ & $f_{0.3-8.0}$ $(10^{-14}\,{\rm erg}$ ${\rm cm}^{-2}\,{\rm s}^{-1})$ & [$8.12_{-0.23}^{+0.22}$]{} ------------------------------------------------------------------------ \ & $L_{0.3-8.0}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.2_{-0.4}^{+0.4}$]{}\ & $L_{\rm bol}$ $(10^{39}\,{\text{erg s$^{-1}$}})$ & [$2.9_{-0.5}^{+0.5}$]{} ------------------------------------------------------------------------ \ \[tab\_ulx2\_stacked\] The different role played by an optically-thick thermal component in the modelling of ULX-1 and ULX-2 exemplifies the confusion sometimes found in the literature about the properties of ULX disks. In ULX-1, the “disk" emission is much cooler ($kT \sim 0.1$–0.2 keV) and comes from a large area, with characteristic size $\approx$2000–3000 km (as inferred from the normalization of the [*diskir*]{} component in Table \[tab\_ulx1\_diskir\] and/or the normalization of the [*diskbb*]{} component in Table \[tab\_ulx1\_stack\]). This is much further out than the innermost stable circular orbit around a BH; it is probably located at, or just outside, the spherization radius, where massive radiation-driven outflows are predicted to be launched. This thermal component represents what is sometimes referred to as the “soft excess" in ULXs [[*e.g.*]{}, @2003ApJ...585L..37M; @2005MNRAS.357.1363R; @2006MNRAS.368..397S; @2009MNRAS.397.1836G]. In M51 ULX-1 and in many other similar ULXs, it contributes $\la 10\%$ of the continuum flux in the [*Chandra*]{} band. Despite being often modelled with a [*diskbb*]{} component for practical purposes, it is by no means clear whether or not it originates from the disk; it could come instead from the more optically thick parts of the outflow [@2015MNRAS.447.3243M; @2016MNRAS.456.1859U]. On the other hand, the “disk" in ULX-2 is the dominant continuum emission component. It is probably emitted by a non-standard, geometrically thicker disk with advection, photon trapping and outflows (slim disk model), extending all the way down to the innermost stable circular orbit and possibly even a little further inside it [@2008PASJ...60..653V]. This state is the natural progression from the high/soft state of stellar-mass BHs , to the apparently standard regime [@2004ApJ...601..428K] and the super-Eddington regime. It is also sometimes referred to as the “broadened disk" ultraluminous regime [@2013MNRAS.435.1758S]. A temperature $kT_e \approx 0.8$ keV for the Comptonizing region in ULX-1 is certainly unusually low for a ULX, but not unique. The ULX NGC55 X-1 has a similar Comptonizaton temperature, similar seed photon temperature $kT_0 \approx 0.2$ keV, similar optical depth $\tau \approx 10$ and similar luminosity $L_{\rm X} \approx 2 \times 10^{39}$ erg s$^{-1}$ [@2009MNRAS.397.1836G]. It is a classic example of a ULX in the soft ultraluminous regime [@2013MNRAS.435.1758S]. NGC55 X-1 is also viewed at high inclination, as proved by X-ray dips attributed to clumps of obscuring material in the outer disk [@2004MNRAS.351.1063S]. As for M51 ULX-1, NGC55 X-1 gets softer during the dips: this is consistent with the obscuration of the harder emission from the inner disk region, while a more extended source of soft X-ray photons remains partially unocculted [@2004MNRAS.351.1063S]. It is important to underline the detection of the thermal-plasma emission in ULX-1 out of eclipse, with a luminosity $L_{\rm X,mekal} \approx 2 \times 10^{38}$ erg s$^{-1}$ and an emission measure $\sim n_e^2 V$ $\approx 10^{61}$ cm$^{-3}$ (as fitted to the spectra of ObsIDs 13812, 13813 and 13814). The detection of residual soft emission in eclipse, with a luminosity $L_{\rm X} \approx 2 \times 10^{37}$ erg s$^{-1}$, is consistent with a fraction of the emitting hot gas (perhaps the outer part of the same outflow responsible for the Comptonized component) extending on a scale similar to, or larger than, the size of the companion star; namely, a radius $\ga$ a few $\times 10^{12}$ cm (as we shall discus in Section \[donor\_sec\]). Conversely, the fact that $\approx$90% of the thermal-plasma emission seen out of eclipse also disappears in eclipse is evidence that the emission comes directly from a region of comparable size to the binary system, and not for example from the hot spots of a compact jet on a scale of a few pc (which would still be unresolved by [*Chandra*]{} but unaffected by eclipses). An extended hot halo is a characteristic feature of the best-studied eclipsing X-ray binary, the low-mass Galactic system EXO0748$-$676 . Soft X-ray residuals consistent with thermal plasma emission (and/or absorption) have been reported in several other (non-eclipsing) ULXs such as NGC5408 X-1 [@2014MNRAS.438L..51M; @2015MNRAS.454.3134M; @2015ApJ...814...73S; @2016arXiv160408593P], NGC6946 X-1 [@2014MNRAS.438L..51M], HoII X-1 [@2001AJ....121.3041M; @2004ApJ...608L..57D], NGC4395 X-1 [@2006MNRAS.368..397S], NGC4559 X-1 [@2004MNRAS.349.1193R], NGC7424 ULX2 [@2006MNRAS.370.1666S], NGC1313 X-1 [@2013ApJ...778..163B; @2016arXiv160408593P] and HoIX X-1 [@2014ApJ...793...21W]; the last two of those ULXs are hard ultraluminous sources, while all the others are classified as soft ultraluminous. ![Top panel: theoretical stellar-population ($I$, $V-I$) isochrones, with the location of the potential donor stars of ULX-1 consistent with the permitted range of binary periods. The dark shaded grey band represents (young) stars with a mean density consistent with a period $12.2\,{\text{d}}\leq P \leq 13.1\,{\text{d}}$. The light shaded grey band represents stars with a mean density consistent with a period $6.1\,{\text{d}}\leq P \leq 6.4\,{\text{d}}$. Bottom panel: as in the the top panel, for the ($V$, $B-V$) isochrones.[]{data-label="col_mag_fig"}](HR_M51_VI.eps "fig:"){width="45.00000%"}\ ![Top panel: theoretical stellar-population ($I$, $V-I$) isochrones, with the location of the potential donor stars of ULX-1 consistent with the permitted range of binary periods. The dark shaded grey band represents (young) stars with a mean density consistent with a period $12.2\,{\text{d}}\leq P \leq 13.1\,{\text{d}}$. The light shaded grey band represents stars with a mean density consistent with a period $6.1\,{\text{d}}\leq P \leq 6.4\,{\text{d}}$. Bottom panel: as in the the top panel, for the ($V$, $B-V$) isochrones.[]{data-label="col_mag_fig"}](HR_M51_BV.eps "fig:"){width="45.00000%"} Constraints on the donor star of ULX-1 {#donor_sec} -------------------------------------- Wind accretion is not an effective mechanism to produce X-ray luminosities $\ga 10^{39}$ erg s$^{-1}$; at such luminosities, stellar-mass BHs require feeding via Roche-lobe overflow, or at the very least, via a focused wind from a donor star that is almost filling its Roche lobe. For the following discussion we will assume that the donor star in ULX-1 (and in ULX-2, although not discussed here for a lack of constraints) is at least close to filling its Roche lobe. Therefore, we will express the radius of the donor star $R_{\ast}$ as a function of binary separation $a$ as, $$R_{\ast}/a \approx \frac{0.49 q^{2/3}}{0.6 q^{2/3} + \ln\left(1+q^{1/3}\right)},$$ valid to better than 1% for any $q$ [@1983MNRAS.204..449E]. We have already shown (Figure \[ec\_test\]) that there are only selected pairs of values for the binary period $P$ and the eclipse duration $\tau_{\rm ecl}$ consistent with the empirical data. Each value of $\phi \equiv \pi \tau_{\rm ecl}/P$ corresponds to one particular solution [Equations 4 and 5 in @1976ApJ...208..512C] for the pair of $(\theta, q)$ where $\theta$ is, as usual, the viewing angle, and $q \equiv M_{\ast}/M_{\bullet}$ is the ratio of donor star mass over compact object mass. Analytic solutions of $q(\phi)$ can be obtained [@2005ApJ...634L..85P] in the limiting case of $\theta = 90^{\circ}$: $$\begin{aligned} \phi &=& \arcsin(R_{\ast}/a) \nonumber \\ & \approx & \arcsin \left[ \frac{0.49 q^{2/3}}{0.6 q^{2/3} + \ln\left(1+q^{1/3}\right)} \right].\end{aligned}$$ As an example, in Figure \[ec\_test\] we labelled 4 representative values of $q(\theta=90^{\circ})$ corresponding to 4 permitted values of $\phi$ (marked as A,B,C,D). For a fixed value of $\phi$, $q$ increases going to lower (less edge-on) values of $\theta$ [Table 1 in @1976ApJ...208..512C]. For example, for $\phi = \tau_{\rm ecl}/P = 0.17$, $q(\theta = 80^{\circ}) \approx 1.3 q(\theta = 90^{\circ})$, and $q(\theta = 70^{\circ}) \approx 2.8 q(\theta = 90^{\circ})$; for $\tau_{\rm ecl}/P = 0.08$, $q(\theta = 80^{\circ}) \approx 2.0 q(\theta = 90^{\circ})$, and $q(\theta = 70^{\circ}) \approx 7.4 q(\theta = 90^{\circ})$. Regardless of the uncertainty in the true value of $\theta$ for ULX-1, the robust result is that permitted periods of $\approx$6 days always correspond to $q(\theta) \geq q(90^{\circ})\ga 4$ (with a more likely range $q \sim 5$–10), while permitted periods of $\approx$12–13 days correspond to $q(\theta) \geq q(90^{\circ}) \approx 0.25$–1.2. In the young stellar environment in which ULX-1 is located, with a likely OB donor star, the higher range of mass ratios (longer eclipse fraction) is indicative of a neutron star accretor, or a low-mass stellar BH seen almost edge-on; instead, the lower range of mass ratios (shorter eclipse fraction) is consistent with a larger range of BH masses, or with a neutron star seen at intermediate angles $\theta \sim 60^{\circ}$–$70^{\circ}$. If $q$ is known or well constrained, we can then derive a period-density relation for the donor star, and constrain its mass and evolutionary stage. In the limiting case of $q \la 0.5$, such a relation reduces to $\bar{\rho} \approx 110 P_{\mathrm hr}^{-2}$ g cm$^{-3}$; however, in the more general case [@1983MNRAS.204..449E], $$\bar{\rho} \approx \frac{10.89}{P_{\rm hr}^2}\left(\frac{q}{1+q}\right)\left[\frac{0.49 q^{2/3}}{0.6 q^{2/3} + \ln\left(1+q^{1/3}\right)}\right]^{-3}\text{g cm}^{-3}.$$ For example, for $q=1$, $\bar{\rho} \approx 99 P_{\mathrm hr}^{-2}$ g cm$^{-3}$; for $q=5$, $\bar{\rho} \approx 65 P_{\mathrm hr}^{-2}$ g cm$^{-3}$. We chose two representative values of $q$ consistent with the 6-day range of period solutions ($q=4$ and $q=10$), and two values of $q$ consistent with periods in the 12-day range ($q=0.5$ and $q=1$). For those four values of $q$, we calculated the average density of the Roche-lobe-filling donor star (Table \[tab\_ulx1\_donor\]). Typical values are $\bar{\rho} \approx 3 \times 10^{-3}$ g cm$^{-3}$ for the shorter period solution, and $\bar{\rho} \approx 10^{-3}$ g cm$^{-3}$ for the longer one. Finally, we used the latest set of Padova isochrones [^15] [@2012MNRAS.427..127B; @2015MNRAS.452.1068C] with metallicity $Z=0.019$, to estimate what types of stars have such densities, for a series of stellar population ages. In practice, we know that both ULXs reside in a region of the M51 disk with recent star formation (Soria et al., in prep.). Therefore, we only focused on population ages $\le$100 Myr as the most likely candidates for the ULX donor stars. We find (Figure \[col\_mag\_fig\]) that both ranges of permitted periods correspond to blue supergiants ($B-V$ color index $\approx$ $-$0.2–0 mag) with absolute brightness $M_V$ spanning the range between $M_V \approx -3$ mag and $M_V \approx -6$ mag, depending on their age; stars corresponding to the longer period approximately half a magnitude brighter than those associated with the shorter period. For the youngest ages ($\approx$5 Myr), the characteristic periods allowed for ULX-1 are consistent with donor stars of mass $\approx$29–31$M_{\odot}$, and radii $\approx$24–35$R_{\odot}$; for an age of $\approx$20 Myr, the predicted mass is $\approx$11$M_{\odot}$, with radii $\approx$17–25$R_{\odot}$; for an age of $\approx$50 Myr, the predicted mass is $\approx$7$M_{\odot}$, with radii $\approx$14–20$R_{\odot}$ (Table \[tab\_ulx1\_donor\]). In follow-up work, we will discuss how the observed optical brightness of the ULX-1 counterpart and of the neighbouring stars overlaps with these predictions. [c|ccccc|ccccc]{} Age & $M_\ast$ & $R_\ast$ & $M_V$ & $T_{\rm eff}$ & $M_\bullet$ & $M_\ast$ & $R_\ast$ & $M_V$ & $T_{\rm eff}$ & $M_\bullet$ ------------------------------------------------------------------------ \ (Myr) & $\left(M_{\odot}\right)$ & $\left(R_{\odot}\right)$ & (mag) & (K) & $\left(M_{\odot}\right)$ & $\left(M_{\odot}\right)$ & $\left(R_{\odot}\right)$ & (mag) & (K) & $\left(M_{\odot}\right)$ ------------------------------------------------------------------------ \ \ & & ------------------------------------------------------------------------ \ 5 & 29.7 & 23.8 & $-6.2$ & 26,800 & 7.4 & 30.3 & 26.9 & $-6.4$& 25,600 & 3.0 ------------------------------------------------------------------------ \ 10 & 17.3 & 19.8 & $-5.4$ & 22,300 & 4.3 & 17.3 & 22.3 & $-5.6$& 21,000 & 1.7\ 15 & 13.1 & 18.1 & $-4.9$ & 19,000 & 3.3 & 13.1 & 20.3 & $-5.1$& 18,000 & 1.3\ 20 & 11.0 & 17.1 & $-4.6$ & 17,100 & 2.8 & 11.0 & 19.2 & $-4.8$& 16,200 & 1.1\ 30 & 8.8 & 15.8 & $-4.2$ & 14,900 & 2.2 & 8.8 & 17.8 & $-4.3$& 14,100 & 0.9\ 40 & 7.7 & 15.1 & $-3.9$ & 13,600 & 1.9 & 7.7 & 17.0 & $-4.1$& 12,800 & 0.8\ 50 & 6.9 & 14.6 & $-3.7$ & 12,600 & 1.7 & 6.9 & 16.4 & $-3.8$& 11,900 & 0.7\ 70 & 5.9 & 13.9 & $-3.4$ & 11,300 & 1.5 & 5.9 & 15.6 & $-3.5$& 10,700 & 0.6\ 100 & 5.1 & 13.2 & $-3.0$ & 10,100 & 1.3 & 5.1 & 14.8 & $-3.1$& 9,400 & 0.5 ------------------------------------------------------------------------ \ \ & & ------------------------------------------------------------------------ \ 5 & 30.8 & 33.2 & $-6.7$ & 23,600 & 61.7 & 30.9 & 35.1 & $-6.8$& 23,000 & 30.9 ------------------------------------------------------------------------ \ 10 & 17.3 & 27.4 & $-5.8$ & 19,100 & 34.6 & 17.3 & 28.9 & $-5.9$& 18,600 & 17.3\ 15 & 13.1 & 25.0 & $-5.3$ & 16,200 & 26.2 & 13.1 & 26.2 & $-5.4$& 15,800 & 13.1\ 20 & 11.0 & 23.6 & $-5.0$ & 14,600 & 22.1 & 11.0 & 24.8 & $-5.1$& 14,200 & 11.0\ 30 & 8.8 & 21.9 & $-4.6$ & 12,700 & 17.7 & 8.8 & 23.0 & $-4.6$& 12,400 & 8.8\ 40 & 7.7 & 20.9 & $-4.3$ & 11,500 & 15.3 & 7.7 & 22.0 & $-4.3$& 11,200 & 7.7\ 50 & 6.9 & 20.1 & $-4.0$ & 10,700 & 13.8 & 6.9 & 21.2 & $-4.1$& 10,400 & 6.9\ 70 & 5.9 & 19.1 & $-3.7$ & 9,500 & 11.9 & 5.9 & 20.2 & $-3.7$& 9,300 & 5.9\ 100 & 5.1 & 18.2 & $-3.2$ & 8,400 & 10.2 & 5.1 & 19.2 & $-3.2$& 8,200 & 5.1 ------------------------------------------------------------------------ \ \[tab\_ulx1\_donor\] The mass $M_\bullet$ of the compact object is still unknown, but from the analysis outlined above we can see how observational constraints on $q$ and $M_\ast$ lead to constraints on the nature of the accretor. For example, for a period in the 6-day range, there are intermediate-age, evolved donor stars that have a mean density consistent with the period-density relation, but would imply (Table \[tab\_ulx1\_donor\]) a mass of the accreting object $\la 2M_{\odot}$, consistent only with a neutron star accretor. On the other hand, an $\approx$6-day period is consistent with a stellar-mass BH accretor only for a narrow range of massive, young ($<$10 Myr) donor stars. Conversely, mass ratios $\la1$ (corresponding to a period in the 12-day range) are consistent only with a BH accretor. Independent observational constraints on the mass and age of the donor star in ULX-1 from the brightness of its optical counterpart will be presented and discussed in follow-up work currently in preparation. Mass transfer from a donor star more massive than the accretor shrinks the binary separation and therefore causes higher, sustained mass transfer rates; this happens for $q > 5/6$ for the conservative mass transfer case, but we must account for possible additional shrinking of the system due to angular momentum losses in a wind [@2002apa..book.....F]. Blue supergiants have radiative envelopes; hence, mass transfer for $q \ga 1$ should proceed on a thermal (Kelvin-Helmholtz) timescale of the envelope, $\sim 10^4$ yr. For $q \la 5/6$ (permitted only for periods in the 12-day range), mass transfer would proceed instead on the nuclear timescale of the donor as it expands to the supergiant state. Therefore, determining the binary period of ULX-1 with future observations may reveal whether thermal-timescale or nuclear-timescale mass transfer is associated with strong ULX outflows. Semi-detached, eclipsing system such as ULX-1 and ULX-2 offer also the best chance to determine the accretor mass from optical spectroscopic observations. Let us assume for example that with future observations we will measure the binary period and strongly constrain the mass $M_\ast$ and radius $R_\ast$ of the donor star, and that we take spectra of the optical counterpart. If the donor star has absorption lines, phase-resolved optical spectroscopy might reveal its radial velocity curve, and hence the mass function $f(M_\bullet)$ of the compact object, $$f(M_\bullet) = \frac{M_\bullet^3 \sin^3 \theta}{\left(M_\bullet + M_\ast \right)^2} \approx \frac{M_\bullet^3}{\left(M_\bullet + M_\ast \right)^2},$$ from which $M_\bullet$ can be determined. Even without phase-resolved spectroscopy (hard to schedule on an 8-m telescope), one can still constrain the accretor mass if double-peaked (disk) emission lines are detected in the optical spectrum (typically, H$\alpha$, H$\beta$ and He $\lambda4686$). Such lines are usually emitted from the outer rings of the accretion disk, and their full-width at half-maximum $V_{\rm fwhm}$ depends on the projected velocity of rotation of the gas at the outer disk radius $R_{\rm d}$: $$V^2_{\rm fwhm} \approx \frac{4GM_\bullet}{R_{\rm d}}\,\sin^2 \theta \approx \frac{4GM_\bullet}{0.7 R_{\rm RL}}, \label{eq_V}$$ where we have used the empirical and theoretical constraint [@1988MNRAS.232...35W] that the accretion disks extends to an outer radius of $\approx$70% of the primary Roche lobe radius $R_{\rm RL}$. The Roche lobe radius is also a function of $q$; for example a useful approximation is $$R_{\rm RL} \approx R_{\ast} (M_\bullet/M_\ast)^{0.45},\label{eq_RL}$$ [@2002apa..book.....F]. From Equations and , $M_\bullet$ can be obtained without the need for phase-resolved spectroscopy. Conclusions =========== Using archival [*Chandra*]{} and [*XMM-Newton*]{} observations, we found X-ray eclipses in two ULXs in the same region of M51. Eclipsing systems among the ULX and luminous BH X-ray binary populations are very rare: finding two of them not only in the same galaxy but a few arcsec from each other is a surprising result. Our serendipitous discovery in the archival data suggests that perhaps other eclipsing sources may have been missed, or mis-classified as variable/transient in previous X-ray source catalogs. If persistent ULXs are stellar-mass BHs fed by Roche-lobe-filling B-type supergiants, with a mass ratio $q \sim 1$, systems seen at inclination angles $\ga 75^{\circ}$ are expected to spend up to $\approx$15% of their time in eclipse, over characteristic binary periods $\sim$10 days. Neutron star accretors are expected to have even longer eclipse fractions when seen edge-on ($\approx$20%), and to have eclipses for viewing angles as low as $55^{\circ}$. Thus, a statistical study of the observed eclipse fractions in ULXs is a possible way to determine whether the ULX population is dominated by BHs or neutron stars. We analyzed the presence and duration of the eclipses in ULX-1 and ULX-2 using a sequence of archival [*Chandra*]{} and [*XMM-Newton*]{} observations. For ULX-1, we argued that the most likely binary period is either $\approx$6.3 days, or $\approx$12.5–13 days. Assuming that the donor star fills its Roche lobe (a plausible assumption in ULXs, given the accretion rate needed to power them), we used the period-density relation to constrain the mass and evolutionary state of the donor star corresponding to those periods. For example, we showed that for a characteristic age $\approx$10 Myr, the donor-star mass is $\approx$17$M_{\odot}$, while for a characteristic age $\approx$20 Myr, $M_{\ast} \approx 11M_{\odot}$. We compared and discussed the X-ray spectral and timing properties of the two eclipsing ULXs. ULX-1 is softer, and has a spectrum well-fitted by Comptonization models in a cool, dense medium. ULX-2 is harder, consistent with either a slim disk or Comptonization in a hotter medium. Both sources are clearly seen at high inclination, given the presence of eclipses; however, neither of them is an ultraluminous supersoft source (ULS). This supports our earlier suggestion [@2016MNRAS.456.1859U] that ULSs require not only a high viewing angle, but also an accretion rate high enough to produce effectively optically thick outflows. ULX-1 has strong spectral residuals around 0.8–1.0 keV: a spectral feature seen in other ULXs (usually those with a softer spectrum, thought to be viewed at higher inclination) but not well understood yet. Its most likely interpretation is a combination of thermal-plasma emission and absorption lines from a dense outflow. In ULX-1, a residual thermal-plasma emission ($\sim$10% of the thermal-plasma emission out of eclipse) is still seen in eclipse, while the continuum component completely disappears. This suggests that the thermal-plasma emission originates from a region slightly larger than the size of the eclipsing star (that is, from a characteristic size of a few $10^{12}$ cm), rather than from the inner disk (which would be completely eclipsed) or from pc-scale shock-ionized gas (which would not be eclipsed at all). Instead, ULX-2 does not show significant thermal-plasma emission, although it does show residual emission in eclipse. In conclusion, ULX-1 and ULX-2 are important sources to help us disentangle the effects of inflow/outflow structure versus viewing angle, and deserve further follow-up multiband studies. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the anonymous referee for several useful suggestions. We thank Manfred Pakull, Christian Motch and Fabien Grisé for their contribution to the multiband analysis and interpretation, Kiki Vierdayanti for clarifications on slim disk models, Megumi Shidatsu, Yoshihiro Ueda, Andrew Sutton and Matt Middleton for useful discussions on Comptonization models in stellar-mass BHs and ULXs, Douglas Swartz and Allyn Tennant for consultations on the [*Chandra*]{} data analysis, and Duncan Galloway for his suggestions on X-ray dips. We also thank our Curtin University colleagues Sam McSweeney, Vlad Tudor, Bradley Meyers, Steven Tremblay, Peter Curran, James Miller-Jones, Gemma Anderson, Richard Plotkin and Thomas Russell for their useful suggestions and comments. This paper benefited from discussions at the 2015 and 2016 International Space Science Institute workshops ‘The extreme physics of Eddington and super-Eddington accretion onto black holes’ in Bern, Switzerland (team PIs: Diego Altamirano & Omer Blaes). RU acknowledges support from Curtin University’s 2015 Department of Applied Physics Scholarship and the Australian Postgraduate Award. RS acknowledges hospitality and scientific discussions at Kyoto University and at the Strasbourg Observatory during part of this work. \[lastpage\] natexlab\#1[\#1]{} , P. J., & [Livio]{}, M. 1996, , 470, 1024 , K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. 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[^3]: The [*mekal*]{} normalizations ($N_1$ and $N_2$) are in units of $10^{-14}/(4\pi d^2)\int n_e\,n_{\rm H}\,{\rm d}V$. [^4]: The [*diskbb*]{} normalization is in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$, where $r_{\rm in}$ is the apparent inner-disk radius. [^5]: The seed photon temperature for the Comptonizing medium, $kT_{0}$, is locked to peak color temperature of the disk, $kT_{\rm in}$. [^6]: Disk normalization in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$. [^7]: Disk normalization in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$. [^8]: The [*mekal*]{} normalizations ($N_1$, $N_2$ and $N_3$) are in units of $10^{-14}/(4\pi d^2)\int n_e\,n_{\rm H}\,{\rm d}V$. [^9]: The [*diskbb*]{} normalization is in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$, where $r_{\rm in}$ is the apparent inner-disk radius. [^10]: The seed photon temperature for the Comptonizing medium, $kT_{0}$, is locked to peak color temperature of the disk, $kT_{\rm in}$. [^11]: Disk normalization in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$. [^12]: The [*diskbb*]{} normalization is in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$, where $r_{\rm in}$ is the apparent inner-disk radius. [^13]: The seed photon temperature $kT_{0}$ is locked to the peak temperature of the disk, $kT_{\rm in}$. [^14]: Disk normalization in units of $(r_{\rm in}/{\rm km})^2\cos{\theta}\,(d/10\,{\rm kpc})^{-2}$. [^15]: Available at http://stev.oapd.inaf.it/cgi-bin/cmd

--- author: - | C. Afonso, J.N. Albert, C. Alard, J. Andersen, R. Ansari, É. Aubourg, P. Bareyre, F. Bauer, J.P. Beaulieu, G. Blanc, A. Bouquet, S. Char[^1], X. Charlot, F. Couchot, C. Coutures, F. Derue[^2], R. Ferlet, P.Fouqué J.F. Glicenstein, B. Goldman, A. Gould, D. Graff, M. Gros, J. Haissinski, C. Hamadache, J.C. Hamilton[^3], D. Hardin[^4], J. de Kat, A. Kim[^5], T. Lasserre, L.LeGuillou, É. Lesquoy, C. Loup, C. Magneville, B. Mansoux, J.B. Marquette, É. Maurice, A. Maury, A. Milsztajn , M. Moniez, N. Palanque-Delabrouille, O. Perdereau, L. Prévot, N. Regnault, J. Rich, M. Spiro, P. Tisserand, A. Vidal-Madjar, L. Vigroux, S. Zylberajch\ The EROS collaboration date: 'Received;accepted' title: 'Bulge Microlensing Optical Depth from EROS 2 observations[^6]' --- [ccc]{} 1 & 1 & 1\ 1 & 1 & 1\ 4 & 3 & 8 Introduction {#section:introduction} ============ When microlensing surveys toward the Galactic bulge were first proposed by Paczyński (1991) and Griest (1991), it was expected that the optical depth to microlensing in the Baade Window ($l=1^\circ,b=-3.\hskip-2pt^\circ9$) due to ordinary disc lenses would be $\tau \sim 4\times 10^{-7}$. In the presence of brown dwarfs in the disc with a total mass density equal to that of ordinary stars, the microlensing optical depth toward the bulge would rise to $\tau \sim 8\times 10^{-7}$. The initial detections by OGLE reporting six microlensing events ([@UDA94a]) seemed to indicate that the optical depth was higher than the predicted values, although no estimate was published. Kiraga & Paczyński (1994) then realized that the contribution of lenses in the bulge had also to be considered and that the density of disc lenses had to be reevaluated. They concluded that the bulge itself would most likely dominate the event rate. Nevertheless, when OGLE obtained an optical depth in the Baade Window of $\tau=(3.3\pm1.2)\times 10^{-6}$ with a sample of 9 microlensing events ([@UDA94b]) and MACHO made the first formal estimate $\tau_{bulge}=3.9^{+1.8}_{-1.2}\times 10^{-6}$ (at $l=2.\hskip-2pt^\circ55, b=-3.\hskip-2pt^\circ64$) based on 13 events with clump-giant sources ([@ALC97]), the community found it quite surprising. In the same paper MACHO also derived $\tau = 2.4\pm 0.5\times 10^{-6}$, based on 41 events, including not only clump giants, but all sources. They argued, however, that the determination of the optical depth for fainter source stars is less straightforward than for bright ones due to blending problems in crowded fields, where a source star can be a blend of two or more stars. Hence the entire luminosity function has to be modeled to account for both resolved and unresolved sources. Gould (1994) and Kuijken (1997) showed that the expected maximum optical depth generated by axisymmetric mass distributions of the Galaxy was surpassed by the observations. Indeed, attention was immediately focused on the possibility that the high microlensing rate represented yet another detection of a (non-axisymmetric) bar in the central regions of the Galaxy. At this time, a “bar consensus” was developing based on gas kinematics ([@BIN91]), infrared light measurements ([@DWEK95]), and star counts ([@NIK97]). However, even barred bulge models, with various values for the bar mass and the orientation to our line of sight, predict optical depths systematically lower than the observed values: Han & Gould (1995b) found $1.5\times 10^{-6}< \tau_{bulge} <2\times 10^{-6}$ at the Baade Window for bulge-giant sources; Zhao et al. (1996) determined $\tau_{bulge}=2.2\times 10^{-6}$ for clump-giant sources at the MACHO field positions ($l=2.\hskip-2pt^\circ.55, b=-3.\hskip-2pt^\circ64$); Zhao & Mao (1996) showed that several boxy and ellipsoidal-type bar models constrained by the COBE maps produce optical depths at the Baade Window $2\sigma$ lower than MACHO and OGLE measured values, even with a massive bar $M_{bar}=2.8\times10^{10}\msun$ and a small orientation angle $\theta<20^\circ$ to the line of sight. Moreover, Binney, Bissantz & Gerhard (2000) recently showed that $\tau\sim 4\times 10^{-6}$ cannot be produced by any plausible non-axisymmetric model of the Galaxy. Up to the present there have been several other estimates of $\tau$. Alcock et al. (2000) analyzed a subset of three years of MACHO data using difference imaging. This method increases the number of detected microlensing events. The mean optical depth (to the heterogeneous collection of bulge and disc sources in the MACHO fields) based on the 99 events found by this technique is estimated to be $\tau=2.43^{+0.39}_{-0.38}\times 10^{-6}$ at ($l=2.\hskip-2pt^\circ 68, b=-3.\hskip-2pt^\circ 35$). MACHO corrected this value to the true optical depth to bulge sources by assuming that 25% of the sources lay in the foreground and therefore did not contribute significantly to the observed microlensing events. They found the optical depth to be $\tau_{bulge}=3.23^{+0.52}_{-0.50}\times 10^{-6}$. Another optical depth value is given by Popowski et al. (2000). Their analysis of 5 years of MACHO data revealed 52 microlensing events with clump-giant sources. The corresponding optical depth is $\tau_{bulge}=(2.0\pm0.4)\times 10^{-6}$ averaged over 77 fields centered at ($l=3.\hskip-2pt^\circ 9, b=-3.\hskip-2pt^\circ 8$). A large fraction, perhaps a majority, of events detected toward the bulge have been found by OGLE II ([@UDA00], [@WOZ01]) but so far these have not been used to estimate $\tau$, as the OGLE II experimental detection efficiencies, necessary for the determination of microlensing optical depths, have not been made available yet. In this paper we present the first estimate of the EROS 2 optical depth to microlensing toward the Galactic Center. The EROS 2 bulge survey, begun in July 1996, was specifically designed to find events with bright sources including the extended clump area (see Fig. \[fig:cmd\])and other giants because, as discussed above, these can be interpreted unambiguously ([@GOU95b]). Data {#section:data} ==== The data were acquired at the EROS 2 team 1 m MARLY telescope at La Silla, Chile. The imaging was done simultaneously by two cameras, using a dichroic beam-splitter. Each camera is composed of a mosaic of eight 2K$\times$2K LORAL CCDs, with a pixel size of $0.\hskip-2pt''6$ and a corresponding field of $0.\hskip-2pt^\circ 7(\alpha)\times 1.\hskip-2pt^\circ 4(\delta)$. One camera observes in the so-called EROS-blue filter and the other in the so-called EROS-red filter, these filters having been specifically designed to cover a broad passband to collect as many photons as possible. Thus, the EROS filters are non-standard: EROS-red (620-920 nm) is roughly equivalent to Cousins $I$, but larger, while EROS-blue (420-720 nm) is a band that overlaps Johnson/Cousins $V$ and $R$. Details about the instrument can be found in Bauer et al. (1997). For information about the acquisition pipeline see Palanque-Delabrouille (1997). ![Galactic plane map of the EROS 2 bulge fields. A total of 82 fields are monitored. The high priority fields are supposed to be observed at least every other night (solid lines). The lower priority fields (dashed lines) are only monitored if some observation time is still available after the high prioriy sequence. The analysis and the results reported in this paper concern 15 deg$^2$ (bold lines).[]{data-label="fig:eros2_bulge_fields"}](fig/3031f1){width="7.8cm"} Although the total sky area covered by the EROS bulge survey is $82\,\rm deg^2$, the observations reported here concern only 15 of these fields, monitored between mid-July 1996 and 31 May 1999. Fig.\[fig:eros2\_bulge\_fields\] shows the location of the 82 fields in the galactic plane $(l,b)$. We also indicate the fields classified as high-priority (solid line), with the largest number of red clump giants, which we attempt to observe every other night. The lower priority fields (dashed line) are monitored only if there is still enough time left after the high-priority sequence, taking into account the compromise between the bulge survey and other EROS targets: Spiral Arms ([@DER01]), LMC ([@LASS00]), SMC ([@AFO99]), proper motion survey ([@GOLD02]), supernova search ([@HAR00]). The 15 fields whose analysis is presented here are marked in bold. The corresponding data set contains $2.3\times 10^6$ light curves, of which $1.4\times 10^6$ are bulge clump giants of the extended clump area (see Fig. \[fig:cmd\]). As we mentioned above, our bulge program was specifically designed to select events with bright stars as sources to avoid blending problems. Since one of the red CCDs was not functioning well during a large fraction of the observation period, it was not included in our analysis, since we require two colours. Event Selection {#section:event_selection} =============== The image photometry was performed with software specifically designed for crowded fields, PEIDA (Photométrie et Étude d’Images Destinées à l’Astrophysique) ([@ANS96]). After the production of the light curves and removal of defective data points due to images with a specific problem (bad atmospheric conditions, temporary instrumental deficiencies), several cuts were applied to the data set. The selection criteria explained in detail below are based on the characteristics of microlensing events light curves, which follow the Paczyński (1986) function $$\begin{aligned} F(t) & = & F_{\rm s} A[u(t)] \\ {\rm where\;\;\;} A(u) & = & \frac{u^{2}+2}{u\sqrt{u^{2}+4}} \\ {\rm and\;\;\;\;}u^{2}(t) & = & u_{0}^{2} + \frac{(t-t_0)^2}{t_\e^2} \; .\end{aligned}$$ These equations contain 5 parameters (which are obtained for each star by fitting the Paczyński profile to the corresponding light curve): two baseline fluxes ($F_{\rm s}$) of the source star in the red and blue EROS filters, $F_{\rm R_{EROS}}$ and $F_{\rm B_{EROS}}$, the date of maximum amplification $t_0$, the impact parameter $u_0=u(t_0)$ (i.e. the minimum lens-source separation projected in the lens plane, normalized by the Einstein ring radius $R_\e$) and finally, the microlensing event duration, i.e. the Einstein ring radius crossing time $t_\e=R_\e/v_{\rm t}$. The time scale $t_\e$ contains a 3-fold degeneracy between the transverse velocity $v_{\rm t}$ of the lens, its mass $M$, and its distance from the observer $D_{\rm OL}$. The $t_\e$ dependence on the mass and the distance comes through $R_\e^2=4GMD_{\rm LS}D_{\rm OL}/c^2D_{\rm OS}$, where $D_{\rm OS}$ is the distance between the observer and the source and $D_{\rm LS}$ the distance between the lens and the source. The main characteristics of an amplified light curve of a source star during gravitational microlensing are a symmetric shape in time, with an increasing light intensity as the foreground lens approaches the line of sight to the background source star, and then decreases as the lens moves away (assuming a constant transverse velocity $v_t$ of the lens). When blending is neglected, the amplification $A[u(t)]$ is the same in the two observing bands (the imaging being done simultaneously in both bands) and therefore achromatic, since microlensing is a purely geometrical phenomenon and should thus not depend on the observing wavelength. When the reconstructed star is a blend of two or more stars, the observed baseline flux $F_{\rm s}=F_1+F_2$ is the sum of the flux $F_1$ of the main component of star and the flux $F_2$ due to unresolved background stars. Assuming that the main component is amplified by a factor $A(t)$, the observed flux during the amplification is given by $F(t)=F_1A(t)+F_2=F_{\rm s}A_{\rm obs}(t)=ACF_{\rm s}A(t)+(1-C)F_{\rm s}$ where $C=(A_{\rm obs}(t)-1)/(A(t)-1)$ is the blending coefficient and $A_{\rm obs}(t)$ is the observed amplification. Finally, another main characteristic is that the amplification peak should be unique for a given source star, as the probablitity of a star to be lensed is extremely low, of the order of one per $10^6$ stars. If two or more variations occur, the source star is more likely to be variable. Thus, several of our cuts concern the rejection of variable stars. Hereafter we describe the selection criteria, which are similar to those used in the other EROS microlensing programs. 1. [**Rejecting stable stars**]{}\ 1. The main fluctuations in both the red and blue light curves should be positive[^7], and should overlap in time by at least 10%. 2. To select light curves with a significant main fluctuation we use the discriminant $LP$[^8], to which no true statistical meaning is assigned. It is rather used in an empirical and relative manner, in the sense that the light curves with higher $LP$ values than other light curves have a more significant variation. Thus we require that the main fluctuation in both colours is significant: $LP(\rm main\;fluct.)>40$.\ 2. [**Eliminating variable stars**]{}\ 1. To reject the scattered light curves of short period variable stars, which vary on time scales shorter than the average time sampling of our fields, the following requirement is made: the distribution of the difference, in units of $\sigma_{\rm i}$ (the error for the i-th flux measurement) , between each flux $F(t_{\rm i})$ and the linear interpolation of the two adjacent neighbors $F(t_{\rm i-1})$ and $F(t_{\rm i+1})$, should have an RMS lower than 2.5. $$\sigma_{int} < 2.5 \;,$$ 2. Longer period variable stars display variations in both red and blue bands. They are likely to show such correlated variations outside the principal fluctuation. Such correlations are searched for using the Fisher variable (FV) which is a function of the correlation coefficient $\rho$ between the red and blue fluxes. This variable allows one to distinguish, with a better resolution, between correlation values very close to each other and thus to tune more precisely the cut. We require $$FV(\rho) = 0.5\times\sqrt{N-3}\times \ln{\left(\frac{1+\rho}{1-\rho}\right)} < 13\;,$$ where $N$ is the number of pairs of simultaneous measurements in the red and blue bands, belonging to the unamplified part of the light curve. The exclusion of the principal fluctuation (plus a security time margin) guarantees the survival of the microlensing candidates, which as expected exhibit a strong correlation within the amplification peak. 3. The following rejection criterion is similar to 2(a) and 2(b), eliminating the variable stars that passed these cuts. We keep only the light curves that have a stable baseline outside the principal fluctuation in both bands $$\chi^2(baseline)=\frac{\chi^2_{\rm ml}(baseline)}{d.o.f.(baseline)}<5\;,$$ where $\chi^2_{ml}(baseline)$ and $d.o.f.(baseline)$ are respectively the chi square of the microlensing fit (carried out separately in each band) and the number of degrees of freedom of the fit, both values concerning the unamplified part of the light curve.\ 3. [**Selecting high S/N events**]{}\ 1. To select events with a high signal-to-noise ratio (S/N) a cut is applied to a semi-empirical estimator, whose value will increase as a microlensing fit (ml) improves over a constant-flux fit (cst) $$\Delta\chi^2 = \frac{\chi^2_{\rm cst} - \chi^2_{\rm ml}}{\chi^2_{\rm ml}/d.o.f.}\frac{1}{\sqrt{2 d.o.f.(peak)}}\;,$$ where $d.o.f.$ is the number of degrees of freedom of the fit over the entire light curve and $d.o.f.(peak)$ refers to the number of degrees of freedom of the fit within the amplification peak. For the fits we use simultaneously the data points of both red and blue light curves. We require $\Delta\chi^2>70$. 2. Candidates with low fitted maximum amplifications $A(u_0)$, may be due to statistical fluctuations or systematic photometry biases, or may be impossible to distinguish from these when the photometric precision of the stars (which for clump giants is of the order of 2-3%) does not allow it. To remove these candidates from the remaining set, we demand for each star that its maximum amplification be greater than 5 times the photometric precision of the star (calculated from the unamplified part of the light curve). For a 2-3% photometric precision, this cut allows the detection of maximum amplifications as low as 10%.\ 4. [**Date of maximum amplification and time span allowing to validate a candidate**]{}\ 1. Although the above criteria select candidates that [*a priori*]{} are microlensing events, some exhibit their date of maximum amplification $t_0$ just before or after the observation period. The confirmation of a candidate for which we have only the decreasing or increasing part of the amplification peak on the light curve is extremely difficult, if not impossible. Thus, we require that the fitted date of maximum amplification $t_0$ is within the observation period $$T_{\rm first}-\frac{t_{\e}}{3}<t_{0}<T_{\rm last}+\frac{t_{\e}}{3}\;,$$ where $t_\e$ is the event time scale, $T_{first}$ is the first day of the observations and $T_{last}$ the last one. A margin is allowed ($t_{E}/3$) due to the uncertainty of the fitted $t_0$ value. 2. As for the previous cut, it is also difficult to confirm candidates with time scales $t_\e$ too long compared to the observation period ($T_{\rm obs}\sim 3$ years), even if the date of maximum amplification is contained in the light curve. We demand that the observation period be at least 3 times greater than the Einstein ring radius crossing time $t_\e$, so that the starting or ending points of the amplification are visible on the light curves. The candidates removed by this cut from the final set (with $t_\e>400$ days), are kept on a list for regular follow-up and checking, as they could be due to black holes or neutron stars. To be chosen as a candidate, the light curve must satisfy each one of the above listed criteria. These are tuned by applying the same selection criteria to the data and to a set of simulated microlensing events (generated on top of the real light curves, see §\[section:efficiencies\]). One tries to eliminate a maximum of false candidates, while keeping the greatest possible number of simulated events. In order to detect also non-standard microlensing events (source size effect, caustic crossing), the selection criteria have been tuned sufficiently loosely. EROS 2 bulge microlensing candidates {#section:candidates} ==================================== The selection criteria presented in the previous section yield a total of 33 microlensing candidates, of which 25 have clump-giant sources (belonging to the extended clump area). Fig. \[fig:cmd\] shows the location of the source stars for the 33 microlensing candidates in an instrumental colour-magnitude diagram (CMD). This diagram was obtained by splitting up the EROS $0.7\times1.4\;{\rm deg}^2$ field into 32 $0.17\times0.17\;{\rm deg}^2$ sub-fields, finding the center of the clump of each of the sub-fields and then aligning them independently to an arbitrary common position on an instrumental CMD, which was chosen to be the EROS field centered on the Baade Window. To define $R_{\rm EROS}$ and $B_{\rm EROS}$ magnitudes, stars in the OGLE Baade Window catalog (with field coordinates $\alpha(J2000)=18^h03^m37,\delta(J2000)=-30^\circ05'00''$) ([@PAC99]) were matched with EROS stars $$\begin{aligned} R_{\rm Eros} & = & 26.95 - 2.5\times\log(F_{\rm R_{\rm EROS}}) \\ B_{\rm Eros} & = & 27.86 - 2.5\times\log(F_{\rm B_{\rm EROS}}) \; .\end{aligned}$$ where $F_{\rm R_{\rm EROS}}$ and $F_{\rm B_{\rm EROS}}$ are the red and blue fluxes (in ADU/120 s) of the center of the clump in the EROS sub-field corresponding to the Baade Window. The source stars of the microlensing candidates believed to be clump giants of the extended clump area are marked with solid circles, and sources other than clump giants are depicted with crosses. The markers surrounded by open circles refer to microlensing candidates with a maximum amplification $A_0>1.34$, i.e. an impact parameter $u_0<1$. The hatched area indicates the variation from field to field of the CMD adopted apparent magnitude cuts. Indeed, as we already mentioned in §\[section:introduction\], the purpose of the EROS bulge program was to find events with bright sources so as to avoid uncertainties due to blending. The selection of these sources was made by determining the center of the clump in the CMD of each sub-field with a special search algorithm, and rejecting all the stars below the lower limit of the clump minus 0.5 magnitude. The lower limit of the clump is defined as being 1.5$\sigma$ away from the mean of a Gaussian fitted along the magnitude axis of the CMD. Finally, the dashed lines delimit the extended clump area. ![Colour-magnitude diagram (CMD) for the source stars of the EROS 2 microlensing candidates superimposed (after alignment) on the CMD of the stars of the EROS sub-field centered on the Baade Window. Clump-giant sources (of the extended clump area) of the microlensing candidates are indicated with solid circles. Sources other than clump giants are marked with crosses. The empty circles surrounding the markers refer to microlensing candidates with a maximum amplification $A_0>1.34$. The magnitude cut in the CMD for the selection of bright reference source stars varies from field to field. This variation is indicated by the hatched area. Finally, the dashed lines surround the extended clump area containing the stars used for the optical depth calculation.[]{data-label="fig:cmd"}](fig/3031f2){width="7.8cm"} To obtain a reliable value for the bulge optical depth to microlensing, the least affected by systematic errors due to blending, we decided to consider only events with clump-giant sources (of the extended clump area), and to make a final cut requiring $u_0<1$, because it is difficult to totally rule out other forms of stellar variability for lower amplification events. This selection yielded 16 events. In 2 cases, we found that the microlensing fit was improved by adding two additional parameters for parallax ([@GOU92]), $\pi_{\rm E}$, the amplitude of the displacement in the Einstein ring due to the Earth’s orbital motion, and $\phi$, the phase of that displacement (see Table \[tab:cand\], Fig. \[fig:cdl\_candidates4\] and Fig. \[fig:cdl\_candidates5\]). We also searched for blending effects on the selected sample of 16 candidates. Two light curves seem to be affected, showing a significant improvement of the microlensing fit when blending is taken into account, particularly for candidate \#9 (see Table \[tab:cand\], Fig. \[fig:cdl\_candidates2\] and Fig. \[fig:cdl\_candidates3\]). Fig. \[fig:cdl\_candidates\] to \[fig:cdl\_candidates5\] show the light curves for the 16 events. In Table \[tab:cand\] we present the characteristics of the 16 microlensing candidates with clump-giant sources (of the extended clump area) and $u_0<1$. The mean and standard deviation of the time scales distribution (see Fig. \[fig:distri\_evt\_te\]) for these events are $$\begin{aligned} \langle t_\e\rangle & = & 33.3 \;\rm days\\ \sigma(t_\e) & = & 39.6 \;\rm days\;. \label{eqn:meantime}\end{aligned}$$ ![Time scales distribution of the 16 microlensing candidates with clump-giant sources (of the extended clump area) and $u_0<1$ $(A_0>1.34)$. The dashed line shows the raw data, while the solid curve is corrected for the detection efficiency. For the sake of comparison, the distribution of the corrected data was scaled so that the two histograms have the same area.[]{data-label="fig:distri_evt_te"}](fig/3031f3){width="7.8cm"} In order to check whether the experimental distribution of the observed impact parameters are drawn from the same distribution as the one expected for microlensing events, we use a Kolmogorov-Smirnov test. The theoretical cumulative distributions are calculated by selecting the Monte Carlo (MC) simulated events (generated randomly, see §\[section:efficiencies\]) with the same order of time scales as the observed ones and that were chosen by our analysis cuts. This method takes implicitly into account the detection efficiency, which will be presented in the next section. Fig. \[fig:testks\_u0\] shows the cumulative distribution of the impact parameters for the 16 candidates with clump-giant sources (of the extended clump area) and $u_0<1$. The dotted line refers to the expected $u_{\rm 0MC}$ distribution for microlensing. The Kolmogorov-Smirnov probability $P_{\rm KS}$ indicates the significance of the similarity of two distributions at distance $D_{\rm max}$ from each other. We obtain $D_{\rm max}=0.23$ which corresponds to $P_{\rm KS}=34\%$, which shows a good agreement between the measured and expected distributions. ![Kolmogorov-Smirnov test for the impact parameter of the 16 candidates with clump-giant sources (of the extended clump area) and $u_0<1$. The maximal distance between the experimental cumulative distribution of $u_0$ (solid line) and the expected one (dashed line) is $D_{\rm max}$=0.23. This yields a Kolmogorov-Smirnov probability $P_{KS}(D_{\rm max})$=34%.[]{data-label="fig:testks_u0"}](fig/3031f4){width="7.8cm"} Detection efficiency {#section:efficiencies} ==================== To determine the optical depth (see Eq. \[eqn:theo\_opt\]), we first evaluate the detection efficiency for each field as a function of time scale by using Monte Carlo simulated light curves. We superimpose artificial microlensing events, with randomly generated parameters (impact parameter, date of maximum amplification and time scale), on each of the real monitored light curves, and find the fraction that are recovered by our detection algorithm. Thus, the detection efficiency is given by $$\epsilon(t_{\rm EMC}\;\in\;bin\;i)=\frac{N_{\rm DE}(t_\e\;\in\;bin\;i)}{N_{\rm GE, u_{0MC}<1}(t_{\rm EMC}\;\in\;bin\;i)} \label{eqn:eff}$$ where $t_{\rm EMC}$ is the generated time scale, $N_{\rm DE}$ is the number of simulated events detected by our analysis, $t_\e$ is the time scale obtained by the microlensing fit, and $N_{\rm GE,u_{0MC}<1}$ is the number of generated events with an impact parameter $u_{\rm 0MC}<1$. The microlensing parameters of the simulated events are drawn uniformly: the impact parameter $u_{\rm 0MC}$ in the interval \[0,2\] and the date of maximum amplification $t_{\rm 0MC}$ in the observation period, with a margin of 180 days before and after respectively the first and last day of the observations $[T_{\rm first}-180,T_{\rm last}+180]$. The time period for the detection efficiency determination, equal to 1418 days, corresponds to the observation period (1058 days) extended by a 180 days margin on both sides, in order to check whether we are sensitive to microlensing events with maximum magnification occurring just before or after the actual observation period. Finally the Einstein ring radius crossing time $t_{\rm EMC}$ is drawn uniformly from a $log(t_{\rm EMC})$ distribution (to enhance the efficiency precision at small time scales) over the interval \[1,180\] days. Efficiencies were calculated only until $t_\e=180$ days because there were no events detected longer than 145 days. Fig. \[fig:eff\_cg2\_moy\_1\] shows these efficiencies averaged over two sub-groups of 10 fields (solid line) and 5 fields (dashed line), as well as the global detection efficiency which is the average over all 15 fields (bold line). These sub-groups refer to the most and least densely sampled light curves, with $\sim$ 350 data points and $\sim$ 180 points respectively within the observation period, which is the same for all fields (1058 days). For the high signal to noise events used in this paper, the efficiency is affected mostly by time gaps in the data. For example, for long events with $t_\e\sim100$ days, the 60% efficiency reflects the non-observability of the galactic center during the southern summer. Shorter time scale events are affected by periods of bad weather and instrumental failures. ![Detection efficiency as a function of the event time scale (in days) averaged over all 15 fields (solid line) and two sub-groups of 10 fields (dashed line) and 5 fields (dotted line), with different time sampling: $\sim~350$ and $\sim~180$ data points respectively.[]{data-label="fig:eff_cg2_moy_1"}](fig/3031f5){width="7.8cm"} Optical Depth {#section:optical_depth} ============= The microlensing optical depth can be defined as the probability that a given star, at a given time $t$, is magnified by at least 1.34, i.e. with an impact parameter $u(t)<1$. The optical depth is then given by $$\tau = {\pi\over 2 N_\star T_{\rm obs}}\sum_{i=1}^{N_{\rm ev}}{t_{\rm E,i}\over \epsilon(t_{\rm E,i})}\;, \label{eqn:theo_opt}$$ where $N_\star$ is the number of monitored stars, $T_{\rm obs}$ is the observation period, $t_{\rm E,i}$ is the measured Einstein ring radius crossing time of the $i$th candidate and $\epsilon(t_{\rm E,i})$ is the corresponding global detection efficiency (see Fig.\[fig:eff\_cg2\_moy\_1\]). Note that the above expression for $\tau$ only applies to objects whose mass and velocity cause events in the time scale range with significant efficiency. There could be more optical depth from events outside this range. In Table \[tab:eff\_te\] we summarize the time scales of the 16 microlensing candidates with clump-giant sources (of the extended clump area) and $u_0<1$, and the detection efficiencies for each measured $t_\e$. [l l c c]{} & Name & $t_\e$ (days) & $\epsilon(t_\e)$ (in %)\ Fig. \[fig:distri\_evt\_te\] shows the time scale distribution of the raw counts (dashed line) and corrected for efficiency (solid line), a rescaling factor having been applied so that the histograms have the same area. For the derivation of the optical depth we replace the parameters of Eq. (\[eqn:theo\_opt\]) by the corresponding values: $N_\star=1.42\times 10^6$, equal to the number of clump giants (of the extended clump area), $T_{\rm obs}=1418$ days which corresponds to the actual time period of the generation of simulated events (for the detection efficiencies determination, see §\[section:efficiencies\]), and finally the time scales and efficiencies found in Table \[tab:eff\_te\]. In the case of the 2 events affected by parallax, we considered the time scale obtained when taking into account this effect, but used the efficiencies for the time scales $t_\e$ determined from a simple microlensing fit without parallax, as initially found by our analysis. Regarding the events with blending, we used the time scale uncorrected for this effect, otherwise we would have had to estimate the number of blended unseen stars to add it to our optical depth equation. We have checked that the measured optical depth depends very little on these assumptions. Moreover, as we will justify below in a study to quantify the effect of blending on bright stars, these are on average unaffected. We obtain a bulge microlensing optical depth of $$\tau = 0.94^{+0.29}_{-0.30} \times 10^{-6} \;\;at\;\; (l,b)=(2.\hskip-2pt^\circ 5,-4.\hskip-2pt^\circ0) \; . \label{eqn:obs_opt}$$ Note that the optical depth, and the associated errors, are valid only for objects within the range of detection $\sim 2\;{\rm days}<t_E<180\;{\rm days}$. The $(l,b)$ position is an average of positions of the clump giants (of the extended clump area) in the 15 fields. The uncertainties are statistical, estimated using the same technique as in Alcock et al. (2000). To do so, a significant number of experiments were simulated. For each experiment we generated the number $n$ of “observed” microlensing events, according to Poisson statistics with a mean of $\mu=16$, equal to the number of actually observed candidates. To each of the $n$ events, one of the 16 measured time scale was assigned randomly (being uniformly drawn), thus obtaining an optical depth estimate for each virtual experiment. The uncertainties are then given by the $\pm1\sigma$ values from the average of the simulated optical depth distribution (see Fig. \[fig:err\_profopt\_err\]). The 2$\sigma$ can be calculated in the same way, yielding $\tau = 0.94^{+0.68}_{-0.46} \times 10^{-6}$. The errors can also be estimated analytically ([@HAN95b]) $$\sigma(\tau) = \tau\frac{\sqrt{<t_\e^2/\epsilon^2>}}{<t_\e/\epsilon>} \frac{1}{\sqrt{N_{\rm ev}}}= 0.29\times 10^{-6}\; ,$$ in very good agreement with the uncertainties given in Eq.  (\[eqn:obs\_opt\]). ![The cumulative distribution of the statistical microlensing optical depth drawn from a significant number of virtual experiments for our sample of 16 microlensing candidates. The optical depth uncertainties are then given by the $\pm1\sigma$ confidence limits (dashed lines).[]{data-label="fig:err_profopt_err"}](fig/3031f6){width="7.8cm"} The contribution of each of the 16 candidates to the measured optical depth is shown in Fig. \[fig:cont\_candidates\], where the area of the cercles is proportional to the individual optical depth due to each event $\tau_i=\pi/(2N_{\star,\rm j}T_{\rm obs})t_{\rm E,i}/\epsilon(t_{\rm E,i})$, $N_{\star,\rm j}$ being the number of stars in field $j$, with the shortest event lasting $\sim5$ days and the longest $\sim146$ days. We also show the measured optical depth in each field. ![The contribution of the 16 candidates to the observed optical depth. The area of the circles is proportional to the optical depth due to each microlensing candidate. The measured optical depth in each field is also shown, as well as the number of the EROS fields.[]{data-label="fig:cont_candidates"}](fig/3031f7){width="7.8cm"} Effect of blending on the measured optical depth {#section:effect_blending} ================================================ In order to check that the measured optical depth given by microlensing events with clump-giant sources is not significantly affected by blending, we created a set of artificial images with simulated microlensing events, and calculated the optical depth from the candidates detected by our selection pipeline on the simulated light curves. Two types of synthetic images, corresponding to the two EROS passbands (see §\[section:data\]) and with a size of $512\times 512$ pixels, were generated from a catalog derived from the Holtzman et al. (1998) luminosity function in the Baade Window. The catalog contained 365,000 stars which were placed randomly over the $512\times 512$ pixels area. The faintest catalog star was 8 magnitudes dimmer than the faintest reconstructed star considered in this paper. On an arbitrarily selected synthetic reference image, 20% (73,000) of the total number of artificial stars were chosen to be microlensed. We then generated in each color a sequence of 3,600 images equally spaced in time, the unit of time being 1 image. On each ensemble of $2\times20$ images (blue and red filters), about 400 microlensing events were generated. To avoid photometric interference between simulated events, only stars at least 20 pixels away from each other and with similar magnitudes were lensed. The microlensing events were generated with impact parameter $u_0$ randomly drawn between 0 and 1.5, date of maximum amplification $t_0$ equal to the center of the ensemble with a margin of 0.5 images, and time scale $t_\e=$ 5 images. For the microlensing fit we used the ensemble containing the fluctuation, plus an additional 20 images generated with no events in order to determine the baseline. Roughly 10,000 stars of the total number of artificial stars were reconstructed by our software on each synthetic image, an example of which is shown in Fig. \[fig:mcbref\]. To define a sample of bright stars a magnitude cut was performed on the CMD of the synthetic reference image (see Fig. \[fig:fake\_cmd\]). A total of 2270 stars were selected, corresponding closely to the mean density of bright stars reconstructed on real EROS images. The analysis pipeline was then applied to the simulated light curves of this sample of reconstructed bright stars. A total of 411 generated microlensing events were found with an impact parameter $u_0<1$ and an average of reconstructed parameters $<t_\e> = 3.55$ images and $<u_0> = 0.56$. From these events, 255 were due to the main star, i.e. the brightest catalog input star in the two pixels around the reconstructed star, with recovered $<t_\e> = 4.67$ images and $<u_0> = 0.49$. The remaining 156 events are due to the fainter, blended, component of the reconstructed star. The average of the recovered time scales for these blended events is $<t_\e> = 1.72$ images, clearly underestimated, and $<u_0> = 0.66$, overestimated. In the absence of blending and with perfect photometric resolution, the number of simulated microlensing events one would expect to recover is $2270\times0.2/1.5=302$, where 2270 is the number of reconstructed bright stars selected by the magnitude cut in the CMD and 0.2 is the fraction of catalog stars microlensed with impact parameters less than 1.5. The optical depth being proportional to the product of the number of events passing the microlensing selection criteria and their mean $t_\e$, the ratio $R$ of the recovered optical depth with the generated one yields $$R = \frac{411}{302}\frac{3.55}{5.0} = 0.97$$ where $411$ is the number of simulated microlensing events found by our analysis pipeline, the value 3.55 is the average of the recovered time scales, 302 is the number of simulated microlensing events one would expect to recover and 5.0 the average of the input time scales. The error of the ratio $R$ is estimated to be 10%. This figure is based on the statistical error and from small differences in results obtained by varying within reason the form of the PSF used to generate synthetic images. The recovery of 97% of the generated optical depth is a reassuring result. Thus, our conclusion is that we can neglect blending effects on the optical depth inferred from microlensing events with bright source stars. ![Example of a synthetic image as described in §\[section:effect\_blending\] (256$\times$256 pixels are shown).[]{data-label="fig:mcbref"}](fig/3031f8){width="7.8cm"} ![Artificial colour-magnitude diagram. The dashed line indicates the magnitude cut for the selection of bright stars.[]{data-label="fig:fake_cmd"}](fig/3031f25){width="7.8cm"} Searching the alerts and microlensing events of the MACHO and OGLE collaborations in the EROS data {#section:check_macho_ogle} ================================================================================================== In view of our low measured optical depth compared to other determinations (see Table \[tab:opt\]), it is important to check that microlensing events had not been lost in the analysis procedure in unsuspected ways that are not taken into account by the Monte Carlo detection efficiency calculation. To do this, we looked for Galactic Center events that had been found independently by the MACHO and OGLE collaborations within our observation period in the 15 fields we analyzed, and whose magnitudes are brighter than our cut in the CMD (see §\[section:candidates\]). We also looked for alerts found by the EROS trigger. From the MACHO collaboration, we considered the 211 online alerts[^9] and 99 published events ([@ALC00]) found by differential photometry. From the OGLE collaboration, we considered the 89 alerts reported during the years 1998 and 1999[^10] and the 214 candidates published by Udalski et al. (2000). Regarding the EROS 2 alert system[^11], although it was only operational after May 1999, beyond the time period of the data analyzed in this paper, a test version was performed during a limited time yielding three alerts to be considered for the search. From these five sources, a total of 22 events occurred within the observation period considered in this paper (from July 1996 to 31 May 1999) and concerned stars bright enough to pass our magnitude cut. Of these 22 events, 13 were identified by our analysis pipeline as microlensing candidates, 8 of which have clump-giant sources and an amplification $A_0>1.34$. The 9 remaining events were not found. Two were rejected by our selection criteria: one because of excessive fluctuations outside the amplification peak and the other event because the improvement of a microlensing fit over a constant-flux fit was not good enough. Another two events occurred on source-stars that do not appear in the EROS catalog and, as such, cannot be considered for the optical depth measurement. These two stars have a magnitude at the limit of our magnitude cut and are at the edge of brighter stars, which explains their non-appearance in the catalog. Finally, 5 events occured during periods that were at best sparsely sampled by EROS due to bad weather or technical problems. Their non-detection is thus normal and corrected for by our Monte Carlo detection efficiency computation. Note that the optical depth estimate presented in this paper is unaffected by these results, since none of the “unseen” MACHO and OGLE candidates were not found without a supporting reason (i.e., the analysis pipeline behaved like we expected it to). Discussion and conclusion ========================= ------------ ------------------------------------- ------------ -------------------- -------- ----------------- ---------- Observed $l,b$ [Optical depth]{} No. No. Bulge Group optical depth [Baade Window]{} of stars seasons ($\times 10^{-6}$) $(^\circ)$ ($\times 10^{-6}$) events ($\times 10^6$) $\tau\pm1\sigma$ 1. EROS 2 $\tau_{bulge}=0.94^{+0.29}_{-0.30}$ $2.5,-4.0$ $1.08\pm0.30$ 16 CG 1.42 $\sim$ 3 2. MACHO $\tau_{bulge}=3.90^{+1.8}_{-1.2}$ $2.6,-3.6$ $3.86\pm1.50$ 13 CG 1.3 190 days $\tau=2.43^{+0.39}_{-0.38}$ $\tau_{bulge}=3.23^{+0.52}_{-0.50}$ $3.11\pm0.51$ 4. MACHO $\tau_{bulge}=2.0\pm0.4$ $3.9,-3.8$ $2.13\pm0.40$ 52 CG 2.1 $\sim$ 5 5. OGLE $\tau=3.30\pm1.2$ $1,-4$ $3.3\pm1.20$ 12 $\sim 1$ $\sim$ 3 6. OGLE II - - - 214 20.5 $\sim$ 3 ------------ ------------------------------------- ------------ -------------------- -------- ----------------- ---------- The optical depth obtained above (Eq. \[eqn:obs\_opt\]) is low compared to other determinations, as can be seen in Table \[tab:opt\]. For direct comparison among these experiments, we also report in this Table the observed optical depths extrapolated to the Baade Window position ($l=1^\circ,b=-4^\circ$), after applying an optical depth gradient in the $l$ and $b$ directions. We deduced a rough estimate for the gradient: $\partial\tau/\partial b = 0.45\times 10^{-6} deg^{-1}$ and $\partial\tau/\partial l = 0.06\times 10^{-6} deg^{-1}$, from several microlensing maps predicted by various non-axisymmetric models ([@HAN95b], [@ZHAO96], [@BISS97], [@EVA02]). The expected opticals depths, for these models, over the interval of Galactic longitude and latitude of our fields ($-6>b>-2, 6>l>0$) ranges roughly from $\tau \sim 1.8 \times 10^{-6}$ to $\tau \sim 0.6 \times 10^{-6}$, as one goes farther away from the Galactic Center. For comparaison with the range of the measured opticals depths in the EROS fields see Fig.\[fig:cont\_candidates\]. The first conclusion that can be drawn is that the quoted measurements are consistent with our optical depth estimate only at the $2\sigma$ level. Moreover, the predicted optical depths seem to be more in agreement with our value. Indeed, $\tau_{bulge}\sim1.3\times 10^{-6}$ is expected at the Baade Window by Han & Gould (1995b), $\tau_{bulge}\sim0.8-0.9\times 10^{-6}$ is the inferred estimation by Bissantz et al. (1997) at the same position, and the predicted optical depths by Evans & Belokurov (2002) with two different models are $\tau_{bulge}\sim1\times 10^{-6}$ and $\tau_{bulge}\sim1.5\times 10^{-6}$, although a third model of these authors gives a higher estimate $\tau_{bulge}\sim2\times 10^{-6}$. All of the above mentioned models consider a barred non-axisymmetric bulge. The MACHO and OGLE optical depth measurements are systematically higher than the predicted values, except for the Popowski et al. (2000) determination which is more in agreement with the models. Furthermore, recently Binney, Bissantz & Gerhard (2000) argued that an optical depth for bulge sources as large as the ones inferred by the MACHO collaboration ([@ALC97], [@ALC00]) is inconsistent with the rotation curve and the local mass-density measurements. We report 3 microlensing candidates with long durations, $t_\e>50$ days: $t_\e=56, 108, 146$ days, all in different fields. These events contribute about 30% to the optical depth. Long time scale events, difficult to reconcile with the known mass functions, were already present in the bulge clump-giant sample from Alcock et al. (1997). They found 3 candidates with $t_\e>75$ days. It was suggested that they might be due to stellar remnants ([@ALC97], [@HAN95a]) or to directions where there is a spiral arm concentration ([@DER99], [@PEA99]). Popowski et al. (2000) also reported 10 long events, with $t_\e>50$ days, contributing 40% to the measured optical depth, half of them being concentrated in one field. In addition, the Einstein ring radius crossing-time distribution of the 214 microlensing candidates found by the OGLE collaboration ([@UDA00]), has the same type of tail toward long time scales ($t_\e>50$ days) as the distributions found by MACHO, although they are not concentrated in particular fields but rather uniformly scattered. Recently, Evans & Belokurov (2002) pointed out that bar streaming increases significantly the amplification durations, with a growing gradient in the mean time scales from the near-side to the far-side of the bar. In our view, the most robust way to resolve the optical depth issue, reconciling Galactic structure with microlensing observations, is to obtain a larger sample of clump-giant events. We expect to increase our sample of candidates by a factor 5 by the time EROS shuts down in 2002. From the preliminary work of Popowski et al. (2000), one may expect the MACHO sample to be increased by a factor 1.3. Moreover, the OGLE data set represents a potentially rich source of additional events. In the future, the coming of new survey telescopes such as VST and VISTA, will enhance the possibility to distinguish between Galactic models, especially if microlensing observations are done in the $K$ band in the inner $5^\circ\times5^\circ$ region of the Galactic Center ([@GOU95a], [@EVA02]). Thus, the prospects for clarifying this question over the next few years are very promising. We are grateful to D. Lacroix and the technical staff at the Observatoire de Haute Provence and A. Baranne for their help in refurbishing the MARLY telescope and remounting it in La Silla. We are also grateful to the technical staff of ESO, La Silla for the support given to the EROS project. We thank J-F. Lecointe and A. Gomes for the assistance with the online computing. Work by A. Gould was supported by NSF grant AST 02-01266 and by a grant from Le Centre Français pour L’Accueil et Les Échanges Internationaux. 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Astr., 49, 319 Palanque-Delabrouille N. (EROS collaboration) 1997, PhD thesis: Research on Galactic Dark Matter implied by Gravitatinal Lensing, Université de Paris 7 and University of Chicago Peale S. J. 1999, ApJ, 524, L67 Popowski P., Cook K. H., Drake A. J., et al. (MACHO collaboration) 2000, In Menzias J.W., Sackett P.D. eds, Proc. Microlensing 2000, ASP Conference Series, in press (astro-ph/005466) Stanek K. Z., Paczynski B., Udalski A., et al. (OGLE collaboration) 1995, AAS, 186, 304 Udalski A., Szymanski M., Kaluzny J.,et al. (OGLE collaboration) 1994a, ApJ, 426, L69 Udalski A., Szymanski M., Stanek K. Z., et al. (OGLE collaboration) 1994b, Act. Astr., 44, 165 Udalski A., Zebrun K., Szymanski M., et al. (OGLE collaboration) 2000, Act. Astr., 50, 1 Wozniak P. R., Udalski A., Szymanski M. et al. (OGLE collaboration) 2001, Act. Astr., 51, 175 Zhao H., Rich R. M. & Spergel D. N. 1996, MNRAS, 282, 175 Zhao H. & Mao S. 1996, MNRAS, 283, 1197 [l l c c c c c c c c]{} & $\alpha$ (J2000) & $\delta$ (J2000) & $R_{\rm EROS}$ & $B_{\rm EROS}$ & $t_{0}$ & $t_{\e}$ & $A_{0}$ & $\chi^2$/d.o.f.\ ![The light curves of the EROS 2 microlensing candidates \#1 to \#3 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates"}](fig/3031f9a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#1 to \#3 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates"}](fig/3031f9b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#1 to \#3 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates"}](fig/3031f10a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#1 to \#3 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates"}](fig/3031f10b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#1 to \#3 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates"}](fig/3031f11a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#1 to \#3 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates"}](fig/3031f11b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#4 to \#6 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates1"}](fig/3031f12a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#4 to \#6 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates1"}](fig/3031f12b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#4 to \#6 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates1"}](fig/3031f13a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#4 to \#6 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates1"}](fig/3031f13b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#4 to \#6 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates1"}](fig/3031f14a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#4 to \#6 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit.[]{data-label="fig:cdl_candidates1"}](fig/3031f14b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#7 to \#9 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit. For candidate \#9 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates2"}](fig/3031f15a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#7 to \#9 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit. For candidate \#9 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates2"}](fig/3031f15b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#7 to \#9 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit. For candidate \#9 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates2"}](fig/3031f16a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#7 to \#9 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit. For candidate \#9 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates2"}](fig/3031f16b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#7 to \#9 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit. For candidate \#9 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates2"}](fig/3031f17a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#7 to \#9 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span only), as well as the $\chi^2$ values of the fit. For candidate \#9 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates2"}](fig/3031f17b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#10 to \#12 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#11 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates3"}](fig/3031f18a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#10 to \#12 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#11 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates3"}](fig/3031f18b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#10 to \#12 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#11 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates3"}](fig/3031f19a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#10 to \#12 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#11 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates3"}](fig/3031f19b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#10 to \#12 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#11 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates3"}](fig/3031f20a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#10 to \#12 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#11 the dashed line refers to the fit when blending is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without blending and the zoom (right light curve) shows the parameters of the fit with blending.[]{data-label="fig:cdl_candidates3"}](fig/3031f20b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#13 to \#15 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#15 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates4"}](fig/3031f21a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#13 to \#15 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#15 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates4"}](fig/3031f21b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#13 to \#15 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#15 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates4"}](fig/3031f22a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#13 to \#15 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#15 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates4"}](fig/3031f22b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#13 to \#15 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#15 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates4"}](fig/3031f23a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidates \#13 to \#15 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#15 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates4"}](fig/3031f23b){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidate \#16 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#16 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates5"}](fig/3031f24a){width="7.5cm"} ![The light curves of the EROS 2 microlensing candidate \#16 (see Table \[tab:cand\]). In each box the upper light curve refers to the EROS red filter and the lower light curve to the EROS blue filter. Full span of the light curves is shown in the left column and corresponding zoomed light curves are in the right column. The 5 parameters obtained by the fit of the Paczyński profile are shown (on full span), as well as the $\chi^2$ values of the fit. For candidate \#16 the dashed line refers to the fit when parallax is taken into account. The left light curves of this candidate indicate the parameters of the microlensing fit without parallax and the zoom (right light curves) shows the parameters of the fit with parallax.[]{data-label="fig:cdl_candidates5"}](fig/3031f24b){width="7.5cm"} [^1]: deceased [^2]: Presently at Centre de Physique des Particules de Marseille, IN2P3-CNRS, 163 Avenue de Luminy, case 907, 13288 Marseille Cedex 09, France [^3]: Presently at ISN, IN2P3-CNRS-Université Joseph-Fourier, 53 Avenue des Martyrs, 38026 Grenoble Cedex, France [^4]: Presently at LPNHE, IN2P3-CNRS-Université Paris VI et VII, 4 Place Jussieu, F-75252 Paris Cedex, France [^5]: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A. [^6]: Based on observations made with the MARLY telescope at the European Southern Observatory, La Silla, Chile. [^7]: A positive (negative) fluctuation is defined as a set of consecutive points that deviate by at least 1$\sigma$ above (below) the baseline flux. [^8]: The statistical significance of a fluctuation containing $N$ points is given by $LP = - \sum_{i=1}^{i=N}\log\left(\frac{1}{2}\:{\rm Erfc}\left(\frac{x_{i}}{\sqrt{2}}\right)\right)$, with each point $i$ deviating at time $t_i$ by $x_i$ (in units of $\sigma_i$, the error for the i-th flux measurement). [^9]: http://darkstar.astro.washington.edu [^10]: http://www.astrouw.edu.pl/$\sim$ogle/ogle2/ews/ews.html [^11]: http://www-dapnia.cea.fr/Spp/Experiences/EROS/alertes.html

--- abstract: 'Leibniz thought experiment of perception, sensing, and thinking is reconsidered. We try to understand Leibniz picture in view of our knowledge of basic neuroscience. In particular we can see how the emergence of [*consciousness*]{} could in principle be understood.' author: - Markos Maniatis title: Notes on Leibniz thought experiment --- Gottfried Wilhelm Leibniz, [*Monadology*]{} (1714), §[17]{} [@Leibniz]:\ “[*Besides, it must be admitted that perception, and anything that depends on it, cannot be explained in terms of mechanistic causation – that is, in terms of shapes and motions. Let us pretend that there was a machine, which was constructed in such a way as to give rise to thinking, sensing, and having perceptions. You could imagine it expanded in size (while retaining the same proportions), so that you could go inside it, like going into a mill. On this assumption, your tour inside it would show you the working parts pushing each other, but never anything which would explain a perception. So perception is to be sought, not in compounds (or machines), but in simple substances. Furthermore, there is nothing to be found in simple substances, apart from perceptions and their changes. Again, all the internal actions of simple substances can consist in nothing other than perceptions and their changes.”* ]{}\ We would like to reconsider Leibniz thought experiment, presented about three centuries ago. Today we know, that at a [*tour inside*]{} the elementary [*working parts*]{} are the neurons, which [*push each other*]{} by means of electrical activity. Following Leibniz closely it is evident that we can not find at any special location anything from which we could explain perception, sensing or thinking. This hypothetical special location of perception, sensing or thinking we would call [*consciousness*]{} or [*I*]{} or [*self*]{}. However with that notion we would not gain any insight.\ In contrast, it is immediately clear that any special location of perception, sensing and thinking leads to contradictions: suppose, we could detect a special location of perception, sensing and thinking, then, in a further expansion in size we could again go inside and would only find [*working parts*]{}, [*pushing each other*]{}. Indeed, in terms of neurons, we know that the neuronal signals do not converge anywhere. If we consider for example a visual sense, we know that the signals already behind the retina are divided into different neuronal structures and do not come together anywhere. Any kind of convergence at some location of perception would require some new kind of [*inner eye*]{}. We would not be any step forward – this contradiction is usually denoted as infinite regress (see for instance the discussion in [@Rosenthal86]). The crucial point in Leibniz thought experiment is to recognize that there is no special location of perception, sensing or thinking. However, in contrast to Leibniz conclusion, it appears quite natural to explain these phenomena from the interactions of neurons themselves. Hence, we should assume that perceptions, sensing, and thinking do not arise at a special location, but are developed in the whole neuronal system.\ From the picture of working [*parts pushing each other*]{} we draw the conclusion that the processes in our brain are deterministic: in any kind of [*mill*]{}, consisting of mechanical parts any movement is caused by the preceding ones. The cascade of mechanical processes appears to be inescapable. Indeed [*free will*]{} has no meaning in a neurological context since causal processes contradict anything [*free*]{}. Also, trying to employ quantum mechanics to escape from causality [@Bohm; @Pribram; @QM], we do not see any way to explain [*freeness*]{} in terms of the randomness we encounter in quantum mechanics.\ Realizing that perception, sensing, and thinking appear from the cascade of neuronal processes in an unfree manner we will in the following talk about the [*emergence*]{} of these phenomena. Using the expression [*emergence*]{} we stress that there is nothing like an illusionary “inner location” where perception, sensing, and thinking are formed.\ Let us think this thought further. Imagine, under anesthetic, in our brain one neuron after the other would be replaced by an exact copy. Since no neuron would be the special location of perception, sensing or thinking, we would in no step replace this special location. After recovering from anesthesia [*we*]{} would not recognize any change. The neurons would interact in the same manner as before and our perception, sensing, and thinking would appear in the same way.\ Of course, it makes no difference whether we replace the neurons one by one, or at once. Likewise in the latter we would develop perception, sensing, and thinking in the copy in the same way! Hence, suppose that we replace under anesthetic our body by a copy, nothing like [*I*]{} or [*consciousness*]{} or [*self*]{} would be lost. That is, our perception, sensing, and thinking is not attached to certain neurons, but appear from their activities. Let us further imagine that we could replace each neuron in turn by an electronic device, which replicates exactly the same functionality as the original neuron. As before we would not remove in any step a location of perception, sensing or thinking. In this way we finally would be replaced by a machine under anesthetic and this machine would develop the same perception, sensing and thinking and [*we*]{} could not feel any difference!\ Obviously this picture of the emergence of perception, sensing, and thinking is contrary to the accepted opinion. We are convinced to have some kind of [*I*]{} – a location where perception, sensing, and thinking is formed. Why are we subject to this illusion?\ The crucial point here is to see how the usage of [*I*]{} appears in our thoughts: let us consider an example of a perception, for instance the smell of an apple. If we communicate to someone this perception, we say, for instance: “I smell the scent of a fresh apple”. We would use grammatical first-person in order to communicate our own perception, distinguishing it from a perception of someone else. But what happens if we do not communicate this statement but only realize the smell? This thinking must be something emergent, so we can understand it if we suppose that thinking is nothing but silent communication. Hence, we think, in an emergent sense, “I smell the scent of a fresh apple”. Therefore we suppose that thoughts are in this sense always a way of communication and this implicitly seems to originate from a location of perception, likewise denoted by “I” in our example. In this way the illusion of a location of perception, sensing, and thinking is unavoidable – a machine would develop the same illusion of a location of perception, sensing, and thinking (compare with the “zombie” in [@Chalmers97]).\ The question arises whether we, replaced by a machine, could become immortal? We guess the answer is yes, supposing we could exactly represent the circuit of tens of billions of neurons (for an attempt see for instance [@bluebrain]). [99]{} Gottfried Wilhelm Leibniz, in [*Leibniz The Monadology and Other Philosophical Writings*]{}, translated by Robert Latta, Kessinger Publishing Co (Juli 2007) ISBN 978-0548164266. Rosenthal, D. [*Two Concepts of Consciousness*]{}, Philosophical Studies 49: 329-359 (1986). Bohm, D. [*A new theory of the relationship of mind and matter*]{}, Philosophical Psychology, 3: 2, 271—286 (1990). Pribram, K.H. [*Brain and Perception: Holonomy and Structure in Figural Processing*]{}, Taylor & Francis (1991) ISBN 978-0898599954. see also Atmanspacher, H. [*Quantum Approaches to Consciousness*]{}, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL: <http://plato.stanford.edu/archives/sum2011/entries/qt-consciousness>. Chalmersm D., [*The Conscious Mind: In Search of a Fundamental Theory*]{} Oxford University Press. ISBN 019511789, (1997). Markram, H. [*The Blue Brain Project*]{}, Nature Reviews Neuroscience, 7, 153-160 (2006).

--- abstract: 'Berliner (Likelihood and Bayesian prediction for chaotic systems, J. Am. Stat. Assoc. 1991) identified a number of difficulties in using the likelihood function within the Bayesian paradigm for state estimation and parameter estimation of chaotic systems. Even when the equations of the system are given, he demonstrated “chaotic likelihood functions" of initial conditions and parameter values in the 1-D Logistic Map. Chaotic likelihood functions, while ultimately smooth, have such complicated small scale structure as to cast doubt on the possibility of identifying high likelihood estimates in practice. In this paper, the challenge of chaotic likelihoods is overcome by embedding the observations in a higher dimensional sequence-space, which is shown to allow good state estimation with finite computational power. An Importance Sampling approach is introduced, where Pseudo-orbit Data Assimilation is employed in the sequence-space in order first to identify relevant pseudo-orbits and then relevant trajectories. Estimates are identified with likelihoods orders of magnitude higher than those previously identified in the examples given by Berliner. Importance Sampling uses the information from both system dynamics and observations. Using the relevant prior will, of course, eventually yield an accountable sample, but given the same computational resource this traditional approach would provide no high likelihood points at all. Berliner’s central conclusion is supported. “chaotic likelihood functions" for parameter estimation still pose challenge; this fact is used to clarify why physical scientists tend to maintain a strong distinction between the initial condition uncertainty and parameter uncertainty.' author: - | Hailiang Du$^{1,2}$ Leonard A. Smith$^{1,2,3}$\ $^1$Centre for the Analysis of Time Series,\ London School of Economics, London WC2A 2AE. UK\ $^2$Center for Robust Decision Making on Climate and Energy Policy,\ University of Chicago, Chicago, IL 60637, US\ $^3$Pembroke College, Oxford, UK title: Rising Above Chaotic Likelihoods --- Introduction ============ Nonlinear chaotic systems pose several challenges both for state estimation and for parameter estimation. Chaos as a phenomenon implies sensitive dependence on initial condition: initially nearby states will eventually diverge in the future. The bifurcations of various chaotic systems [@Sprott2003] reveal how the behavior of the system differs as a parameter value changes. One might think that likelihood and Bayesian analysis should be able to obtain good estimation both of initial conditions and of parameter values without much trouble. Berliner [@Berliner1991] examined the log-likelihood function of estimates of initial conditions and parameter values for the Logistic Map. He pointed out that chaotic systems can lead to “chaotic likelihood functions”, suggesting that Bayesian analysis would require prohibitively intensive computing. The failure of variational approaches, when applied to long window observations of chaotic systems [@Miller1994; @Judd2001; @DuPDA], supports his point. Sensitivity to initial condition also suggests that information in the observations (even over a relatively short range) can lead to good estimates of the initial condition [@Judd2004]. An importance sampling approach for extracting such information without “intensive computing” is deployed in this paper. Adopting the Pseudo-orbit Data Assimilation (PDA) approach [@Judd2001; @DuPDA] recasts the task into a higher dimensional sequence space, where truly high likelihood states are successfully located near the trajectory manifold. Although statisticians often fail to make a strong distinction between initial conditions and parameter values, the challenges of initial condition estimation and parameter estimation are dissimilar for chaotic systems. PDA does not easily generalize to parameter estimation, as it is unclear how to mathematically define a relevant subspace of parameter space in which high likelihood trajectories might exist. Thus challenges remain in identifying high likelihood parameter values given the initial condition; this asymmetry is used to discuss differences between the initial conditions and parameter values. In terms of estimating initial conditions given the parameter values, however, Berliner’s challenge is met and resolved without prohibitively intensive computing. Chaotic Likelihood Function of Initial Conditions ================================================= Following Berliner [@Berliner1991] the Logistic Map is adopted as the system, assuming that the parameter $a=4$ is known but the true initial state $\tilde{x}_{0}$ is not. In that case, the experiment is said to fall within the perfect model scenario[^1].The evolution of system states $x_{i}\in \mathbb{R}^m$ is then governed by the nonlinear dynamics $f:x_{i+1}=f(x_{i})$, where for the Logistic Map[^2] $$\begin{aligned} \label{eq:model} f(x_{i})=ax_{i}(1-x_{i}).\end{aligned}$$ Assuming additive observational noise $\delta_{i}$ yields observations, $s_{i}=\tilde{x}_{i}+\delta_{i}$ where $\tilde{x}$ is the true system state (Truth) and the observational noise, $\delta_{i}$, is Independent Normally Distributed (IND, $\delta_{i}\sim N(0,\sigma^{2})$). Under this normality assumption, the log-likelihood (LLik) function is: $$\begin{aligned} \label{eq:LLik} LLik(x_{0})=-\sum_{i=1}^{n-1}(s_{i}-f^{i}(x_{0}))^2/2\sigma^{2},\end{aligned}$$ where $f^{i}$ is the $i^{th}$ iteration of $f$, $s_{i}$ is the $i^{th}$ observation, and $n$ is the duration of observations considered. Figure \[fig:LLikNormal\] shows the chaotic likelihood structure of 1024 samples from $U(0,1)$. Panel (a) plots the log-likelihood for $x_{0}$, this can be contrasted with various panels in Berliner [@Berliner1991] Figure 3[^3]. Panel (b) shows the log-likelihood (RLLik) relative to that of the true trajectory of the system states[^4]. For the convenience of illustration, the same normalization is applied in the following three figures in this paper. From Figure \[fig:LLikNormal\], it is clear that no high likelihood states are identified. This is not a case of equifinality[^5]. Given the observational noise distribution, one can add random draws from the inverse of the observational noise distribution to the observation to obtain candidate estimates of initial condition. Figure \[fig:LLikZoom\]a shows the relative log-likelihood of 1024 samples from inverse observational noise. No high likelihood states are identified in this way. To illustrate the impact of making much more precise observations, consider a case where $\tilde{x}_{0}$ is known to be within a region of radius only $\sigma/10$. Figure \[fig:LLikZoom\]b shows the RLLik of 1024 uniformly sampled states in the region around the Truth with $\sigma/10$ radius. Yet again, no high likelihood state are identified. This difficulty here has nothing to do with the Likelihood approaches as there are high likelihood states other than Truth. One may demonstrate that such high likelihood states exist by sampling the points on a logarithmic spiral approaching the Truth (to machine precision)[^6]. Figure \[fig:LLikBlue\] shows that other than Truth there exist high likelihood states, i.e. some of the blue points. A smooth curve of the log-likelihood function is only observed within a radius of $\tilde{x}_{0}$ smaller than $\sim 10^{-7}$, see Figure \[fig:LLikBlue\]b. Without knowing the Truth, of course, this approach to identifying the blue points is inaccessible. The likelihood function is extremely jagged; as Berliner [@Berliner1991] stressed such chaotic likelihoods suggests that finding even one high likelihood state by sampling the state space would be prohibitively costly, making the approach inapplicable. That said there is no sense in which “sensitivity to the initial conditions” can be taken to imply that the information in the initial condition is “forgotten” or “lost". There is sufficient information in the observation segment to identify high likelihood initial states. Candidate states with non vanishing RLLik can be found by extracting the information from the system dynamics using a relatively new approach to data assimilation, which will be interpreted as an Importance Sampling. Importance Sampling via Pseudo-orbit Data Assimilation ======================================================= Methodology ----------- To locate high likelihood states, simply sampling in state space is inefficient. As the dimension of the system increases, this inefficiency makes the task computationally impractical. Importance sampling[^7] (IS) locates high likelihood states in the trajectory manifold by adopting the Pseudo-orbit Data Assimilation approach [@Judd2001; @DuPDA]. PDA takes advantage of the known dynamics in a higher dimensional sequence space. A brief introduction of the PDA approach is given in the following paragraph (see [@Judd2001; @DuPDA] for additional details). Given a perfect dynamical model of dimension $m$, a perfect knowledge of the observational noise model, and a sequence of $n$ observations $s_{i}, i=0,...,n-1$, define a sequence space as the $m\times n$ dimensional space consisting of all sequences of $n$ states $u_{i}$.[^8] Most points in sequence space do not correspond to a trajectory of the system. Define a [*pseudo-orbit*]{}, $\textbf{U}\equiv \{u_{0},...,u_{n-2},u_{n-1}\}$, to be a point in the $m\times n$ dimensional sequence space for which $u_{i+1}\neq f(u_{i})$ for one or more components of $\textbf{U}$. Thus a pseudo-orbit corresponds to a sequence of system states which is not a trajectory of the system. Define the [*mismatch*]{} to be: $$\begin{aligned} \label{eq:mismatch} e_{i}=\mid f(u_{i})-u_{i+1} \mid\end{aligned}$$ By construction, system trajectories have a mismatch of zero. The mismatch cost function is then given by: $$\begin{aligned} \label{eq:miscost} C(\textbf{U})=\sum_{i=0}^{n-1} e_{i}^{2}\end{aligned}$$ [*Pseudo-orbit Data Assimilation*]{} minimizes the mismatch cost function for $\textbf{U}$ in the $m\times n$ dimensional sequence space. If a gradient descent (GD) approach is adopted[^9], then a minimum of the mismatch cost function can be obtained by solving the ordinary differential equation $$\begin{aligned} \label{eq:odegd} \frac{d\textbf{U}}{d\tau}=-\nabla C(\textbf{U}),\end{aligned}$$ where $\tau$ denotes algorithmic time.[^10] A sequence of observations in the system state space define an initial pseudo-orbit, so called [*observation-based pseudo-orbit*]{}, $\textbf{S}\equiv \{s_{0},...,s_{n-2},s_{n-1}\}$, which with probability one will not be a trajectory. In practice, the minimization is initialized with the observation-based pseudo-orbit, i.e. $^{0}\textbf{U}=\textbf{S}$ where the pre-super-script $^0$ on $\textbf{U}$ denotes the initial stage of the GD. The pseudo-orbit is a point in sequence space, under Equation \[eq:odegd\] this point moves towards the manifold of all trajectories. The mismatch cost function has no local minima other than on the manifold, for which $C(\textbf{U})=0$, (i.e. the trajectory manifold[^11]) and every segment of trajectory lies on this manifold [@Judd2001]. Let the result of the GD minimization at time $\alpha$ be $^{\alpha}\textbf{U}$. Here $\alpha$ indicates algorithmic time in GD (i.e. the number of iterations of the GD minimization). As $\alpha \rightarrow \infty$, the pseudo-orbit $^{\alpha}\textbf{U}\equiv ^{\alpha}\textbf{u}_{0},...,^{\alpha}\textbf{u}_{n-1}$ approaches a trajectory of the model asymptotically. In other words, the GD minimization takes us from the observation-based pseudo-orbit towards a system trajectory (a point in sequence space, $^{\infty}\textbf{U}$, which is on the trajectory manifold). In practice, the GD minimization is run for a finite time and thus a trajectory is not obtained. The result of these large $\alpha$ GD runs, $^{\alpha}u_{0}$ provide candidates for the initial state, based on information from the observations with $i<n$. For $i>0$, the $i$-step preimage of the relevant component of $^{\alpha}u_{i}$ are calculated to obtain additional candidates for the initial state. The Logistic Map is a two-to-one map, and in most cases[^12] only one of the two preimages for each $^{\alpha}u_{i}$ is relevant to $\tilde{x}_{i-1}$. In practice a criteria to discard irrelevant preimages must be defined, a simple example would be to discard (with high probability) those preimages whose distance from the corresponding previous observation exceeds some threshold based on the standard deviation of the observational noise (a $3\sigma$ criteria is used to generate the results presented in the following section). Results ------- The green points in Figure \[fig:LLikGreen\] are located using the IS approach; Note that some have RLLik close to $0$. As expected, the observations do not contain sufficient information to identify the state of the system at the time of the final observation with the same degree of precision. This is reflected in the fact that the green points are much less close to the true state at time $31$ (Figure \[fig:LLikGreen\]b) than those at time $0$ (Figure \[fig:LLikGreen\]a). Two experiments were conducted to test the robustness of the IS approach. The first is based on $2048$ different realizations of observations for $\tilde{x}_{0}=\sqrt{2}/2$ to examine consistency. The second is based on $2048$ different true initial conditions to examine robustness. Three different observation window lengths were used in each experiment. Table \[tab:LLikO\] and Table \[tab:LLikS\] shows the results. Given uncertain observations, one can never identify the Truth of a chaotic system unambiguously as was noted by Lalley [@Lalley99; @Lalley00] and later explored by Judd and Smith [@Judd2001]. Using the IS approach, high likelihood states (IS states) are indeed found, as the states whose RLLik$>-1$ are found in every single experimental run. The fact that some IS states have greater likelihood than the Truth supports the expectation that the Truth is not expected to be the most likely system state given the observations. For each experimental run, the minimum distance between those IS states (whose RLLik$>-1$) and the Truth is recorded. The minimum, maximum and median statistical values of the minimum distance from the Truth are reported in Table \[tab:LLikO\] and \[tab:LLikS\]. It is clear that the quality of the IS states improves (the minimum distance from the Truth decreases) as the observation window length increases. This is expected inasmuch as more information from the system dynamics becomes available when using a longer window. It is shown in Table \[tab:LLikO\] that the maximum value of the minimum distance among the $2048$ different realizations is $1.49\times 10^{-10}$ for a window length of 32 and in Table \[tab:LLikS\] the maximum value of the minimum distance among different true initial conditions is $2.02\times 10^{-10}$. PDA importance sampling appears both robust and efficient[^13]. -------- ------- ------- ---------- ------- ------- ---------- ----------------------- ----------------------- ----------------------- Window length `Min` `Max` `Median` `Min` `Max` `Median` `Min` `Max` `Median` 32 6 15 8 0 14 6 $2.00\times 10^{-15}$ $1.49\times 10^{-10}$ $9.98\times 10^{-12}$ 16 2 11 8 0 11 6 $1.57\times 10^{-10}$ $7.63\times 10^{-6}$ $5.13\times 10^{-7}$ 8 2 7 7 0 7 5 $8.58\times 10^{-8}$ $4.55\times 10^{-2}$ $2.56\times 10^{-4}$ -------- ------- ------- ---------- ------- ------- ---------- ----------------------- ----------------------- ----------------------- : Statistics of high likelihood states located by IS based on 2048 different realizations of observations (of $\tilde{x}_{0}=\sqrt{2}/2$) for the Logistic Map, i) statistics of the number of states (whose $RLLik>-1$) ii) statistics of the number of states (whose $RLLik>0$) iii) statistics of the minimum distance between the states (whose $RLLik>-1$) and the Truth.[]{data-label="tab:LLikO"} -------- ------- ------- ---------- ------- ------- ---------- ----------------------- ----------------------- ----------------------- Window length `Min` `Max` `Median` `Min` `Max` `Median` `Min` `Max` `Median` 32 6 360 28 0 338 16 $4.77\times 10^{-15}$ $2.02\times 10^{-10}$ $1.50\times 10^{-11}$ 16 3 56 14 0 48 9 $3.04\times 10^{-10}$ $1.54\times 10^{-5}$ $9.93\times 10^{-7}$ 8 2 20 7 0 20 7 $6.36\times 10^{-8}$ $5.60\times 10^{-3}$ $3.32\times 10^{-4}$ -------- ------- ------- ---------- ------- ------- ---------- ----------------------- ----------------------- ----------------------- : Statistics of high likelihood states located by IS based on 2048 different true initial states for the Logistic Map, i) statistics of the number of states (whose $RLLik>-1$) ii) statistics of the number of states (whose $RLLik>0$) iii) statistics of the minimum distance between the states (whose $RLLik>-1$) and the Truth.[]{data-label="tab:LLikS"} Complications arise from the fact that the Logistic Map is two-to-one; these have nothing to do with chaos per se (beyond the fact that one-to-one maps in one-dimension cannot display chaotic dynamics). Moving to higher dimensional[^14] one-to-one maps, the calculation of preimages becomes straightforward. The experiments above demonstrate that truly high likelihood points can be located using dynamical information. This eases Berliner’s identification problem of initial condition with the appearance of chaotic likelihoods. Selecting an ensemble from this high likelihood set allows for informative forecasts which do not become useless until long after those from the point forecasts illustrated by Berliner [@Berliner1991] become uninformative. Relative likelihoods ==================== Maximum Likelihood Estimation has been widely used for estimation [@Pawitan01] since it was introduced by Fisher [@Fisher1922] in 1922. The “best” estimate is often chosen from a set of samples and only the relative likelihood in that sample is considered. Figure \[fig:LLikRef\]a shows the log-likelihood of 1024 states (the same set used in Figure \[fig:LLikZoom\]b), the grey dashed line is the median log-likelihood of those states. In this case, it is not the problem of which estimate one shall pick, but how to show that they are all “bad” estimates. In practice, the Truth is unknown therefore it cannot be used as a reference like the cross plotted in Figure \[fig:LLikZoom\]. Given the observations and the noise model, however, the expected log-likelihood of the Truth[^15] can be derived and serve as a reference. Figure \[fig:LLikRef\]b, the log-likelihood of 1024 states are plotted along with the expected log-likelihood of the Truth (black dashed line). Figure \[fig:LLikRef\]b shows that it is not a case of equifinality in Figure \[fig:LLikRef\]a but a case of equidismality. In cases where it is observed that all traditional candidate states have vanishingly small log-likelihood relative to the expected log-likelihood of the Truth, approaches like those suggested above might prove valuable. Difference between initial conditions and parameters ==================================================== Statisticians often treat estimating initial conditions and estimating parameter values as similar problems. Although similar behaviors of likelihood functions of initial conditions and that of parameter values are observed [@Berliner1991], there are fundamental differences in the information available to address these two distinct estimation problems. Given the structure of the model class, the model parameter value determines the dynamical behaviour of the model (e.g. the natural measure) which is not changed by the initial condition. Given the model and its parameter value(s), the invariant measure constrains the relevant set(s) of initial conditions in the state space (and thereby trajectories in the sequence space). It is unclear how to construct similar constraints (if they exist[^16]) on the parameter values in the parameter space given the “true” initial state. Uncertainty in initial state differs from uncertainty in the parameter value. The information in a measurement of the initial condition uncertainty will decay with time and eventually becomes statistically indistinguishable from a random sample of the natural measure, while the information on each member from an ensemble under parameter uncertainty is preserved (and can be straight forwardly extracted given a trajectory segment), arguably forever. While assuming the parameter value is perfect may not be ideal, it is not so nonsensical given that one has already assumed that the model structure is perfect. Assuming the initial state is perfect indicates a noise free observation is possible. Let the model’s parameters be contained in the vector $\textbf{a}\in \mathbb{R}^{l}$. A set of $l+1$ sequential noise free observations $s_{i}, s_{i+1},...,s_{i+l}$ would, in general, be sufficient to determine $\textbf{a}$ [@McSharry1999]. If one noise free observation is obtainable, obtaining only a few more noise free observations would define the true parameter value precisely. A more realistic way to put the problem is to estimate the parameter value(s) given the observations without assuming the “true” initial condition is known or even exists. In that case, the goal is to locate high likelihood trajectories (Smith et al. [@Smith2010] call these shadowing trajectories) defined by the parameter values. Unfortunately it is not clear how to solve such a problem. In fact, it is not clear how to constrain the solution in the parameter space in a manner that reflects the constraints in the space of initial condition achieved by using trajectory manifold in the state space. Given a perfect model structure and knowing the true parameter value(s), the true initial state is a well defined goal of the identification. Inasmuch as structural model errors imply no true parameter value exists [@Judd2004; @DuPDA2], it is unclear how one might define “true” initial state and the goal of estimation must be rethought. Despite the importance of model parameters, there is no general method of parameter estimation outside linear systems. Methods have been developed to obtain useful parameter values with some success: McSharry and Smith [@McSharry1999; @Sornette] estimate model parameters by incorporating the global behaviour of the model into the selection criteria; Creveling et al. [@Creveling2008] have exploited synchronization for parameter estimation; Smith et al. [@Smith2010] focused on the geometric properties of trajectories; Du and Smith [@Du2011] select parameter values based on the Ignorance Score of ensemble forecasts. Each of these methods, however, require a large set of observations. Challenges remain when only a short sequence of observations is available. Conclusion ========== Berliner illustrated that even in the perfect model scenario traditional likelihood methods are unable to provide good estimates of the initial condition for nonlinear chaotic systems. In large part, the failure is due to the inability of those approaches to skillfully meld the information in the dynamics of the nonlinear system itself with that in the observations. The importance sampling approach presented here better combines information from both observations and dynamics, thereby locating high likelihood initial states; this achieves an aim Berliner (1991) argued to be impossible by traditional methods. Despite the similarity of state estimation and parameter estimation, there are fundamental differences between uncertainty in the initial state and uncertainty in parameter value. Significant challenges remain in solving the challenge chaotic likelihood functions pose in parameter estimation. Acknowledgment {#acknowledgment .unnumbered} ============== This research was supported by the LSE’s Grantham Research Institute on Climate Change and the Environment and the ESRC Centre for Climate Change Economics and Policy, funded by the Economic and Social Research Council and Munich Re. Additional support for H.D. was also provided by the National Science Foundation Award No. 0951576 “DMUU: Center for Robust Decision Making on Climate and Energy Policy (RDCEP)". L.A. S. gratefully acknowledges the continuing support of Pembroke College, Oxford. [990]{} C. Beck. Effects of roundoff errors on chaotic dynamics. In J. Vandewalle, R. Boite, M. Moonen, and A. Oosterlinck, editors, Signal Processing VI - Theories and Applications Proceedings of EUSIPCO-92, volume 1, page 183186, 1992. M.L. Berliner. Likelihood and bayesian prediction for chaotic systems. J. Am. Stat. Assoc., 86:938–952, 1991. D.R. Creveling, P.E. Gill, and H.D.I. Abarbanel. State and parameter 258 estimation in nonlinear systems as an optimal tracking problem. Physics Letters A, 372:2640–2644, 2008. H. Du and L.A. Smith. Parameter estimation through ignorance. Phys. Rev. E, 016213, 2012. H. Du and L.A. Smith. Pseudo-orbit data assimilation. II: Assimilation with imperfect models. Journal of the Atmospheric Sciences, 71:483495, 2014. H. Du and L.A. Smith. Pseudo-orbit data assimilation part I: The perfect model scenario. Journal of the Atmospheric Sciences, 71:469–482, 2014. R.A. Fisher. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, 222:309–368, 1922. K. Judd and L.A. Smith. Indistinguishable states I: The perfect model scenario. Physica D, 151:125–141, 2001. K. Judd and L.A. Smith. Indistinguishable states II: the imperfect model scenario. Physica D, 196:224–242, 2004. Kevin Judd, Leonard Smith, and Antje Weisheimer. Gradient free descent: shadowing, and state estimation using limited derivative information. Physica D: Nonlinear Phenomena, 190(3–4):153–166, April 2004. S.P. Lalley. Beneath the noise, chaos. The Annals of Statistics, 27(2):461–479, 1999. S.P. Lalley. Removing the noise from chaos plus noise. In A.I. Mees, editor, Nonlinear Dynamics and Statistics, Boston, 2000. P. McSharry and L.A. Smith. Better nonlinear models from noisy data: Attractors with maximum likelihood. Phys. Rev. Lett, 83(21):4285–4288, 1999. R.N. Miller, M. Ghil, and F. Gauthiez. Advanced data assimilation in strongly nonlinear dynamical systems. J. Atmos. Sci., 51:1037–1056, 1994. Y. Pawitan. In all likelihood : statistical modelling and inference 283 using likelihood. Oxford science publications. Clarendon press, Oxford, 2001. V.F. Pisarenko and D. Sornette. Statistical methods of parameter estimation for deterministically chaotic time series. Phys. Rev. E, 69:036122, 2004. L.A. Smith. Lacunarity and Chaos in Nature. PhD thesis, Columbia University, New York, 1987. L.A. Smith. What might we learn from climate forecasts? In Proc. National Acad. Sci., volume 4, pages 2487-2492, USA, 2002. L.A. Smith, M.C. Cuellar, H. Du, and K. Judd. Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models. Physics Letters A, 374:2618–2623, 2010. J.C. Sprott. Chaos and Time-Series Analysis. Oxford University Press, Oxford, 2003. [^1]: By assuming a perfect model and generating the data on a digital computer, one avoid the issue of “round-off error”: the fitted model is evaluated on the same computer. For discussion, see [@Smith87; @Beck92; @Smith02] and references thereof. [^2]: For $m = 1$, the state $x_i$ is a scalar. [^3]: Here LLik based on a sequence of 32 observations is computed. Berliner examined 15 & 10 observations. The problem becomes more obvious when more observations are used. Shorter sequences of observations are examined below. [^4]: Note: given that only finite observations are considered the true state of the system is, with probability 1, not the maximum likelihood state [^5]: Equifinality occurs when many potential solutions to a task are equally good, making it impossible to identify the true solution given the information in hand. In this case the sampled likelihood function is relatively flat. Equidismality arises when the sampled relative likelihood function is flat yet all solutions tested have vanishingly small likelihood given the information. Examining the relative likelihood obscures the difference; fortunately the expected (distribution of) likelihood can be computed from the noise model alone without knowledge of the true initial condition. [^6]: In this experiments 1024 points are generated by $\tilde{x}_{0}+2^{-(10+\frac{60i}{1024})}\epsilon_{i}, i=1,2,...,1024$ where $\epsilon_{i}$ is random drawn from $U(0,1)$. [^7]: In high-dimensional space the sampler targets the relevant low-dimensional trajectory manifold which is more efficient than sampling a hypersphere. Even in the one dimensional Logistic Map this approach succeeds by using PDA to sample the trajectory manifold in the n-dimensional sequence space. [^8]: For the Logistic Map where $m=1$, both $s_i$ and $u_i$ are scalars. [^9]: Other methods for this minimization are available, GD is discussed here due to its simplicity. [^10]: The approach can be generalized to situations where gradient of the model is not known analytically [@2004JuddB]; improving the ability to work without gradient information would widen the applications of the approach significantly. [^11]: All points on the trajectory manifold have zero mismatch (are trajectories) and only points on the trajectory manifold have zero mismatch. [^12]: Not in all cases, however. For discussion of the point see [@Lalley99]. [^13]: Drawing samples uniformly from within a distance of $0.1$ of Truth would require $\sim 10^8$ candidates in order to find a candidate within $\sim 2\times 10^{-10}$ of Truth. Such a procedure need not identify any high likelihood states. The results of Table \[tab:LLikO\] and \[tab:LLikS\] were obtained with only 1024 GD minimization iterations in each realization (each and every one of which identified high likelihood states close to Truth). [^14]: where the model state becomes a state vector [^15]: The log-likelihood of the Truth is $\frac{\sum_{i=0}^{n-1}\delta_{i}^{2}}{2\sigma^{2}}$ (from Eq. \[eq:LLik\]) where $\delta_{i}$ (observation noise) is $IID\sim N(0,\sigma^{2})$ distributed. Let $Z=\frac{\sum_{i=0}^{n-1}\delta_{i}^{2}}{\sigma^{2}}$, $Z$ is a random variable following chi-squared distribution with $n$ degrees of freedom. Statistics of the log-likelihood of the Truth can therefore simply derived from $Z$. [^16]: It is not clear either how to construct the set of parameter values whose corresponding invariant measure contains the “true” initial state, or how to exploit this set, while it is clear how to exploit the existence of trajectory manifold given a particular value of the parameter.

--- abstract: 'From an experimental point of view, clear signatures of multifragmentation have been detected by different experiments. On the other hand, from a theoretical point of view, many different models, built on the basis of totally different and often even contrasting assumptions, have been provided to explain them. In this contribution we show the capabilities and the shortcomings of one of this models, a QMD code developed by us and coupled to the nuclear de-excitation module taken from the multipurpose transport and interaction code FLUKA, in reproducing the multifragmentation observations recently reported by the INDRA collaboration for the reaction Nb + Mg at a 30 MeV/A projectile bombarding energy. As far as fragment production is concerned, we also briefly discuss the isoscaling technique by considering reactions characterized by a different isospin asymmetry, and we explain how the QMD + FLUKA model can be applied to obtain information on the slope of isotopic yield ratios, which is crucially related to the symmetry energy of asymmetric nuclear matter.' author: - 'A. Ferrari$^1$, M.V. Garzelli$^{2,3 }$, P.R. Sala$^2$' title: 'A model for multifragmentation in heavy-ion reactions' --- Features of multifragmentation {#section1} ============================== When studying heavy-ion collisions at non-relativistic energies, multifragmentation can be observed for the most central ones, in a range of projectile-ion bombarding energies from tens MeV/A up to a few hundreds MeV/A, depending of the properties of the nuclei under consideration. Many issues of this phenomenon are still under discussion, in particular concerning the stage at which it occurs in the evolution of a reaction, e.g. if a nuclear system undergoing multifragmentation is or not equilibrated, how a simultaneous break-up in multiple fragments can occur, and if the multifragmentation is the result of a phase-transition. According to the currently most believable scenario, during the overlapping stage of heavy-ion collisions (typical time $\simeq$ 100 fm/c) matter can undergo compression, leading to large excitation energies. As a consequence, the blob of nuclear matter starts to expand and can go on expanding down to sub-saturation densities ($\rho$ $\simeq$ 0.1 - 0.3 $\rho_0$, where $\rho_0$ is the normal nuclear matter density) and reach temperatures $\simeq$ 3 - 8 MeV, where it becomes unstable and breaks up into multiple fragments. These conditions are typical of a liquid-gas coexistence region [@bondorf; @buyuk]. As already mentioned, one of the open issues is if equilibration is reached in these reactions. A statistical description of multifragmentation is based on this assumption. A dynamical description of multifragmentation instead is not based on this assumption. Difficulties in coming to a non-controversial conclusion are largely due to the fact that most of the experimental data refer to patterns of particles detected just in the last stage of the reactions, involving channels feeded by the sequential decays which heavily affect and modify the primary fragment distribution. Multifragmentation can be distinguished from other decay channels on the basis of the excitation energy: a typical scenario of small excitation energies (E $<$ 2 - 3 MeV/A) is characterized by the formation of a compound-like system and by its evolution through binary sequential decays (evaporation/fission), whereas for high excitation energies (E $>$ 3 MeV/A) multifragmentation in a finite volume and a simultaneous break-up into multiple fragments can occur. The excitation energy is indeed related to the mass asymmetry $(A_{proj} - A_{target})$: in case of symmetric central reactions the compression is responsible of the high excitation energy, whereas in case of asymmetric reactions only a partial compression can occur and a large part of the excitation energy appears in the form of thermal energy [@singh]. In all cases, multifragmentation is tipically driven by the following expansion. Due to its mentioned features, multifragmentation, occurring during the phase of expansion of the nuclear system formed by an ion-ion (central) collision, allows to study the nuclear Equation of State (EoS) at subnormal nuclear densities. In particular, it is possible to infer useful information concerning the symmetry energy and its density dependence, by investigating the isotopic yield distributions of the emitted fragments. The isoscaling technique, based on the analysis of isotopic yield ratios obtained in reactions with a different isospin asimmetry, has been developed with this purpose. After an overview of the models to study multifragmentation in Section \[section2\], and a brief presentation of the one used in this work in Subsection \[subsection2.1\], examples of its application in the isoscaling technique and in the reconstruction of multifragmenting sources at energies of a few tens MeV/A are provided in Subsection \[subsection3.1\] and \[subsection3.2\], respectively. Finally, our perspectives on further applications of our model are drawn in Section \[section4\]. Models to study multifragmentation {#section2} ================================== One can distinguish between - Dynamical Models: some of them are 1-body approaches, inspired to the BUU/BNV/Landau-Vlasov transport theory. Alternatively, n-body approaches have been developed, such as the QMD/AMD/FMD. n-body approaches are very powerful in the description of the simultaneous break-up of a nuclear system in multiple fragments, since they preserve correlations among nucleons. - Statistical Models: they assume to work with an equilibrated excited source at freeze-out (thermal equilibrium). Taking into account that the nuclear system undergoes an expansion, leading to decreasing densities, down to subnormal values, the freeze-out [@trautmann] occurs when the mutual nuclear interaction among fragments can be neglected. Statistical models have been worked out both in the grand-canonical framework (see e.g. Ref. [@chauduri]) and in the micro-canonical framework. The most widespread among the last ones is the SMM [@botvina; @bondorf; @gupta] and its modifications ISMM [@tan] and SMM-TF [@souza]. We emphasize that the onset of multifragmentation according to dynamical models is different from the description of multifragmentation according to statistical models. In fact, in the statistical models a source in thermal equilibrium is assumed to fragment. This means that memory effects concerning how the source has been originated are neglected. On the other hand, in the dynamical models multifragmentation is a fast process: the involved nucleons have not the time to come to equilibrium. Fragments originate from the density fluctuations (nucleon-nucleon correlations) due to collisions in the ion-ion overlapping stage, which survive the expansion phase (memory effects). The chemical composition of hot fragments is expected to play a role in helping to disentangle the nature (dynamical / statistical) of the multifragmentation mechanism [@milazzo]. Different models reproduce different features of the collisions with different success. A mixed model, inspired to the QMD dynamical approach to describe the fast stage of ion-ion collisions and to a statistical approach to describe the further decay of the multiple primary excited fragments produced by QMD down to their ground state, has been used to obtain the results presented in this work. Due to the crucial role of dynamics, as supported by our results, in the following we mainly concentrate on the description of the dynamical aspects of multifragmentation. QMD/AMD approaches {#subsection2.1} ------------------ In these microscopic models a nucleus is considered a set of mutually interacting nucleons. The propagation of each nucleon occurs according to a classical Hamiltionian with quantum effects [@aiche]. In particular, nucleons are described by gaussian wave packets. Each of them moves under the effects of a potential given by the sum of the contribution of all other nucleons (2-body effects). Furthermore, when two nucleons come very close to each other, they can undergo elastic collisions (nucleon-nucleon stochastic scattering cross-sections) with Pauli blocking. A proper treatment of antisymmetrization is implemented in AMD [@ono]. On the other hand, QMDs do not provide any antisymmetrization of the nuclear wave-function. An approximate effect can be obtained through the inclusion in the Hamiltonian of a Pauli potential term, or through the implementation of specific constraints. Still open questions in molecular dynamics approaches concern the functional form of the nucleon-nucleon potential (each working group who developed a molecular dynamics code has its preferred choice of terms), the potential parameters and their relation to the nuclear matter EoS. Nowaday, many groups prefer parameter sets leading to a soft EoS. Anyway, there are open questions concerning the symmetry term [@li; @baran]. In particular, a stiff dependence for this term means that the symmetry energy always increases with increasing densities. On the other hand, a soft dependence means that the symmetry energy decreases at high densities. At present, a stiff dependence seems more reliable than a soft one. Many uncertainties come from the fact that our observations are mainly based on symmetric nuclear matter (N / Z $\simeq$ 1) near normal nuclear density ($\simeq$ 0.16 $\mathrm{fm}^{-3}$), since it is difficult to obtain highly asymmetric nuclear matter in terrestrial laboratories. On the other hand these studies are crucial to understand features of astrophysical objects (such as neutron star formation and structure), where conditions of extreme neutron-proton asymmetry can be present. Other open issues concern the gaussian width, the use of in-medium nucleon-nucleon cross-sections instead of free nucleon-nucleon cross-sections (in QMD the free choice is usually implemented, whereas in the AMD the in-medium choice has been implemented), the question of how long the dynamical simulation has to be carried over and the problem of the development of a fully relativistic approach (on the last point see e.g. Ref. [@mancusi]). A QMD code has been developed by us [@mvg] in fortran 90. It includes a 3-body repulsive potential and a surface term (attractive at long distances and repulsive at short distances). Pauli blocking is implemented by means of the CoMD constraint [@papa]. Neutron and proton are fully distinguished by means of a simmetry term and an isospin dependent nucleon-nucleon stochastic scattering cross-section. The kinematics is relativistic and attention is paid to the conservation of key quantities (total energy/momentum, etc.) in each ion-ion collision. Simulations are performed by means of our code from the ion-ion overlapping stage up to t $\simeq$ 200 - 300 fm/c (fast stage of the reaction). The description of the de-excitation of the excited fragments present at the end of the fast stage is obtained through the coupling of our QMD with the statistical model taken from the PEANUT module available in the FLUKA Monte Carlo code [@fluka0; @fluka1; @fluka2; @fluka3] in a version for the g95 compiler. Up to now, the QMD + FLUKA interface has been tested in the collisions of ions with charge up to Z=86 (radon isotopes), providing interesting results (see e.g. Ref. [@nd2007] and references therein). Results ======= Isospin dependence in fragment production: application of the isoscaling technique {#subsection3.1} ---------------------------------------------------------------------------------- The isoscaling technique, already mentioned in Section \[section1\], is based on ratio of yields taken in multifragmentation reactions with similar total size, but different isospin asymmetries ($N - Z$) / ($N + Z$) [@tsang; @ono2]: $$\begin{aligned} R_{21}(N,Z) = Y_2(N,Z) / Y_1(N,Z) = Const \,\, \exp( A_{coeff} N + B_{coeff} Z ) \, . \label{eq1}\end{aligned}$$ The numerator of this formula refers to the yield of a given fragment ($N,Z$) obtained from a neutron rich nucleus-nucleus reaction system, whereas the denominator refers to the yield of the same fragment from a neutron poor (more symmetric) reaction at the same energy. $A_{coeff}$ is related to the symmetry energy and is increasingly larger for couple of reactions with increasingly different isospin composition $N/Z$. In particular, we have considered the neutron rich systems Ar + Fe ($N/Z$ = 1.18) and Ar + Ni ($N/Z$ = 1.13) with respect to the neutron poor system Ca + Ni ($N/Z$ = 1.04). Among other authors, these systems have been previously studied by [@shetty] (see also Ref. [@shetty2; @wuenschel]). Isotopic yield ratios for light fragment (Z $\le$ 8) emission have been obtained from our QMD + FLUKA simulations for the couple of reactions Ar + Ni / Ca + Ni and Ar + Fe / Ca + Ni at 45 MeV/A projectile bombarding energy. When plotted in the logarithmic plane, isotopic yield ratios for each fixed $Z$ turn out to be approximately linear, with a slope given by a $A_{coeff}$, as expected from Eq. (\[eq1\]). As for the isoscaling parameter $A_{coeff}$, our simulations give the following insights: - The results of our analysis are quite sensitive to the number of isotopes included in the linear fit, at fixed $Z$ (i.e. to the goodness of the gaussian approximation to the fragment isotopic distribution). - $A_{coeff}$ differs with $Z$, in agreement with [@tsang], which claims that isoscaling is observed for a variety of reaction mechanisms, from multifragmentation to evaporation to deep inelastic scattering, with different slopes in the logarithmic plane. - $A_{coeff}$ is larger for the couple of reactions with larger difference in the isospin compositions ($N_1/Z_1$ - $N_2/Z_2$). - Our average values $A_{coeff}$ = 0.18 for Ar + Ni / Ca + Ni and $A_{coeff}$ = 0.31 for Ar + Fe / Ca + Ni are larger than the experimental values [@shetty], but the comparison is not so meaningful, since it is largely affected by the fact that we include fragments emitted in all directions in our preliminary analysis, whereas in the experiment only fragments emitted at 44$^o$ were selected. - $A_{coeff}$ turns out to be affected by the choice of the impact parameter and decreases significantly when selecting only the most central events. - $A_{coeff, hot}$ at the end of the overlapping stage can be larger than $A_{coeff}$ at the end of the full simulation by no more than 20%, at least for the reaction systems under study. As far as the emissions at preequilibrium are concerned, our simulations lead to the following results: - For central collisions of Ca + Ni the yield of emitted protons turns out to be larger than the yield of emitted neutrons by 20%. For central collisions of Ar + Ni and Ar + Fe, on the other hand, the yield of emitted protons turns out to be lower than the yield of emitted neutrons by 10 - 15%. - For each of the three systems under study, the fragment asymmetry of the liquid phase $(Z/A)_{liq}$ at the end of the preequilibrium stage turns out to be lower than the corresponding value at $t=0$, in qualitative agreement with the AMD simulations [@shetty]. - No traces of isospin fractionation appear, expected indeed for systems with an higher $N/Z$ content (e.g. $^{60}$Ca + $^{60}$Ca). The dependence of our results on the projectile bombarding energy is currently under study, by considering the same reactions at different bombarding energies. Multifragmenting source reconstruction in Nb + Mg reactions at 30 MeV/A {#subsection3.2} ----------------------------------------------------------------------- Multifragmentation has been observed in Nb + Mg reactions at a 30 MeV/A projectile bombarding energy in an experiment performed at the INDRA detector by the INDRA + CHIMERA collaborations [@manduci]. Event selection has been performed, according to experimental cuts on the momentum along the beam axis, $p_{z,det} > 0.6 \, p_{z,tot}$, and on the angular acceptance of the INDRA detector, $4^o$ &lt; $\theta$ &lt; $176^o$. The selected events have then been assigned to different regions, corresponding to portions of the plane identified by the total transverse energy and the total multiplicity of charged particles detected in each event. Three regions have been singled out this way, as shown in Fig. 2 of Ref. [@manduci]. We have applied the same selection procedure by implementing proper cuts and filters on the simulated events obtained by our QMD + FLUKA. The selected theoretical events are plotted in Fig. \[nostrafigura1\], which can be directly compared with Fig. 2 of Ref. [@manduci] and turns out to be in good agreement. The events plotted in the T1 region (red) are the less dissipative ones (more peripheral collisions), whereas the events in the T3 region (blue) correspond to more dissipative (central) collisions. ![Multifragmentation of Nb + Mg at 30 MeV/A: event selection and identification of different regions T1 (red), T2 (green) and T3 (blue) by our QMD + FLUKA simulations. Each point corresponds to a different ion-ion reaction event in the plane identified by the total multiplicity of detected charged particles and the detected total transverse energy.[]{data-label="nostrafigura1"}](cut2.eps){width="8cm"} For each of the three regions, average values of interesting quantities have been obtained both in the experiment and in the theoretical simulations. Our results, concerning the transverse energy, the multiplicity of charged particles, the velocity and the charge of the biggest residual averaged over all events belonging respectively to the T1, T2 and T3 regions are shown in Table \[tabellagenerale\]. As far as the average transverse energy and multiplicity of charged particle are concerned, the results of our simulations turn out to be in good agreement with the experimental data, within the experimental uncertainties, in the region T1 and T2, corresponding respectively to peripheral and semiperipheral collisions, whereas in case of central collisions the theoretical average transverse energy underestimates the experimental one and the theoretical multiplicity of charged particles slightly overestimates the experimental result. On the other hand, as for the properties of the largest residual, the results of the theoretical simulations show good agreement with experimental data especially for the most central collisions, belonging to the T3 region, whereas for the more peripheral ones the theory overestimates the velocity and the charge of the largest residual. These results, considered all together, seem to point out to the fact that in the experiment the interacting nuclei are slightly more stopped than in the simulation. [p[5cm]{}cccc]{} **Region** & **$<E_{trasv}>$** & **$<M_{tot}>$** & **$<v_{maxres}>$** & **$<Z_{maxres}>$**\ \ T1 & 72.9 (10.1) & 8.1 (0.8) & 6.6 (0.2) & 34.1 (2.3)\ T2 & 120.9 (10.6) & 11.2 (0.9) & 6.4 (0.2) & 31.9 (2.7)\ T3 & 176.4 (13.8) & 13.4 (1.0) & 6.3 (0.2) & 30.5 (2.8)\ \ T1 & 74.8 & 8.4 & 7.0 & 38.8\ T2 & 115.8 & 12.0 & 6.6 & 35.7\ T3 & 155.1 & 15.3 & 6.2 & 31.1\ Furthermore, the considered experiment aim at the recostruction of the properties of the so-called source, the blob of matter formed by compression in the ion-ion overlapping stage, which undergoes multifragmentation. Since the experiment detects final cold fragments, i.e. fragments in their ground state after the de-excitation, a procedure has been established to reconstruct the properties of the source by using the observed properties of the final fragments and emitted protons. In particular, the source is isolated by a selection in parallel velocity of different fragments (velocity cuts), by considering different cuts in different regions. The velocity cuts implemented are summarized in Table 2 of Ref. [@manduci]. As far as the proton are concerned, the parallel velocity cut is fixed to 3 cm/ns, whereas for increasingly heavier fragments the velocity cuts are fixed to increasing values. The velocities of the emitted fragments are easily obtained even from our simulation by QMD + FLUKA, so it is possible to apply the same procedure for the reconstruction of the source properties even in case of our simulation. As an example, the velocities of the emitted protons obtained by our simulation for events in each of the three regions are shown in Fig. \[nostrafigura2\], by plotting their perpendicular component $v_{perp}$ vs. their component along the beam axis $v_{par}$. This figure can be compared with Fig.3 of Ref. [@manduci]. The vertical line in each panel corresponds to the $v_{par}$ cut implemented in the reconstruction of the source. ![Multifragmentation of Nb + Mg at 30 MeV/A: $v_{perp}$ vs. $v_{par}$ for protons emitted in the region T1 (left panel), T2 (central panel) and T3 (right panel) , respectively, as obtained by our QMD + FLUKA simulations. Each point in each panel correspond to a different emitted proton. The vertical lines correspond to the velocity cuts implemented for the reconstruction of the multifragmenting sources. []{data-label="nostrafigura2"}](vel2.eps){width="15cm"} Since the experiment is able to detect the charge of the emitted fragments but not their mass, the velocity cuts can be directly used just to obtain the charge of the source $Z_s$. To calculate the mass of the source $A_s$ a further hypothesis is needed. The author of Ref. [@manduci] assume that the source has the same isotopic ratio as the projectile, i.e. $A_s/Z_s = A_{proj}/Z_{proj}$. Source properties in the three regions T1, T2 and T3, as reconstructed both from the experiment and from our simulation, are shown in Table \[tabella123\]. Since the theoretical model allows to simulate even the process of source formation in a straightforward way, the properties of the source can be directly obtained before its de-excitation and break-up into multiple fragments, without using an a-posteriori reconstruction based on velocity cuts. If we identify the source with the biggest fragment present just at the end of the QMD simulation, we obtain a very good agreement with the experimental results of Ref. [@manduci], especially in the region T1 and T2, even if the experimental results are based on the a-posteriori reconstruction, as can be inferred from Table \[tabella123\]. On the other hand, if we use a reconstruction procedure analogous to the one adopted by the authors of Ref. [@manduci], we overestimate the size of the source, especially for the most central collisions. Finally, as for the average multiplicity of IMF fragments (Z $\ge$ 3) subsequently emitted from the source, we obtain good agreement with the experiment in all regions. [p[5cm]{}ccccc]{}\ **Region** & **$<Z_s>$** & **$<A_s>$** & **$<M_p>$** & **$<M_\alpha>$** & **$<M_{frag}>$**\ T1 & 40.7 (2.0) & 91.2 (4.7) & 2.0 (0.6) & 1.1 (0.5) & 1.2 (0.4)\ T2 & 42.8 (2.1) & 96.0 (4.9) & 2.7 (0.7) & 1.8 (0.6) & 1.4 (0.3)\ T3 & 45.1 (2) & 101.3 (4.6) & 3.1 (0.7) & 2.5 (0.7) & 1.6 (0.3)\ \ **Region** & **$<Z_s>$** & **$<A_s>$** & **$<M_p>$** & **$<M_\alpha>$** & **$<M_{frag}>$**\ \ T1 & 42.6 & 96.2 & 2.2 & 0.4 & 1.0\ T2 & 45.1 & 101.5 & 3.5 & 1.6 & 1.25\ T3 & 48.5 & 109.5 & 4.2 & 3.8 & 1.5\ \ T1 & 41.0 & 91.0 & $\,$& $\,$& $\,$\ T2 & 43.3 & 96.6 & $\,$& $\,$& $\,$\ T3 & 47.5 & 106.1 & $\,$ & $\,$ & $\,$\ Conclusions and perspectives {#section4} ============================ The QMD model developed in Milano and coupled to the de-excitation module of the Monte Carlo FLUKA code has been used to study reactions between ions of intermediate mass which exhibit multifragmentation features. The results presented in this paper are encouraging, and can be further refined by more precisely investigating up to which extent the statistical de-excitation process from FLUKA modifies the pattern of primary fragments originated dynamically by QMD, and how the results of the simulation change when the time of the transition from the dynamical description of the nuclear system to a statistical description is modified. Further studies at non relativistic energies that we are going to perform with our theoretical simulation tool concern: - the isospin distillation effect: it occurs in the multifragmentation of charge-asymmetric systems, and leads to IMF fragments (liquid) more symmetric with respect to the initial matter, and light fragments (gas) more neutron rich. This effect is related to the density dependence of the symmetry energy. - The bimodality in the probability distribution of the largest fragment as a function of the mass number $A_{max}$ of the largest fragment, as a signature of a phase transition. Experimental data on this effect have been obtained by the CHIMERA collaboration (see e.g. Ref. [@pichon]). - (Complete) fusion cross-sections (this kind of analysis has already been performed by other groups, e.g. by means of the ImQMD model [@zhao]). Acknowledgements {#acknowledgements .unnumbered} ================ We wish to thank L. Manduci for enlightening comments on the data on the reaction Nb + Mg at 30 MeV/A collected at the INDRA detector. The QMD code developed by us and used in this study is the fruit of a collaboration involving many people along the years. In particular, we would like to mention F. Ballarini, G. Battistoni, F. Cerutti, A. Fassò, E. Gadioli, A. Ottolenghi, M. Pelliccioni, L.S. Pinsky and J. 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--- abstract: 'Probabilistic transition system specifications using the rule format  provide structural operational semantics for Segala-type systems and guarantee that probabilistic bisimilarity is a congruence. Probabilistic bisimilarity is for many applications too sensitive to the exact probabilities of transitions. Approximate bisimulation provides a robust semantics that is stable with respect to implementation and measurement errors of probabilistic behavior. We provide a general method to quantify how much a process combinator expands the approximate bisimulation distance. As a direct application we derive an appropriate rule format that guarantees compositionality with respect to approximate bisimilarity. Moreover, we describe how specification formats for non-standard compositionality requirements may be derived.' author: - Daniel Gebler - Simone Tini bibliography: - 'concur.bib' title: Compositionality of Approximate Bisimulation for Probabilistic Systems --- Introduction {#sec:introduction} ============ Over the last decade a number of researchers have started to develop a theory of structural operational semantics (SOS) [@Plo04] for probabilistic transition systems (PTSs). Several rule formats for various PTSs were proposed that ensure compositionality (in technical terms congruence) of probabilistic bisimilarity [@Bar04; @LT05; @LT09; @DL12; @LGD12; @BM12]. The rule format [@DL12; @LGD12] subsumes all earlier formats and can be understood as the probabilistic variant of the  format [@Gro93]. Probabilistic bisimilarity is very sensitive to the exact probabilities of transitions. The slightest perturbation of the probabilities can destroy bisimilarity. Two proposals for a more robust semantics of probabilistic processes have been put forward. The *metric bisimulation* approach [@GJS90; @DGJP04; @BW05] is the quantitative analogue of the relational notion of probabilistic bisimulation. It assigns a distance to each pair of processes, which measures the proximity of their quantitative properties. Another approach is the *approximate bisimulation* (also called $\epsilon$-bisimulation) approach [@GJS90; @DLT08; @TDZ11]. Approximate bisimulations are probabilistic bisimulations where the transfer condition is relaxed, namely two processes are related by an $\epsilon$-bisimulation if their probability to reach a set of states related by that $\epsilon$-bisimulation differs by at most $\epsilon$. Processes that are related by an $\epsilon$-bisimulation with $\epsilon$ being small are “almost bisimiliar”. Approximate bisimulations are not transitive in general, as two states with quite different behaviors could be linked by a sequence of states, which pairwise have only little behavioral difference. Approximate bisimulations have been characterized in operational terms [@GJS90], by a modal logic [@DLT08; @TDZ11], and in terms of games [@DLT08]. The metric and approximate bisimulation approach are in general not comparable (see [@Trac10; @Tin10; @TDZ11]). The main difference is that in the approximate bisimulation approach (contrary to the metric bisimulation approach) the differences along paths are neither accumulated nor weighted by the probability of the realization of that path. In this paper we consider approximate bisimulations. In order to allow for compositional specification and reasoning, it is necessary that the considered behavioral semantics is compatible with all operators of the language of interest. For behavioral equivalences (e.g. probabilistic bisimulations) this is the well-known congruence property. For approximate bisimulations the quantitative analogue to the congruence property requires that when different processes are combined by a process combinator (i.e., an operator of the language), then the distance between the resulting combined processes is (reasonably) bounded. A natural notion for this bound is the sum of the distances between the processes to be combined [@DGJP04]. A process combinator respecting this specific bound is called *non-expansive*. Intuitively, this bound expresses that a process combinator does not increase the behavioral distance of the processes to be combined. The congruence property and non-expansivity property of an $n$-ary process combinator $f$ can be expressed by the following proof rules (with $\bisim$ denoting the probabilistic bisimilarity and $d$ denoting the approximate bisimulation distance): $$\begin{gathered} \SOSrule{s_i \bisim t_i \ \text{ for all }\ i=1,\ldots,n} {f(s_1,\ldots,s_n) \bisim f(t_1,\ldots,t_n)} \qquad\qquad \SOSrule{d(s_i,t_i) \le \epsilon_i \ \text{ for all }\ i=1,\ldots,n} {d(f(s_1,\ldots,s_n),f(t_1,\ldots,t_n)) \le \sum_{i=1}^n \epsilon_i}\end{gathered}$$ However, for specific applications, alternative compositionality requirements are required that allow for more or less variance (than the linear sum used in non-expansivity) of the combined processes. For instance, a process combinator that combines a number of distributed systems with a measurement unit may allow for some variance in the combined distributed systems, but must enforce that the measurement unit itself is strict. In this paper we report a substantial first step towards a theory of robust specifications for probabilistic processes. As an operational model for probabilistic processes, we consider Segala-type PTSs that exhibit both probabilistic and nondeterministic behavior. The probabilistic processes are specified by probabilistic transition system specifications (PTSS) with simple  rules. By simple  rules we mean  rules without lookahead. In order to facilitate compositional specification and reasoning, we study how the distance between two terms with the same topmost function symbol depends on the distances of the arguments. In detail, we characterize the *expansivity* of a process combinator, which gives an upper bound on the distance of the combined processes given the distance between their components. Formally, the expansivity of a process combinator $f$ with $n$ arguments is defined as a mapping $\R^{n} \to \R$ taking distances of the arguments $\epsilon_1, \ldots \epsilon_n$ to $\epsilon$, with $\epsilon$ defined as the maximal distance between all $f(s_1,..,s_n)$ and $f(t_1,..,t_n)$ whenever all $s_i$ and $t_i$ are in approximate bisimulation distance $\epsilon_i$. The first contribution of our paper is the characterization of the expansivity of each process combinator. The expansivity of a process combinator is defined as the least fixed point of a monotone function that counts recursively how often the processes are copied. Our second contribution is to deduce, from the expansivity of process combinators, an appropriate rule format that guarantees non-expansivity of all operators specified in this format. The rule format is derived from the simple  rule format by prohibiting that source processes or derivatives are copied. Finally, we demonstrate how the expansivity of process combinators can be used to derive rule formats for alternative compositionality requirements. We consider in this paper approximate bisimulations because the relaxed transfer condition preserves the basic relational nature of probabilistic bisimulations and allows us to apply (adapted and extended) known proof techniques developed for congruence rule formats. Moreover, the new techniques introduced in this paper to quantify the expansivity of process combinators translate naturally to bisimulation metrics. In this sense, we are also opening the door to develop a theory of robust process specifications with respect to bisimulation metrics. This is the first paper that explores systematically the approximate bisimulation distance of probabilistic processes specified by transition system specifications. Tini already proposed a rule format for reactive probabilistic processes [@Tin08; @Tin10]. Our format significantly generalizes and extends that format. First of all, we apply the more general Segala-type systems that admit, besides probabilistic behavior (probabilistic choice), nondeterministic reactive behavior (internal nondeterministic branching). Furthermore, while Tini used a notion of approximate bisimulation, which is an equivalence but not closed under union, we are using the (by now) standard notion [@DLT08; @TDZ11], which is only reflexive and symmetric but closed under union. Finally, the novel rule format based on counting of copies of processes and their derivatives in its defining rules allows us to handle a wider class of process combinators that ensure non-expansivity. Preliminaries {#sec:preliminaries} ============= We assume an infinite set of (state) variables $\TVar$. We let $x, y, z$ range over $\TVar$. A *signature* is a structure $\Sigma = (F, \rank)$, where $F$ is a set of *function names* (operators) disjoint from $\TVar$, and $\rank : F \to \N$ is a *rank function*, which gives the arity of a function name. An operator $f \in F$ is called a *constant* if $\rank(f)=0$. We write $f\in\Sigma$ for $f\in F$. Let $W \subseteq \TVar$ be a set of variables. The set of $\Sigma$-terms (also called state terms) over $W$, denoted by $T(\Sigma, W)$, is the least set satisfying: $W \subseteq T(\Sigma, W)$, and if $f \in \Sigma$ and $t_1, \cdots, t_{\rank(f)} \in T(\Sigma, W)$, then $f (t_1, \cdots, t_{\rank(f)}) \in T(\Sigma, W)$. $T(\Sigma, \emptyset)$ is the set of all *closed terms* and abbreviated as $\closedTerms$. $T(\Sigma, \TVar)$ is the set of *open terms* and abbreviated as $\openTerms$. We may refer to operators as process combinators, and refer to terms as processes. $\Var(t) \subseteq \TVar$ denotes the set of variables in $t$. $\MVar\!: \openTerms \to (\TVar \to \N)$ denotes for $\MVar(t)(x)$ how often the variable $x$ occurs in $t$. A (state variable) *substitution* is a mapping $\sigma_\TVar: \TVar \to \openTerms$. A substitution is closed if it maps each variable to a closed term. A substitution extends to a mapping from terms to terms as usual. Let $\Delta(\closedTerms)$ denote the set of all (discrete) probability distributions on $\closedTerms$. We let $\pi$ range over $\Delta(\closedTerms)$. For $S \subseteq \closedTerms$ we define $\pi(S)=\sum_{t\in S}\pi(t)$. For each $t \in \closedTerms$, let $\delta_t$ denote the *Dirac distribution*, i.e., $\delta_t(t)=1$ and $\delta_t(t')=0$ if $t$ and $t'$ are not syntactically equal. The convex combination $\sum_{i \in I}p_i \pi_i$ of a family $\{\pi_i\}_{i \in I}$ of probability distributions with $p_i \in (0,1]$ and $\sum_{i \in I} p_i = 1$ is defined by $(\sum_{i \in I}p_i \pi_i)(t) = \sum_{i \in I} (p_i \pi_i(t))$. By $f(\pi_1,\dots,\pi_{\rank(f)})$ we denote the distribution that is defined by $f(\pi_1,\dots,\pi_{\rank(f)})(f(t_1,\ldots,t_{\rank(f)})) = \prod_{i=1}^{\rank(f)}\pi_i(t_i)$. We may use the infix notation where appropriate. In order to describe probabilistic behavior, we need expressions that denote probability distributions. We assume an infinite set of distribution variables $\PVar$. We let $\mu$ range over $\PVar$ and $\zeta$ range over $\PVar \cup \TVar$. Let $D \subseteq \PVar$ be a set of distribution variables and $V\subseteq\TVar$ be a set of state variables. The set of *distribution terms* over $D$ and $V$, notation $\DT(\Sigma, D, V)$, is the least set satisfying: \[def:DT:var\_and\_inst\_dirac\] $D \cup \{\delta_t \mid t\in T(\Sigma, V)\} \subseteq \DT(\Sigma, D, V)$, \[def:DT:sum\] ${\textstyle \sum_{i\in I} p_i \theta_i \in \DT(\Sigma, D, V)}$ if $\theta_i\in \DT(\Sigma, D, V)$ and $p_i \in (0,1]$ with $\sum_{i\in I} p_i = 1$, and \[def:DT:prod\] $f(\theta_1,\ldots,\theta_{\rank(f)}) \in \DT(\Sigma, D, V)$ if $f\in\Sigma$ and $\theta_i\in \DT(\Sigma, D, V)$.[^1] A *distribution variable* $\mu\in D$ is a variable that takes values from $\Delta(\closedTerms)$. An *instantiable Dirac distribution* $\delta_t$ with $t\in\openTerms$ is a symbol that takes value $\delta_{t'}$ when variables in $t$ are substituted so that $t$ becomes the closed term $t'\in\closedTerms$. Case \[def:DT:sum\] allows one to construct convex combinations of distributions. For concrete terms we use the infix notation, e.g., $[p_1]\theta_1 \oplus [p_2]\theta_2$ for $\theta = \sum_{i\in \{1,2\}} p_i\theta_i$. Case \[def:DT:prod\] lifts the structural inductive construction of state terms to distribution terms. $\DT(\Sigma, \PVar, \TVar)$ is abbreviated as $\openDTerms$. $\MVar\!: \openDTerms \to (\TVar \cup \PVar \to \N)$ denotes for $\MVar(\theta)(\zeta)$ how often the variable $\zeta$ occurs in $\theta$. For convex combinations $\sum_{i\in I} p_i \theta_i$ the maximal occurrence in some $ \theta_i$ is considered because the probabilistic choice selects (probabilistically) exactly one of the summands. Formally, we have $\MVarMax(\mu)(\mu) = 1$, $\MVarMax(\mu)(\zeta) = 0$ if $\mu\neq\zeta$, $\MVarMax(\delta_t)(x) = \MVar(t)(x)$, $\MVarMax(\delta_t)(\mu) = 0$, $\MVarMax(\sum_{i\in I} p_i \theta_i)(\zeta) = \max_{i\in I} \MVarMax(\theta_i)(\zeta)$, and $\MVarMax(f(\theta_1,\ldots,\theta_{\rank(f)}))(\zeta)$ = $\sum_{i=1}^{\rank(f)} \MVarMax(\theta_i)(\zeta)$. A substitution on state and distribution variables is a mapping $\sigma: (\TVar \cup \PVar) \to (\openTerms \cup \openDTerms)$ such that $\sigma(x) \in \openTerms$ if $x\in \TVar$, and $\sigma(\mu) \in \openDTerms$ if $\mu\in \PVar$. A substitution extends to distribution terms by $\sigma(\delta_t)=\delta_{\sigma(t)}$, $\sigma(\sum_{i\in I} p_i \theta_i) = \sum_{i\in I} p_i \sigma(\theta_i)$ and $\sigma(f(\theta_1,\ldots,\theta_{\rank(f)})) = f(\sigma(\theta_1),\ldots,\sigma(\theta_{\rank(f)}))$. Notice that closed instances of distribution terms are probability distributions. Probabilistic Transition System Specifications {#sec:ptss} ============================================== Probabilistic transition systems (PTSs) generalize labelled transition systems (LTSs) by allowing for probabilistic choices in the transitions. We consider nondeterministic PTSs (Segala-type systems) [@Seg95a] with countable state spaces. [ *(PTS)***[.]{}**]{} A *probabilistic labeled transition system* (PTS) is a triple $(\closedTerms,\Act,{\trans})$, where $\Sigma$ is a signature, $\Act$ is a countable set of actions, and ${\trans} \subseteq \closedTerms \times \Act \times \Delta(\closedTerms)$ is a transition relation. We write $s \trans[a] \pi$ for ${(s,a,\pi)} \in {\trans}$. PTSs are specified by means of transition system specifications [@Plo04; @Gro93; @GV92; @LGD12]. [ *(Simple -rule)***[.]{}**]{} \[def:simple\_ntmuft\] A *simple -rule* has the form: $$\SOSrule{\{ t_k \trans[a_k] \mu_k \mid k \in K\} \qquad \{ {t_l \ntrans[b_l]} \mid {l \in L}\} } {f(x_1,\ldots,x_{\rank(f)}) \trans[a] \theta}$$ with $t_k,t_l\in \openTerms$, $a_k,b_l,a \in \Act$, $\mu_k\in\PVar, f\in\Sigma,x_1,\ldots,x_{\rank(f)}\in\TVar$, $\theta\in\openDTerms$, and constraints: 1. \[cond:simple\_ntmufxt:pairwise\_difference\_positive\_literal\] all $\mu_k$ for $k \in K$ are pairwise different; 2. \[cond:simple\_ntmufxt:pairwise\_difference\_source\] all $x_1,\ldots,x_{r(f)}$ are pairwise different. A *simple -rule* is as above with source of its conclusion $x\in\TVar$ instead of $f(x_1,\ldots,x_{\rank(f)})$. A *simple -rule* is either a simple -rule or a simple -rule. The expressions $t_k \trans[a_k] \mu_k$ (resp. $t_l \ntrans[b_l]$) above the line are called *positive* (resp. *negative*) *premises*. We call $\mu_k$ in $t_k \trans[a_k] \mu_k$ a *derivative* for each $x \in \Var(t_k)$. For rule $\rho$ we denote the set of positive (resp. negative) premises by $\pprem{\rho}$ (resp. $\nprem{\rho}$), and the set of all premises by $\prem{\rho}$. A rule without premises is called an *axiom*. We allow the sets of positive and negative premises to be infinite. The expression $f(x_1,\ldots,x_{\rank(f)}) \trans[a] \theta$ below the line is called *conclusion*, notation $\conc{\rho}$. The term $f(x_1,\ldots,x_{\rank(f)})$ is called the *source* of $\rho$, notation $\source(\rho)$, and $x_i$ are the *source variables*, notation $x_i \in \source(\rho)$. $\theta$ is the *target* of $\rho$, notation $\target(\rho)$. An expression $t \trans[a] \theta$ (resp. $t \ntrans[a]$) is called a *positive* (resp. *negative*) *literal*. Hence, premises and conclusions are literals. We denote the set of variables in $\rho$ by $\Var(\rho)$, *bound variables* by $\bound(\rho) = \{x_1,\dots,x_{\rank(f)}\} \cup \{\mu_k \mid k\in K\}$, and *free variables* by $\free(\rho) = \Var(\rho) \setminus \bound(\rho)$. A *probabilistic transition system specification* (PTSS) in simple -format, called simple -PTSS for short, is a triple $P = (\Sigma,A,R)$ with $\Sigma$ a signature, $\Act$ a set of action labels, and $R$ a set of simple -rules. As PTSS have negative premises, there are multiple approaches to assign a meaning (see [@vG04] for an overview). We will use the stratification approach presented in [@DL12] to assign to each PTSS $P = (\Sigma,\Act,R)$ (if possible) a PTS $(\closedTerms,\Act,{\trans}_P)$. A closed literal $t \trans[a]\pi$ (resp. $t \ntrans[a]$) *holds in* ${\trans}_P$, notation ${{\trans}_P} \models t \trans[a]\pi$ (resp. ${{\trans}_P} \models t \ntrans[a]$), if $(t, a, \pi) \in {{\trans}_P}$ (resp. there is no $\pi \in \Delta(\closedTerms)$ s.t. $(t, a, \pi) \in {{\trans}_P}$). A substitution $\sigma$ extends to literals by $\sigma(t \trans[a] \mu) = \sigma(t) \trans[a] \sigma(\mu)$, and $\sigma(t \ntrans[a]) = \sigma(t) \ntrans[a]$, and to rules as expected. [ *(Stratification [@DL12])***[.]{}**]{} \[def:stratification\] Let $P = (\Sigma, \Act, R)$ be a PTSS. A function $S:\closedTerms \times \Act \times \Delta(\closedTerms) \to \alpha$, where $\alpha$ is an ordinal, is called a *stratification* of $P$ if for every rule $\rho$ $$\SOSrule{\{ t_k \trans[a_k] \mu_k \mid k \in K\} \quad \{ {t_l \ntrans[b_l]} \mid {l \in L}\} } {f(x_1,\ldots,x_{\rank(f)}) \trans[a] \theta}$$ in $R$ and substitution $\sigma : (\TVar\cup\PVar) \to (\closedTerms\cup\Delta(\closedTerms))$ we have: $S(\sigma(t_k \trans[a_k] \mu_k)) \leq S(\conc{\sigma(\rho)})$ for all $k\in K$, and $S(\sigma(t_l \trans[b_l] \mu)) < S(\conc{\sigma(\rho}))$ for all $l\in L, \mu \in \PVar$. The set $S_{\beta} = \{\psi \mid S(\psi)=\beta\}$, with $\beta<\alpha$, is called a *stratum*. We call $P$ *stratifiable* if $P$ has some stratification. A transition relation is constructed stratum by stratum in an increasing manner. [ *(Induced PTS [@DL12])***[.]{}**]{} \[def:pts\_induced\_by\_ptss\] Let $P = (\Sigma, \Act, R)$ be a PTSS with stratification $S:\closedTerms \times \Act \times \Delta(\closedTerms) \to \alpha$. For all rules $\rho$, let $\degPTSS(\rho)$ be the smallest regular cardinal greater than $|\pprem{\rho}|$, and let $\degPTSS(P)$ be the smallest regular cardinal such that $\degPTSS(P)\geq\degPTSS(\rho)$ for all $\rho\in R$. The *induced PTS* $(\closedTerms, \Act, \trans_{P, S})$ is defined by ${\trans_{P,S}} = \bigcup_{\beta < \alpha} {\trans_{P_\beta}}$, where $\trans_{P_\beta} = \bigcup_{j \leq \degPTSS(P)} {\trans_{P_{\beta,j}}}$ and ${\trans_{P_{\beta,j}}}$ is $$\begin{aligned} {\trans_{P_{\beta,j}}} = \left\{ \psi \,\left|\, \begin{array}{l} S(\psi) = \beta \text{ and } \exists \rho \in R \text{ and substitution } \sigma \text{ s.t. } \psi = \conc{\sigma(\rho)}, \text{ and} \\[.7ex] \textstyle(\bigcup_{ \gamma < \beta} {\trans_{P_\gamma}}) \cup (\bigcup_{ j' < j} {\trans_{P_{\beta,j'}}}) \models {\pprem{\sigma(\rho)}}, \text{ and} \\[.7 ex] \textstyle(\bigcup_{ \gamma < \beta} {\trans_{P_\gamma}}) \models \nprem{\sigma(\rho)} \end{array} \right.\right\}\end{aligned}$$ The induced PTS is independent from the chosen stratification [@DL12]. We can construct for each simple -PTSS $(\Sigma, \Act, R)$ a PTSS $(\Sigma, \Act, R')$ with only simple -rules that induces the same PTS [@LGD12]. The construction defines $R'$ as $R$ where each rule with a source of the form $x$ is replaced by a set of rules where $x$ is substituted by $f(x_1,\ldots,x_{\rank(f)})$ for each $f\in \Sigma$ . Hence, all our results below for simple -PTSS generalize to simple -PTSS. Given a relation ${\relR} \subseteq \closedTerms \times \closedTerms$, a set $X \subseteq \closedTerms$ is $\closed{\relR}$, denoted by $\closed{\relR}(X)$, if ${\relR}(X) \subseteq X$ where ${\relR}(X) = \{y \in \closedTerms \mid \exists x \in X. x \relR y\}$. [ *(Probabilistic Bisimulation [@LS91; @DGJP03])***[.]{}**]{} \[def:bisimulation\] Let $(\closedTerms,\Act,{\trans})$ be a PTS. A symmetric relation ${\relR} \subseteq \closedTerms \times \closedTerms$ is a *probabilistic bisimulation* if whenever $t \relR t'$ and $t \trans[a] \pi$ then there exists a transition $t' \trans[a] \pi'$ such that $\pi \relR \pi'$, where $$\pi \relR \pi' \quad\text{iff}\quad\text{for all } X\subseteq \closedTerms \text{ with } \closed{\relR}(X) \text{ we have } \pi(X) = \pi'(X).$$ Notice that this standard definition can be slightly reformulated to relate it to the later introduced $\epsilon$-bisimulation (Definition \[def:epsilon\_bisim\]) by requiring that $\pi \relR \pi'$ iff $\pi(X) \le \pi'({\relR}\,(X))$ for all $X\subseteq \closedTerms$ [@DLT08]. The union of all probabilistic bisimulations is the largest probabilistic bisimulation, called probabilistic bisimilarity, and denoted by $\bisim$. We shall refer to probabilistic bisimulation as strict bisimulation to distinguish it from the later introduced relaxed notion of $\epsilon$-bisimulation. A crucial property of process description languages to ensure compositional modelling and verification is the compatibility of process operators with the behavioral relation chosen for the application context. In algebraic terms the compatibility of a behavioral equivalence $\relR$ with operator $f\in \Sigma$ is expressed by the congruence property which is defined as $f(t_1,\ldots,t_{\rank(f)}) \relR f(u_1,\ldots,u_{\rank(f)})$ whenever $t_i \relR u_i$ for $i=1,\ldots,\rank(f)$. The rule format of Definition \[def:simple\_ntmuft\] is an instance of the  rule format [@LGD12], which ensures that bisimilarity is a congruence. [ *(Probabilistic Bisimilarity as a congruence [@LGD12])***[.]{}**]{} \[thm:bisimilarity\_as\_congruence\] Let $P=(\Sigma,\Act,R)$ be a stratifiable simple -PTSS. Then probabilistic bisimilarity is a congruence for all operators defined in $P$. In order to allow for robust reasoning on PTSs, the behavioral relations should allow for (limited) perturbation of probabilities [@GJS90]. $\epsilon$-bisimulation is a behavioral relation based on strict probabilistic bisimulation, where the transfer condition is relaxed by some upper bound on the pertubation of probabilities. [ *($\epsilon$-Bisimulation [@DLT08])***[.]{}**]{} \[def:epsilon\_bisim\] Let $(\closedTerms,\Act,{\trans})$ be a PTS and $\epsilon \in [0,1]$. A symmetric relation ${\relR} \subseteq \closedTerms \times \closedTerms$ is an *$\epsilon$-bisimulation* if whenever $t \relR t'$ and $t \trans[a] \pi$ then there exists a transition $t' \trans[a] \pi'$ such that $\pi \relR \pi'$, where $$\pi \relR \pi' \quad\text{iff}\quad \text{for all } X\subseteq \closedTerms \text{ we have } \pi(X) \le \pi'({\relR}\,(X)) + \epsilon.$$ We call $t$ and $t'$ (resp. $\pi$ and $\pi'$) $\epsilon$-bisimilar if $t \relR t'$ (resp. $\pi \relR \pi'$) for some $\epsilon$-bisimulation $\relR$. Notice that $\epsilon$-bisimulations are reflexive and symmetric but not necessarily transitive. $\epsilon$-bisimulations are closed under union. We denote the largest $\epsilon$-bisimulation, called $\epsilon$-bisimilarity, by $\bisim_\epsilon$. According to [@DLT08], $\epsilon$-bisimulations induce a pseudo-metric over the set of closed terms $d:\closedTerms \times \closedTerms \rightarrow [0,1]$ with $d(t,t') = \inf\{\epsilon\in[0,1] \mid t \bisim_{\epsilon} t'\}$, where $\inf\, \emptyset = 1$. We say that $t$ and $t'$ are within the approximation bisimulation distance $\epsilon$ if $d(t,t')=\epsilon$. Expansivity of Process Combinators {#sec:aptss} ================================== The expansivity of an operator $f \in \Sigma$ is defined as the maximal approximate bisimulation distance of terms with an outermost function symbol $f$ in relation to the approximate bisimulation distances of its arguments. In this section we quantify the expansivity of operators defined by a PTSS. We start by showing that the expansivity of an operator $f$ defined by a rule $\rho$ depends on the *multiplicity* (i.e. number of occurrences) of source variables and their derivatives in the target of $\rho$; the *expansivity power* of operators (i.e. how much does the operator multiply the distance of its arguments) that define a context around the source variables or their derivatives; and the (reactive behavior) *discriminating power* of the premises of $\rho$. [(Factors of Expansivity)**[.]{}**]{} \[ex:expansivity\_of\_operators\_approx\_bisim\] Let $(\Sigma, \Act, R)$ be a PTSS with a signature $\Sigma$ that contains constants $r,s,0$, unary function symbols $f,f_2$, binary function symbols $g,g_2,g_3$ and a quaternary function symbol $h$, action set $\Act=\{a\}$, and axioms $R = \{r \trans[a] \delta_r, s \trans[a] [1-\epsilon]\delta_s \oplus [\epsilon]\delta_0\}$ for some fixed $\epsilon\in(0,1)$. It is not hard to see that $d(r,s)=\epsilon$ in the PTS induced by $(\Sigma,\Act,R)$. Consider the rules: $$\begin{gathered} \SOSrule{x \trans[a] \mu}{f(x) \trans[a] g(\mu,\mu)} \qquad\qquad \SOSrule{x_1 \trans[a] \mu_1 \quad x_2 \trans[a] \mu_2}{g(x_1,x_2) \trans[a] g(\mu_1,\mu_2)}\end{gathered}$$ These rules together with $R$ define $R_2$. In the first rule the derivative $\mu$ of source variable $x$ appears twice in the rule target $g(\mu,\mu)$. The induced PTS of $(\Sigma,\Act,R_2)$ contains the following transitions: \(r) at (0,0) [$r$]{} ; (pir) at ($ (r) - (0,1) $) [$\circ$]{}; (r) edge node \[right\] [[$a$]{}]{} (pir); (pir) \[bend left = 70,dotted\] edge node \[left\] [[$1.0$]{}]{} (r); \(s) at ($ (r) + (1.7,0) $) [$s$]{} ; (pis) at ($ (s) - (0,1) $) [$\circ$]{}; (s’) at ($ (pis) - (0,1) $) [$0$]{}; \(s) edge node \[left\] [[$a$]{}]{} (pis); (pis) \[bend right = 70,dotted\] edge node \[right\] [[$1.0-\epsilon$]{}]{} (s); (pis) \[dotted\] edge node \[right\] [[$\epsilon$]{}]{} (s’); \(r) at ($ (s) + (2.8,0) $) [$f(r)$]{} ; (pir1) at ($ (r) - (0,1) $) [$\circ$]{}; (r’) at ($ (pir1) - (0,1) $) [$g(r,r)$]{} ; \(r) edge node \[left\] [[$a$]{}]{} (pir1); (pir1) \[dotted\] edge node \[right\] [[$1.0$]{}]{} (r’); (r’) \[bend left = 70\] edge node \[left\] [[$a$]{}]{} (pir1); (fs) at ($ (r) + (3.9,0) $) [$f(s)$]{} ; (pis) at ($ (fs) - (0,1) $) [$\circ$]{}; (hs1) at ($ (pis) - (1.875,1) $) [$g(s,s)$]{} ; (hs2) at ($ (pis) - (0.625,1) $) [$g(s,0)$]{} ; (hs3) at ($ (pis) - (-0.625,1) $) [$g(0,s)$]{} ; (hs4) at ($ (pis) - (-1.875,1) $) [$g(0,0)$]{} ; (fs) edge node \[right\] [[$a$]{}]{} (pis); (pis) \[dotted, bend right = 25\] edge node \[above\] [[$(1-\epsilon)^2$]{}]{} (hs1); (pis) \[dotted\] edge node \[left\] [[$\epsilon - \epsilon^2$]{}]{} (hs2); (pis) \[dotted\] edge node \[right\] [[$\epsilon - \epsilon^2$]{}]{} (hs3); (pis) \[dotted, bend left = 25\] edge node \[above\] [[$\epsilon^2$]{}]{} (hs4); (hs1) \[bend left = 85,distance=1.05cm\] edge node \[left,xshift=-0.1cm\] [[$a$]{}]{} (pis); Observe that $d(f(r),f(s))=1-(1-\epsilon)^2$. The power of $2$ in the distance reflects directly the multiplicity of $2$ of the derivative $\mu$ in the rule target. The same effect can be observed for multiple occurrences of source variables in the rule target, e.g. consider for the $f$-defining rule $g(\delta_x,\delta_x)$ instead of $g(\mu,\mu)$ as target. Furthermore, the expansivity power of operators used in the rule target determine the expansivity of the operator defined by that rule. A simple example is the axiom $f_2(x) \trans[a] \delta_{f(x)}$. While the variable $x$ occurs only once in the rule target, we still have $d(f_2(r),f_2(s))=1-(1-\epsilon)^2$, because the operator $f$ has an expansivity power of 2 wrt. its single argument. This indicates that the expansivity power of (arguments of) operators need to be defined recursively. The multiplicity of source variables and their derivatives and the expansivity power of operators applied on those variables multiply. Consider the rules: $$\begin{gathered} \SOSrule{x_1 \trans[a] \mu_1 \quad x_2 \trans[a] \mu_2}{g(x_1,x_2) \trans[a] h(\mu_1,\mu_1,\mu_2,\mu_2)} \qquad\qquad \SOSrule{\{x_i \trans[a] \mu_i \mid i=1,\ldots,4\}}{h(x_1,x_2,x_3,x_4) \trans[a] h(\mu_1,\mu_2,\mu_3,\mu_4)}\end{gathered}$$ These rules together with $R_2$ define $R_3$. Now $d(f(r),f(s))=1-(1-\epsilon)^4$. As explained above for $R_2$, in the rule defining operator $f$ the derivative $\mu$ appears twice in the rule target. Additionally the operator $g$ that is applied to $\mu$ has for both of its arguments an expansivity power of two because in the $g$-defining rule the derivatives $\mu_1, \mu_2$ of both arguments $x_1, x_2$ appear twice in the rule target. The expansivity power of an operator may be unbounded. Consider the recursive unary operator $f$ defined by the rules: $$\begin{gathered} \SOSrule{x \trans[a] \mu}{f(x) \trans[a] g(f(\mu),f(\mu))} \qquad\qquad \SOSrule{x_1 \trans[a] \mu_1 \quad x_2 \trans[a] \mu_2}{g(x_1,x_2) \trans[a] g(\mu_1,\mu_2)}\end{gathered}$$ These rules together with $R$ define $R_4$. In the rule that defines the operator $f$ the derivative $\mu$ occurs twice in the target. Moreover, each occurrence of $\mu$ is put in the context of that operator $f$, which is defined by this rule (recursive call). Additionally both occurrences of $f(\mu)$ are put in the binary context $g$, which enforces that the distances of the two copies of $\mu$ multiply. Recursive multiplication of the distances leads to an approximate bisimulation distance of $d(f(r),f(s)) = 1$. The expansivity power of $f$ will in this case be denoted by $\infty$. On the other hand, an operator may also absorb the approximate bisimulation distance. Consider the rules: $$\begin{gathered} \SOSrule{x \trans[a] \mu}{f(x) \trans[a] g_2(\mu,\mu)} \qquad\qquad \SOSrule{x \trans[a] \mu}{f_2(x) \trans[a] g_3(\mu,\mu)} \qquad\qquad \SOSrule{}{g_3(x_1,x_2) \trans[a] \delta_0}\end{gathered}$$ These rules together with $R$ define $R_5$. The first rule applies the undefined operator $g_2$ to the two copies of the derivative $\mu$. As $g_2$ has no rules, we get $d(f(r),f(s))=0$. Similarly, the rule defining $f_2$ applies operator $g_3$ in the target. The operator $g_3$ allows one to derive an unconditional move to the idle process $0$. Hence, $d(f_2(r),f_2(s))=0$. However, if the reactive behavior of the process associated to a source variable is tested by some premise, then the operator defined by this rule may discriminate states with different reactive behavior. Consider the rules: $$\begin{gathered} \SOSrule{x \trans[a] \mu}{f(x) \trans[a] g(\mu,\mu)} \qquad\qquad \SOSrule{x_1 \trans[a] \mu_1 \quad x_2 \trans[a] \mu_2}{g(x_1,x_2) \trans[a] \delta_0}\end{gathered}$$ These rules together with $R$ define $R_6$. We get $d(f(r),f(s))=1-(1-\epsilon)^2$ because the $a$-transition of term $r$ leads to a distribution where all states can perform the action $a$, but the $a$-transition of term $s$ leads to a distribution where only states with a total probability mass of $1-\epsilon$ can perform the action $a$. We denote by $R_f$ those rules of $R$ that define the operator $f$. We define by $\reactdist{f}{i} \in \{0,1\}$ the (reactive behavior) discriminating power of argument $i$ of $f$. Formally, $\reactdist{f}{i} = 1$ if the source variable $x_i$ appears in a premise of some $\rho \in R_f$, i.e., if for some $\rho \in R_f$ there is a $t_k \trans[a_k] \mu_k \in \pprem{\rho}$ with $x_i \in \Var(t_k)$ or a ${t_l \ntrans[b_l]} \in {\nprem{\rho}}$ with $x_i \in \Var(t_l)$. Otherwise, $\reactdist{f}{i} = 0$. With $\Ninfty$ we denote $\N \cup \{\infty\}$, with the natural ordering extended by $n < \infty$ for each $n \in \N$, and the usual arithmetic extended for summation by $\infty + n = n + \infty = \infty + \infty = \infty$ for $n\ge 0$ and multiplication by $0 \cdot \infty = \infty \cdot 0 = 0$ and $n \cdot \infty = \infty \cdot n = \infty \cdot \infty = \infty$ for $n \ge 1$. We quantify the expansivity power of operators $f \in \Sigma$ as least fixed point of a monotone function. Let $(\Sigma, \Act, R)$ with $\Sigma = (F, \rank)$ be a PTSS. We define a poset $\mathcal{S} = (S,\sqsubseteq)$ with $S = S_{\FF} \times S_{\VV}$, $S_{\FF} = F \times \N \to \Ninfty$, $S_{\VV}=(\openTerms\,\cup\,\openDTerms) \to ((\TVar\,\cup\,\PVar) \to \Ninfty)$, equipped with the point-wise partial order $(\GF, \GV) \sqsubseteq (\GF',\GV')$ iff $\GF(f,i) \le \GF'(f,i)$, for all $f \in F, i \in \N$, and $\GV(t)(\zeta) \le \GV'(t)(\zeta)$ for all $t \in \openTerms \cup \openDTerms, \zeta \in \TVar \cup \PVar$. Elements of $S$ are pairs of maps $(\GF,\GV)$. $\GF(f,i)$ denotes the expansivity power of argument $i$ in operator $f$, i.e., how much the operator $f$ multiplies the approximate bisimulation distance of argument $i$. $\GV(t)(\zeta)$ defines the frequency of variable $\zeta \in \TVar \cup \PVar$ in the state or distribution term $t \in \openTerms \cup \openDTerms$ weighted by the expansivity power of the operators applied on top of $\zeta$. $\mathcal{S}$ forms a complete lattice with bottom element $\bot$ and top element $\top$, defined by constant maps $\bot((f,i),(t,\zeta)) = (0,0)$ and $\top((f,i),(t,\zeta))=(\infty,\infty)$ for each $f \in F$, $i \in \N$, $t \in \openTerms \cup \openDTerms$, $\zeta \in \TVar \cup \PVar$. \[prop:s\_complete\_lattice\] $\mathcal{S}$ is a complete lattice. The function $\functor: S \to S$ defined in Fig. \[fig:functor\_multiplicity\_approx\_bisim\] computes in parallel the expansivity power of arguments of operators, and the multiplicities of variables in terms weighted by the expansivity power of the operators applied on top of them. The expansivity power $\GF'(f,i)$ of argument $i$ of operator $f$ is defined as the maximum expansivity power over each $f$-defining rule $\rho \in R_f$. For $\rho \in R_f$ the expansivity power is defined as the sum of the multiplicity of $x_i$ in the rule target $\target(\rho)$ and of the multiplicity of $x_i$ in some premise $t_k \trans[a] \mu_k \in \pprem{\rho}$ weighted by the multiplicity of the derivative $\mu_k$ in the rule target $\target(\rho)$. Note that source variables and derivatives in the rule target contribute equally to the expansivity power of an argument. The multiplicity $\GV'(t)(\zeta)$ of $\zeta$ in a state term $t$ counts the occurrences of variable $\zeta$ in $t$ and weights them by the expansivity power of the operators applied on top of $\zeta$. The multiplicity $\GV'(\theta)(\zeta)$ of $\zeta$ in a distribution term $\theta$ counts the occurrences of variable $\zeta$ in $\theta$ and weights them by the expansivity power of the operators applied on top of $\zeta$, but at least by the discriminating power of those operators. Note that the discriminating power of operators is considered only for distribution terms. To understand this, consider the reactive behavior of $\epsilon$-bisimilar state and distribution terms. For a state term $f(t_1,\ldots,t_{\rank(f)})$ we have that $\sigma(t_i) \bisim_{\epsilon_i} \sigma'(t_i)$ implies $\sigma(t_i) \trans[a]$ iff $\sigma(t_i') \trans[a]$ for each $a\in\Act$, i.e., $\sigma(t_i)$ and $\sigma(t'_i)$ agree on their immediate reactive behavior. However, for a distribution term $f(\theta_1,\ldots,\theta_{\rank(f)})$ we have that if $\sigma(\theta_i) \sim_{\epsilon_i} \sigma'(\theta_i)$ then $\sigma(\theta_i)$ and $\sigma'(\theta_i)$ may have states with different reactive behavior (cf. $R_6$ in Example 1). $\functor$ is order-preserving. This ensures the existence and uniqueness of the least fixed point of $\functor$ by the Knaster-Tarski fixed point theorem. \[prop:fb\_order\_preserving\] $\functor$ is order-preserving. [align\*]{}\ &: S S (,) = (’,’)\ ’(f,i) &= \_[R\_f]{} ( (())(x\_i) + \_[t\_k \_k ]{} (t\_k)(x\_i) (())(\_k) )\ ’(t)() &= 1 & t=\ \_[i=1]{}\^[(f)]{} ( (f,i) (t\_i)()) & t=f(t\_1,…,t\_[(f)]{})\ 0 & \ ’()() &= 1 & =\ (t)() & = \_t\ \_[iI]{} ((\_i)()) & =\_[iI]{} p\_i \_i\ \_[i=1]{}\^[(f)]{} ( ((f,i), ) (\_i)()) & =f(\_1,…,\_[(f)]{})\ 0 & \ We denote the least fixed point of $\functor$ by $(\lfpF,\lfpT)$. We call $\lfpF(f,i)$ the expansivity power of argument $i$ of operator $f$, and $\lfpT(t)(\zeta)$ the weighted multiplicity of variable $\zeta$ in term $t$. The expansivity power of $f$ allows us to derive an upper bound on the approximate bisimulation distance between terms $f(t_1,\ldots,t_{\rank(f)})$ and $f(t_1',\ldots,t_{\rank(f)}')$ expressed in relation to the approximate bisimulation distances $\epsilon_i$ between the arguments $t_i$ and $t_i'$. [ *(Expansivity bound)***[.]{}**]{} \[def:upper\_bound\_expansion\_approx\_bisim\] The *expansivity bound* $\expbound^f$ of operator $f \in \Sigma$ wrt. the approximate bisimulation distances $\epsilon_i$ of its arguments $i=1,\ldots,\rank(f)$ is defined by $$ \expbound^f(\epsilon_1,\ldots,\epsilon_{\rank(f)}) = 1 -\prod_{i=1}^{\rank(f)} (1-\epsilon_i)^{\lfpF(f,i)}$$ Notice that $\expbound^f(\epsilon_1,\ldots,\epsilon_{\rank(f)}) = 0$ if $\epsilon_i=0$ for all arguments $i$ with $\lfpF(f,i) > 0$. In particular, we have $\expbound^f(\epsilon_1,\ldots,\epsilon_{\rank(f)}) = 0$ if all $\epsilon_i=0$. We call an argument $i$ of operator $f \in \Sigma$ (behavioral distance) *absorbing* if $\lfpF(f,i) = 0$. We demonstrate first the application of the expansivity bound and prove later its correctness. [(continued)**[.]{}**]{} For the PTSS $(\Sigma,A,R_2)$ we have $\lfpF(f,1) = 2$ because $\lfpF(g,1)=\lfpF(g,2)=1$. Terms $r$ and $s$ with approximate bisimulation distance $d(r,s)=\epsilon$ agree by $1-\epsilon$ on their behavior. Thus, the pair of processes $(r,r)$ and $(s,s)$ agree by $(1-\epsilon)^2$ on their behavior. Hence, they disagree by $1-(1-\epsilon)^2$ on their behavior. This gives a behavioral distance of $d(f(r),f(s))=1-(1-\epsilon)^2$. We continue with PTSS $(\Sigma,A,R_3)$. For operator $h$ we have $\lfpF(h,1)=\lfpF(h,2)=\lfpF(h,3)=\lfpF(h,4)=1$, for $g$ we have $\lfpF(g,1)=\lfpF(g,2)=2$ and thus for $f$ we get $\lfpF(f,1)=4$. For PTSS $(\Sigma,A,R_4)$ the recursive definition of $f$ applied to the two occurrences of the derivative $\mu$ in the rule target gives $\lfpF(f,1)=\infty$. The (behavioral distance) absorbing effect of $f$ and $f_2$ in $(\Sigma,A,R_5)$ results in $\lfpF(f,1)=\lfpF(f_2,1)=\lfpF(g_2,1)=\lfpF(g_2,2)=\lfpF(g_3,1)=\lfpF(g_3,2)=0$. In $(\Sigma,\Act,R_6)$ the (reactive behavior) discriminating power $\reactdist{g}{1} = \reactdist{g}{2} = 1$ of operator $g$ leads to $\lfpF(f,1)=2$. Now we can show that the approximate bisimulation distance between terms $f(t_1,\ldots,t_{\rank(f)})$ and $f(t_1',\ldots,t_{\rank(f)}')$ is bounded by the expansivity bound. [ *(Expansivity bound of simple -PTSS)***[.]{}**]{} \[thm:bounded\_expansion\_ntmuft\_bisim\] Let $(\Sigma, \Act, R)$ be a stratifiable simple -PTSS. Then for each operator $f \in \Sigma$ we have $$f(t_1,\dots,t_{\rank(f)}) \bisim_{\epsilon} f(t_1',\dots,t_{\rank(f)}') \quad\text{whenever}\quad t_i \bisim_{\epsilon_i} t_i' \ \text{ for } i=1,\ldots,\rank(f)$$ with $\epsilon = \expbound^f(\epsilon_1,\ldots,\epsilon_{\rank(f)})$. Theorem \[thm:bounded\_expansion\_ntmuft\_bisim\] implies Theorem \[thm:bisimilarity\_as\_congruence\] by considering $\epsilon_i=0$ for all $i=1,\ldots,\rank(f)$ and exploiting that $\bisim_0$ is in fact the strict probabilistic bisimilarity. Our target was to define the expansivity power $\lfpF(f)(i)$ of the argument $i$ of operator $f \in \Sigma$ in order to characterize the behavioral distance of terms with outermost function symbol $f$. We conclude this section by outlining how the expansivity bound could be further refined. Sequential composition $\_\, ; \_$ is defined by the following rules [@DL12] $$\begin{gathered} \SOSrule{x\trans[a]\mu}{{x; y}\trans[a] \mu; \delta_y}{\ a\neq\tick} \qquad\qquad \SOSrule{x\trans[\tick]\mu \quad y\trans[a]\mu'}{{x; y}\trans[a]\mu'}\end{gathered}$$ Action $\tick$ denotes successful termination. The expansivity power $\lfpF(;)(1) = \lfpF(;)(2) = 1$ gives an expansivity bound $\expbound^;(\epsilon_1,\epsilon_2) = 1-(1-\epsilon_1)(1-\epsilon_2)$. However, the sequential composition describes separate moves of either process $x$ or process $y$. Hence, the expansivity of $\_\, ; \_$ is actually bounded by $1-\min(1-\epsilon_1,1-\epsilon_2)$. In general, if multiple rules define an operator $f$, then the expansivity power and weighted multiplicity should be quantified per rule instead of per operator. In detail, the expansivity power $\lfpF(f)(i)$ should take a rule $\rho$ instead of $f$ as argument, and the weighted multiplicity $\lfpT(t,x)$ should take a tree of rules instead of term $t$ as argument. We leave this as future work. Specification of Non-expansive Process Combinators {#sec:non-exp} ================================================== Non-expansivity is the quantitative analogue of the congruence property of (strict) probabilistic bisimulation. Intuitively, non-expansivity means that different processes are not more different when they are put in the same context. [ *(Non-expansivity)***[.]{}**]{} \[def:non\_expansivity\] Let $(\closedTerms, \Act, {\trans}_{P})$ be the PTS induced by the PTSS $P=(\Sigma, \Act, R)$. An operator $f \in \Sigma$ is *non-expansive* if $$f(t_1,\dots,t_{\rank(f)}) \bisim_{\epsilon} f(t_1',\dots,t_{\rank(f)}') \quad\text{whenever}\quad t_i \bisim_{\epsilon_i} t_i'\, \text{ for all }\, i=1,\ldots,\rank(f)$$ with $\epsilon = \min(\sum_{i=1}^{\rank(f)} \epsilon_i,1)$. We call $f$ *expansive* if $f$ is not non-expansive. Argumentation for this linear upper bound and a discussion on alternative upper bounds like maximum norm or Euclidean norm can be found in [@Tin10]. From the expansivity bound $\expbound^f$ (Definition \[def:upper\_bound\_expansion\_approx\_bisim\]) of operator $f \in \Sigma$ it follows that $f$ is non-expansive if $\lfpF(f,i) \le 1$ for all $i=1,\ldots,\rank(f)$. This yields the following rule format. [ *( rule format)***[.]{}**]{} \[def:entmuftRuleFormat\] A simple -rule $\rho$ is an *-rule* if for each $x_i \in \source(\rho)$ we have $$\MVarMax(\target(\rho))(x_i) + \displaystyle \!\!\!\!\!\!\!\sum_{t_k \trans[a_k] \mu_k \in \atop \pprem{\rho}}\!\!\!\!\!\!\! \MVar(t_k)(x_i) \cdot \MVarMax(\target(\rho))(\mu_k) \le 1.$$ A PTSS $P = (\Sigma,A,R)$ is in  format, -PTSS for short, if all rules in $R$ are in the  rule format. [ *(Non-expansivity of -PTSS)***[.]{}**]{} \[thm:nonexp\_entmuft\_ptss\] Let $(\Sigma, \Act, R)$ be a stratifiable -PTSS. Then all operators $f \in \Sigma$ are non-expansive. The constraints of the  rule format are easy to verify. It suffices to count the occurrences of source variables and derivatives in the rule target. There is no need for recursive reasoning over other rules. We deliberately decided against the (slightly more general) rule format which could be given as simple -rules $\rho$ that define some operator $f\in\Sigma$ and for which the only requirement would be $\lfpF(f,i) \le 1$ for all $i=1,\ldots,\rank(f)$. We justify this by considering the extension of a PTSS $P=(\Sigma,\Act,R)$ to $P'=(\Sigma,\Act,R')$ with $R \subseteq R'$. If $P$ is in  format, then in order to decide if $P'$ is in  format only the rules in $R' \setminus R$ need to be verified wrt. the  format constraints. On the contrary, the generalized rule format would require that whenever a rule is added, all other rules are again validated with respect to the format constraints. For instance, consider the set of rules $R_6$ in Example 1. The rule defining operator $f$ alone would be non-expansive. However, by adding the rule defining operator $g$ (even though $g$ is non-expansive), operator $f$ becomes expansive. Applications {#sec:applications} ============ The standard process combinators sequential composition, (probabilistic and non-probabilistic) choice, and (probabilistic and non-probabilistic) CCS and CSP like parallel composition [@Bar04; @DL12] are all in the -format. On the other hand, recursion and iteration operators may be expansive if they replicate (some of) their arguments. We consider the replication operator of $\pi$-calculus. The nondeterministic variant $!\_$ and the probabilistic variant $!^p\_$ with $p \in (0,1) \cap \Q$ are defined by the rules[@MS13]: $$\begin{gathered} \SOSrule{x \trans[a] \mu}{!x \trans[a] \mu \parallel \delta_{!x}} \qquad \SOSrule{x \trans[a] \mu}{!^px \trans[a] \mu \oplus_p (\mu \parallel \delta_{!^px})} \qquad \SOSrule{x_1 \trans[a] \mu_1 \quad x_2 \trans[a] \mu_2}{x_1 \parallel x_2 \trans[a] \mu_1 \parallel \mu_2}\end{gathered}$$ The first two rules defining both variants of the replication operator are not in -format. The expansivity power of both operators is unbounded with $\lfpF(!)(1)=\lfpF(!^p)(1)=\infty$. Hence, both operators are expansive. However, if the synchronous parallel composition defined in the third rule above is replaced by the non-communicating asynchronous parallel composition, then both variants of the replication operator would become non-expansive.         \ -------------------------------------------------------------------------------------------------------------------------------------------- Description Rule ------------------------------------------- ------------------------------------------------------------------------------------------------ $1\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Non-linearity of the rule target}\\ \SOSrule{x \trans[a] \mu} \text{wrt.\ a source variable} {f(x) \trans[a] \delta_x \parallel \delta_x} \end{array}$ }$ $2\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Non-linearity of the rule target}\\ \SOSrule{x \trans[a] \mu} \text{wrt.\ a derivative} {f(x) \trans[a] \mu \parallel \mu} \end{array}$ }$ $3\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Non-linearity of a state term}\\ \SOSrule{x \trans[a] \mu} \text{in the rule target} {f(x) \trans[a] \delta_{x \parallel x}} \end{array}$ }$ $4\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Non-linearity of a term in a}\\ \SOSrule{x \parallel x \trans[a] \mu} \text{premise} {f(x) \trans[a] \mu} \end{array}$ }$ $5\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Multiple derivatives of a source}\\ \SOSrule{x \trans[a] \mu_1 \quad x \trans[b] \mu_2} \text{variable in the rule target} {f(x) \trans[a] \mu_1 \parallel \mu_2} \end{array}$ }$ $6\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Source and derivative in the}\\ \SOSrule{x \trans[a] \mu} \text{rule target} {f(x) \trans[a] \delta_x \parallel \mu} \end{array}$ }$ $7\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Multiple derivatives weighted by}\\ \SOSrule{x \trans[a] \mu} \text{convex combination in rule target} {f(x) \trans[a] [0.5] \mu \parallel \mu \oplus [0.5]\delta_{s'}} \end{array}$ }$ $8\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Lookahead by existential test}\\ \SOSrule{x \trans[a] \mu_1 \quad \mu_1(y)>0 \quad y \trans[c] \mu_2} \text{in quantitative premise} {f(x) \trans[a] \mu_2} \end{array}$ }$ $9\hspace{0.15cm}\begin{array}{l} $\displaystyle{ \text{Lookahead by universal test}\\ \SOSrule{x \trans[a] \mu \quad \mu(Y) \ge 1 \quad \{ y \trans[a] \mu_y \mid y\in Y \}} \text{in quantitative premise} {f(x) \trans[a] \delta_r} \end{array}$ }$ -------------------------------------------------------------------------------------------------------------------------------------------- : SOS rules that specify expansive operators[]{data-label="tab:counterexample_rule_restrictions_entmufxt"} We summarize the structural patterns of rules that may lead to expansive behavior in Table \[tab:counterexample\_rule\_restrictions\_entmufxt\]. None of these rules is in the  format. For cases 1 to 7 the expansivity power of $f$ is $\lfpF(f)(1) = 2$ and, therefore, the expansivity bound is $\expbound^f(\epsilon) = d(f(r),f(s)) = 1 - (1-\epsilon)^2$. Cases 8 and 9 indicate that lookahead cannot be admitted and we need to employ simple -rules (Definition \[def:simple\_ntmuft\]) instead of -rules [@LGD12]. The expressions $\mu_1(y)>0$ and $\mu_1(Y) \ge 1$ (with $y\in\TVar$, $Y \subseteq \TVar$) are quantitative premises as introduced by the  format [@DL12]. As argued above, $\epsilon$-bisimilar instances may have states with different reactive behavior. For instance, in case 8, while distributions $\pi_r$ and $\pi_s$ are $\epsilon$-bisimilar, only $\pi_s$ has in its support a state that can perform a $c$-move. Similarly, in case 9, only for $\pi_r$ we have that all states in the support can perform an $a$-move. We conjecture that a notion of *non-expansivity up to $\epsilon$* for some $\epsilon \in [0,1]$ (bounded non-expansivity) would allow for limited lookahead. An operator is non-expansive up to $\epsilon$ if it is non-expansive whenever its arguments have an approximate bisimulation distance of at most $\epsilon$. In this case, quantitative premises $\mu(Y) \ge p$ that measure the probability of $Y$ and test against the boundary $p$ could be allowed, if $p$ is in the interval $p \in [\epsilon,1-\epsilon]$. On the other hand, quantitative premises with $p < \epsilon$ or $p > 1-\epsilon$ cannot be permitted because they allow for lookahead with respect to probabilistic choices that do not mimic each other’s reactive behavior. The expansivity bound (Definition \[def:upper\_bound\_expansion\_approx\_bisim\]) allows rule formats to be derived for alternative compositionality requirements. For instance, consider an $n$-ary process combinator $\otimes$ with the compositionality requirements that the approximate bisimulation distance of the combined processes should not depend on the approximate bisimulation distance of processes at some argument $i \in \{1,\ldots,n\}$. From the expansivity bound we derive that either argument $i$ of operator $\otimes$ is behavioral distance absorbing ($\lfpF(\otimes)(i) = 0$), or the application context guarantees that processes for argument $i$ are strictly bisimilar ($\epsilon_i = 0$). The non-expansivity requirement (Definition \[def:non\_expansivity\]) is in fact the Manhattan norm, or more general the $p$-norm $(\sum_{i=1}^{\rank(\otimes)}\epsilon_i^p)^{1/p}$ with $p=1$. Consider the alternative compositionality requirement that the expansivity of a process combinator $\otimes$ should be bounded by the $p$-norm with $p>1$ (which includes the Euclidean norm by $p=2$ and the maximum norm by $p \to \infty$). From the expansivity bound we derive that $\lfpF(\otimes)(i) = 1$ for at most one argument $i$, and all other arguments $j \ne i$ are behavioral distance absorbing with $\lfpF(\otimes)(j) = 0$. Conclusion and Future Work {#sec:conclusion} ========================== We studied structural specifications of probabilistic processes that are robust with respect to bounded implementation and measurement errors of probabilistic behavior. We provided for each process combinator an upper bound on the distance between the combined processes using the structural specification of the process combinator (Theorem \[thm:bounded\_expansion\_ntmuft\_bisim\]). We derived an appropriate rule format that guarantees non-expansivity (standard compositionality requirement) of process combinators (Theorem \[thm:nonexp\_entmuft\_ptss\]). All standard process algebraic operators are compatible for approximate reasoning and satisfy the rule format, except operators which replicate processes and combine them by synchronous parallel composition. We exemplified how rule formats for non-standard compositionality requirements can be derived. Our work can be extended in several directions. In Section \[sec:aptss\] and Section \[sec:applications\] we sketched already how the expansivity bound can be further refined and how a restricted form of lookahead in the rules specifying the process combinators could be admitted. The techniques and results developed in this paper for approximate bisimulation can be carried over to bisimulation metrics. Initial work in this direction suggests that the  format presented in this paper ensures also non-expansivity for the bisimulation metric based on the Kantorovich and Hausdorff metric. Moreover, for the bisimulation metric the rule format can be further generalized because in this case convex combinations weigh the distance and multiplicity of processes (unlike approximate bisimilarity, see case 7 of Table \[tab:counterexample\_rule\_restrictions\_entmufxt\]). Furthermore, we will investigate the expansivity of process combinators and rule formats for variants of bisimulation metrics and $\epsilon$-bisimulation that discount the influence of future transitions [@DGJP04; @TDZ11]. #### Acknowledgements We are grateful to Josée Desharnais for discussions on $\epsilon$-bisimulation, Matteo Mio for discussions on approximate semantics of structurally defined probabilistic systems, and Wan Fokkink and David Williams for feedback on earlier versions of this paper. Furthermore, we thank the anonymous referees for thorough reviews and very helpful comments. [^1]: This fixes a flaw in [@LGD12; @DL12] where arbitrary functions $f\!\!:\closedTerms^n \to \closedTerms$ were allowed. In this case probabilistic bisimilarity (Definition \[def:bisimulation\]) may not be a congruence (Theorem \[thm:bisimilarity\_as\_congruence\]). Example: PTSS $(\Sigma,A,R)$, constants $r,r',s$ in $\Sigma$, $A=\{a\}$, $R=\left\{\SOSrule{}{s \trans[a] \delta_s}, \SOSrule{x \ntrans[a]}{g(x) \trans[a] f(\delta_x)}\right\}$ with $f(r)=r, f(r')=s$. Now $r \bisim r'$ but $g(r) \not\bisim g(r')$.

--- abstract: 'The (proper) power graph of a group is a graph whose vertex set is the set of all (nontrivial) elements of the group and two distinct vertices are adjacent if one is a power of the other. Various kinds of planarity of (proper) power graphs of groups are discussed.' address: - 'Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran' - 'Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran' author: - 'A. Doostabadi' - 'M. Farrokhi D. G.' title: 'Embeddings of (proper) power graphs of finite groups' --- Introduction ============ Recently, there have been an increasing interest in associating graphs to algebraic structures and studying how the properties of the associated graphs influence the structure of the given algebraic structures. If $G$ is a group (or a semigroup), then the *power graph* of $G$, denoted by ${\mathcal{P}}(G)$, is a graph whose vertex set is $G$ in which two distinct vertices $x$ and $y$ are adjacent if one is a power of the other, in other words, $x$ and $y$ are adjacent if $x\in{\langle y\rangle}$ or $y\in{\langle x\rangle}$. Chakrabarty, Ghosh and Sen [@ic-sg-mks] investigated the power graph of semigroups and characterized all semigroups with connected or complete power graphs. In the case of groups, Cameron and Ghosh [@pjc-sg] showed that two finite abelian groups are isomorphic if and only if they have isomorphic power graphs. As a generalization, Cameron [@pjc] proves that finite groups with isomorphic power graphs, have the same number of elements of each order. Further properties of power graphs including planarity, perfectness, chromatic number and clique number are discussed by Doostabadi, Erfanian and Jafarzadeh in [@ad-ae-aj]. Since the identity element in a group $G$ is adjacent to all other vertices in the power graph ${\mathcal{P}}(G)$, we may always remove the identity element and study the resulting graph called the *proper power graph* of $G$. The proper power graph of $G$ is denoted by ${\mathcal{P}}^*(G)$. For further results concerning power graphs and proper power graphs, we may refer the interested reader to [@ad-ae-mfdg] and [@ad-mfdg]. The aim of this paper is to study various kinds of planarity of (proper) power graphs. Indeed, we shall classify all groups whose (proper) power graphs are planar, outerplanar, ring graph, $1$-planar, almost planar, maximal planar, toroidal and projective. In what follows, $\omega(G)$ stands for the set of orders of all elements of a given group $G$, i.e., $\omega(G)=\{|x|:x\in G\}$. Also, a Frobenius group with kernel $K$ and a complement $H$ is denoted by $K\rtimes_F H$. The *dot product* of two vertex transitive graphs $\Gamma_1$ and $\Gamma_2$, is the graph obtained from the identification of a vertex of $\Gamma_1$ with a vertex of $\Gamma_2$ and it is denoted by $\Gamma_1\cdot\Gamma_2$. Planarity of (proper) power graphs ================================== We begin with the usual notion of planarity. A graph $\Gamma$ is called *planar* if there is an embedding of $\Gamma$ in the plane in which the edges intersect only in the terminals. A famous theorem of Kuratowski states that a graph is planar if and only if it has no subgraphs as a subdivision of the graphs $K_5$ or $K_{3,3}$ (see [@kk]). The following results give a characterization of all planar (proper) power graphs and have central roles in the proofs of our subsequent results. We note that a graph is said to be $\Gamma$-free if it has no induced subgraphs isomorphic to $\Gamma$. \[planar\] Let $G$ be a group. Then ${\mathcal{P}}(G)$ is planar if and only if $\omega(G)\subseteq\{1,2,3,4\}$. \[properplanar\] Let $G$ be a group. Then the following conditions are equivalent: - ${\mathcal{P}}^*(G)$ is planar, - ${\mathcal{P}}^*(G)$ is $K_5$-free, - ${\mathcal{P}}^*(G)$ is $K_6$-free, - ${\mathcal{P}}^*(G)$ is $K_{3,3}$-free, - $\omega(G)\subseteq\{1,2,3,4,5,6\}$. We just prove the equivalence of (1) and (5). The other equivalences can be establish similarly. First assume that ${\mathcal{P}}^*(G)$ is a planar graph. If $G$ has an element of infinite order, then clearly ${\mathcal{P}}^*(G)$ has a subgraph isomorphic to $K_5$, which is a contradiction. Thus $G$ is a torsion group. Now, let $x\in G$ be an arbitrary element. If $|x|=p^m$ is a prime power, then $p^m-1\leq4$ and hence $|x|\leq5$ for ${\langle x\rangle}\setminus\{1\}$ induces a complete subgraph of ${\mathcal{P}}^*(G)$. Also, if $|x|$ is not prime power and $p^mq^n$ divides $|x|$, then the elements of ${\langle x\rangle}\setminus\{1\}$ whose orders divide $p^m$ together with elements whose orders equal $p^mq^n$ induce a complete subgraph of ${\mathcal{P}}^*(G)$ of size $p^m-1+\varphi(p^mq^n)$, where $\varphi$ is the Euler totient function. Since ${\mathcal{P}}^*(G)$ is planar, this is possible only if $|x|\leq6$, as required. Conversely, assume that $G$ is a torsion group with $\omega(G)\subseteq\{1,2,3,4,5,6\}$. A simple verification shows that ${\mathcal{P}}^*(G)$ is a union (not necessary disjoint) of some $K_1$, $K_2$, $K_4$, friendship graphs and families of complete graphs on $4$ vertices sharing an edge in such a way that any two such graphs have at most one edge in common and any three such graphs have no vertex in common. Hence, the resulting graph is planar, which completes the proof. An *$n$-coloring* of a graph $\Gamma$ is an assignment of $n$ different colors to the vertices of $\Gamma$ such that adjacent vertices have different colors. The *chromatic number* $\chi(\Gamma)$ is the minimal number $n$ such that $\Gamma$ has an $n$-coloring. An *$n$-star coloring* of $\Gamma$ is an $n$-coloring of $\Gamma$ such that no path on four vertices in $\Gamma$ is $2$-colored. The *star chromatic number* $\chi_s(\Gamma)$ is the minimal number $n$ such that $\Gamma$ has an $n$-star coloring. Utilizing the above theorems we have: If $G$ is a group with planar (proper) power graph, then $\chi({\mathcal{P}}^*(G))=\chi_s({\mathcal{P}}^*(G))$. A *chord* in a graph $\Gamma$ is an edge joining two nonadjacent vertices in a cycle of $\Gamma$ and a cycle with no chord is called a *primitive cycle*. A graph $\Gamma$ in which any two primitive cycles intersect in at most one edge is said to admit the *primitive cycle property* (PCP). The *free rank* of $\Gamma$, denoted by ${\mathrm{frank}}(\Gamma)$, is the number of primitive cycles of $\Gamma$. Also, the *cycle rank* of $\Gamma$, denoted by ${\mathrm{rank}}(\Gamma)$, is the number $e-v+c$, where $v,e,c$ are the number of vertices, the number of edges and the number of connected components of $\Gamma$, respectively. Clearly, the cycle rank of $\Gamma$ is the same as the dimension of the cycle space of $\Gamma$. By [@ig-er-rhv Proposition 2.2], we have ${\mathrm{rank}}(\Gamma)\leqslant {\mathrm{frank}}(\Gamma)$. A graph $\Gamma$ is called a *ring graph* if one of the following equivalent conditions holds (see [@ig-er-rhv]). - ${\mathrm{rank}}(\Gamma)={\mathrm{frank}}(\Gamma)$, - $\Gamma$ satisfies the PCP and $\Gamma$ does not contain a subdivision of $K_4$ as a subgraph. Also, a graph is *outerplanar* if it has a planar embedding all its vertices lie on a simple closed curve, say a circle. A well-known result states that a graph is outerplanar if and only if it does not contain a subdivision of $K_4$ and $K_{2,3}$ as a subgraph (see [@gc-fh]). Clearly, every outerplanar graph is a ring graph and every ring graph is a planar graph. Let $G$ be a group. Then ${\mathcal{P}}(G)$ (resp. ${\mathcal{P}}^*(G)$) is ring graph if and only if $\omega(G)\subseteq\{1,2,3\}$ (resp. $\omega(G)\subseteq\{1,2,3,4\}$). If ${\mathcal{P}}^*(G)$ is a ring graph, then by Theorem \[planar\], $\omega(G)\subseteq\{1,2,3,4,5,6\}$. If $G$ has an element of order $5$ or $6$, then ${\langle x\rangle}$ contains a subgraph isomorphic to $K_4$, which is impossible. Thus $\omega(G)\subseteq\{1,2,3,4\}$. Clearly, $w(G)\subseteq\{1,2,3\}$ when ${\mathcal{P}}(G)$ is a ring graph. The converse is obvious. Let $G$ be a group. Then ${\mathcal{P}}(G)$ (resp. ${\mathcal{P}}^*(G)$) is outerplanar if and only if it is a ring graph. A graph is called *$1$-planar* if it can be drawn in the plane such that its edges each of which is crossed by at most one other edge. \[1-planar\] If $\Gamma$ is a $1$-planar graph on $v$ vertices and $e$ edges, then $e\leq4v-8$. The complete graph $K_7$ is not $1$-planar. Suppose on the contrary that $K_7$ is $1$-planar. Then, by Theorem \[1-planar\], we should have $21=e\leq 4v-8=20$, which is a contradiction. To deal with the case of $1$-planar power graphs, we need to decide on the $1$-planarity of a particular graph, which is provided by the following lemma. \[K9-K6+3K2\] Let $\Gamma$ be the graph obtained from $K_9\setminus K_6$ by adding three new disjoint edges. Then $\Gamma$ is not $1$-planar. Le $u_1,u_2,u_3$ be the vertices adjacent to all other vertices, and $\{v_1,v_2\}$, $\{v_3,v_4\}$ and $\{v_5,v_6\}$ be the three disjoint edges whose end vertices are different from $u_1,u_2,u_3$. Suppose on the contrary that $\Gamma$ is $1$-planar and consider a $1$-planar embedding ${\mathcal{E}}$ of $\Gamma$ with minimum number of crosses. If ${\mathcal{E}}$ has an edge crossing itself or two crossing incident edges, then one can easily unknot the cross and reach to a $1$-planar embedding of $\Gamma$ with smaller number of crosses contradicting the choice of ${\mathcal{E}}$. Hence, ${\mathcal{E}}$ has neither an edge crossing itself nor two crossing incident edges. This implies that the subgraph $\Delta$ induced by $\{u_1,u_2,u_3\}$ is simply a triangle. Using a direct computation one can show, step-by-step, that - the edges incident to each of the vertices $v_1,\ldots,v_6$ crosses no more than one edges of $\Delta$, - at most one edge of $\Delta$ is crossed by an edge of $\Gamma$ different from $\{v_1,v_2\}$, $\{v_3,v_4\}$ and $\{v_5,v_6\}$, - the only edges of $\Gamma$ that can cross $\Delta$ are $\{v_1,v_2\}$, $\{v_3,v_4\}$ and $\{v_5,v_6\}$. According to the above observations, we reach to the following $1$-planar drawing of $\Gamma$ in the interior region of $\Delta$ with maximum number of vertices (see Figure 1). Clearly, there must exists a vertex $v_i$ outside $\Delta$ adjacent to some vertex $v_j$ inside $\Delta$ where $1\leq i,j\leq 6$. Hence, by Figure 1, we must have an edge in the interior region of $\Delta$ crossed more than once, leading to a contradiction. \(A) at ([0.7\*cos(90)]{},[0.7\*sin(90)]{}) ; (B) at ([0.7\*cos(210)]{},[0.7\*sin(210)]{}) ; (C) at ([0.7\*cos(330)]{},[0.7\*sin(330)]{}) ; (D) at ([2\*cos(90)]{},[2\*sin(90)]{}) ; (E) at ([2\*cos(210)]{},[2\*sin(210)]{}) ; (F) at ([2\*cos(330)]{},[2\*sin(330)]{}) ; (A)–(D)–(E)–(F)–(D)–(B) –(E)–(A)–(F)–(C)–(D); (B)–(F); (C)–(E); \ Figure 1 Utilizing the same method as in the proof of Lemma \[K9-K6+3K2\], we obtain a new minimal non-$1$-planar graph, which is of independent interest. \[K9-K6+2K2\] Let $\Gamma$ be the graph obtained from $K_9\setminus K_6$ by adding two new disjoint edges. Then $\Gamma$ is a minimal non-$1$-planar graph. Let $G$ be a group. Then ${\mathcal{P}}(G)$ is $1$-planar if and only if $\omega(G)\subseteq\{1,2,3,4,5,6\}$ and any two cyclic subgroups of $G$ of order $6$ have at most two elements in common. The same as in the proof of Theorem \[planar\], we can show that $\omega(G)\subseteq\{1,2,3,4,5,6\}$. If $\omega(G)\subseteq\{1,2,4,5\}$, then we are done. Thus we may assume that $3\in\omega(G)$. Let $g\in G$ be an element of order $3$. If $C_G(g)$ has two distinct involutions $x,y$, then it must have one more involution, say $z$, for ${\langle x,y\rangle}$ is a dihedral group. But then, by Lemma \[K9-K6+3K2\], the subgraph induced by elements of orders $1,3,6$ in ${\langle xg\rangle}\cup{\langle yg\rangle}\cup{\langle zg\rangle}$ is not $1$-planar giving us a contradiction. Thus no two distinct cyclic subgroups of $G$ of order $6$ have three elements in common. Conversely, if all the conditions are satisfied, then ${\mathcal{P}}(G)$ is a combination of induced subgraphs as drawn in Figure 2 in such a way that any these subgraphs have pairwise disjoint edges except possibly for a common edge whose end vertices are the trivial element and an involution. Therefore ${\mathcal{P}}(G)$ is $1$-planar, as required. Note that in Figure 2, $a,b,c,d,e$ denote elements of orders $2,3,4,5,6$, respectively. [ccc]{} \(A) at (0,0) ; (B) at (2,0) ; (A)–(B); & \(A) at (0,0) ; (B) at (2,0) ; (C) at (1,[2\*sin(60)]{}) ; (A)–(B)–(C)–(A); & \(A) at (0,0) ; (B) at (2,0) ; (C) at (1,[2\*sin(60)]{}) ; (D) at (1,[0.666\*sin(60)]{}) ; (B)–(C)–(A)–(B)–(D)–(C); (A)–(D); [cc]{} \(A) at ([2\*cos(90)]{},[2\*sin(90)]{}) ; (B) at ([2\*cos(210)]{},[2\*sin(210)]{}) ; (C) at ([2\*cos(330)]{},[2\*sin(330)]{}) ; (D) at ([cos(210)]{},[sin(210)]{}) ; (E) at ([cos(330)]{},[sin(330)]{}) ; (A)–(B)–(C)–(D)–(E)–(A)–(C)–(E)–(B)–(D)–(A); & \(A) at ([cos(90)]{},[sin(90)]{}) ; (B) at ([cos(210)]{},[sin(210)]{}) ; (C) at ([cos(330)]{},[sin(330)]{}) ; (D) at ([2\*cos(90)]{},[2\*sin(90)]{}) ; (E) at ([2\*cos(210)]{},[2\*sin(210)]{}) ; (F) at ([2\*cos(330)]{},[2\*sin(330)]{}) ; (B)–(F)–(A)–(D)–(E)–(F)–(D)–(C)–(A)–(B)–(C)–(E)–(F)–(C); (A)–(E); \ Figure 2 Let $G$ be a group. Then ${\mathcal{P}}^*(G)$ is $1$-planar if and only if $\omega(G)\subseteq\{1,2,3,4,5,6,7\}$. The same as in the proof of Theorem \[properplanar\], we can show that $\omega(G)\subseteq\{1,2,3,4,5,6,7\}$. The converse is obvious since every element of order $7$ of $G$ along with its nontrivial powers gives a complete connected component of ${\mathcal{P}}^*(G)$ isomorphic to $K_6$ and the remaining elements of $G$, by Theorem \[properplanar\], induce a planar graph. An *almost-planar* graph $\Gamma$ is a graph with an edge $e$ whose removal is a planar graph. Let $G$ be a group. Then ${\mathcal{P}}(G)$ is almost-planar if and only if $w(G)\subseteq\{1,2,3,4\}$ or $G$ is isomorphic to one of the groups ${\mathbb{Z}}_5$, ${\mathbb{Z}}_6$, $D_{10}$, $D_{12}$, ${\mathbb{Z}}_3\rtimes{\mathbb{Z}}_4$ or ${\mathbb{Z}}_5\rtimes_F{\mathbb{Z}}_4$. Clearly, $w(G)\subseteq\{1,2,3,4,5,6\}$. If $w(G)\subseteq\{1,2,3,4\}$, then ${\mathcal{P}}(G)$ is planar and we are done. Thus we may assume that $5\in w(G)$ or $6\in w(G)$. First suppose that $5\in w(G)$. If $G$ has two distinct cycles ${\langle x\rangle}$ and ${\langle y\rangle}$ or order $5$. Then the subgraph induced by ${\langle x\rangle}\cup{\langle y\rangle}$ is isomorphic to $K_5\cdot K_5$, which is not almost-planar. Hence $G$ has a unique cyclic subgroup ${\langle x\rangle}$ of order $5$. Then ${\langle x\rangle}{\trianglelefteq}G$ and $G/C_G(x)$ is a cyclic group of order dividing $4$. However, $C_G(x)={\langle x\rangle}$ from which it follows that $|G|$ divides $20$ and hence $G\cong{\mathbb{Z}}_5$, $D_{10}$, ${\mathbb{Z}}_5\rtimes_F{\mathbb{Z}}_4$. Now, suppose that $5\notin w(G)$ but $6\in w(G)$. If $G$ has two distinct cyclic subgroups ${\langle x\rangle}$ and ${\langle y\rangle}$ of order $6$, then a simple verification shows that ${\langle x\rangle}\cup{\langle y\rangle}$ is never almost-planar in either of cases ${\langle x\rangle}\cap{\langle y\rangle}$ has one, two or three elements. Thus $G$ has a unique cyclic subgroup ${\langle x\rangle}$ of order $6$. Clearly, ${\langle x\rangle}{\trianglelefteq}G$ and $C_G(x)={\langle x\rangle}$, which implies that $|G|$ divides $12$. Therefore, $G\cong{\mathbb{Z}}_6$, $D_{12}$ or ${\mathbb{Z}}_3\rtimes{\mathbb{Z}}_4$. The converse is straightforward. Let $G$ be a group. Then ${\mathcal{P}}^*(G)$ is almost planar if and only if $\omega(G)\subseteq\{1,2,3,4,5,6\}$. If ${\mathcal{P}}^*(G)$ is almost planar, then since $K_6$ is not almost planar, by Theorem \[properplanar\], $\omega(G)\subseteq\{1,2,3,4,5,6\}$. The converse is obvious for by Theorem \[properplanar\], ${\mathcal{P}}^*(G)$ is planar. A simple graph is called *maximal planar* if it is planar but the graph obtained by adding any new edge is not planar. Let $G$ be a group. Then ${\mathcal{P}}(G)$ is maximal planar if and only if $G$ is a cyclic group of order at most four. Suppose that ${\mathcal{P}}(G)$ is maximal planar. Since ${\mathcal{P}}(G)$ is planar, by Theorem \[planar\], $\omega(G)\subseteq\{1,2,3,4\}$. If $G$ has two different maximal cycles ${\langle x\rangle}$ and ${\langle y\rangle}$ of order $2$, $3$ or $4$, then the addition of the edge $\{x,y\}$ results in a planar graph, which is a contradiction. Therefore, the maximal cycles in $G$ give rise to a partition for $G$. Since ${\mathcal{P}}(G)$ is connected, it follows that $G$ is cyclic, from which the result follows. The converse is clear. Let $G$ be a group. Then ${\mathcal{P}}^*(G)$ is maximal planar if and only if $G$ is a cyclic group of order at most five. Suppose that ${\mathcal{P}}^*(G)$ is maximal planar. Since ${\mathcal{P}}^*(G)$ is planar, by Theorem \[properplanar\], $\omega(G)\subseteq\{1,2,3,4,5,6\}$. If $G$ has two different cycles ${\langle x\rangle}$ and ${\langle y\rangle}$ of order $4$ or $6$ whose intersection is nontrivial, then by Theorem \[properplanar\], the addition of the edge $\{x,y\}$ results in a planar graph, which is a contradiction. Therefore, the maximal cycles in $G$ give rise to a partition of $G$. Since ${\mathcal{P}}^*(G)$ is connected, it follows that $G$ is cyclic, from which the result follows. The converse is clear. It is worth noting that the structure of groups with elements of orders at most six is known and we refer the interested reader to [@rb-wjs; @ndg-vdm; @tlh-wjs; @fl-blw; @dvl; @vdm; @bhn; @wjs-cy; @wjs-wzy; @cy-sgw-xjz; @akz-vdm] for details. Toroidal (proper) power graphs ============================== Let $S_k$ be the sphere with $k$ handles (or connected sum of $k$ tori), where $k$ is a non-negative integer, that is, $S_k$ is an oriented surface of genus $k$. The genus of a graph $\Gamma$, denoted by $\gamma(\Gamma)$, is the minimal integer $k$ such that the graph can be embedded in $S_k$ such that the edges intersect only in the endpoints. A graph with genus $0$ is clearly a planar graph. A graph with genus $1$ is called a *toroidal graph*. We note that if $\Gamma'$ is a subgraph of a graph $\Gamma$, then $\gamma(\Gamma')\leq\gamma(\Gamma)$. For complete graph $K_n$ and complete bipartite graph $K_{m,n}$, it is well known that $$\gamma(K_n)=\left\lceil\frac{(n-3)(n-4)}{12}\right\rceil$$ if $n\geq 3$ and $$\gamma(K_{m,n})=\left\lceil\frac{(m-2)(n-2)}{4}\right\rceil$$ if $m,n\geq 2$. (See [@gr-2] and [@gr-1], respectively). Thus - $\gamma(K_n)=0$ for $n=1,2,3,4$, - $\gamma(K_n)=1$ for $n=5,6,7$, - $\gamma(K_n)\geq 2$ for $n\geq 8$, - $\gamma(K_{m,n})=0$ for $m=0,1$ or $n=0,1$, - $\gamma(K_{m,n})=1$ for $\{m,n\}=\{3\}$, $\{3,4\}$, $\{3,5\}$, $\{3,6\}$, $\{4\}$, - $\gamma(K_{m,n})\geq2$ for $\{m,n\}=\{4,5\}$, or $m,n\geq3$ and $m+n\geq10$. Given a connected graph $\Gamma$, we say that a vertex $v$ of $\Gamma$ is a cut-vertex if $\Gamma-v$ is disconnected. A block is a maximal connected subgraph of $\Gamma$ having no cut-vertices. The following result of Battle, Harary, Kodama, and Youngs gives a powerful tool for computing genus of various graphs. \[blocks\] Let $\Gamma$ be a graph and $\Gamma_1,\ldots,\Gamma_n$ be blocks of $\Gamma$. Then $$\gamma(\Gamma)=\gamma(\Gamma_1)+\cdots+\gamma(\Gamma_n).$$ \[toroidal\] Let $G$ be a group. Then ${\mathcal{P}}(G)$ is a toroidal graph if and only if $G\cong{\mathbb{Z}}_5$, ${\mathbb{Z}}_6$, ${\mathbb{Z}}_7$, $D_{10}$, $D_{12}$, $D_{14}$, ${\mathbb{Z}}_3\rtimes{\mathbb{Z}}_4$, ${\mathbb{Z}}_5\rtimes_F{\mathbb{Z}}_4$ or ${\mathbb{Z}}_7\rtimes_F{\mathbb{Z}}_3$. First assume that ${\mathcal{P}}(G)$ is a toroidal graph. The same as in the proof of Theorem \[properplanar\], we can show that $\omega(G)\subseteq\{1,2,3,4,5,6,7\}$. If $\omega(G)\subseteq\{1,2,3,4\}$, then by Theorem \[planar\], ${\mathcal{P}}(G)$ is planar, which is a contradiction. Thus $\omega(G)\cap\{5,6,7\}\neq\emptyset$. Since ${\langle x\rangle}$ is a block of ${\mathcal{P}}(G)$ when $|x|=5,7$, and ${\langle x\rangle}$ is a subgraph of a block of ${\mathcal{P}}(G)$ when $|x|=6$, Theorem \[blocks\] shows that $\omega(G)\subseteq\{1,2,3,4,5\}$, $\{1,2,3,4,6\}$ or $\{1,2,3,4,7\}$. Moreover, $G$ has at most one subgroup of order $5$ and $7$. If $7\in\omega(G)$, then $G$ has a unique cyclic subgroup ${\langle x\rangle}$ of order $7$. Clearly, ${\langle x\rangle}{\trianglelefteq}G$ and $C_G(x)={\langle x\rangle}$. Since $G/C_G(x)$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})$ and $6\notin\omega(G)$, $G/C_G(x)$ is a cyclic group of order at most $3$, from which it follows that $G\cong{\mathbb{Z}}_7$, $D_{14}$ or ${\mathbb{Z}}_7\rtimes_F{\mathbb{Z}}_3$. Similarly, if $5\in\omega(G)$, then we can show that $G\cong{\mathbb{Z}}_5$, $D_{10}$ or ${\mathbb{Z}}_5\rtimes_F{\mathbb{Z}}_4$. Finally, suppose that $6\in\omega(G)$. If $G$ has a unique cyclic subgroup of order $6$, say ${\langle x\rangle}$, then ${\langle x\rangle}{\trianglelefteq}G$ and a simple verification shows that $C_G(x)={\langle x\rangle}$. Since $G/{\langle x\rangle}$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})$, it follows that $G\cong{\mathbb{Z}}_6$, $D_{12}$ or ${\mathbb{Z}}_3\rtimes{\mathbb{Z}}_4$. Now suppose that $G$ has at least to distinct cyclic subgroups of order $6$. If $G$ has two distinct cyclic subgroups ${\langle x\rangle}$ and ${\langle y\rangle}$ of order $6$ such that ${\langle x\rangle}\cap{\langle y\rangle}={\langle a\rangle}\cong{\mathbb{Z}}_2$, then the subgraph induced ${\langle x\rangle}\cup{\langle y\rangle}\setminus\{a\}$ is isomorphic to $K_5\cdot K_5$ and by Theorem \[blocks\], ${\mathcal{P}}(G)$ is not toroidal, which is a contradiction. Since the cycles of order $6$ sharing an element of order $3$ are blocks, by Theorem \[blocks\], either $G$ all cyclic subgroups of order $6$ have the same subgroup of order $3$ in common. Clearly, $G$ has at most three cyclic subgroups of order $6$ for otherwise $G$ has four cyclic subgroups ${\langle x\rangle}$, ${\langle y\rangle}$, ${\langle z\rangle}$ and ${\langle w\rangle}$ of order $6$ and the subgraph induced by ${\langle x\rangle}\cup{\langle y\rangle}\cup{\langle z\rangle}\cup{\langle w\rangle}$ has a subgraph isomorphic to $K_{3,8}$, which is a contradiction. If $G$ has three distinct cyclic subgroups ${\langle x\rangle}$, ${\langle y\rangle}$ and ${\langle z\rangle}$ of order $6$, then by using [@en-wm], it follows that the subgraph induced by ${\langle x\rangle}\cup{\langle y\rangle}\cup{\langle z\rangle}$ has genus $2$, which is a contradiction. Hence, $G$ has exactly two distinct subgroups of order $6$, say ${\langle x\rangle}$ and ${\langle y\rangle}$. Let $H=N_G({\langle x\rangle})$. Then $[G:H]\leq2$. Since $G$ has two cyclic subgroups of order $6$, a simple verification shows that $C_H(x)={\langle x\rangle}$. On the other hand, $H/{\langle x\rangle}$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_2$. Hence $|H|$ divides $12$ and consequently $|G|$ divides $24$, which results in a contradiction for there are no such groups. The converse is straightforward. \[propertoroidal\] Let $G$ be a group. Then ${\mathcal{P}}^*(G)$ is a toroidal graph if and only if $G\cong {\mathbb{Z}}_7$, ${\mathbb{Z}}_8$, $D_{14}$, $D_{16}$, $Q_{16}$, $QD_{16}$ or ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_3$. Suppose ${\mathcal{P}}^*(G)$ is toroidal. The same as in the proof of Theorem \[properplanar\], we can show that $\omega(G)\subseteq\{1,2,3,4,5,6,7,8\}$. If $\omega(G)\subseteq\{1,2,3,4,5,6\}$, then by Theorem \[properplanar\], ${\mathcal{P}}^*(G)$ is planar, which is a contradiction. Thus $\omega(G)\cap\{7,8\}\neq\emptyset$. On the other hand, by Theorem \[blocks\], $\{7,8\}\not\subseteq\omega(G)$. Therefore, either $\omega(G)\subseteq\{1,2,3,4,5,6,7\}$ or $\omega(G)\subseteq\{1,2,3,4,5,6,8\}$. If $7\in\omega(G)$, then again by Theorem \[blocks\], $G$ has a unique cyclic subgroup ${\langle x\rangle}$ of order $7$. Clearly, ${\langle x\rangle}{\trianglelefteq}G$ and $C_G(x)={\langle x\rangle}$. Since $G/{\langle x\rangle}$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_6$, it follows that $G\cong{\mathbb{Z}}_7$, $D_{14}$, ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_3$ or ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_6$. Now, suppose that $8\in\omega(G)$. If $G$ has two distinct cyclic subgroups ${\langle x\rangle}$ and ${\langle y\rangle}$ of order $8$, then the subgraph induced by ${\langle x\rangle}\cup{\langle y\rangle}$ has a subgraph isomorphic to $K_5\cdot K_5$ or $2K_5$, which is a contradiction by Theorem \[blocks\]. Therefore, $G$ has a unique cyclic subgroup ${\langle x\rangle}$ of order $8$. Then ${\langle x\rangle}{\trianglelefteq}G$ and $C_G(x)={\langle x\rangle}$. Since $G/{\langle x\rangle}$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_2\times{\mathbb{Z}}_2$, by using GAP [@tgg], we obtain $G\cong{\mathbb{Z}}_8$, $D_{16}$, $Q_{16}$ or $QD_{16}$. The converse is obvious. Projective (proper) power graphs ================================ The *real projective plane* is a non-orientable surface, which can be represented on plane by a circle with diametrically opposed points identified. Let $N_k$ be the connected sum of $k$ projective planes, where $k$ is a non-negative integer. The corsscap number of a graph $\Gamma$, denoted by $\overline{\gamma}(\Gamma)$, is the minimal integer $k$ such that $\Gamma$ can be embedded in $N_k$ such that the edges intersect only in the endpoints. A graph with crosscap $0$ is clearly a planar graph. A graph with crosscap $1$ is called a *projective graph*. Clearly, if $\Gamma'$ is a subgraph of a graph $\Gamma$, then $\overline{\gamma}(\Gamma')\leq\overline{\gamma}(\Gamma)$. For complete graph $K_n$ and complete bipartite graph $K_{m,n}$, it is well known that $$\overline{\gamma}(K_n)=\begin{cases}\left\lceil\frac{(n-3)(n-4)}{6}\right\rceil,&n\geq3\emph{ and }n\neq7,\\3,&n=7,\end{cases}$$ and $$\overline{\gamma}(K_{m,n})=\left\lceil\frac{(m-2)(n-2)}{2}\right\rceil$$ if $m,n\geq 2$. (See [@gr-2] and [@gr-1], respectively). Thus - $\overline{\gamma}(K_n)=0$ for $n=1,2,3,4$, - $\overline{\gamma}(K_n)=1$ for $n=5,6$, - $\overline{\gamma}(K_n)\geq 2$ for $n\geq 7$, - $\overline{\gamma}(K_{m,n})=0$ for $m=0,1$ or $n=0,1$, - $\overline{\gamma}(K_{m,n})=1$ for $\{m,n\}=\{3\}$, $\{3,4\}$, - $\overline{\gamma}(K_{m,n})\geq2$ for $m,n\geq3$ and $m+n\geq8$. A graph $\Gamma$ is *irreducible* for a surface $S$ if $\Gamma$ does not embed in $S$ but any proper subgraph of $\Gamma$ embeds in $S$. Kuratowski’s theorem states any graph which is irreducible for the plane is homomorphic to either $K_5$ or $K_{3,3}$. Glover, Huneke and Wang [@hg-ph-cw] constructed a list of $103$ pairwise non-homomorphic graphs which are irreducible for the real projective plane. Also, Archdeacon in [@da] proved this list is complete in the sense that a graph can be embedded on the real projective plane if and only if it has no subgraph homomorphic to any of the $103$ given graphs. For example, the graphs $K_5\cdot K_5$, $2K_5$, $K_{3,3}\cdot K_{3,3}$, $2K_{3,3}$, $K_{3,3}\cdot K_5$ and $K_{3,3}\cup K_5$ are irreducible for the real projective plane. Furthermore, the irreducible graphs in [@hg-ph-cw] show that the graph $K_7$ is not projective. \[projective\] Let $G$ be a finite group. Then ${\mathcal{P}}(G)$ is projective if and only if $G\cong{\mathbb{Z}}_5$, ${\mathbb{Z}}_6$, $D_{10}$, $D_{12}$, ${\mathbb{Z}}_3\rtimes{\mathbb{Z}}_4$ or ${\mathbb{Z}}_5\rtimes_F{\mathbb{Z}}_4$. Suppose ${\mathcal{P}}(G)$ is projective. The same as in the proof of Theorem \[properplanar\], it can be easily seen that $\omega(G)\subseteq\{1,2,3,4,5,6\}$. If $\omega(G)\subseteq\{1,2,3,4\}$, then by Theorem \[planar\], ${\mathcal{P}}(G)$ is planar, which is a contradiction. If $5,6\in\omega(G)$, then the subgraph induced by ${\langle x\rangle}\cup{\langle y\rangle}$, where ${\langle x\rangle}$ and ${\langle y\rangle}$ are distinct subgroups of orders $5$ and $5$, or $5$ and $6$, respectively, contains a subgraph isomorphic to $K_5\cdot K_5$, which is a contradiction. Thus $\omega(G)\subseteq\{1,2,3,4,5\}$ or $\{1,2,3,4,6\}$. If $5\in\omega(G)$, then $G$ has a unique normal cyclic subgroup ${\langle x\rangle}$ of order $5$. Clearly, $C_G(x)={\langle x\rangle}$. Hence $G/C_G(x)$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_4$, which implies that $|G|$ divides $20$. Therefore $G\cong{\mathbb{Z}}_5$, $D_{10}$ or ${\mathbb{Z}}_5\rtimes_F{\mathbb{Z}}_4$. Finally, suppose that $6\in\omega(G)$. If ${\langle x\rangle}$ and ${\langle y\rangle}$ are two distinct subgroups of order $6$, then ${\langle x\rangle}\cap{\langle y\rangle}\cong{\mathbb{Z}}_3$ for otherwise the subgraph induced by ${\langle x\rangle}\cup{\langle y\rangle}$ has a subgraph isomorphic to $K_5\cdot K_5$, which is a contradiction. Since $G$ has no subgraphs isomorphic to $K_{3,6}$ it follows that $G$ has at most two cyclic subgroups of order $6$. If $G$ has two distinct cyclic subgroups ${\langle x\rangle}$ and ${\langle y\rangle}$ of order $6$, then since $N_G({\langle x\rangle})/C_G(x)$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_2$ and $C_G(x)={\langle x\rangle}$, it follows that $|N_G({\langle x\rangle})|$ divides $12$. On the other hand, ${\langle x\rangle}$ has at most two conjugates, namely ${\langle x\rangle}$ and ${\langle y\rangle}$, which implies that $[G:N_G({\langle x\rangle})]\leq2$. Thus $|G|$ divides $24$. A simple verification by GAP [@tgg] shows that there are no groups of order dividing $24$ which admit exactly two cyclic subgroups of order $6$, which contradicts our assumption. Therefore, $G$ has a unique cyclic subgroup of order $6$, say ${\langle x\rangle}$. Then ${\langle x\rangle}{\trianglelefteq}G$ and $C_G(x)={\langle x\rangle}$. Hence $G/{\langle x\rangle}$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_2$, which implies that $G\cong{\mathbb{Z}}_6$, $D_{12}$ or ${\mathbb{Z}}_3\rtimes{\mathbb{Z}}_4$. The converse is straightforward. \[properprojective\] Let $G$ be a finite group. Then ${\mathcal{P}}^*(G)$ is projective if and only if $G\cong{\mathbb{Z}}_7$, $D_{14}$, ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_3$ or ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_6$. Suppose ${\mathcal{P}}^*(G)$ is projective. The same as in the proof of Theorem \[properplanar\], it can be easily seen that $\omega(G)\subseteq\{1,2,3,4,5,6,7\}$. If $\omega(G)\subseteq\{1,2,3,4,5,6\}$, then by Theorem \[properplanar\], ${\mathcal{P}}(G)$ is planar, which is a contradiction. Thus $7\in\omega(G)$. If $G$ has two distinct cyclic subgroups of order $7$, then ${\mathcal{P}}^*(G)$ has a subgraph isomorphic to $2K_6$, which is impossible. Thus $G$ has a unique cyclic subgroup ${\langle x\rangle}$ of order $7$. Then ${\langle x\rangle}{\trianglelefteq}G$ and $G/C_G(x)$ is isomorphic to a subgroup of ${\mathrm{Aut}}({\langle x\rangle})\cong{\mathbb{Z}}_6$. However, $C_G(x)={\langle x\rangle}$, which implies that $|G|$ divides $42$. Therefore $G\cong{\mathbb{Z}}_7$, $D_{14}$, ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_3$ or ${\mathbb{Z}}_7\rtimes{\mathbb{Z}}_6$. The converse is obvious. The authors would like to thank prof. Myrvold for providing us with the toroidal graph testing algorithm used in the proof of Theorem \[toroidal\]. [0]{} A. T. White, *Graphs, Groups and Surfaces*, North-Holland, Amsterdam, 1973. D. 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--- abstract: 'Let $K/k$ be an abelian extension of number fields with a distinguished place of $k$ that splits totally in $K$. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in $K$, called the Stark unit, constructed from the values of the $L$-functions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a result we deduce a weak version of the conjecture (“up to absolute values”) in these cases and precise results on when the Stark unit, if it exists, is a square.' address: 'Tokyo Institute of Technology, Department of Mathematics, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan' author: - 'Xavier-François Roblot' bibliography: - 'refs.bib' date: ' (draft v1)' title: Index formulae for Stark units and their solutions --- [^1] Introduction ============ Let $K/k$ be an abelian extension of number fields. Denote by $G$ its Galois group. Let $S_\infty$ and $S_\textrm{ram}$ denote respectively the set of infinite places of $k$ and the set of finite places of $k$ ramified in $K/k$. Let $S(K/k) := S_\infty \cup S_\mathrm{ram}$. Fix a finite set $S$ of places of $k$ containing $S(K/k)$ and of cardinality at least $2$. Assume that there exists at least one place in $S$, say $v$, that splits totally in $K/k$ and fix a place $w$ of $K$ dividing $v$. Let $e$ be the order of the group of roots of unity in $K$. In this setting Stark [@stark:4] made the following conjecture.  \ There exists an $S$-unit $\varepsilon_{K/k,S}$ in $K$ such that 1. For all characters $\chi$ of $G$ $$L'_{K/k,S}(0, \chi) = \frac{1}{e} \sum_{g \in G} \chi(g) \log |\varepsilon_{K/k,S} ^g|_w$$ where $L_{K/k,S}(s, \chi)$ denotes the $L$-function associated to $\chi$ with Euler factors at prime ideals in $S$ deleted. 2. The extension $K( \varepsilon_{K/k,S}^{\,1/e})/k$ is abelian. 3. If furthermore $|S| \geq 3$ then $\varepsilon$ is a unit of $K$. The unit $\varepsilon_{K/k,S}$ is called the Stark unit associated to the extension $K/k$, the set of places $S$ and the infinite place $v$.[^2] It is unique up to multiplication by a root of unity in $K$. A good reference for this conjecture is [@tate:book Chap. IV]. The starting point of this research is the conjectural method used in [@cohen-roblot] and [@stark12] (and inspired by [@stark:example]) to construct totally real abelian extensions of totally real fields. Let $L/k$ be such an extension. The idea is to construct a quadratic extension $K/L$, abelian over $k$, satisfying some additional conditions similar to the assumptions (A1), (A2) and (A3) below. Assuming the Stark conjecture for $K/k$, $S(K/k)$ and a fixed real place $v$ of $k$, one can prove that $K = k(\varepsilon)$ and $L = k(\alpha)$ where $\alpha := \varepsilon + \varepsilon^{-1}$ and $\varepsilon := \varepsilon_{K/k, S(K/k)}$ is the corresponding Stark unit. Using part (1) of the conjecture, one computes the minimal polynomial $A(X)$ of $\alpha$ over $k$. The final step is to check unconditionally that the polynomial $A(X)$ does indeed define the extension $L$. One notices in that setting that the rank of the units of $K$ is equal to the rank of units of $L$ plus the rank of the module generated by the Stark unit and its conjugates over $k$. A natural question to ask is wether the index of the group generated by the units of $L$ and the conjugates of the Stark unit has finite index inside the group of units of $K$ and, if so, if this index can be computed. A positive answer to the first question is given by Stark in [@stark:3 Th. 1]. In [@arakawa], Arakawa gives a formula for this index when $k$ is a quadratic field. Using similar methods, we obtain a general result (Theorem \[th:globindex\]) in the next section. Then we derive a “relative” index formula (Theorem \[th:main\]) that relates the index of the subgroup generated over $\mathbb{Z}[G]$ by the Stark unit inside the “minus-part” of the group of units of $K$ to the cardinality of the “minus-part” of the class group of $K$.[^3] In the third section, we use results of Rubin [@rubin:index] on some type of Gras conjecture for Stark units to show that the relative index formula implies local relative index formulae (Theorem \[th:localindex\]). Starting with the fourth section, we stop assuming the abelian rank one Stark conjecture and study directly the solutions to the index formulae. In section 4, we look at how much these index formulae characterize the Stark unit (Proposition \[prop:prodform\] and Corollary \[cor:unicity\]). In the next section, we introduce the algebraic tools that will be needed to prove the existence of solutions in some cases in the following sections. We also reprove in that section the abelian rank one Stark conjecture for quadratic extensions (Theorem \[th:quadratic\]). Finally, sections 6 and 7 are devoted to a proof that solutions to the index formulae always exist for quartic extensions (Theorem \[th:quartic\]) and sextic extensions (Theorem \[th:sextic\]) with some additional conditions in that case. We show that the existence of solutions in those cases imply a weak version of the conjecture where part (1) is satisfied only up to absolute values.[^4] We also obtain results on when the Stark unit, if it exists, is a square (Corollary \[cor:whensquare\], Theorem \[th:quadratic\], Corollary \[cor:square4\] and Corollary \[cor:square6\]). The index formulae {#sec:index} ================== We assume from now on that the place $v$ is infinite[^5] and that $k$ has at least two infinite places. Therefore we can always apply the conjecture for any finite set $S$ containing $S(K/k)$. The cases that we are excluding are $k = \mathbb{Q}$ and $k$ a complex quadratic field. In both cases the conjecture is proved and the Stark unit is strongly related to cyclotomic units and elliptic units respectively. Fix a finite set $S$ of places of $k$ containing $S(K/k)$. We make the following additional assumptions. 1. $k$ is totally real and the infinite places of $K$ above $v$ are real, the infinite places of $K$ not above $v$ are complex. 2. The maximal totally real subfield $K^+$ of $K$ satisfies $[K:K^+] = 2$. 3. All the finite primes in $S$ are either ramified or inert in $K/K^+$. If $S$ contains more than one place that splits totally in $K/k$ then the conjecture is trivially true with the Stark unit being equal to $1$. Therefore the only non trivial case excluded by (A1) is the case when $k$ has exactly one complex place and $K$ is totally complex. It is likely that most of the methods and results in this paper can be adapted to cover also that case. Assumptions (A2) and (A3) are necessary to ensure that the rank of the group generated by the units of $K^+$ and the conjugate of the Stark unit has finite index inside the group of units of $K$. Without these assumptions, global index formulae for Stark units as they are stated in this article cannot exist although it is still possible to prove index formulae for some $p$-adic characters if one takes also into account Stark units coming from subextensions (see [@rubin:index] or Section 3). **We assume until further notice that the conjecture is true for the extension $K/k$, the set of places $S$ and the distinguished place $v$.**[^6] Denote by $\varepsilon := \varepsilon_{K/k, S}$ the corresponding Stark unit. From now on, all subfields of $K$ (including $K$ itself) are identified with their image in $\mathbb{R}$ by $w$. We make the Stark unit unique by imposing that $\varepsilon > 0$. It follows that $\varepsilon^g > 0$ for all $g \in G$, see [@tate:book §IV.3.7]. One can also prove under these hypothesis, see [@stark12 Lem. 2.8], that $|S(K/k)| \geq 3$ and therefore $\varepsilon$ is a unit of $K$ by part (3) of the Conjecture, and that $|\varepsilon|_{w'} = 1$ for any place $w'$ of $K$ not above $v$. Let $m$ be the degree of $K^+/k$ and $d$ be the degree of $k/\mathbb{Q}$. Thus we have $[K:k] = 2m$ and $[K:\mathbb{Q}] = 2md$. Let $\tau$ denote the non trivial element of ${\mathrm{Gal}}(K/K^+)$. It is the complex conjugation of the extension $K$ and, by the above remark, we have $\varepsilon^\tau = \varepsilon^{-1}$. Let $G^+$ denote the Galois group of $K^+/k$, thus $G^+ \cong G/\langle \tau \rangle$. It follows from (A1) that the signatures of $K^+$ and $K$ are respectively $(dm,0)$ and $(2m, m(d-1))$. Therefore the rank of $U_{K^+}$ and $U_K$, the group of units of $K^+$ and $K$, are respectively $dm - 1$ and $2m + m(d-1) - 1 = (dm-1)+m$. Let $U_{\rm Stark}$ be the multiplicative $\mathbb{Z}[G]$-module generated by $\pm 1$, $\varepsilon$ and $U_{K^+}$. Let $R := \{\rho_1, \dots, \rho_m\}$ be a fixed set of representatives of $G$ modulo $\langle \tau \rangle$. Set $\varepsilon_\ell := \rho_\ell^{-1}(\varepsilon)$ for $\ell = 1, \dots, m$. Since $\tau(\varepsilon) = \varepsilon^{-1}$, the group $U_{\rm Stark}$ is generated over $\mathbb{Z}$ by $\{\pm 1, \eta_1, \dots, \eta_{dm-1}$, $\varepsilon_1, \dots, \varepsilon_m\}$ where $\eta_1, \dots, \eta_{dm-1}$ is a system of fundamental units of $K^+$. Let $|\cdot|_j$, $1 \leq j \leq (d+1)m$ denote the infinite normalized absolute values of $K$ ordered in the following way. The $2m$ real absolute values of $K$, corresponding to the places over $v$, are $|\cdot|_j := |\rho_j(\cdot)|$ and $|\cdot|_{j+m} := |\rho_j\tau(\cdot)|$ for $1 \leq j \leq m$. The complex absolute values, corresponding to the infinite places not above $v$, are $|\cdot|_j$ for $2m+1 \leq j \leq (d+1)m$. The regulator of $U_{\rm Stark}$ is the absolute value of the determinant of the following matrix[^7] $$\left(\begin{array}{c|c} \log |\eta_i|_j & \log |\varepsilon_\ell|_j \end{array}\right)_{j,(i,\ell)}$$ where $1 \leq j \leq (d+1)m-1$, $1 \leq i \leq dm-1$ and $1 \leq \ell \leq m$. For $1 \leq j \leq (d+1)m$, let $|\cdot|^+_j$ denote the restriction of the absolute value $|\cdot|_j$ to $K^+$. For $1 \leq j \leq m$, the places corresponding to $|\cdot|_j$ and $|\cdot|_j^+$ are real and $\log |\eta_i|^+_j = \log |\eta_i|_j = \log |\eta_i|_{j+m}$. For $2m + 1 \leq j \leq (d+1)m$, the places corresponding to $|\cdot|_j$ and $|\cdot|_j^+$ are respectively complex and real, thus $\log |\eta_i|^+_j = 2 \log |\eta_i|_j$. Note also that $|\varepsilon_\ell|_{j+m} = |\varepsilon_\ell|^{-1}_m$ for $1 \leq j \leq m$ and $|\varepsilon_\ell|_j = 1$ for $2m+1 \leq j \leq (d+1)m$. Therefore the matrix is equal to $$\renewcommand{\arraystretch}{1.2} \left(\begin{array}{c|c} \log |\eta_i|^+_j & \log |\varepsilon_\ell|_j \\[2pt] \hline \log |\eta_i|^+_j & -\log |\varepsilon_\ell|_j \\[2pt] \hline 2 \log |\eta_i|^+_{j'} & 0 \\ \end{array}\right)_{(j,j'),(i,\ell)}$$ where $1 \leq j \leq m$, $2m+1 \leq j' \leq (d+1)m-1$, $1 \leq i \leq dm-1$ and $1 \leq \ell \leq m$. Now we add the $j$-th row to the $(m+j)$-th row for $1 \leq j \leq m$ and we obtain finally the following matrix with the same determinant $$\renewcommand{\arraystretch}{1.2} \left(\begin{array}{c|c} \log |\eta_i|^+_j & \log |\varepsilon_\ell|_j \\[2pt] \hline 2\log |\eta_i|^+_j & 0 \\[2pt] \hline 2 \log |\eta_i|^+_{j'} & 0 \end{array}\right)_{(j,j'),(i,\ell)}.$$ Therefore the regulator of $U_{\rm Stark}$ is $$\label{eq:eq1} \textrm{Reg}(U_{\rm Stark}) = \left|\det(\log |\varepsilon_\ell|_j)_{j,\ell}\det\left(2\log |\eta_i|_{j'}^+\right)_{j',i}\right|$$ where $1 \leq \ell, j \leq m$, $1 \leq i \leq dm-1$ and $j'$ runs through the set $\{1, \dots, m, 2m+1, \dots, (d+1)m-1\}$. The absolute values $|\cdot|^+_1, \dots, |\cdot|^+_m, |\cdot|^+_{2m+1}, \dots, |\cdot|^+_{(d+1)m-1}$ are the absolute values corresponding to all the infinite places of $K^+$ but one. Thus the second term is $2^{dm-1} R_{K^+}$. For the first term, we have $$|\det(\log |\varepsilon_\ell|_j)_{j,\ell}| = |\det(\log |\varepsilon^{\rho\lambda^{-1}}|)_{\rho,\lambda \in R}|.$$ We say that a character $\chi$ of $G$ is even if $\chi(\tau) = 1$, otherwise $\chi$ is odd and $\chi(\tau) = -1$. The even characters of $G$ are the inflations of characters of $G^+$. We have the following modification of the classical determinant group factorization. \[lem:detgroup\] Let $a_g\in \mathbb{C}$, for $g \in G$, be such that $a_{\tau g} = -a_{g}$ for all $g \in G$. Then $$\det(a_{\rho\lambda^{-1}})_{\rho, \lambda \in R} = \prod_{\chi \textup{ odd}} \sum_{\rho \in R} \chi(\rho) a_\rho.$$ Let $E$ be the $\mathbb{C}$-vector space of functions $f : G \to \mathbb{C}$ such that $f(\tau g) = -f(g)$ for all $g \in G$. Clearly it has dimension $m$ and admits $(\chi)_{\chi \text{ odd}}$ has a basis. Another basis is given by the functions $(\delta_\rho)_{\rho \in R}$ defined by $$\delta_\rho(\rho) = 1,\ \delta_\rho(\tau\rho) = -1 \text{ and } \delta_\rho(g) = 0 \text{ for all } g \in G \text{ with } g \not= \rho, \tau\rho.$$ The group $G$ acts on $E$ by $f^\sigma : g \mapsto f(g\sigma)$ for $f \in E$ and $\sigma \in G$. In particular, we have $f^\tau = -f$. We extend this action linearly to give $E$ a structure of $\mathbb{C}[G]$-module. Now consider the endomorphism defined by $T := \sum\limits_{g \in G} a_g g$. We have $$\begin{aligned} T(\delta_\rho) & = \sum_{\substack{g \in G \\ \rho g^{-1} \in R}} a_g \delta_\rho^g + \sum_{\substack{g \in G \\ \rho g^{-1} \not\in R}} a_g \delta_\rho^g = \sum_{\substack{g \in G \\ \rho g^{-1} \in R}} a_g \delta_{\rho g^{-1}} - \sum_{\substack{g \in G \\ \rho g^{-1} \not\in R}} a_g \delta_{\tau\rho g^{-1}}. \\ \intertext{We write $\lambda = \rho g^{-1}$ in the first sum and $\lambda = \tau\rho g^{-1}$ in the second one. We get} T(\delta_\rho) & = \sum_{\lambda \in R} a_{\rho\lambda^{-1}} \delta_{\lambda} - \sum_{\lambda \in R} a_{\tau\rho\lambda^{-1}} \delta_{\lambda} = 2 \sum_{\lambda \in R} a_{\rho\lambda^{-1}} \delta_{\lambda}.\end{aligned}$$ Therefore the determinant of $T$ is $2^m \det(a_{\rho\lambda^{-1}})_{\rho, \lambda \in R}$. On the other hand, for $\chi$ odd, we compute $$\begin{aligned} T(\chi) = \sum_{g \in G} a_g \chi^g = \sum_{g \in G} a_g \chi(g) \chi.\end{aligned}$$ Thus $\chi$ is an eigenvector for $T$ with eigenvalue $\sum\limits_{g \in G} a_g \chi(g) = 2\sum\limits_{\rho \in R} \chi(\rho) a_\rho$. Therefore $\det(T) = 2^m \prod\limits_{\chi \text{ odd}} \sum\limits_{\rho \in R} \chi(\rho) a_\rho$ and the result follows. By the lemma, we get $$\begin{aligned} \label{eq:eq2} \det(\log |\varepsilon^{\rho\lambda^{-1}}|)_{\rho,\lambda \in R} & = \prod_{\chi \text{ odd}} \sum_{\rho \in R} \chi(\rho) \log |\varepsilon^\rho| = \frac{1}{2} \prod_{\chi \text{ odd}} \sum_{g \in G} \chi(g) \log |\varepsilon^g| \notag \\ & = \prod_{\chi \text{ odd}} L'_{K/k,S}(0, \chi)\end{aligned}$$ using part (1) for the last equality and the fact that the number of roots of unity in $K$ is $2$ since $K$ is not totally complex by (A1). On the other hand, we have $$\label{eq:deczet} \prod_{\chi \text{ odd}} L_{K/k,S}(s, \chi) = \frac{\zeta_{S,K}(s)}{\zeta_{S,K^+}(s)}$$ where $\zeta_{S,K}(s) := \zeta_{S_K, K}(s)$ and $\zeta_{S,K^+}(s) := \zeta_{S_{K^+}, K^+}(s)$ denote respectively the Dedekind zeta functions of $K$ and $K^+$ with the Euler factors at primes in $S_K$ and $S_{K^+}$ removed. Here $S_K$ and $S_{K^+}$ denote respectively the set of places of $K$ and of $K^+$ above the places in $S$. We will often use by abuse the subscript $S$ instead of $S_K$ or $S_{K^+}$ to simplify the notation. Taking the limit when $s \to 0$ in and using the expression for the Taylor development at $s = 0$ of Dedekind zeta functions, see [@tate:book Cor. I.1.2], we get $$\label{eq:eq3} \prod_{\chi \text{ odd}} L'_{K/k,S}(0, \chi) = 2^{t_S} \frac{h_K R_K}{h_K^+ R_K^+}$$ where $t_S$ is the number of prime ideals in $S_{K^+}$ that are inert in $K/K^+$ and $h_K$, $R_K$, $h_{K^+}$ and $R_{K^+}$ are respectively the class numbers and regulators of $K$ and $K^+$. Putting together equations , and , we get the following result. \[th:globindex\] The index of $U_{\rm Stark}$ in the group of units of $K$ is $$(U_K: U_{\rm Stark}) = 2^{t_S+dm-1} \frac{h_K}{h_{K^+}}$$ where $t_S$ is the number of prime ideals in $S_{K^+}$ that are inert in $K/K^+$. Let ${\mathrm{Cl}}_K$ and ${\mathrm{Cl}}_{K^+}$ denote respectively the class groups of $K$ and $K^+$. Define ${\mathrm{Cl}}^-_K$ and $U^-_K$ as the kernel of the following maps induced by the norm $\mathcal{N} := 1 + \tau$ of the extension $K/K^+$ $${\mathrm{Cl}}_K^- := {\mathrm{Ker}}(\mathcal{N} : {\mathrm{Cl}}_K \to {\mathrm{Cl}}_{K^+}) \quad\text{and}\quad U_K^- := {\mathrm{Ker}}(\mathcal{N} : \bar{U}_K \to \bar{U}_{K^+})$$ where $\bar{U}_K$ and $\bar{U}_{K^+}$ are respectively $U_K/\{\pm 1\}$ and $U_{K^+}/\{\pm 1\}$. From now on, we use the additive notation to denote the action of $\mathbb{Z}[G]$, and other group rings, on $\bar{U}_K$ and its subgroups $U_K^-, \bar{U}_{K^+}, \dots$. For $x \in U_K$, we denote by $\bar{x}$ its class in $\bar{U}_K$ and adopt the following convention: if $\bar{x} \in \bar{U}_K$, we let $x$ denote the unique element in the class $\bar{x}$ such that $x > 0$. Note that $\mathcal{N}(x) = \mathcal{N}(-x) = 1$ since $K/K^+$ is ramified at at least one real place. \[th:main\] We have $$\big(U^-_K : \mathbb{Z}[G] \cdot \bar\varepsilon\big) = 2^{e+t_S} |{\mathrm{Cl}}_K^-|$$ where $2^e = (\bar{U}_{K^+}:\mathcal{N}(\bar{U}_K))$. By class field theory the map $\mathcal{N} : {\mathrm{Cl}}_K \to {\mathrm{Cl}}_{K^+}$ is surjective. Therefore $|{\mathrm{Cl}}_K^-| = h_K/h_{K^+}$. On the other hand, if we let $\bar{U}_{\rm Stark} := U_{\rm Stark}/\{\pm 1\}$, we have $${\mathrm{Ker}}\left(\mathcal{N} : \bar{U}_\mathrm{Stark} \to \bar{U}_{K^+}\right) = \mathbb{Z}[G] \cdot \bar\varepsilon \quad\text{and}\quad \mathrm{Im} \left(\mathcal{N} : \bar{U}_\mathrm{Stark} \to \bar{U}_{K^+}\right) = 2 \cdot \bar{U}_{K^+}.$$ Therefore we get $$(\bar{U}_K: \bar{U}_{\rm Stark}) = (\mathcal{N}(\bar{U}_K) : 2 \cdot \bar{U}_{K^+}) \, \big(U_K^- : \mathbb{Z}[G] \cdot \bar\varepsilon\big).$$ Since $(\bar{U}_K: \bar{U}_{\rm Stark}) = (U_K: U_{\rm Stark})$, it follows from Theorem \[th:globindex\] that $$\big(U_K^- : \mathbb{Z}[G] \cdot \bar\varepsilon\big) = \frac{2^{t_S+dm-1} |{\mathrm{Cl}}_K^-|}{\big(\mathcal{N}(\bar{U}_K) : 2 \cdot \bar{U}_{K^+}\big)}.$$ We conclude by noting that $$\big(\mathcal{N}(\bar{U}_K) : 2 \cdot \bar{U}_{K^+}\big) = \frac{\big(\bar{U}_{K^+} : 2 \cdot \bar{U}_{K^+}\big)}{\big(\bar{U}_{K^+}: \mathcal{N}(\bar{U}_K) \big)} = \frac{2^{dm-1}}{\big(\bar{U}_{K^+}: \mathcal{N}(\bar{U}_K) \big)}.\qedhere$$ It has been observed that the Stark unit is quite often a square. The theorem provides us with a necessary condition for that to happen. \[cor:whensquare\] Let $c$ be the $2$-valuation of the order of ${\mathrm{Cl}}_K^-$. A necessary condition for the Stark unit $\varepsilon$ to be a square in $K$ is $$e+t_S+c \geq m.$$ Assume that $\varepsilon = \eta^2$ with $\eta \in K$. Then it is easy to see that $\eta \in U_K^-$ and therefore $\big(\mathbb{Z}[G] \cdot \bar\varepsilon : \mathbb{Z}[G] \cdot \bar\eta\big) = 2^m$ divides $2^{e+t_S} |{\mathrm{Cl}}_K^-|$. We will see below, see , that $e \geq (d-1)m-2$. Therefore the inequality in the theorem is always satisfied for $d \geq 2 + 2/m$. However, this is not enough to ensure that the Stark unit is a square in general. Indeed at the end of the paper we give an example of a cyclic sextic extension $K/k$ satisfying (A1), (A2) and (A3), and with $k$ a totally real cubic field where the Stark unit, assuming it exists, is not a square even though $e > m$. But, in all the cases that we study, we can prove that for $d$ sufficiently large the Stark unit is always a square. Of course theses cases are quite specific and it is difficult to draw from them general conclusions, but still we are lead to ask the following question. Fix a relative degree $m$. Does there exist a constant $D(m)$, depending only on $m$, such that for any extensions $K/k$ of degree $2m$ and any finite set of places $S$ containing $S(K/k)$ satisfying (A1), (A2) and (A3), and with $d \geq D$, the corresponding Stark unit, assuming that it exists, is always a square in $K$? Rubin’s index formula ===================== In [@rubin:index], Rubin proves Gras conjecture type results for Stark units using Euler systems. His results are generalized by Popescu [@popescu:survey]. In this section, we use the results of Rubin to get a similar result in our settings. To be able to use Rubin’s results we need to make the following additional assumption: 1. $K$ contains the Hilbert Class Field $H_k$ of $k$. **We assume in this section that the conjecture is true for the extensions and set of places as described in [@rubin:index].** We first introduce the results of Rubin. Let $\mathfrak{f}$ be the conductor of $K/k$. For any modulus $\mathfrak{g}$ dividing $\mathfrak{f}$, let $K_\mathfrak{g} = K \cap k(\mathfrak{g})$ be the intersection of $K$ with the ray class field of $k$ of conductor $\mathfrak{g}$. Since $v$ is totally split in $K/k$, one can apply the conjecture to the extension $K_\mathfrak{g}/k$, the set of places $S(K_\mathfrak{g}/k)$ and the place $v$, and get a Stark unit that we denote by $\varepsilon_\mathfrak{g}$. Let $G_\mathfrak{g}$ be the Galois group of $K_\mathfrak{g}/k$. Note that by (A1) the group of roots of unity in $K_\mathfrak{g}$ is $\{\pm 1\}$. Part (2) of the conjecture is equivalent to the fact that $\varepsilon_\mathfrak{g}^{g - 1} \in U_{K_\mathfrak{g}}^2$ for all $g \in G_\mathfrak{g}$, see [@tate:book Prop. IV.1.2]. Define $R_{\rm Stark}$ as the following $\mathbb{Z}[G]$-module $$R_{\rm Stark} = \langle \pm 1,\ (\varepsilon_\mathfrak{g}^{g - 1})^{1/2} \text{ for } \mathfrak{g} \mid \mathfrak{f} \text{ and } g \in G_\mathfrak{g} \rangle_{\mathbb{Z}[G]}.$$ Let $p$ be a prime number that does not divide the order of $G$. In particular, $p$ is an odd prime. Denote by $\hat{G}_p$ the set of irreducible $\mathbb{Z}_p$-characters of $G$. For $\psi \in \hat{G}_p$ and $M$ a $\mathbb{Z}[G]$-module, we set $$M^\psi := M \otimes_{\mathbb{Z}[G]} \mathbb{Z}_p[\psi]$$ where $\mathbb{Z}_p[\psi]$ is the ring generated over $\mathbb{Z}_p$ by the values of $\psi$ and $G$ acts on $\mathbb{Z}_p[\psi]$ via the character $\psi$. The following result is a direct consequence of Theorem 4.6 of [@rubin:index]. If $\psi \in \hat{G}_p$ is odd then $$\left|(U_K/R_{\rm Stark})^\psi\right| = \left|{\mathrm{Cl}}_K^\psi\right|.$$ From this we deduce an analogue statement for our case. \[th:localindex\] For all $\psi \in \hat{G}_p$, we have $$\left|(U_K^-/\mathbb{Z}[G] \cdot \bar\varepsilon)^\psi\right| = \left|({\mathrm{Cl}}_K^-)^\psi\right|.$$ For $M$ a $\mathbb{Z}[G]$-module and $\psi \in \hat{G}_p$, it is direct to see that $M^\psi = (M^{1+\tau})^\psi$ if $\psi$ is even and $M^\psi = (M^{1-\tau})^\psi$ if $\psi$ is odd. In particular, if $\psi$ is even, we get $|(U_K^-/\mathbb{Z}[G] \cdot \bar\varepsilon)^\psi| = |({\mathrm{Cl}}_K^-)^\psi| = 1$ and the result follows trivially in that case. Assume now that $\psi$ is odd. Let $\varepsilon_0$ be the Stark unit corresponding to the extension $K/k$, the set of places $S(K/k)$ and the distinguished place $v$. Assume first that $S = S(K/k) \cup \{\mathfrak{p}\}$ for some finite prime ideal $\mathfrak{p}$ of $k$ not in $S(K/k)$. It follows from [@tate:book Prop. IV.3.4] that $\bar\varepsilon = (1-{\mathrm{F}}_\mathfrak{p}(K/k)) \cdot \bar\varepsilon_0$ where ${\mathrm{F}}_\mathfrak{p}(K/k)$ is the Frobenius at $\mathfrak{p}$ for the extension $K/k$. By (A3), $\tau$ is a power of ${\mathrm{F}}_\mathfrak{p}(K/k)$ and thus $\psi({\mathrm{F}}_\mathfrak{p}(K/k))$ is a non trivial root of unity of order dividing $|G|$. Then $1 - \psi(F_\mathfrak{p}(K/k))$ is a $p$-adic unit and therefore $(\mathbb{Z}[G] \cdot \bar\varepsilon)^\psi = (\mathbb{Z}[G] \cdot \bar\varepsilon_0)^\psi$. By repeating this argument if necessary, we see that this last equality also holds in the general case. Now, by taking $\mathfrak{g} = \mathfrak{f}$ and $\sigma = \tau$ in the definition of $R_{\rm Stark}$, we see that $\varepsilon_0^{(\tau-1)/2} = \varepsilon_0^{-1} \in R_{\rm Stark}$. Therefore we have $\varepsilon_0^{\mathbb{Z}[G]} \subset R_{\rm Stark} \subset U_K$, and thus $$\varepsilon_0^{2\mathbb{Z}[G]} \subset R_{\rm Stark}^{\tau - 1} \subset U_K^{\tau - 1}.$$ We take the $\psi$-component, by the above remarks and the theorem, we get $$\begin{gathered} |(U_K^-/\mathbb{Z}[G] \cdot \bar\varepsilon)^\psi| = |(U_K^-/\mathbb{Z}[G] \cdot \bar\varepsilon_0)^\psi| = |(U_K^{\tau - 1}/\varepsilon_0^{2\mathbb{Z}[G]})^\psi| \geq \\ |(U_K^{\tau-1}/R_{\rm Stark}^{\tau-1})^\psi| = |(U_K/R_{\rm Stark})^\psi| = |{\mathrm{Cl}}_K^\psi| = |({\mathrm{Cl}}_K^-)^\psi|.\end{gathered}$$ Assume there exists a character $\psi$ for which this is a strict inequality. Multiplying over all characters in $\hat{G}_p$, we get $|(U_K^-/\mathbb{Z}[G] \cdot \bar\varepsilon) \otimes \mathbb{Z}_p| > |{\mathrm{Cl}}_K^- \otimes \mathbb{Z}_p|$, a contradiction with Theorem \[th:main\]. Therefore the equality holds for all $\psi \in \hat{G}_p$ and the theorem is proved. The index property ================== **From now on, we do not assume any more that the conjecture is true.** From the results of the previous sections, we see that the conjecture implies that there exists a unit $\bar\varepsilon \in U_K^-$ such that[^8] 1. $\left(U_K^- : \mathbb{Z}[G] \cdot \bar\varepsilon\right) = 2^{e+t_S} |{\mathrm{Cl}}_K^-|$, 2. $\left|(U_K^-/\mathbb{Z}[G] \cdot \bar\varepsilon)^\psi\right| = \left|({\mathrm{Cl}}_K^-)^\psi\right|$ for all $p \nmid [K:k]$ and $\psi \in \hat{G}_p$. A priori the existence of a solution to (P1) and (P2) does not imply in return the conjecture (except for quadratic extensions, see Theorem \[th:quadratic\] below). Indeed, in general, properties (P1) and (P2) do not even characterize the Stark unit $\varepsilon$. To see that assume that $\bar\eta$ is a solution to (P1) and (P2), and let $\bar\eta' := u \cdot \bar\eta$ where $u \in \mathbb{Z}[G]^\times$ is a unit of $\mathbb{Z}[G]$. Then $\bar\eta'$ also satisfies (P1) and (P2). If $u$ belongs to $\{\pm \gamma : \gamma \in G\} \subset \mathbb{Z}[G]^\times$, the group of trivial units of $\mathbb{Z}[G]$, then $\bar\eta'$ is essentially the *same solution* since it is a conjugate of $\bar\eta$ or the inverse of a conjugate of $\bar\eta$. However there may be some non trivial units in $\mathbb{Z}[G]$ (see the end of this section) and thus solutions to (P1) and (P2) that are not related in any obvious way to the Stark unit. In any case, we have the following result that shows that solutions to (P1) satisfy a very weak version of part (1) of the conjecture. \[prop:prodform\] Let $\bar\eta$ be an element of $U_K^-$ satisfying . Then we have $$\label{eq:factform} \prod_{\chi\ \mathrm{odd}} \frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g| = \pm \prod_{\chi\ \mathrm{odd}} L'_{K/k,S}(0, \chi).$$ Let $\bar x \in U_K^-$. Using the notations of Section \[sec:index\], we have $|x^\tau|_j = |x|_j$ for $2m+1 \leq j \leq (d+1)m$ since these absolute values are complex and $\tau$ is the complex conjugation. Since, by construction, we have $x^\tau = x^{-1}$, it follows that $|x|^2_j = |x^{1+\tau}|_j = 1$ and $|x|_j = 1$ for $2m+1 \leq j \leq (d+1)m$. We can therefore reproduce the determinant computation done in Section \[sec:index\] replacing $\varepsilon$ by $\eta$ and $U_\mathrm{Stark}$ by the subgroup $U_0$ of $U_K$ generated by $U_{K^+}$ and the conjugates of $\eta$. We get $$(U_K:U_0) = \pm 2^{dm-1} \frac{R_{K^+}}{R_K} \prod_{\chi\ \mathrm{odd}} \frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|.$$ We then proceed as in Theorem \[th:main\] by looking at the kernel of the norm map acting on $U_0/\{\pm 1\}$. Since $\bar\eta$ satisfies (P1), it follows that $$2^{dm-1} \frac{R_{K^+}}{R_K} \prod_{\chi\ \mathrm{odd}} \frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g| = \pm 2^{dm-1+t_S} |{\mathrm{Cl}}_K^-|.$$ Then, by , we get the result $$\prod_{\chi\ \mathrm{odd}} \frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g| = \pm 2^{t_S} \frac{h_K R_K}{h_{K^+} R_{K^+}} = \pm \prod_{\chi\ \mathrm{odd}} L'_{K/k,S}(0, \chi). \qedhere$$ We now turn to the study of the structure of the $\mathbb{Q}[G]$-module $U_K^- \otimes \mathbb{Q}$. Since $U_K^-$ is killed by $1+\tau$, it is a $\mathbb{Q}[G]^-$-module where $\mathbb{Q}[G]^- := e^- \mathbb{Q}[G]$ and $e^- := \frac{1}{2}(1 - \tau)$ is the sum of the idempotents of odd characters of $G$.[^9] Since $U_K^-$ injects into $U_K^- \otimes \mathbb{Q}$, we will identify it with its image. The following result describes the structure of $U_K^- \otimes \mathbb{Q}$ as a Galois module. \[prop:iso\] The module $U_K^- \otimes \mathbb{Q}$ is a free $\mathbb{Q}[G]^-$-module of rank $1$. Let $\mathcal{Y}_K$ be the $\mathbb{Q}$-vector space with basis the elements $z$ in the set $S_\infty(K)$ of infinite places of $K$. The group $G$ acts on $\mathcal{Y}_K$ in the following way: $z^g$ for $g \in G$ and $z \in S_\infty(K)$ is the infinite place defined by $x \mapsto z(x^g)$ for all $x \in K$. Denote by $\mathcal{X}_K$ the subspace of elements $\sum_z a_z \, z \in \mathcal{Y}_K$ such that $\sum_z a_z = 0$. Then the two $\mathbb{Q}[G]$-modules $\mathcal{X}_K$ and $U_K\otimes \mathbb{Q}$ are isomorphic by a result of Herbrand and Artin [@artin:units]. Fix an isomorphism $f : U_K \otimes \mathbb{Q} \to \mathcal{X}_K$. A direct computation shows that $\mathcal{X}_K^- := f(U_K^- \otimes \mathbb{Q})$ is spanned by the vectors $\{w^\rho - w^{\rho\tau}\}_{\rho \in R}$ where $w$ is the fixed place of $K$ above $v$. In particular, $\mathcal{X}_K^-$ is generated as a $\mathbb{Q}[G]^-$-module by the vector $w - w^\tau$. This proves the result. There exist $\bar\theta \in \bar{U}_K^-$ and $q \in \mathbb{Q}^\times$ such that $$\prod_{\chi\ \mathrm{odd}} \frac{1}{2} \sum_{g \in G} \chi(g) \log |\theta^g| = q \prod_{\chi\ \mathrm{odd}} L'_{K/k,S}(0, \chi).$$ From the proposition, there exists $u \in U_K^- \otimes \mathbb{Q}$ such that $U_K^- \otimes \mathbb{Q} = \mathbb{Q}[G]^- \cdot u$. We let $\bar\theta := n \cdot u$ where $n \in \mathbb{N}$ is large enough so that $\bar\theta \in U_K^-$. Then we set $$q := \frac{(U_K^- : \mathbb{Z}[G] \cdot \bar\theta)}{2^{e+t} |{\mathrm{Cl}}_K^-|}.$$ The result follows by the proof of Proposition \[prop:prodform\] *mutatis mutandis* and replacing $q$ by $-q$ if necessary. Thanks to Proposition \[prop:iso\], it is enough to study the structure of $\mathbb{Q}[G]^-$ to understand that of $U_K^- \otimes \mathbb{Q}$. Let $X$ be the set of irreducible $\mathbb{Z}$-characters of $G$. Each $\xi \in X$ is the sum of the irreducible characters in a conjugacy class $C_\xi$ of $\hat{G}$ under the action of ${\mathrm{Gal}}(\bar{\mathbb{Q}}/\mathbb{Q})$. For $\xi \in X$, we let $e_\xi := \sum_{\chi \in C_\xi} e_\chi \in \mathbb{Q}[G]$ be the corresponding rational idempotent where $e_\chi$ denotes the idempotent associated to the character $\chi$. We have $$\mathbb{Q}[G] = \bigoplus_{\xi \in X} e_\xi \mathbb{Q}[G] \simeq \bigoplus_{\xi \in X} \mathbb{Q}(\xi)$$ where $\mathbb{Q}(\xi)$ is the cyclotomic field generated by the values of any character in $C_\xi$. Let $X_\mathrm{odd}$ be the set of $\mathbb{Z}$-characters $\xi \in X$ such that one, and thus all, characters in $C_\xi$ are odd. We have $e^- = \sum_{\xi \in X_\mathrm{odd}} e_\xi$ and from the above decomposition, we get $$\label{eq:QG-} \mathbb{Q}[G]^- = \bigoplus_{\xi \in X_\mathrm{odd}} e_\xi \mathbb{Q}[G] \simeq \bigoplus_{\xi \in X_\mathrm{odd}} \mathbb{Q}(\xi).$$ We now define $\mathbb{Z}[G]^ - := e^- \mathbb{Z}[G]$ and let $\mathcal{O}_G^-$ be the maximal order of $\mathbb{Q}[G]^-$. We have $$\label{eq:OG-} \mathcal{O}_G^- = \bigoplus_{\xi \in X_\mathrm{odd}} e_\xi \mathbb{Z}[G] \simeq \bigoplus_{\xi \in X_\mathrm{odd}} \mathbb{Z}[\xi].$$ Now let $p$ be a prime number. By , we get $$\label{eq:QpG-} \mathbb{Q}_p[G]^- \simeq \bigoplus_{\xi \in X_\mathrm{odd}} \mathbb{Q}(\xi) \otimes_\mathbb{Q} \mathbb{Q}_p \simeq \bigoplus_{\xi \in X_\mathrm{odd}} \bigoplus_{\mathfrak{p} \in S_{\xi, p}} \mathbb{Q}(\xi)_\mathfrak{p}$$ where $S_{\xi, p}$ is the set of prime ideals of $\mathbb{Q}(\xi)$ above $p$ and $\mathbb{Q}(\xi)_\mathfrak{p}$ is the completion of $\mathbb{Q}(\xi)$ at the prime ideal $\mathfrak{p}$. On the other hand, each rational character $\xi \in X$ is the sum of irreducible $\mathbb{Z}_p$-characters, say $\xi = \sum_{\psi \in C_{\xi, p}} \psi$, and we have $$\mathbb{Q}_p[G]^- = \bigoplus_{\xi \in X_\mathrm{odd}} \bigoplus_{\psi \in C_{\xi, p}} e_\psi \mathbb{Q}_p[G]^-.$$ Therefore there is a bijection between the prime ideals in $S_{\xi, p}$ and the characters in $C_{\xi, p}$. For $\mathfrak{p}$ a prime ideal in $S_{\xi, p}$, we denote by $\psi_{\xi,\mathfrak{p}}$ the corresponding irreducible $\mathbb{Z}_p$-character. Before stating the first result, we need one more notation. Let $T$ be a set of primes. We say that an element $u \in \mathbb{Q}[G]$ is a $T$-unit if $u \in \mathbb{Z}_p[G]^{-,\times}$ for all $p \not\in T$ where $ \mathbb{Z}_p[G]^{-,\times}$ is the group of units of $\mathbb{Z}_p[G]^-$. \[prop:unicity\] Let $M$ be a sub-$\mathbb{Z}[G]^-$-module of $\mathbb{Q}[G]^-$ of finite index. Let $x$ be an element of $M$ such that $x\mathbb{Z}[G]^-$ has finite index inside $M$. Assume that $y$ is another element of $M$ such that $$(M: x\mathbb{Z}[G]^-) = (M: y\mathbb{Z}[G]^-) \quad\text{and}\quad (M^\psi: (x\mathbb{Z}_p[G]^-)^\psi) = (M^\psi : (y\mathbb{Z}_p[G]^-)^\psi)$$ for all $p \nmid |G|$ and all $\psi \in \hat{G}_p$ with $\psi$ odd. Then there exists a unique $B$-unit $u \in \mathbb{Q}[G]^-$ such that $y = ux$ where $B$ is the set of primes dividing both $|G|$ and $(M: x\mathbb{Z}[G]^-)$. Since $\mathbb{Q}[G]^- = x\mathbb{Q}[G]^-$, there exists $u \in \mathbb{Q}[G]^-$ such that $y = ux$. Assume $y = vx$ for another $v \in \mathbb{Q}[G]^-$. Then, for all $\xi \in X_\mathrm{odd}$, we have $\xi(u)\xi(x) = \xi(v)\xi(x)$. Since $\xi(x) \not= 0$, it follows that $\xi(u) = \xi(v)$ and thus by , we get $u = v$ which proves that $u$ is unique. Let $p$ be a prime. Assume first that $p$ does not divide $|G|$. Let $\xi \in X_\mathrm{odd}$ and $\mathfrak{p} \in S_{\xi, p}$. Write $\psi := \psi_{\xi, \mathfrak{p}}$ and denote by $\mathbb{Z}[\xi]_{\mathfrak{p}} := \psi(\mathbb{Z}_p[G]^-)$ the ring of integers of $\mathbb{Q}(\xi)_{\mathfrak{p}}$. Then $M^\psi$ is an ideal of $\mathbb{Z}[\xi]_{\mathfrak{p}}$ and we have $$\frac{(M^\psi: (x\mathbb{Z}_p[G]^-)^\psi)}{(M^\psi : (y\mathbb{Z}_p[G]^-)^\psi)} = \frac{(M^\psi : \psi(y) \mathbb{Z}[\xi]_{\mathfrak{p}})}{(M^\psi : \psi(x) \mathbb{Z}[\xi]_{\mathfrak{p}})} = |\psi(u)|_{\mathfrak{p}}.$$ Thus $\psi_{\xi, \mathfrak{p}}(u)$ is a unit in $\mathbb{Z}[\xi]_{\mathfrak{p}}$ for all $\xi \in X_\mathrm{odd}$ and $\mathfrak{p} \in S_{\xi, p}$ and thus $u$ lies in $\mathbb{Z}_p[G]^{-,\times}$. Assume now that $p$ does not divide the index $(M: x\mathbb{Z}[G]^-)$. We have $$(M \otimes \mathbb{Z}_p: x\mathbb{Z}_p[G]^-) = (M \otimes \mathbb{Z}_p: y\mathbb{Z}_p[G]^-) = 1.$$ Therefore $x\mathbb{Z}_p[G]^- = M \otimes \mathbb{Z}_p = y\mathbb{Z}_p[G]^-$ and $u \in \mathbb{Z}_p[G]^{-,\times}$. By Propositions \[prop:iso\] and \[prop:unicity\], we get the following result. \[cor:unicity\] Let $B$ be the set of primes that divide both $|G|$ and $|{\mathrm{Cl}}_K^-|$. Assume there exist $\bar\eta$ and $\bar\eta'$ two elements of $U_K^-$ satisfying and . Then there exists a unique $B$-unit $u \in \mathbb{Q}[G]^-$ such that $\bar\eta' = u \cdot \bar\eta$. From this result and the discussion at the beginning of the section, one cannot expect the properties (P1) and (P2) to characterize the Stark unit if $\mathbb{Z}[G]^-$ has some non trivial $B$-units and a fortiori if $\mathbb{Z}[G]^-$ has some non trivial units.[^10] It follows from the methods of [@higman] that $\mathbb{Z}[G]^-$ has some non trivial units if and only if $\mathcal{O}_G^-$ does. By , this is the case if and only if there exists an odd character of $G$ divides $6$. In particular, for $G$ a cyclic group, $\mathbb{Z}[G]^-$ has only trivial units if and only if the order of $G$ is at most $6$. We will prove in the next sections that there exist solutions to (P1) and (P2) in these cases (with some additional conditions for sextic extensions). From this we will deduce another proof of the conjecture for quadratic extensions and a weak version of the conjecture for quartic and sextic extensions. Algebraic tools =============== In this section we introduce some algebraic tools and results that will be useful in the next sections. We start with the properties of Fitting ideals. Let $R$ be a commutative ring with an identity element. Let $M$ be a finitely generated $R$-module. Therefore there exists a surjective homomorphism $f : R^a \to M$ for some $a \geq 1$. The Fitting ideal of $M$ as an $R$-module, denoted ${\mathrm{Fitt}}_R(M)$, is the ideal of $R$ generated by $\det(\vec{v}_1, \dots, \vec{v}_a)$ where $\vec{v}_1, \dots, \vec{v}_a$ run through the elements of the kernel of $f$. One can prove that it does not depend on the choice of $f$. We will use the following properties of Fitting ideals, see [@northcott Chap. 3] or [@eisenbud Chap. 20]. - If there exist ideals $A_1, \dots, A_t$ of $R$ such that $$M \simeq R/A_1 \oplus \cdots \oplus R/A_t,$$ then we have $${\mathrm{Fitt}}_R(M) = A_1 \cdots A_t.$$ - Let $T$ be an $R$-algebra. We have $${\mathrm{Fitt}}_T(M \otimes_R T) = {\mathrm{Fitt}}_R(M) T.$$ - Let $N$ be another finitely generated $R$-module. We have $${\mathrm{Fitt}}_R(M \oplus N) = {\mathrm{Fitt}}_R(M) {\mathrm{Fitt}}_R(N).$$ \[lem:p1\] Let $M$ be a finite $\mathcal{O}_G^-$-module. Then $$|M| = |(\mathcal{O}_G^-/{\mathrm{Fitt}}_{\mathcal{O}_G^-}(M))|.$$ We have $$(\mathcal{O}_G^- : {\mathrm{Fitt}}_{\mathcal{O}_G^-}(M)) = \prod_{\xi \in X_{\rm odd}} (e_\xi \mathbb{Z}[G] : e_\xi {\mathrm{Fitt}}_{\mathcal{O}_G^-}(M)) = \prod_{\xi \in X_{\rm odd}} (\mathbb{Z}[\xi] : {\mathrm{Fitt}}_{\mathbb{Z}[\xi]}(e_\xi M)).$$ Fix $\xi \in X_{\rm odd}$. Since $e_\xi M$ is a finite $\mathbb{Z}[\xi]$-module, there exist ideals $\mathfrak{a}_1, \dots, \mathfrak{a}_r$ such that $$e_\xi M = \mathbb{Z}[\xi]/\mathfrak{a}_1 \oplus \cdots \oplus \mathbb{Z}[\xi]/\mathfrak{a}_r.$$ Therefore ${\mathrm{Fitt}}_{\mathbb{Z}[\xi]}(e_\xi M) = \mathfrak{a}_1 \cdots \mathfrak{a}_r$ and $$(\mathbb{Z}[\xi] : {\mathrm{Fitt}}_{\mathbb{Z}[\xi]}(e_\xi M)) = N_{\mathbb{Q}(\xi)/\mathbb{Q}}(\mathfrak{a}_1 \cdots \mathfrak{a}_r) = |e_\xi M|.$$ It follows that $(\mathcal{O}_G^- : {\mathrm{Fitt}}_{\mathcal{O}_G^-}(M)) = \prod_{\xi \in X_{\rm odd}} |e_\xi M| = |M|$. \[lem:p2\] Let $M$ be a finite $\mathbb{Z}[G]^-$-module. Let $p$ be a prime number not dividing $|G|$ and let $\psi$ be an odd irreducible $\mathbb{Z}_p$-character. Then $$|M^\psi| = |(\mathbb{Z}[G]^-/{\mathrm{Fitt}}_{\mathbb{Z}[G]^-}(M))^\psi| = |(\mathcal{O}_G^-/{\mathrm{Fitt}}_{\mathcal{O}_G^-}(M))^\psi|.$$ We have $({\mathrm{Fitt}}_{\mathbb{Z}[G]^-}(M))^\psi = {\mathrm{Fitt}}_{\mathbb{Z}_p[\psi]}(M^\psi)$. Since $M^\psi$ is a finite $\mathbb{Z}_p[\psi]$-module, there exist integers $c_1, \dots, c_r \geq 1$ such that $$M^\psi \simeq \bigoplus_{i=1}^r \mathbb{Z}_p[\psi]/\mathfrak{p}^{c_i}$$ where $\mathfrak{p}$ is the prime ideal of $\mathbb{Z}_p[\psi]$. Then ${\mathrm{Fitt}}_{\mathbb{Z}[G]^-}(M)^\psi = \mathfrak{p}^{c}$ with $ c := c_1 + \cdots + c_r$ and therefore $|(\mathbb{Z}[G]^-/{\mathrm{Fitt}}_{\mathbb{Z}[G]^-}(M))^\psi| = (\mathbb{Z}_p[\xi] :\mathfrak{p}^c) = |M^\psi|$. The last equality is clear since $(\mathcal{O}_G^-)^\psi = \mathbb{Z}_p[\psi]$. In what follows we will also use repeatedly the Tate cohomology of finite cyclic groups, see [@lang:book §IX.1]. Let $A$ be a finite cyclic group with generator $a$ and let $M$ be a $\mathbb{Z}[A]$-module. The zero-th and first group of cohomology are defined by $$\hat{H}^0(A, M) := M^A/N_A(M) \quad\text{and}\quad \hat{H}^1(A, M) := {\mathrm{Ker}}(N_A: M \to M)/(1-a) M$$ where $N_A := \sum_{b \in A} b$ and $M^A$ is the submodule of elements in $M$ fixed by $A$. Let $N$ and $P$ be two other $\mathbb{Z}[A]$-modules such that the following short sequence is exact: $$\xymatrix{ 1 \ar[r] & M \ar[r] & N \ar[r] & P \ar[r] & 1. }$$ Then the hexagon below is also exact. $$\label{eq:hexa}\xymatrix{ & \hat{H}^0(A, M) \ar[rd] & \\ \hat{H}^1(A, P) \ar[ru] & & \hat{H}^0(A, N) \ar[d] \\ \hat{H}^1(A, N) \ar[u] & & \hat{H}^0(A, P) \ar[ld] \\ & \hat{H}^1(A, M) \ar[lu] & \\ }$$ The Herbrand quotient of $M$ is defined by $$Q(A, M) := \frac{|\hat{H}^0(A, M)|}{|\hat{H}^1(A, M)|}.$$ The Herbrand quotient is multiplicative, that is for an exact short sequence as above, we have $Q(A, N) = Q(A, M) \, Q(A, P)$. The following result plays a crucial rôle in the next sections. It is a direct consequence of [@lang:book Cor. IX.4.2]. \[lem:qu\] Let $E/F$ be a quadratic extension with Galois group $T$. Let $R \geq 0$ be the number of real places in $F$ that becomes complex in $E$. Then we have $$Q(T, U_E) = 2^{R-1}. \tag*{\hspace{-1em}\qed}$$ We use this result in the following way. Assume that $R \geq 1$. Write $\bar{U}_F$ and $\bar{U}_E$ for the group of units of $F$ and $E$ respectively modulo $\{\pm 1\}$. Then we have $$\hat{H}^0(T, U_E) = \frac{U_F}{\mathcal{N}_{E/F}(U_E)} = \{\pm 1\} \frac{\bar{U}_F}{\mathcal{N}_{E/F}(\bar{U}_E)}$$ since $-1$ cannot be a norm in $E/F$. It follows from the lemma that $|\hat{H}^0(T, U_E)|$ is divisible by $2^{R-1}$ and therefore $$\label{eq:bounde} 2^{R-2} \mid (\bar{U}_F:\mathcal{N}_{E/F}(\bar{U}_F)).$$ In some cases we will not be able to get non trivial lower bounds with that method, but still be able to deduce that $\hat{H}^1(T, U_E)$ is trivial. In this situation, we have the following lemma. \[lem:h11\] Let $E/F$ be a quadratic extension with Galois group $T$. Assume that $\hat{H}^1(T, U_E)$ is trivial. Then either $E/F$ is unramified at finite places or there exists an element of order $2$ in the kernel of the norm map from ${\mathrm{Cl}}_E$ to ${\mathrm{Cl}}_F$. Consider the submodules of elements fixed by $T$ in the short exact sequence $$\xymatrix{ 1 \ar[r] & U_E \ar[r] & E^\times \ar[r] & P_E \ar[r] & 1. }$$ We get $$\xymatrix{ 1 \ar[r] & U_F \ar[r] & F^\times \ar[r] & P_E^T \ar[r] & \hat{H}^1(T, U_E) \ar[r] & \cdots. }$$ Since $\hat{H}^1(T, U_E) = 1$ by hypothesis, it follows that $P_F \simeq P_E^T$. The isomorphism is the natural map that sends $\mathfrak{a} \in P_F$ to $\mathfrak{a}\mathbb{Z}_E \in P_E^T$. Assume that there is a prime ideal $\mathfrak{p}$ of $F$ that ramifies in $E/F$. Let $\mathfrak{P}$ be the unique prime ideal of $E$ above $\mathfrak{p}$ and let $h \geq 1$ be the order of $\mathfrak{P}$ in ${\mathrm{Cl}}_E$. Since $\mathfrak{P}^h \in P_E^T$, there exists a principal ideal $\mathfrak{a} \in P_F$ such that $\mathfrak{P}^h = \mathfrak{a}\mathbb{Z}_E$. Clearly $\mathfrak{a}$ is a power of $\mathfrak{p}$. Looking at valuations at $\mathfrak{P}$, it follows that $h$ is even. We set $\mathfrak{C} := \mathfrak{P}^{h/2}$. Its class is an element of order $2$ in ${\mathrm{Cl}}_E$. But $\mathcal{N}_{E/F}(\mathfrak{C}) = \mathfrak{p}^{h/2} = \mathfrak{a}$ is a principal ideal. This concludes the proof. To conclude this section we prove the conjecture in our settings when $K/k$ is a quadratic extension. This result is proved in full generality in [@tate:book Th. IV.5.4]. \[th:quadratic\] Let $K/k$ be a quadratic extension and $S \supset S(K/k)$ be a finite set of places of $k$ satisfying , and . Then the abelian rank one Stark conjecture is satisfied for the extension $K/k$ and the set $S$ with the Stark unit being the unique solution, up to trivial units, of and . Moreover the Stark unit is a square in $K$ if and only if $e+t_S+c \geq 1$ where $c$ is the $2$-valuation of the order of ${\mathrm{Cl}}_K^-$. In particular, if $d \geq 4$ then it always a square and, in fact, it is a $2^{d-3}$-th power. It is also a square if $d = 3$ and the extension $K/k$ is ramified at some finite prime. The only non trivial element of $G$ is $\tau$. Let $\chi$ be the character that sends $\tau$ to $-1$. It is the only non trivial character of $G$ and also the only odd character. We have $\mathbb{Z}[G]^- = \mathcal{O}_G^- = e^- \mathbb{Z} \simeq \mathbb{Z}$. In particular, using Proposition \[prop:iso\], it is direct to see that there exists $\bar\theta \in U_K^-$ such that $U_K^- = \mathbb{Z} \cdot \bar\theta$. Define $$\bar\eta := 2^{e+t_S} |{\mathrm{Cl}}_K^-| \cdot \bar\theta.$$ From its construction, it is clear that $\bar\eta$ satisfies (P1) and (P2). It follows from Proposition \[prop:prodform\], and replacing $\eta$ by $\eta^{-1}$ if necessary, that $$\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g| = L'_{K/k}(0, \chi).$$ This proves part  (1) of the conjecture. Part (3) is direct by construction. It remains to prove part (2). But $(\tau - 1) \cdot \bar\eta = -2 \cdot \bar\eta$ so part (2) follows and the conjecture is proved in this case. Finally, from its definition, it is clear that $\eta$ is a $2^r$-th power in $K^\times$ if and only if $e+t_S+c \geq r$. Now, by , we have $e \geq d - 3$ and therefore the Stark unit is always a square if $d \geq 4$. Assume that $d = 3$ and that $\eta$ is not a square. Then $e = 0$ and $|\hat{H}^0(G, U_K)| = 2$. From Lemma \[lem:qu\], we get $\hat{H}^1(G, U_K) = 1$ and therefore, since $c = 0$, the extension $K/k$ is unramified at finite places by Lemma \[lem:h11\] When $d = 2$, there exist extensions for which the Stark unit is a square and extensions for which it is not a square. Using the PARI/GP system [@PARI], we find the following examples.[^11] Let $k := \mathbb{Q}(\sqrt{5})$ and let $v_1, v_2$ denote the two infinite places of $k$ with $v_1(\sqrt{5}) < 0$ and $v_2(\sqrt{5}) > 0$. Let $K$ be the ray class field modulo $\mathfrak{p}_{11} v_2$ where $\mathfrak{p}_{11} := (1/2 + 3\sqrt{5}/2)$ is one of the two prime ideals above $11$. Then $K/k$ is a quadratic extension that satisfies (A1), (A2) and (A3) with $S := S(K/k)$, and one can prove that the corresponding Stark unit is not a square. Now, on the other hand, let $K$ be the ray class field modulo $\sqrt{5}\mathfrak{q}_{11} v_1$, where $\mathfrak{q}_{11} := (1/2 - 3\sqrt{5}/2)$ is the other prime ideal above $11$. Then $K/k$ is a quadratic extension that satisfies (A1), (A2) and (A3) with $S := S(K/k)$ and, in this case, the Stark unit is a square. When $d = 3$ and $K/k$ is unramified both cases are possible. Indeed, let $k := \mathbb{Q}(\alpha)$ where $\alpha^3 - \alpha^2 - 13\alpha + 1 = 0$. It is a totally real cubic field. Let $v_1, v_2, v_3$ be the three infinite places of $k$ with $v_1(\alpha) \approx -3.1829$, $v_2(\alpha) \approx 0.0765$ and $v_3(\alpha) \approx 4.1064$. Let $K$ be the ray class field of $k$ of conductor $\mathbb{Z}_k v_2v_3$. Then $K/k$ is a quadratic extension that satisfies (A1), (A2) and (A3) with $S := S(K/k)$, and that is unramified at finite places. One can prove in this setting that the Stark unit is not a square. On the other hand, let $k := \mathbb{Q}(\beta)$ with $\beta^3 - \beta^2 - 24\beta - 35 = 0$. It is a totally real cubic field. Let $v_1, v_2, v_3$ be the three infinite places of $k$ with $v_1(\alpha) \approx -3.0999$, $v_2(\alpha) \approx -1.8861$, and $v_3(\alpha) \approx 5.9860$. Let $K$ be the unique quadratic extension $k$ of conductor $\mathbb{Z}_k v_2v_3$. Then $K/k$ satisfies (A1), (A2) and (A3) with $S := S(K/k)$ and is unramified at finite places. One can prove that $k$ is principal and the class number of $K$ is $2$. Therefore the Stark unit in this case is a square. Cyclic quartic extensions ========================= The goal of this section is to prove the following result. \[th:quartic\] Let $K/k$ be a cyclic quartic extension and $S \supset S(K/k)$ be a finite set of places of $k$ satisfying , and . Then there exists $\bar\eta \in U_K^-$ satisfying and . Furthermore, $\bar\eta$ is unique up to a trivial unit, satisfies for all $\chi \in \hat{G}$ $$\left|L'_{K/k,S}(0, \chi)\right| = \frac{1}{2} \left|\sum_{g \in G} \chi(g) \log |\eta^g|\right|$$ and the extension $K(\sqrt{\eta})/k$ is abelian. Denote by $\gamma$ a generator of $G$, therefore $\tau = \gamma^2$. Let $\chi$ be the character of $G$ such that $\chi(\gamma) = i$ and let $\xi := \chi + \chi^3$ be the only element in $X_\mathrm{odd}$. From the results of Section 4, we have $$\mathbb{Q}[G]^- = e^- \mathbb{Q}[G] \simeq \mathbb{Q}(i)$$ where the isomorphism sends any element of $x \in \mathbb{Q}[G]^-$, written uniquely as $x = e^- (a + b\gamma)$ for $a, b \in \mathbb{Q}$, to $\chi(x) = a+bi$. In particular, we have $\mathbb{Z}[G]^- = \mathcal{O}_G^- \simeq \mathbb{Z}[i]$ and $\mathbb{Z}[G]^-$ is a principal ring. By Proposition \[prop:iso\], this implies that there exists $\bar\theta \in U_K^-$ such that $U_K^- = \mathbb{Z}[G]^- \cdot \bar\theta$. We now prove the unicity of the solution. Assume that $\bar\eta$ and $\bar\eta'$ are two solutions to (P1) and (P2). By Corollary \[cor:unicity\], there exists a unique $2$-unit $u$ in $\mathbb{Q}[G]^-$ such that $\bar\eta' = u \cdot \bar\eta$. Let $\mathfrak{p}_2 := (i+1)\mathbb{Z}[i]$ be the unique prime ideal above $2$ in $\mathbb{Z}[i]$. Let $n := v_{\mathfrak{p}_2}(\chi(u))$. Assume, without loss of generality, that $n \geq 0$ (otherwise, exchange $\bar\eta$ and $\bar\eta'$ and replace $u$ by $u^{-1}$) and therefore $\bar\eta' \in \mathbb{Z}[G]^- \cdot \bar\eta$. Let $x \in \mathbb{Z}[G]^-$ be such that $\bar\eta = x \cdot \bar\theta$. We have $$\begin{aligned} (\mathbb{Z}[G]^- \cdot \bar\eta : \mathbb{Z}[G]^- \cdot \bar\eta') & = (x \mathbb{Z}[G]^- : u x \mathbb{Z}[G]^-) \\ & = (\chi(x) \mathbb{Z}[i] : \chi(u) \chi(x) \mathbb{Z}[i]) \\ & = |\chi(u)| = 2^n.\end{aligned}$$ Therefore $n = 0$ and $u$ is a unit. Since $\mathbb{Z}[G]^-$ has only trivial units, it follows that $u$ is a trivial unit. This proves the unicity statement. Next we prove that there exist solutions to (P1) and (P2). Let $\mathcal{F} := {\mathrm{Fitt}}_{\mathbb{Z}[G]^-}({\mathrm{Cl}}_K^-)$ be the Fitting ideal of $\mathrm{Cl}_K^-$ as a $\mathbb{Z}[G]^-$-module. Let $f$ be a generator of $\mathcal{F}$. We set $\bar\eta := f \, (\gamma+1)^{e+t_S} \cdot \bar\theta$. We have by Lemma \[lem:p1\] $$(U_K^- : \mathbb{Z}[G] \cdot \bar\eta) = 2^{e+t_S} (\mathbb{Z}[G]^- : \mathcal{F}) = 2^{e+t_S} |{\mathrm{Cl}}_K^-|.$$ Thus $\bar\eta$ is a solution to (P1). In the same way it follows directly from Lemma \[lem:p2\] that it is a solution to (P2). Now, since $\bar\eta \in U_K^-$, we have for $\nu = \chi_0$, the trivial character, or $\nu = \chi^2$ that $$\frac{1}{2} \sum_{g \in G} \nu(g) \log |\eta^g| = 0.$$ On the other hand, $L'_{K/k, S}(\nu, 0) = 0$ follows directly from [@tate:book Prop. I.3.4]. From Proposition \[prop:prodform\], using the fact that $\chi^3 = \bar\chi$, we get[^12] $$\begin{gathered} \left|L'_{K/k,S}(0, \chi)\right|^2 = L'_{K/k,S}(0, \chi) L'_{K/k,S}(0, \chi^3) \\ = \left(\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|\right) \left(\frac{1}{2} \sum_{g \in G} \chi^3 (g) \log |\eta^g|\right) \\ = \left|\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|\right|^2\end{gathered}$$ and the equality to be proved follows by taking square-roots. Finally, to prove that $K(\sqrt{\eta})/k$ is abelian, we need to prove that $(\gamma - 1) \cdot \bar\eta \in 2 \cdot U_K^-$ by [@tate:book Prop. IV.1.2]. This is equivalent to prove that $$(i-1)(i+1)^{e+t_S} \chi(f) \subset 2 \mathbb{Z}[i],$$ that is one of the following assertions is satisfied: $e \geq 1$, $t_S \geq 1$ or $2$ divides $|{\mathrm{Cl}}_K^-|$. We have $e \geq 2d-4$ by  and therefore the result is proved if $d \geq 3$. Assume that $d = 2$ and $e = 0$. Then it follows by Lemma \[lem:qu\] that $\hat{H}^1(T, U_K) = 1$ where $T := \langle \tau\rangle$. By Lemma \[lem:h11\] this implies that either $2$ divides $|{\mathrm{Cl}}_K^-|$ and the result is proved, or $K/K^+$ is unramified at finite places. Assume the latter. At least one prime ideal of $k$ ramifies in $K$ by the proof of [@stark12 Lem. 2.8] since $k$ is a quadratic field. By (A3) this prime ideal is inert in $K/K^+$, thus $t_S \geq 1$. This concludes the proof. A consequence of this result is that we can say quite precisely when the Stark unit, it it exists, is a square in that case. The result is very similar to the situation in the quadratic case (see Theorem \[th:quadratic\]). \[cor:square4\] Under the hypothesis of the theorem and assuming that the Stark unit exists, then it is a square in $K$ if and only if $e+t_S+c \geq 2$ where $c$ is the $2$-valuation of $|{\mathrm{Cl}}_K^-|$. In particular, if $d \geq 3$ then it is always a square and, in fact, it is a $2^{d-2}$-th power. We prove the equivalence. The inequality is satisfied when the Stark unit $\varepsilon$ is a square by Corollary \[cor:whensquare\]. Now assume that the inequality is satisfied. By the unicity statement of the theorem, we have $\bar\varepsilon = \bar\eta$ (replacing $\eta$ by one of its conjugate if necessary). From the proof of the theorem, we see that $\bar\eta$ belongs to $2^r \cdot U_K^-$ if and only if $(i+1)^{e+t_S} \chi(f) \in 2^r \mathbb{Z}[i]$. Taking valuation at $\mathfrak{p}_2$, the only prime ideal above $2$, we see that it is equivalent to $e+t_S+c \geq 2r$. This proves the first assertion. Now, to prove the second assertion, we see that $e \geq 2d-4$ by . Therefore $\bar{\eta}$ lies in $2^{d-2} \cdot U_K^-$. This proves the result. When $d = 2$ it is possible to find examples for which the Stark unit, if it exists, is a square and examples for which it is not a square. For example, let $k := \mathbb{Q}(\sqrt{5})$ and let $v_1, v_2$ denote the two infinite places of $k$ with $v_1(\sqrt{5}) < 0$ and $v_2(\sqrt{5}) > 0$. Let $K$ be the ray class field modulo $\mathfrak{p}_{29} v_1$ where $\mathfrak{p}_{29} := (11/2 - \sqrt{5}/2)$ is one of the two prime ideals above $29$. Then $K/k$ is a cyclic quartic extension that satisfies (A1), (A2) and (A3) with $S := S(K/k)$ and one can prove that, if it exists, the Stark unit is not a square. Now, on the other hand, let $K$ be the ray class field modulo $\sqrt{5}\mathfrak{p}_{41} v_1$ where $\mathfrak{p}_{41} := (13/2 - \sqrt{5}/2)$ is one of the two prime ideals above $41$. Then $K/k$ is a cyclic quartic extension that satisfies (A1), (A2) and (A3) with $S := S(K/k)$, but one can prove that, in this case, the Stark unit, if it exists, is a square. Cyclic sextic extensions ======================== In this final section we study the case when $K/k$ is a cyclic sextic extension. We will need some additional assumptions to be able to prove that there exists solutions to (P1) and (P2). \[th:sextic\] Let $K/k$ be a cyclic sextic extension such that , and are satisfied with $S := S(K/k)$ . Assume also that $3$ does not divide the order of ${\mathrm{Cl}}_K$ and that no prime ideal above $3$ is wildly ramified in $K/k$. Let $F$ be the quadratic extension of $k$ contained in $K$. Then there exists $\bar\eta \in U_K^-$ satisfying and and such that $\mathcal{N}_{K/F}(\eta)$ is the Stark unit for the extension $F/k$ and the set of places $S$. Furthermore, $\bar\eta$ is unique up to multiplication by an element of ${\mathrm{Gal}}(K/F)$, satisfies for all $\chi \in \hat{G}$ $$\left|L'_{K/k,S}(0, \chi)\right| = \frac{1}{2} \left|\sum_{g \in G} \chi(g) \log |\eta^g|\right|,$$ and the extension $K(\sqrt{\eta})/k$ is abelian. Let $\gamma$ be a generator of the Galois group $G$, thus $\tau = \gamma^3$. Let $\chi$ be the character that sends $\gamma$ to $-\omega$ where $\omega$ is a fixed primitive third root of unity. It is a generator of the group of characters of $G$. We have $X_{\rm odd} = \{\xi_2, \xi_6\}$ where $\xi_2 := \chi^3$ and $\xi_6 := \chi + \chi^5$. The corresponding idempotents are $$e_{\xi_2} = \frac{1}{6}(1-\gamma^3)(1+\gamma^2+\gamma^4)\quad \text{and}\quad e_{\xi_6} = \frac{1}{6}(1-\gamma^3)(2-\gamma^2-\gamma^4).$$ We have the isomorphism $$\label{eq:q6iso} \mathbb{Q}[G]^- = e_{\xi_2} \mathbb{Q}[G] + e_{\xi_6} \mathbb{Q}[G] \cong \mathbb{Q} \oplus \mathbb{Q}(\omega).$$ Let $\sigma := \gamma^2$ and let $H$ be the subgroup of order $3$ generated by $\sigma$. Then we have $G = \langle \tau \rangle \times H$ and the projection map on $H$ extends to an isomorphism between $\mathbb{Q}[G]^-$ and $\mathbb{Q}[H]$, that restricts to an isomorphism between $\mathbb{Z}[G]^-$ and $\mathbb{Z}[H]$. From now on, we will identify $\mathbb{Q}[G]^-$ and $\mathbb{Q}[H]$. Note that, with that identification, both act in the same way on $U_K^-$, $U_K^- \otimes \mathbb{Q}$, ${\mathrm{Cl}}_K^-$, etc. Let $e_0$ and $e_1$ be the image by the projection map of $e_{\xi_2}$ and $e_{\xi_6}$. Then $e_0 = \frac{1}{3}(1+\sigma+\sigma^2)$ is the idempotent of the trivial character of $H$ and $e_1 = \frac{1}{3}(2-\sigma-\sigma^2)$ is the sum of the idempotents of the two non trivial characters of $H$. The main difference between this case and the quartic case is the fact that the isomorphism between $\mathbb{Q}[H]$ and $\mathbb{Q} \oplus \mathbb{Q}(\omega)$ does not restrict to an isomorphism between $\mathbb{Z}[G]^-$ and $\mathbb{Z} \oplus \mathbb{Z}[\omega]$. In particular, $\mathbb{Z}[G]^-$ is not a principal ring. Because of that the proof is somewhat more intricate than in the quartic case. We will therefore proceed by proving a series of different claims. First, we define $$\label{eq:oiso} \mathcal{O} := e_0 \mathbb{Z}[H] + e_1 \mathbb{Z}[H] \simeq \mathbb{Z} \oplus \mathbb{Z}[\omega].$$ Note that, by the above identification, we have $\mathcal{O}_G^- \cong \mathcal{O}$. \[claim:oprincipal\] The ring $\mathcal{O}$ is principal and contains $\mathbb{Z}[H]$ with index 3. Let $\mathcal{I}$ be an ideal of $\mathcal{O}$. Then $e_0\mathcal{I}$ is an ideal of $e_0\mathbb{Z} \simeq \mathbb{Z}$. Thus there exists $a \in \mathbb{Z}$ such that $e_0\mathcal{I} = a e_0\mathbb{Z}[H]$. In the same way, $e_1\mathcal{I}$ is an ideal of $e_1\mathbb{Z} \simeq \mathbb{Z}[\omega]$. Since $\mathbb{Z}[\omega]$ is a principal ring, there exists $b, c \in \mathbb{Z}$ such that $e_1\mathcal{I} = e_1(b+c\sigma)\mathbb{Z}[H]$. One verify readily that $e_0 a + e_1 (b + c\sigma)$ is a generator of $\mathcal{I}$. To conclude the proof of Claim \[claim:oprincipal\], we note that $\mathcal{O}/\mathbb{Z}[H] = \langle e_0+\mathbb{Z}[H]\rangle = \langle e_1+\mathbb{Z}[H]\rangle$ clearly has order $3$. \[claim:mainid\] Let $\mathcal{A}$ be an ideal of $\mathbb{Z}[H]$ of finite index. Then there exists $g \in \mathcal{A}$ such that $$\label{eq:index} \mathcal{O}/\mathcal{AO} \simeq \mathbb{Z}[H]/g\mathbb{Z}[H].$$ Furthermore, $\mathcal{A} = g\mathbb{Z}[H]$ if $\mathcal{A}$ is a principal ideal. Otherwise $(\mathcal{A}:g\mathbb{Z}[H]) = 3$. We prove this claim by considering the two cases: $\mathcal{AO} \not= \mathcal{A}$ and $\mathcal{AO} = \mathcal{A}$. \[claim:aprincipal\] Assume that $\mathcal{AO} \not= \mathcal{A}$. Then $\mathcal{A}$ is a principal ideal. Let $g' = e_0a + e_1(b+c\sigma)$ be a generator of the principal ideal $\mathcal{AO}$ of $\mathcal{O}$. If $e_1(b+c\sigma) \in \mathcal{A}$, then $e_1\mathcal{A} = e_1(b+c\sigma)\mathbb{Z}[H] \subset \mathcal{A}$ and it follows that $\mathcal{A} = \mathcal{AO}$, a contradiction. Therefore $\mathcal{AO}/\mathcal{A} = \langle e_1(b+c\sigma) + \mathcal{A} \rangle$ has order $3$. Thus one of the three elements: $e_0a + e_1(b+c\sigma)$, $e_0a - e_1(b+c\sigma)$ or $e_0a$ belongs to $\mathcal{A}$. It cannot be $e_0a$ since that would imply, as above, that $\mathcal{A} = \mathcal{AO}$. Denote by $g$ the one element between $e_0a \pm e_1(b+c\sigma)$ that lies in $\mathcal{A}$. Clearly we still have $g\mathcal{O} = \mathcal{AO}$. Now, $g$ is not a zero divisor since $\mathcal{A}$ has finite index in $\mathbb{Z}[H]$, so we have $(g\mathcal{O}:g\mathbb{Z}[H]) = (\mathcal{O}:\mathbb{Z}[H]) = 3$. Therefore we get $$(\mathcal{A}:g\mathbb{Z}[H]) = \frac{(g\mathcal{O}:g\mathbb{Z}[H])}{(\mathcal{AO}:\mathcal{A})} = 1$$ and $\mathcal{A}=g\mathbb{Z}[H]$. Equation follows in that case from the equality $$\label{eq:indexAO} (\mathcal{O}:\mathcal{AO}) (\mathcal{AO}:\mathcal{A}) = (\mathcal{O}: \mathbb{Z}[H]) (\mathbb{Z}[H]: \mathcal{A})$$ and the fact that $(\mathcal{AO}:\mathcal{A}) = (\mathcal{O}: \mathbb{Z}[H])$ by the above. \[claim:anonprincipal\] Assume that $\mathcal{AO} = \mathcal{A}$. Then $\mathcal{A}$ is not a principal ideal, but there exists $g \in \mathcal{A}$ such that $(\mathcal{A}: g\mathbb{Z}[H]) = 3$. Let $g$ be a generator of the principal ideal $\mathcal{AO}$ of $\mathcal{O}$. Since $\mathcal{AO} =\mathcal{A}$, $g$ lies in $\mathcal{A}$ and we compute as above $$(\mathcal{A}:g\mathbb{Z}[H]) = (\mathcal{AO}:g\mathcal{O})(g\mathcal{O}:g\mathbb{Z}[H]) = 3.$$ Since $(\mathcal{O}: \mathbb{Z}[H])(\mathbb{Z}[H]:\mathcal{A}) = 3(\mathbb{Z}[H]:\mathcal{A}) = (\mathcal{A}:g\mathbb{Z}[H])(\mathbb{Z}[H]:\mathcal{A}) = (\mathbb{Z}[H]:g\mathbb{Z}[H])$ and $(\mathcal{AO}:\mathcal{A}) = 1$, Equation  follows from . It remains to prove that $\mathcal{A}$ cannot be principal in that case. In order to prove this, we need another result. Let $x \in \mathcal{O}$. By the isomorphism in , it corresponds to a pair $(x_0, x_1)$ in $\mathbb{Z} \oplus \mathbb{Z}[\omega]$. We define the norm of $x$ as the following quantity $$\text{Norm}(x) := |x_0| \, N_{\mathbb{Q}(\omega)/\mathbb{Q}}(x_1).$$ Note that we recover the usual definition of the norm of $\mathbb{Q}[H]$ as a $\mathbb{Q}$-algebra. The proof of the following claim is straightforward and is left to the reader. \[claim:norm\] Let $x \in \mathcal{O}$ with $\text{Norm}(x) \not= 0$. Then $(\mathcal{O}:x\mathcal{O}) = \text{Norm}(x)$. If furthermore $x \in \mathbb{Z}[H]$ then $(\mathbb{Z}[H]:x\mathbb{Z}[H]) = \text{Norm}(x)$. We now finish the proof of Claim \[claim:anonprincipal\]. Assume that $\mathcal{A}$ is principal, say $\mathcal{A} = h\mathbb{Z}[H]$. Then there exists $z \in \mathbb{Z}[H]$ such that $g = hz$ and we have $(\mathcal{O}:z\mathcal{O}) = 3$. Thanks to the above claim, we can explicitly compute all the elements $z \in \mathcal{O}$ such that $(\mathcal{O}:z\mathcal{O}) = 3$. There are the elements $z = e_0 a + e_1 (b + c\sigma)$ with $a = \pm 1$ and $b + c\sigma \in \{\pm(1+2\sigma), \pm(2+\sigma), \pm (1-\sigma)\}$, or $a = \pm 3$ and $b + c\sigma \in \{\pm 1, \pm \sigma, \pm (1+\sigma)\}$. One can compute all possibilities and check that none of those belong to $\mathbb{Z}[H]$. This gives a contradiction and concludes the proof of Claim \[claim:anonprincipal\] and of Claim \[claim:mainid\]. We now turn to the $\mathbb{Z}[H]$-structure of $U_K^-$. The principal result is the following claim that we will prove in several steps. \[claim:ukprincipal\] There exists $\bar\theta \in U_K^-$ such that $U_K^- = \mathbb{Z}[H] \cdot \bar\theta$. Let $\bar\theta' \in U_K^-$ be such that $U_K^- \otimes \mathbb{Q} = \mathbb{Q}[H] \cdot \bar\theta'$. Note that the existence of $\bar\theta'$ follows from Proposition \[prop:iso\]. We define $$\Lambda := \left\{ x \in \mathbb{Q}[H] : x \cdot \bar\theta' \in U_K^- \right\}.$$ It is a fractional ideal of $\mathbb{Z}[H]$. The above claim is satisfied if and only if it is a principal ideal. Assume that this is not the case. Then, by the above, we have[^13] $\Lambda\mathcal{O} = \Lambda$. Recall that $F$ denotes the subfield of $K$ fixed by $H$. It is a quadratic extension of $k$ and ${\mathrm{Gal}}(F/k) = \langle \tau\rangle$. We define $U_F^-$ as the kernel of norm map from $U_F/\langle \pm 1\rangle$ to $U_k/\langle \pm 1\rangle$. We have also $U_F^- = U_K^- \cap (F^\times/\langle \pm 1\rangle)$. Let $N_H := 1 + \sigma + \sigma^2$. It is the group ring element corresponding to the norm of the extensions $K/F$ and $K^+/k$. \[claim:nuk=3uf\] $\Lambda\mathcal{O} = \Lambda$ if and only if $N_H \cdot U_K^- = 3 \cdot U_F^-$. If $\Lambda\mathcal{O} \not= \Lambda$, then $N_H \cdot U_K^- = U_F^-$. We have $\Lambda\mathcal{O} = \Lambda$ if and only if $e_0\Lambda \subset \Lambda$, that is $N_H \cdot U_K^- \subset 3 \cdot U_K^-$. Assume that it is the case. Let $\bar\delta \in U_K^-$ and set $\kappa := N_{K/F}(\delta) \in U_F$. Then the polynomial $X^3 - \kappa$ has a root, say $\nu$, in $U_K$. If $\nu$ does not belong to $F$ then all the roots of $X^3 - \kappa$ belongs to $K$ since $K/F$ is a Galois extension. It follows that $K$ contains the third roots of unity, a contradiction. Therefore $\bar\nu \in U_F^-$ and $N_H \cdot U_K^- \subset 3 \cdot U_F^-$. The other inclusion is trivial and the first assertion of the claim is proved. If $\Lambda\mathcal{O} \not= \Lambda$ then $3 \cdot U_F^- \varsubsetneq N_H \cdot U_K^- \subset U_F^-$. Since $U_F^-$ is a $\mathbb{Z}$-module of rank $1$, it follows that $N_H \cdot U_K^- = U_F^-$. The claim is proved. Let $\mathcal{S}$ be the set of prime ideals of $K$ that are totally split in $K/k$. Denote by $I_{K,\mathcal{S}}$ the subgroup of $I_K$, the group of ideals of $K$, generated by the prime ideals in $\mathcal{S}$. Then, by Chebotarev’s theorem, the following short sequence is exact $$\xymatrix{ 1 \ar[r] & P_K \cap I_{K,\mathcal{S}}^{1-\tau} \ar[r] & I_{K,\mathcal{S}}^{1-\tau} \ar[r] & {\mathrm{Cl}}_K^{1-\tau} \ar[r] & 1 }$$ where $P_K$ is the group of principal ideals of $K$. We take the Tate cohomology of this sequence for the action of $H$. Since $3$ does not divide the order of ${\mathrm{Cl}}_K$, it does not divide the order of ${\mathrm{Cl}}_K^{1-\tau}$ and $\hat{H}^0({\mathrm{Cl}}_K^{1-\tau}) = \hat{H}^1({\mathrm{Cl}}_K^{1-\tau}) = 1$. Note that here and in what follows, to simplify the presentation, we drop the group $H$ in the notation of the cohomology groups and write $\hat{H}^i(M)$ instead of $\hat{H}^i(H, M)$ for $M$ a $\mathbb{Z}[H]$-module. It follows from the exact hexagon for the above exact sequence that $\hat{H}^i(P_K \cap I_{K,\mathcal{S}}^{1-\tau}) \simeq \hat{H}^i(I_{K,\mathcal{S}}^{1-\tau})$ for $i = 0, 1$. Let $\mathfrak{A} \in P_K \cap I_{K,\mathcal{S}}^{1-\tau}$. There exist $\alpha \in K_\mathcal{S}^\times$, the subgroup of elements of $K^\times$ supported only by prime ideals in $\mathcal{S}$, and $\mathfrak{B} \in I_{K,\mathcal{S}}$ such that $$\mathfrak{A} = (\alpha) = \mathfrak{B}^{1-\tau}.$$ We apply $1-\tau$ to this equation $$\mathfrak{A}^{1-\tau} = (\alpha)^{1-\tau} = \mathfrak{B}^{(1-\tau)^2} = (\mathfrak{B}^{1-\tau})^2 = \mathfrak{A}^2.$$ Therefore we have $(P_K \cap I_{K,\mathcal{S}}^{1-\tau})^2 \subset P_{K,\mathcal{S}}^{1-\tau}$ where $P_{K,\mathcal{S}}$ is the subgroup of principal ideals generated by the elements of $K^\times_\mathcal{S}$. It follows that the quotient $(P_K \cap I_{K,\mathcal{S}}^{1-\tau})/P_{K,\mathcal{S}}^{1-\tau}$ is killed by $2$ and therefore $\hat{H}^i(P_K \cap I_{K,\mathcal{S}}^{1-\tau}) = \hat{H}^i(P_{K,\mathcal{S}}^{1-\tau})$ for $i = 0$ or $1$. We have proved the following claim. \[claim:isohi\] $\hat{H}^0(P_{K,\mathcal{S}}^{1-\tau}) \simeq \hat{H}^0(I_{K,\mathcal{S}}^{1-\tau})$ and $\hat{H}^1(P_{K,\mathcal{S}}^{1-\tau}) \simeq \hat{H}^1(I_{K,\mathcal{S}}^{1-\tau})$. Let $u \in U_K \cap (K_\mathcal{S}^\times)^{1-\tau}$. There exists $\alpha \in K_\mathcal{S}^\times$ such that $u = \alpha^{1-\tau}$. Therefore we get $$u^{1-\tau} = \alpha^{(1-\tau)^2} = (\alpha^{1-\tau})^2 = u^2.$$ Reasoning as above, this implies that $\hat{H}^i(U_K \cap (K_\mathcal{S}^\times)^{1-\tau}) = \hat{H}^i(U_K^{1-\tau}) = \hat{H}^i(U_K^-)$ for $i = 0, 1$. We now consider the short exact sequence $$\xymatrix{ 1 \ar[r] & U_K \cap (K_\mathcal{S}^\times)^{1-\tau} \ar[r] & (K_\mathcal{S}^\times)^{1-\tau} \ar[r] & P_{K,\mathcal{S}}^{1-\tau} \ar[r] & 1. }$$ Taking the Tate cohomology and using the above equalities, we extract the following exact sequence from the exact hexagon  corresponding to this exact sequence $$\label{eq:exact}\xymatrix{ \cdots \ar[r] & \hat{H}^1(P_{K,\mathcal{S}}^{1-\tau}) \ar[r] & \hat{H}^0(U^-_K) \ar[r] & \hat{H}^0((K_\mathcal{S}^\times)^{1-\tau}) \ar[r] & \cdots \\ }$$ The next claim is just a reformulation of the first part of Claim \[claim:nuk=3uf\].\  \[claim:h1uk=3\] $\Lambda\mathcal{O} = \Lambda$ if and only if $\hat{H}^0(U_K^-) \simeq \mathbb{Z}/3\mathbb{Z}$. Assume the two followings claims for the moment. \[claim:H0PK=1\] $\hat{H}^1(P_{K,\mathcal{S}}^{1-\tau})$ is trivial. \[claim:H0K=1\] $\hat{H}^0((K_\mathcal{S}^\times)^{1-\tau})$ is trivial. By we get that $\hat{H}^0(U_K^-) = 1$. Thus $\Lambda\mathcal{O} \not= \mathcal{O}$ by Claim \[claim:h1uk=3\] and therefore $\Lambda$ is principal by Claim \[claim:aprincipal\], and Claim \[claim:ukprincipal\] follows. It remains to prove Claims \[claim:H0PK=1\] and \[claim:H0K=1\]. We start with the proof of Claim \[claim:H0PK=1\]. By Claim \[claim:isohi\], this is equivalent to prove that $\hat{H}^1(I_{K,\mathcal{S}}^{1-\tau})$ is trivial. We have as $\mathbb{Z}[H]$-modules $$I_{K,\mathcal{S}}^{1-\tau} = \prod_{\mathfrak{p}_0 \in \mathcal{S}_0}{\!\!\!}^{'} \Big(\prod_{\mathfrak{P} \mid \mathfrak{p}_0} \mathfrak{P}^{\mathbb{Z}}\Big)^{1-\tau} \simeq \prod_{\mathfrak{p}_0 \in \mathcal{S}_0}{\!\!\!}^{'} (1-\tau) \mathbb{Z}[G]$$ where $\mathcal{S}_0$ is the set of prime ideals of $k$ that splits completely in $K/k$, $\mathfrak{P}$ runs through the prime ideals of $K$ dividing $\mathfrak{p}_0$ and the $'$ indicates that it is a restricted product, that is the exponent of $\mathfrak{P}$ is zero for all but finitely many prime ideals. The isomorphism comes from fixing a prime ideal above $\mathfrak{p}_0$ and the fact that $\mathfrak{p}_0$ is totally split in $K/k$. Therefore we have $$\hat{H}^1(I_{K,\mathcal{S}}^{1-\tau}) = \prod_{\mathfrak{p}_0 \in \mathcal{S}_0}{\!\!\!}^{'} \hat{H}^1((1-\tau)\mathbb{Z}[G]) \simeq \prod_{\mathfrak{p}_0 \in \mathcal{S}_0}{\!\!\!}^{'} \hat{H}^1(\mathbb{Z}[H]).$$ It is well-known that $\hat{H}^1(\mathbb{Z}[H]) = 1$, thus Claim \[claim:H0PK=1\] is proved. To prove Claim \[claim:H0K=1\], we prove that the norm from $(K_\mathcal{S}^\times)^{1-\tau}$ to $(F_\mathcal{S}^\times)^{1-\tau}$ is surjective. Let $\alpha^{1-\tau} \in (F_\mathcal{S}^\times)^{1-\tau}$. By the Hasse Norm Principle, $\alpha^{1-\tau}$ is a norm in $K/F$ if and only if it is a norm in $K_\mathfrak{P}/F_\mathfrak{p}$ for all prime ideals $\mathfrak{P}$ of $K$ where $\mathfrak{p}$ denotes the prime ideal of $F$ below $\mathfrak{P}$. If $\mathfrak{p}$ splits in $K/F$, then $\alpha^{1-\tau}$ is trivially a norm in $K_\mathfrak{P}/F_\mathfrak{p}$. Assume now that $\mathfrak{p}$ is inert. It follows from the theory of local fields, see [@lang:book §XI.4], that the norm of $K_\mathfrak{P}/F_\mathfrak{p}$ is surjective on the group of units of $F_\mathfrak{p}$. But $\alpha^{1-\tau}$ is a unit at $\mathfrak{P}$ since $\mathfrak{P} \not\in \mathcal{S}$, and therefore it is a norm also in this case. Finally we assume that $\mathfrak{P}$ is ramified in $K/F$. Let $p$ be the rational prime below $\mathfrak{P}$. By hypothesis, $p \not= 3$ since $3$ is not wildly ramified in $K/k$. Write $\mu_\mathfrak{P}$, $\mathbb{U}_\mathfrak{P}$, $\mu_\mathfrak{p}$ and $\mathbb{U}_\mathfrak{p}$ for the group of roots of unity of order prime to $p$ and the group of principal units of $K_\mathfrak{P}$ and $F_\mathfrak{p}$ respectively. We have $\mu_\mathfrak{P} = \mu_\mathfrak{p}$ and therefore $\mathcal{N}_{K_\mathfrak{P}/F_\mathfrak{p}}(\mu_\mathfrak{P}) = \mu_\mathfrak{p}^3$. On the other hand $\mathcal{N}_{K_\mathfrak{P}/F_\mathfrak{p}}(\mathbb{U}_\mathfrak{p}) = \mathbb{U}^3_\mathfrak{p} = \mathbb{U}_\mathfrak{p}$ and the norm is surjective on principal units. Since $\mathfrak{P} \not\in \mathcal{S}$, $v_\mathfrak{p}(\alpha) = 0$ and $\alpha = \zeta u$ with $\zeta \in \mu_\mathfrak{p}$ and $u \in \mathbb{U}_\mathfrak{p}$. It follows from the above discussion that $\alpha^{1-\tau}$ is a norm in $K_\mathfrak{P}/F_\mathfrak{p}$ if and only if $\zeta^{1-\tau} \in \mu_\mathfrak{p}^3$. Let $\mathfrak{p}_0$ be the prime ideal of $k$ below $\mathfrak{p}$. Assume first that $\mathfrak{p}_0$ is ramified in $F/k$. Then $\mu_\mathfrak{p} \subset k_{\mathfrak{p}_0}$ and $\zeta^{1-\tau} = 1$, thus $\alpha^{1-\tau}$ is a norm in $K_\mathfrak{P}/F_\mathfrak{p}$. Assume now that $\mathfrak{p}_0$ is inert[^14] in $F/k$. Denote by $f$ the residual degree of $\mathfrak{p}_0$. The group $\mu_{\mathfrak{p}_0}$ of roots of unity in $k$ of order prime to $p$ has order $p^f-1$. Let $\mathfrak{P}^+ := \mathfrak{P} \cap K^+$. The extension $K^+_{\mathfrak{P}^+}/k_{\mathfrak{p}_0}$ is a tamely ramified cyclic cubic extension. Therefore it is a Kummer extension by and $k_{\mathfrak{p}_0}$ contains the third roots of unity, that is $3$ divides $p^f-1$. Since $\tau$ is the Frobenius element at $\mathfrak{p}_0$ of the extension $F/k_0$, we have $\zeta^{1-\tau} = \zeta^{1-p^f} = (\zeta^{(1-p^f)/3})^3 \in \mu_\mathfrak{p}^3$ and therefore $\alpha^{1-\tau}$ is a norm in $K_\mathfrak{P}/F_\mathfrak{p}$. We have proved that $\alpha^{1-\tau}$ is a norm everywhere locally. It follows by the Hasse Norm Principle that there exists $\beta \in K^\times$ such that $\mathcal{N}_{K/F}(\beta) = \alpha^{1-\tau}$. Let $\mathfrak{P}$ be a prime ideal of $K$ not in $\mathcal{S}$ and, as above, let $\mathfrak{p}$ be the prime ideal of $F$ below $\mathfrak{P}$. Assume first that $\mathfrak{P}$ is ramified or inert in $K/F$, then $v_\mathfrak{P}(\beta) = v_\mathfrak{p}(\alpha^{1-\tau})$ or $\frac{1}{3} v_\mathfrak{p}(\alpha^{1-\tau})$ respectively. In both cases we get $v_\mathfrak{P}(\beta) = 0$ since $\alpha \in K^\times_{\mathcal{S}}$. If $\mathfrak{P}$ is split in $K/F$ then it must be inert or ramified in $K/K^+$ by (A3). It follows that $v_\mathfrak{P}(\beta^{1-\tau}) = 0$. Therefore $\delta := \beta^{1-\tau} \in K_\mathcal{S}^\times$. We now compute $$\mathcal{N}_{K/F}(\delta^{1-\tau}) = \mathcal{N}_{K/F}(\beta)^{(1-\tau)^2} = (\alpha^{1-\tau})^{2(1-\tau)} = (\alpha^{1-\tau})^4.$$ Thus $\alpha^{1-\tau}$ is the norm of $(\delta/\alpha)^{1-\tau} \in (K_\mathcal{S}^\times)^{1-\tau}$. This concludes the proof of Claim \[claim:H0K=1\] and therefore also the proof of Claim \[claim:ukprincipal\]. The next claim follows from Claim \[claim:nuk=3uf\] and the fact that $\Lambda\mathcal{O} \not= \Lambda$. \[claim:usurj\] $N_H \cdot U_K^- = U_F^-$. Let $\mathcal{F} := {\mathrm{Fitt}}_{\mathbb{Z}[H]}({\mathrm{Cl}}_K^-)$ be the Fitting ideal of ${\mathrm{Cl}}_K^-$ as a $\mathbb{Z}[H]$-module. Apply Claim \[claim:mainid\] to the ideal $\mathcal{F}$ and call $f$ the element of $\mathcal{F}$ such that $\mathcal{O}/\mathcal{FO} \simeq \mathbb{Z}[H]/f\mathbb{Z}[H]$. Set $\bar\eta' := f \cdot \bar\theta$. We have $$\label{eq:indetap} (U_K^- : \mathbb{Z}[H] \cdot \bar\eta') = (\mathbb{Z}[H] : f\mathbb{Z}[H]) = (\mathcal{O}:\mathcal{FO}) = |{\mathrm{Cl}}_K^-|.$$ The last equality follows from Lemma \[lem:p1\] and the fact that $\mathcal{FO}$ is the Fitting ideal of ${\mathrm{Cl}}_K^-$ as an $\mathcal{O}$-module. \[claim:kappa\] Let $n, m \geq 0$ be two integers. Then there exists $\kappa_{n,m} \in \mathbb{Z}[H]$, unique up to a trivial unit, such that $$\label{eq:propkappa} \text{Norm}(\kappa_{n,m}) = 2^{n+2m} \quad\text{and}\quad e_0 \kappa_{n,m} = e_0 2^n.$$ We define $$\kappa_{n,m} := 2^n e_0 - (-1)^{n+m} 2^m e_1.$$ It is clear from its construction that $\kappa_{n,m}$ satisfies . One can see also directly that $\kappa_{n,m} \in \mathbb{Z}[H]$ since $2 \equiv -1 \pmod{3}$. It remains to prove the unicity statement. Clearly $e_0 \kappa_{n,m}$ is fixed by construction. On the other hand $e_1\kappa_{n,m}$ is an element of norm $2^{2m}$ in $e_1\mathbb{Z}[H] \simeq \mathbb{Z}[\omega]$. Since $2$ is inert in $\mathbb{Z}[\omega]$, there exists only one element in $\mathbb{Z}[\omega]$ of norm $2^{2m}$ up to units. This concludes the proof of the claim. Let $e' \in \mathbb{N}$ be such that $2^{e'} = (\bar{U}_k : \mathcal{N}(\bar {U}_F))$. We now prove the following claim. The integer $e-e'$ is non-negative and even. We consider the natural map $\bar{U}_k \to \bar{U}_{K^+}/\mathcal{N}(\bar{U}_K)$ that comes from the inclusion $U_k \subset U_{K^+}$. Let $\bar{u} \in \bar{U}_k$ be in the kernel of this map. Thus there exists $\bar{x} \in \bar{U}_K$ such that $\bar{u} = \mathcal{N}(\bar{x})$. Set $\bar{y} := N_H \cdot \bar{x} - \bar{u} \in \bar{U}_F$. We have $$\mathcal{N}(\bar{y}) = N_H \cdot \mathcal{N}(\bar{x}) - \mathcal{N}(\bar{u}) = 3 \cdot \bar{u} - 2 \cdot \bar{u} = \bar{u}.$$ Therefore the kernel of the above map is $\mathcal{N}(\bar{U}_F)$ and there is a well-defined injective group homomorphism from $\bar{U}_k/\mathcal{N}(\bar{U}_F)$ to $\bar{U}_{K^+}/\mathcal{N}(\bar{U}_K)$. This proves that[^15] $e \geq e'$. The cokernel of this map is $$\label{eq:defcok} \frac{\bar{U}_{K^+}/\mathcal{N}(\bar{U}_K)}{\bar{U}_k/\mathcal{N}(\bar{U}_F)} \simeq \bar{U}_{K^+}/(\bar{U}_k + \mathcal{N}(\bar{U}_K)).$$ It is a finite $\mathbb{Z}[H]$-module of order $2^{e-e'}$. In particular, the idempotents $e_0$ and $e_1$ act on it. We have $e_0 \cdot \bar{U}_{K^+}/(\bar{U}_k + \mathcal{N}(\bar{U}_K)) = N_H \cdot \bar{U}_{K^+}/(\bar{U}_k + \mathcal{N}(\bar{U}_F)) = 1$. It follows that $\bar{U}_{K^+}/(\bar{U}_k + \mathcal{N}(\bar{U}_K)) = e_1 \cdot \bar{U}_{K^+}/(\bar{U}_k + \mathcal{N}(\bar{U}_K))$ is a $\mathbb{Z}[\omega]$-module. Since $2$ is inert in $\mathbb{Z}[\omega]$, the order of $\bar{U}_{K^+}/(\bar{U}_k + \mathcal{N}(\bar{U}_K))$ is an even power of $2$. This concludes the proof of the claim. Let $\kappa := \kappa_{e'+t_S, (e-e')/2}$. We define $$\label{eq:defeta} \bar\eta := \pm \kappa \cdot \bar\eta'.$$ The choice of the sign will be done during the proof of the next claim. By Claims \[claim:norm\], \[claim:kappa\] and , it is direct to see that $\bar\eta$ satisfies (P1). It follows directly from its construction, the fact that $\kappa$ is a $2$-unit, and Lemma \[lem:p2\] that it is also a solution of (P2). The next step is to prove the following result. \[claim:chi3\] Up to the right choice of sign in , we have $$\frac{1}{2} \sum_{g \in G} \chi^3(g) \log |\eta^g|_w = L'_{K/k,S}(0, \chi^3).$$ The $\mathbb{Z}[H]$-module $U_K^-/(\mathbb{Z}[H] \cdot \bar\eta)$ has order not divisible by $3$ since $\bar\eta$ satisfies (P1). Thus it is a $\mathcal{O}$-module and we can split it into two parts corresponding to the two idempotents $e_0$ and $e_1$. On one side, using Claim \[claim:usurj\], we have $$e_0 \cdot \big(U_K^-/\mathbb{Z}[H] \cdot \bar\eta\big) = N_H \cdot \big(U_K^-/\mathbb{Z}[H] \cdot \bar\eta\big)\simeq U_F^-/\mathbb{Z} \cdot \bar\eta_F$$ where $\bar\eta_F := N_H \cdot \bar\eta \in U^-_F$. On the other side, we compute $$e_1 \cdot \big(U_K^-/\mathbb{Z}[H] \cdot \bar\eta\big) \simeq e_1 \big(\mathbb{Z}[H]/\kappa f \mathbb{Z}[H]\big) \simeq \mathbb{Z}[\omega]/2^{(e-e')/2} \mathcal{F}_1$$ where $\mathcal{F}_1$ is the Fitting ideal of $({\mathrm{Cl}}_K^-)^{e_1}$ viewed as an $\mathbb{Z}[\omega]$-module. Indeed, we have by construction $$e_1 f \mathbb{Z}[H] = e_1 \mathcal{F}\mathcal{O} \simeq {\mathrm{Fitt}}_{\mathbb{Z}[\omega]}(({\mathrm{Cl}}_K^-)^{e_1}).$$ Since ${\mathrm{Cl}}_K^- = ({\mathrm{Cl}}_K^-)^{e_0} \oplus ({\mathrm{Cl}}_K^-)^{e_1}$ and $({\mathrm{Cl}}_K^-)^{e_0} \simeq \mathcal{N}_{K/F}({\mathrm{Cl}}_K^-)$, we have$$(\mathbb{Z}[\omega] : \mathcal{F}_1) = |({\mathrm{Cl}}_K^-)^{e_1}| = \frac{|{\mathrm{Cl}}_K^-|}{|\mathcal{N}_{K/F}({\mathrm{Cl}}_K^-)|}.$$ \[claim:nclk\] $\mathcal{N}_{K/F}({\mathrm{Cl}}_K^-) = {\mathrm{Cl}}^-_F$. Consider the composition of maps ${\mathrm{Cl}}_F^- \to {\mathrm{Cl}}_K^- \to {\mathrm{Cl}}_F^-$ where the first map is the map induced by the lifting of ideals from $F$ to $K$ and the second map is the norm $\mathcal{N}_{K/F}$. The map constructed in that way is the multiplication by $3$ and therefore, if the order of ${\mathrm{Cl}}_F^-$ is not divisible by $3$, it is a bijection and the claim is proved. Assume that $3$ divides the order of ${\mathrm{Cl}}_F^-$. Let $h_E$ denote the class number of a number field $E$. Thus $h_K^- := |{\mathrm{Cl}}_K^-| = h_K/h_{K^+}$ and $h_F^- := |{\mathrm{Cl}}_F^-| = h_F/h_k$. If $K/F$ is ramified at some finite prime then $h_F$ divides $h_K$. As $h_F^-$ divides $h_F$, it follows that $3 \mid h_K$, a contradiction. Assume now that $K/F$ is unramified at finite primes. Therefore $3$ divides $h_F$ and $h_F/3$ divides $h_K$. In the same way, $K^+/k$ is unramified and therefore $3$ divides $h_k$. Since $3 \mid h_F^-$, this implies that $9$ divides $h_F$ and therefore $3$ divides $h_K$, a contradiction. It follows that $3$ does not divide $|{\mathrm{Cl}}_F^-|$ and the claim is proved. Putting together the claim and the computation that precedes it, we find that $$\label{eq:eqnf} (U_F^- : \mathbb{Z} \cdot \bar\eta_F) = \frac{(U_K^- : \mathbb{Z} \cdot \bar\eta)}{2^{e-e'}} \frac{|{\mathrm{Cl}}_F^-|}{|{\mathrm{Cl}}_K^-|} = 2^{e'+t_S} |{\mathrm{Cl}}_F^-|.$$ Let $\mathfrak{P}^+ \in S_{K^+}$ and denote by $\mathfrak{p}_0$ the prime ideal of $k$ below $\mathfrak{P}^+$. Then $\mathfrak{P}^+$ is inert in $K/K^+$ if and only if $\mathfrak{p}_0$ is inert in $F/k$. Furthermore, if $\mathfrak{P}^+$ is inert in $K/K^+$, then it is ramified[^16] in $K^+/k$ and it is the only prime ideal in $S_{K^+}$ above $\mathfrak{p}_0$. It follows that the number $t_S$ of prime ideals in $S_{K^+}$ that are inert in $K/K^+$ is equal to the number of prime ideals in $S$ that are inert in $F/k$. Therefore $\bar\eta_F$ satisfy the properties (P1) and (P2) for the extension $F/k$ and the set of primes $S$. As a consequence of Theorem \[th:quadratic\], we see that either $\eta_F$ or $\eta_F^{-1}$ is the Stark unit for the extension $K/F$ and the set of places $S$. By choosing the right sign in , we can assume that $\eta_F$ is the Stark unit. Therefore we have $$\frac{1}{2} (\log |\eta_F|_w + \nu(\tau) \log |\eta_F^\tau|_w) = L'_{F/k,S}(0, \nu)$$ where $\nu$ is the non trivial character of $F/k$. It follows from the functorial properties of $L$-functions that $L_{F/k, S}(s, \nu) = L_{K/k, S}(s, \chi^3)$, and from the definition of $\eta_F$ that $$\log |\eta_F|_w + \nu(\tau) \log|\eta_F^\tau|_w = \sum_{g \in G} \chi^3(g) \log|\eta^g|_w.$$ This completes the proof of the claim. Now, by Proposition \[prop:prodform\], we know that $$\begin{gathered} \left(\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|\right) \left(\frac{1}{2} \sum_{g \in G} \chi^3(g) \log |\eta^g|\right) \left(\frac{1}{2} \sum_{g \in G} \chi^5(g) \log |\eta^g|\right) \\ = \pm L'_{K/k,S}(0, \chi) L'_{K/k,S}(0, \chi^3) L'_{K/k,S}(0, \chi^5).\end{gathered}$$ We cancel the non zero terms corresponding to $\chi^3$ using Claim \[claim:chi3\] and, since $\chi$ and $\chi^5$ are conjugate, we get $$\begin{gathered} \left|\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|\right|^2 = \left(\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|\right) \left(\frac{1}{2} \sum_{g \in G} \chi^5(g) \log |\eta^g|\right) \\ = L'_{K/k,S}(0, \chi) L'_{K/k,S}(0, \chi^5) = |L'_{K/k,S}(0, \chi)|^2.\end{gathered}$$ Taking square-roots, we get $$\left|\frac{1}{2} \sum_{g \in G} \chi(g) \log |\eta^g|\right| = \left|\frac{1}{2} \sum_{g \in G} \chi^5(g) \log |\eta^g|\right| = |L'_{K/k,S}(0, \chi)| = |L'_{K/k,S}(0, \chi^5)|.$$ Note that we have directly using [@tate:book Prop. I.3.4] $$\begin{gathered} \frac{1}{2} \sum_{g \in G} \chi_0(g) \log |\eta^g| = \frac{1}{2} \sum_{g \in G} \chi^2(g) \log |\eta^g| = \frac{1}{2} \sum_{g \in G} \chi^4(g) \log |\eta^g| \\ = L'_{K/k,S}(0, \chi_0) = L'_{K/k,S}(0, \chi^2) = L'_{K/k,S}(0, \chi^4) = 0. \end{gathered}$$ We now prove that $\bar\eta$ is unique up to multiplication by an element of $H$. Assume that $\bar\eta'$ is another element of $U_K^-$ satisfying (P1), (P2) and such that $N_H \cdot \bar\eta'$ is the Stark unit for the extension $F/k$ and the set of places $S$. Let $u \in \mathbb{Q}[H]$ be such that $\bar\eta' = u \cdot \bar\eta$. By Corollary \[cor:unicity\], $u$ is a $2$-unit. Now, by hypothesis, $\bar\eta_F = N_H \cdot (u \cdot \bar\eta) = u \cdot (N_H \cdot \bar\eta) = u \cdot \bar\eta_F$ and thus $e_0u = e_0$. Write $u_1$ for the element of $\mathbb{Q}(\omega)$ such that $(1, u_1)$ corresponds to $u$ by the isomorphism in . Since both $\bar\eta$ and $\bar\eta'$ satisfy (P1), we have $\text{Norm}(u) = 1$ and thus $N_{\mathbb{Q}(\omega)/\mathbb{Q}}(u_1) = 1$. But $u_1$ is a 2-unit in $\mathbb{Q}(\omega)$ and there is only prime ideal above $2$ in $\mathbb{Q}(\omega)$. Therefore $u_1$ is in fact a unit and $u \in H$. Finally, it remains to prove that $K(\sqrt{\eta})/k$ is an abelian extension. As noted before this is equivalent to prove that $(\gamma - 1) \cdot \bar{\eta} \in 2 \cdot \bar{U}^-_K$ by [@tate:book Prop. IV.1.2]. Now $\gamma$ acts on $U_K^-$ as $-\sigma^2$. Thus, by the definition of $\bar{\eta}$ and the isomorphism between $U^-_K$ and $\mathbb{Z}[H]$, this is equivalent to prove that $$\label{eq:abcond} (\sigma^2 + 1) \kappa f \in 2 \mathbb{Z}[H].$$ \[eq:2zh\] Let $x \in \mathbb{Z}[H]$. Then $x \in 2\mathbb{Z}[H]$ if and only if $x e_0 \in 2e_0\mathbb{Z}[H]$ and $xe_1 \in 2e_1\mathbb{Z}[H]$. If $x \in 2\mathbb{Z}[H]$ then clearly $x e_0 \in 2e_0\mathbb{Z}[H]$ and $xe_1 \in 2e_1\mathbb{Z}[H]$. Reciprocally, assume that $x e_0 = 2e_0a_0$ and $x e_1 = 2e_1a_1$ with $a_0, a_1 \in \mathbb{Z}[H]$. Let $a := e_0a_0 + e_1a_1$. We have by construction $2a = x \in \mathbb{Z}[H]$ and $3a = (3e_0)a_0 + (3e_1)a_1 \in \mathbb{Z}[H]$. Therefore $a$ belongs to $\mathbb{Z}[H]$ and the claim is proved. We prove using the claim. On one hand, we have $$e_0 (\sigma^2 + 1) \kappa f = 2^{e'+t_S+1} e_0 f \in 2e_0\mathbb{Z}[H].$$ On the other hand, we have $$e_1 (\sigma^2 + 1) \kappa f = 2^{(e-e')/2} e_1 (\sigma^2 + 1) f.$$ The proof will be complete if we prove that $e - e' > 0$. For that we use the following claim. \[claim:h0iso\] $|\hat{H}^0(T, U_K/U_F)| = 2^{e-e'}.$ Let $U_K^\circ$ be the subgroup of elements $u \in U_K$ such that $u^{1 - \tau} \in U_F$. We have $$\begin{aligned} \hat{H}^0(T, U_K/U_F) & = \frac{(U_K/U_F)^T}{\mathcal{N}(U_K/U_F)} = \frac{U_K^\circ/U_F}{\mathcal{N}(U_K)/\mathcal{N}(U_K) \cap U_F} \\ & \simeq \frac{U_K^\circ/U_F}{(\mathcal{N}(U_K) \, U_F)/U_F} \simeq U_K^\circ/(\mathcal{N}(U_K) \, U_F) \\ & \simeq \bar{U}_K^\circ/(\mathcal{N}(\bar{U}_K) + \bar{U}_F). \end{aligned}$$ By , it is enough to prove the following isomorphism $$\label{eq:prove} \bar{U}_{K^+}/(\mathcal{N}(\bar{U}_K) + \bar{U}_k) \simeq \bar{U}_K^\circ/(\mathcal{N}(\bar{U}_K) + \bar{U}_F).$$ Since $\bar{U}_{K^+} \cap (\mathcal{N}(\bar{U}_K) + \bar{U}_F) = \mathcal{N}(\bar{U}_K) + \bar{U}_k$, there is a natural injection of the LHS of in the RHS induced by the inclusion $\bar{U}_{K^+} \subset \bar{U}_K^\circ$. We prove now that this map is surjective. Let $\bar{u} \in \bar{U}_K^\circ$. Thus $\bar{x} := (1 - \tau) \cdot \bar{u} \in \bar{U}_F$. Note that $(1 - \tau) \cdot \bar{x} = 2 \cdot \bar{x}$. Define $\bar{y} := N_H \cdot \bar{u} - \bar{x} \in \bar{U}_F$ and $\bar{z} := \bar{u} - \bar{y}$. We have $$(1 - \tau) \cdot \bar{z} = (1 - \tau) \cdot \bar{u} - (1 - \tau) N_H \cdot \bar{u} + (1 - \tau) \cdot \bar{x} = \bar{x} - N_H \cdot \bar{x} + 2 \cdot \bar{x} = 0.$$ Thus $\bar{z} \in \bar{U}_{K^+}$. This proves that $\bar{u} = \bar{z} + \bar{y} \in \bar{U}_{K^+} + \bar{U}_F$. Equation follows and the proof of the claim is finished. Now by the multiplicativity of the Herbrand quotient and Lemma \[lem:qu\], we find that $$\label{eq:qukf} Q(T, U_K/U_F) = \frac{Q(T, U_K)}{Q(T, U_F)} = 2^{2d-2}.$$ Therefore $e - e' \geq 2d - 2 \geq 2$. This concludes the proof that $K(\sqrt{\eta})$ is abelian over $k$ and the proof of the theorem. \[cor:square6\] Under the hypothesis of the theorem and assuming that the Stark unit exists, then it is a square in $K$ if and only if the Stark unit for the extension $F/k$ and the set $S$ is a square and $(e - e')/2 + c - c' \geq 1$ where $c$ is the $2$-valuation of $|{\mathrm{Cl}}_K^-|$, $c'$ is the $2$-valuation of $|{\mathrm{Cl}}_F^-|$ and $(\bar{U}_k : \mathcal{N}(\bar U_F)) = 2^{e'}$. In particular, if $d \geq 4$ then it is always a square and, in fact, it is a $2^{d-3}$-th power. It is also a square if $d = 3$ and the extension $K/k$ is ramified at some finite prime. We use the notations and results of the proof of the theorem. By the unicity statement, the Stark unit, if it exists, is equal to $\eta$ or one of this conjugate over $F$. In particular, the Stark unit is a $2^r$-th power in $K$ if and only if $\bar\eta \in 2^r \cdot U_K^-$. By Claim \[eq:2zh\] and the construction of $\bar\eta$, this is equivalent to $$2^{e'+t_S} e_0 f \in 2^r e_0\mathbb{Z} \quad\text{and}\quad 2^{(e-e')/2} e_1 f \in 2^r e_1\mathbb{Z}[H].$$ Now $e_0 f \mathbb{Z} = e_0 |{\mathrm{Cl}}_F^-| \mathbb{Z}$ by the definition of $f$, Claim \[claim:nclk\] and the discussion that precedes it. Thus the first condition is equivalent to $e'+t_S+c' \geq r$. For $r = 1$, this is equivalent to the fact that the Stark unit for $F/k$ and the set $S$ is a square by Theorem \[th:quadratic\] and the discussion that follows on the number of primes in $S$ that are inert in $F/k$. For the second condition, recall that $e_1f\mathbb{Z}[H] \simeq {\mathrm{Fitt}}_{\mathbb{Z}[\omega]}(({\mathrm{Cl}}_K^-)^{e_1})$ and therefore $e_1f \in 2^v \mathbb{Z}[H]$ where $v$ is the $2$-valuation of the index $(\mathbb{Z}[\omega]:{\mathrm{Fitt}}_{\mathbb{Z}[\omega]}(({\mathrm{Cl}}_K^-)^{e_1}))$. By Claim \[claim:nclk\] and the computation before it, this index is equal to $|{\mathrm{Cl}}_K^-|/|{\mathrm{Cl}}_F^-|$. Therefore the second condition is equivalent to $(e-e')/2 + c-c' \geq r$. This proves the first assertion: the Stark unit for $K/k$ and $S = S(K/k)$ is a square if and only if the Stark unit for the extension $F/k$ and the set $S$ is a square and $(e - e')/2 + c - c' \geq 1$. For the second assertion, we have $e' \geq d-3$ by and $(e-e')/2 \geq d-1$ by Claim \[claim:h0iso\] and . Thus $\bar\eta \in 2^{d-3} \cdot U_K^-$ for $d \geq 4$ and we have that the Stark unit is a $2^{d-3}$-th power if $d \geq 4$. Finally, for $d = 3$, the condition $2^{(e-e')/2} e_1 f \in 2 e_1\mathbb{Z}[H]$ is always satisfied. Assume that the extension $K/k$ is ramified at some finite prime. If $F/k$ is also ramified at some finite prime then the Stark unit for the extension $F/k$ and the set $S$ is a square by Theorem \[th:quadratic\]. If $F/k$ is unramified at finite primes then any prime ideal that ramifies in $K/k$ is inert in $F/k$ by (A3). Therefore $t_S \geq 1$ and the Stark unit for the extension $F/k$ and the set $S$ is a also square by Theorem \[th:quadratic\]. It follows that the Stark unit for $K/k$ is a square by the first part. This concludes the proof. Note that the condition in the case $d = 3$ is sharp. Indeed let $k := \mathbb{Q}(\alpha)$, where $\alpha^3 + \alpha^2 - 9\alpha - 8 = 0$, be the smallest totally real cubic field of class number $3$. Let $v_1, v_2, v_3$ be the three infinite places of $k$ with $v_1(\alpha) \approx -3.0791$, $v_2(\alpha) \approx -0.8785$ and $v_3(\alpha) \approx 2.9576$. Let $K$ be the ray class field of $k$ of modulus $\mathbb{Z}_k v_2 v_3$. The extension $K/k$ is cyclic of order $3$, satisfies (A1), (A2) and (A3) with $S := (S/k)$, and is unramified at finite places. One can check that, if it exists, the Stark unit is not a square in $K$. [^1]: Supported by the JSPS Global COE *CompView*. [^2]: In fact the place $w$ but changing the place $w$ just amounts to replace the Stark unit by one of its conjugate. [^3]: Similar in some way to the index formulae for cyclotomic units, see [@washington:book Chap. 8]. [^4]: Unfortunately, in most cases the values are complex and there does not appear to be any obvious way to remove these absolute values. [^5]: For $v$ a finite place, the abelian rank one Stark conjecture is basically equivalent to the Brumer-Stark conjecture, see [@tate:book §IV.6]. Recent results of Greither and Popescu [@gp:iwasawa] imply the validity of the Brumer-Stark conjecture away from its $2$-part and under the hypothesis that an appropriate Iwasawa $\mu$-invariant vanishes. [^6]: Since $v$ is the only real place of $k$ that stays real in $K$, we will usually not specify it. [^7]: We discard the last absolute value $|\cdot|_{(d+1)m}$. [^8]: Although assumptions (A1) to (A4) are necessary to prove that the Stark unit is a solution of (P2), it is not necessary to assume (A4) to prove that solutions exist in the cases that we study below. It is an interesting question whether or not one could prove that the Stark unit is a solution to (P2) without having first to assume (A4). [^9]: Note that $\mathbb{Q}[G]^-$ is a ring with identity $e^-$. [^10]: If $\bar\eta$ is a solution to (P1) and (P2) and $u$ is a $B$-unit then $u \cdot \bar\eta$ is not necessarily a solution to (P1) and (P2). A necessary and sufficient condition for that is that the linear map $x \mapsto ux$ of $\mathbb{Q}[G]^-$ has determinant $\pm 1$. This is always true if $u$ is a unit of $\mathbb{Z}[G]^-$. [^11]: PARI/GP was also used to find the examples given in the next two sections. [^12]: Note that is easy to see that both sides of are positive in this case. [^13]: Strictly speaking, the claims above are only for integral ideals of $\mathbb{Z}[H]$ but they admit obvious and direct generalizations to fractional ideals. [^14]: By (A3), it cannot be split in $F/k$. [^15]: This inequality follows also from Claim \[claim:h0iso\] below. [^16]: Recall that $S = S(K/k)$.

--- abstract: | The interval number of a graph $G$ is the minimum $k$ such that one can assign to each vertex of $G$ a union of $k$ intervals on the real line, such that $G$ is the intersection graph of these sets, i.e., two vertices are adjacent in $G$ if and only if the corresponding sets of intervals have non-empty intersection. In 1983 Scheinerman and West \[The interval number of a planar graph: Three intervals suffice. *J. Comb. Theory, Ser. B*, 35:224–239, 1983\] proved that the interval number of any planar graph is at most $3$. However the original proof has a flaw. We give a different and shorter proof of this result. author: - Guillaume Guégan - Kolja Knauer - Jonathan Rollin - Torsten Ueckerdt bibliography: - 'sw.bib' title: The interval number of a planar graph is at most three --- Introduction ============ For a positive integer $k$, a *$k$-interval representation* of a graph $G = (V,E)$ is a set $\{f(v) \mid v \in V\}$ where $f(v)$ is the union of at most $k$ intervals on the real line representing vertex $v\in V$, such that $uv$ ($u \neq v$) is an edge of $G$ if and only if $f(u)\cap f(v) \neq \emptyset$. In 1979 Trotter and Harary [@Tro-79] introduced the *interval number* of $G$, denoted by $i(G)$, as the smallest $k$ such that $G$ has a $k$-interval representation. In 1983 Scheinerman and West [@Sch-83] constructed planar graphs with interval number at least $3$ and proposed a proof for the following: \[thm:planar-3-intervals\] If $G$ is planar, then $i(G) \leq 3$. In this paper we point out a flaw in that proof and give an alternative proof of Theorem \[thm:planar-3-intervals\]. The original proof is based on induction, maintaining a very comprehensive description of the (partial) representation and its properties. However in certain cases, one of the required properties can not be maintained during the construction. A precise description of the problematic case is given in Section \[sec:old\]. Let us also remark that in his thesis [@Sch-84], Scheinerman proposes a slightly different argumentation for the considered case which also leads to the same problem. In the remaining part of the introduction we discuss some related work. In Section \[sec:prel\] we introduce the notions used in the new proof, which is then given in Section \[sec:new\]. We conclude with final remarks in Section \[sec:conclusion\]. #### Related work. A graph with $i(G)\leq 1$ is called an *interval graph*. A concept closely related to the interval number is the *track number* of $G$, denoted by $t(G)$, which is the smallest $k$ such that $G$ is the union of $k$ interval graphs. Equivalently, $t(G)$ is the smallest $k$ such that $G$ admits a $k$-interval representation that is the union of $k$ $1$-interval representations, each on a different copy of the real line (called a track) and each containing one interval per vertex. More recently, Knauer and Ueckerdt [@Kna-16] defined the *local track number* of $G$, denoted by $t_\ell(G)$, to be the smallest $k$ such that $G$ is the union of $d$ interval graphs, for some $d$, where every vertex of $G$ is contained in at most $k$ of them. It is easy to see that for every graph $G$ we have $i(G) \leq t_\ell(G) \leq t(G)$. Gonçalves [@Gon-07] proved that the track number of planar graphs is at most $4$, which is best-possible [@Gon-09]. It is an open problem whether there is a planar graph with local track number $4$ [@Kna-16]. Balogh *et al.* [@Bal-04] show that any planar graph of maximum degree at most $4$ has interval number at most $2$. For further recent results about interval numbers of other classes of planar graphs and general graphs, let us refer to [@Bal-99] and [@deQ-16], respectively. Finally, let us mention that West and Shmoys [@Wes-84] proved that deciding $i(G) \leq k$ is NP-complete for each $k \geq 2$, while Jiang [@Jia-13] proved the same for $t(G) \leq k$, and Stumpf [@Stu-15] for $t_\ell(G) \leq k$. Preliminaries {#sec:prel} ============= All graphs considered here are finite, simple, undirected, and non-empty. If $G = (V,E)$ is a graph and $\{f(v) \mid v \in V\}$ is a $k$-interval representation of $G$ for some $k$, we say that a subset $S$ of the real line is *intersected* by some $f(v)$ if $S \cap f(v) \neq \emptyset$. For a point $p$ with $p \in f(v)$, we also say that $p$ is *covered* by $f(v)$. A vertex $v$ has a *broken end* $b$ if $b$ is an endpoint of an interval in $f(v)$ and $b$ is not covered by any other $f(w)$ for $w\in V - \{v\}$. We say that a vertex $v$ is *displayed* if $f(v)$ contains a *portion* (i.e., a non-empty open interval) not intersected by any other $f(w)$ for $w\in V - \{v\}$. An edge $uv\in E$ is *displayed* if $f(u)\cap f(v)$ contains a portion not intersected by any other $f(w)$ for $w \in V - \{u,v\}$. The *depth* of the representation $\{f(v) \mid v \in V\}$ is the largest integer $d$ such that some point of the real line is covered by at least $d$ sets $f(v)$ with $v \in V$. For $d \geq 1$, the *depth $d$ interval number* of $G$, denoted by $i_d(G)$, is the smallest $k$ such that $G$ admits a $k$-interval representation of depth $d$. Scheinerman and West [@Sch-83] proved that there are planar graphs $G$ with $i_2(G) \geq 4$, while Gonçalves [@Gon-07] proved that $i_2(G) \leq 4$ for all planar graphs $G$. In Theorem \[thm:4-connected-case\] we obtain that for any $4$-connected planar graph $G$ it holds $i_2(G) \leq 3$. Finally, we shall prove here that $i_3(G) \leq 3$ for all planar graphs $G$, which is also the original claim of Scheinerman and West. An alternative proof of Theorem \[thm:planar-3-intervals\] {#sec:new} ========================================================== In this section we give a new proof for Theorem \[thm:planar-3-intervals\], i.e., that every planar graph $G$ has interval number at most $3$. A *triangulation* is a plane embedded graph in which every face is bounded by a triangle. In other words, a triangulation is a maximally planar graph on at least three vertices with a fixed plane embedding. As every planar graph is an induced subgraph of some triangulation (For example, iteratively adding a new vertex to a non-triangular face together with one edge to each of the face’s incident vertices, eventually results in such a triangulation.) and the interval number is monotone under taking induced subgraphs, we may assume without loss of generality that $G$ is a triangulation. A triangle in $G$ is *non-empty* if its interior contains at least one vertex of $G$. We shall construct a $3$-interval representation of $G$ by recursively splitting $G$ along its non-empty triangles. This leaves us with the task to represent $4$-connected triangulations, i.e., triangulations whose only non-empty triangle is the outer triangle, and to “glue” those representation along the non-empty triangles of $G$. More precisely, we shall roughly proceed as follows: 1. Consider a non-empty triangle $\Delta$ with inclusion-minimal interior and the set $X$ of all vertices in its interior. 2. Call induction on the graph $G_1 = G - X$, obtaining a $3$-interval representation $f_1$ of $G_1$ with additional properties on how inner faces are represented. 3. Since the subgraph $G_2$ of $G$ induced by $V(\Delta) \cup X$ is $4$-connected, we can utilize a recent result of the second and fourth author to decompose $G_2$ into a path and two forests.\[enum:step-decompose\] 4. Using this decomposition, we define a $3$-interval representation $f_2$ of $G_2$ that coincides with $f_1$ on $V(\Delta)$.\[enum:step-glue\] We remark that the essentials of the construction in step \[enum:step-glue\] can be already found in [@Axe-13]. Also, the decomposition of a triangulation along its non-empty triangles is a common method in the field of intersection graphs; see e.g., [@Tho-86; @Cha-09; @Cha-12]. Hence the key to our new proof is the most recent [@Kna-17] decomposition used in step \[enum:step-decompose\], which we shall state next. Let $G = (V,E)$ be a $4$-connected triangulation with outer vertices $x,y,z$. We denote by $u_x$ the unique inner vertex of $G$ adjacent to $y$ and $z$ (if there were several, $G$ would not be $4$-connected), and call the vertex $u_x$ the vertex *opposing $x$*. Similarly, the vertex $u_y$ opposing $y$ and the vertex $u_z$ opposing $z$ are defined. Note that since $G$ is $4$-connected we have that $u_x,u_y,u_z$ all coincide if $|V| = 4$, and are pairwise distinct if $|V| \geq 5$. We use the following recent result of the second and fourth author (Lemma 3.1 in [@Kna-17]), which we have adapted here to the case of $4$-connected triangulations (see Figure \[fig:new-proof\] for an illustration): \[lem:inner-decomposition\] Let $G$ be a plane $4$-connected triangulation with outer triangle $\Delta_{\rm out} = x,y,z$ and corresponding opposing vertices $u_x,u_y,u_z$. Then the inner edges of $G$ can be partitioned into three forests $F_x,F_y,F_z$ such that - $F_x$ is a Hamiltonian path of $G \setminus\{y,z\}$ going from $x$ to $u_x$, - $F_y$ is a spanning tree of $G \setminus\{x,z\}$, - $F_z$ is a spanning forest of $G \setminus\{y\}$ consisting of two trees, one containing $x$ and one containing $z$, unless ${G}\cong K_4$. In this case $F_z=zu_z$. ![The decomposition in Lemma \[lem:inner-decomposition\] for the case $G \not\cong K_4$ (left) and $G \cong K_4$ (right).[]{data-label="fig:new-proof"}](inner-decomposition) Note that the conditions on the decomposition in Lemma \[lem:inner-decomposition\] imply that the edge $xu_y$ is in $F_x$ and the edge $zu_y$ is in $F_z$ (since $x$ and $z$ are in different components of $F_z$ and $z$ is not in $F_x$). We are now ready to give our new proof of Theorem \[thm:planar-3-intervals\]. We have to show that every planar graph $G$ admits a $3$-interval representation of depth at most $3$, in particular that $i(G) \leq 3$. As every planar graph is an induced subgraph of some planar triangulation and the interval number is monotone under taking induced subgraphs, we may assume without loss of generality that $G$ is a triangulation. We proceed by induction on the number $n$ of vertices in $G$, showing that $G$ admits a $3$-interval representation of depth $3$ with the additional invariant that **(I1)** every vertex is displayed, and **(I2)** every inner face contains at least one displayed edge. The base case is $n = 3$, i.e., $G$ is a triangle with vertices $x,y,z$. In this case, it is easy to define a $3$-interval representation of $G$ with invariants **(I1)** and **(I2)**. For example, take $f(x) := [0,3]$, $f(y) := [1,4] \cup [6,7]$, and $f(z) := [2,5]$, and note that edges $xy$ and $yz$ are displayed. Now assume that $n \geq 4$, i.e., $G$ contains at least one non-empty triangle. Let $\Delta$ be a non-empty triangle in $G$ with inclusion-minimal interior among all non-empty triangles in $G$. Let $X \subset V$ be the (non-empty) set of vertices in the interior of $\Delta$, $G_{\rm out} = G - X$ be the triangulation induced by $V - X$, and $G_{\rm in}$ be the $4$-connected triangulation induced by $V(\Delta) \cup X$. By induction hypothesis there exists a $3$-interval representation $\{f(v) \mid v \in V - X\}$ of $G_{\rm out}$ of depth $3$ satisfying the invariant **(I1)** and **(I2)**. In particular, the three vertices $x,y,z$ of $\Delta$, which now form an inner face of $G_{\rm out}$, are already assigned to up to three intervals each. We shall now extend this representation to the vertices in $X$. By invariant **(I2)** at least one edge of $\Delta$, say $xz$, is a displayed edge. Consider the vertices $u_x,u_y,u_z$ opposing $x,y,z$ in $G_{\rm in}$, respectively, and let $F_x$, $F_y$, $F_z$ be the decomposition of the inner edges of $G_{\rm in}$ given by Lemma \[lem:inner-decomposition\] (note that we can choose $x$, $y$, $z$ arbitrarily in Lemma \[lem:inner-decomposition\]). For convenience, if $G_{\rm in} \cong K_4$, let $F_z$ additionally have vertex $x$ as a one-vertex component. We define, based on this decomposition, three intervals for each vertex $v \in X$ as follows. - Represent the path $F_x - \{x\}$ on an unused portion of the real line by using one interval per vertex, and in such a way that every vertex and every edge is displayed. Let this representation be denoted by $\{f_1(v) \mid v \in X\}$. - Consider $y$ to be the root of the tree $F_y$, and for each vertex $v$ in $F_y - \{y\}$ consider the parent $w$ of $v$ in $F_y$, i.e., the neighbor of $v$ on the $v$-to-$y$ path in $F_y$. Create a new interval for $v$ strictly inside the displayed portion of $w$. If $w = y$, such a portion exists in $f(y)$ as **(I1)** holds for $f$, and if $w \neq y$, such a portion exists in $f_1(w)$. Let these new intervals be denoted by $\{f_2(v) \mid v \in X\}$. - Consider the three trees that are the components of $F_z - \{u_yz\}$, that is, of $F_z$ after removing the edge $u_yz$. Say $T_1$ contains vertex $u_y$ (and possibly no other vertex), $T_2$ contains $z$, and $T_3$ contains $x$. Consider $u_y$, respectively $z$ and $x$, to be the root of $T_1$, respectively $T_2$ and $T_3$. For each vertex $v$ in $T_1 - \{u_y\}$, respectively $T_2 - \{z\}$ and $T_3 - \{x\}$, consider the parent $w$ of $v$ in $T_1$, respectively $T_2$ and $T_3$, and create a new interval for $v$ in the displayed portion of $w$. (Again, if $w \in \{x,z\}$, such a portion exists in $f(w)$ as **(I1)** holds for $f$, and if $w \notin \{x,z\}$, such a portion exists in $f_1(w)$.) Let these new intervals be denoted by $\{f_3(v) \mid v \in X - \{u_y\}\}$. - Finally, create a new interval $f_3(u_y)$ in the displayed portion of edge $xz$ (to represent $xu_y$ and $zu_y$). Defining $f(v) = f_1(v) \cup f_2(v) \cup f_3(v)$ for each $v \in X$, it is straightforward to check that $\{f(v) \mid v \in V\}$ is a $3$-interval representation of $G = G_{\rm out} \cup G_{\rm in}$ of depth $3$. Moreover, the edge $xz$, every inner edge of $G_{\rm in}$ except for $u_yx$ and $u_yz$, and every vertex of $X$ is displayed. Hence this representation satisfies our invariants **(I1)** and **(I2)**, which concludes the proof. The approach of Scheinerman and West {#sec:old} ==================================== To explain the proof strategy of Scheinerman and West, we introduce some more terminology from their paper. For a plane embedded graph $G$ let $G^0$ denote the (outerplanar) subgraph induced by its *external vertices*, i.e., those lying on the unbounded, outer face. The edges of $G^0$ that bound the outer face are the *external edges*, while those that bound two inner faces are the *chords*. The graph $G^0$ is considered together with its decomposition into its inclusion-maximal $2$-connected subgraphs, called blocks. Furthermore, we fix for each component of $G^0$ a non-cut-vertex $z$ as the root of that component and say that $z = z_H$ is the root of its corresponding block $H$. For each remaining block $H$ of $G^0$ let $z_H$ be the cut-vertex of $G^0$ in $H$ that is closest to the root of the corresponding component. For a chord $xy$ of $G^0$ contained in block $H$ with root $z_H$ and a third vertex $u \neq x,y$ of $H$, say that $u$ is *on the $z^*_H$-side* of $xy$ if $u$ and $z_H$ lie in different connected components of $H - \{x,y\}$. The argument of Scheinerman and West proceeds by induction on the number of vertices in $G$ with a stronger induction hypothesis on how $G^0$, i.e., the subgraph of $G$ induced by the external vertices, is represented. Every edge $xy$ of $G^0$ shall be represented in such a way that for a possible future vertex $v$ which is adjacent to $x$ and $y$ we can define only one interval $I$ that has at least two of the following three properties: $$\textbf{P1: } I \cap f(x) \neq \emptyset, \qquad \textbf{P2: } I \cap f(y) \neq \emptyset, \qquad \textbf{P3: } I \text{ is displayed}.$$ If $xy$ is displayed, it is easy to find an interval $I$ for $v$ having properties **P1** and **P2**. Moreover if $x$, respectively $y$, has a broken end, it is easy to satisfy **P1**, respectively **P2**, and **P3**. However if $xy$ is not displayed and neither $x$ nor $y$ has a broken end, Scheinerman and West propose to alter the existing representation of another vertex $u$ whose interval covers an endpoint of $x$. As their modification, which is depicted in Figure \[fig:reusable-end\], splits an interval of $u$ into two, it is necessary that $u$ appears at most twice so far, meaning that $f(u)$ consists of at most two intervals. This shall be achieved by specifying the current representation of $G$ quite precisely. ![Figure 3 from [@Sch-83] illustrating how to use a $z^*_H$-reusable endpoint $b$ of $f(x)$, which is covered by some vertex $u$ and assigned to a chord $xy$ of $H$, in order to insert a new vertex $v$ that is adjacent to $x$ and $y$. Note that the interval for $u$ is split into two.[]{data-label="fig:reusable-end"}](Fig-3-from-SW){width="\textwidth"} Given a rooted plane embedded graph $G = (V,E)$, i.e., with a fixed root $z_H$ for each block $H$ of $G^0$, a representation $\{f(v) \mid v \in V\}$ is called *P-special* if every vertex is displayed and each of the following holds: 1. Each root is represented by one interval and every other external vertex is represented by at most two intervals. 2. For each block $H$ of $G^0$ all edges incident to $z_H$ are displayed. 3. Each non-displayed edge $xy$ of any block $H$ of $G^0$ is assigned to an endpoint $b$ of $f(x)$ or $f(y)$, say $f(x)$, such that the following hold. 1. If $xy$ is an external edge, the endpoint $b$ is a broken end. 2. If $xy$ is a chord, the endpoint $b$ is a broken end, or $b$ is covered by $f(u)$ for only one other vertex $u$, where additionally $u$ is on the $z^*_H$-side of $xy$ and $ux$ is an edge of $G^0$. An endpoint $b$ satisfying this condition is called *$z^*_H$-reusable* for edge $xy$. \[enum:reusable\] 3. For each endpoint $b$ there is at most one edge assigned to $b$. 4. Each vertex covers at most one endpoint which has an edge assigned. Now Scheinerman and West propose to show by induction on the number of vertices in $G$ that every rooted plane embedded graph $G$ admits a P-special $3$-interval representation. They remove a small set of carefully chosen external vertices from $G$, induct on the smaller instance to obtain a P-special representation, create intervals for the removed vertex/vertices, extend and/or alter the existing representation, and argue that the result is a P-special $3$-interval representation of $G$. The problem lies in the last step. Some removed vertex $v$ may cover an edge $ux$ that is external in $G - v$, but a chord in $G$. But it may be that some non-displayed chord $xy$ of $G - v$ is assigned to an endpoint $b$ of $f(x)$ that is $z^*_H$-reusable and covered by $u$. This assignment cannot be kept, since $ux$ is no longer external, which however is required for $z^*_H$-reusability as it is defined in \[enum:reusable\] above. Thus, the invariants of the induction cannot be maintained. Let us explain in more detail below (discussing only the relevant cases) on basis of a small example graph that the above problem can indeed occur, and why some straightforward attempt to fix it does not work. To this end, consider the plane embedded $9$-vertex graph $G$ in the top-left of Figure \[fig:counterexample-decompose\]. The thick edges highlight the subgraph $G^0$, which has only one block $H$, and let us pick vertex $x_1$ to be the root of that block, i.e., $z_H = x_1$. The base case of the inductive construction is an independent set of vertices. Otherwise, we identify a set $X$ of external vertices that we want to remove as follows. Consider a leaf-block $H$ of $G^0$, i.e., one that contains no root of another block, its root $z_H$, and distinguish the following cases. ![A plane embedded $9$-vertex graph $G$ with one block $H$ and root $z_H = x_1$, and how it is reduced to an independent set according to the induction rules in [@Sch-83]. In each of the six steps, the set $X$ is highlighted and the smaller graph is obtained by removing all vertices in $X$.[]{data-label="fig:counterexample-decompose"}](counterexample-decompose) Case I: Every chord of $H$ is incident to $z_H$. : In this case label the vertices of $H$ as $z_H,v_1,\ldots,v_k$, $k \geq 1$, in this cyclic order around the outer face of $H$. If $k \geq 2$, let $X = \{v_1,\ldots,v_p\}$, where $p$ is the smallest integer greater than or equal to $2$ for which $v_p$ has degree $2$ in $H$. Case II: Some chord of $H$ is not incident to $z_H$. : Consider an inner face $C$ of $H$ that does not contain $z_H$ and is bounded by only one chord $xy$, and distinguish further. Case IIa: $C$ is a triangle. : Let $X = \{v\}$, where $v \neq x,y$ is the third vertex in $C$. Case IIb: $C$ has at least four vertices. : In this case label the vertices of $C$ as $x,v_1,\ldots,v_k,y$, $k \geq 2$, in this cyclic order around $C$, and let $X = \{v_1,v_2\}$. Figure \[fig:counterexample-decompose\] shows how our example graph $G$ is reduced to an independent set according to these rules. Next, we consider these steps in reverse order and construct an interval representation following the case distinction as proposed in [@Sch-83]. Let us refer to Figure \[fig:counterexample\] for a step-by-step illustration of this construction. The problem arises in Step 5. Step 1: base case. : Represent $x_1$ with a single interval. Step 2: Case I, $X = \{x_2,x_3\}$. : Insert intervals for $x_1,x_2$ into the displayed interval for $x_1 = z_H$. Place overlapping, displayed intervals for $x_1,x_2$ in an unused portion of the real line. Step 3: Case IIa, $X = \{x_4\}$. : As the chord $xy = x_2x_3$ is displayed, called subcase (1), add a displayed interval for $v = x_4$ and place the second interval available for $v$ in the displayed portion for $xy$. Assign one broken end of $v$ to each of the external edges $xv=x_2x_4$ and $yv=x_3x_4$. Step 4: Case IIa, $X = \{x_5\}$. : As the chord $xy = x_3x_4$ is not displayed, but assigned to a broken end $b$ of $x = x_4$, called subcase (2), add the displayed interval for $v = x_5$ so that it overlaps $f(x)$ at $b$. Add a second interval for $v$ in the displayed portion of $f(y)$. Assign the chord $xy$ to the $z^*_H$-reusable endpoint $b$, which is here covered by $u = x_5$ with $ux$ being an external edge. Step 5: Case IIa, $X = \{x_6\}$. : As the chord $xy = x_4x_5$ is displayed, we are in subcase (1) again. We add a displayed interval for $v = x_6$ and place a second interval for $v$ in the displayed portion for $xy$. Now the problem is that the chord $x_3x_4$ can no longer be assigned to the endpoint $b$, since the edge $x_4x_5$ is no longer external. One might be tempted to assume that the problem can be fixed by relaxing the definition of $z^*_H$-reusability as follows. Instead of $ux$ being external, maybe it suffices to require that $u$ appears at most twice so far. Let us call this new concept *almost $z^*_H$-reusability*. However, the assumption that $ux$ is external is crucial for Case IIb, as we will demonstrate in Steps 6 and 7 below. ![The steps of the inductive construction in [@Sch-83] applied to the graph in Figure \[fig:counterexample-decompose\]. Note that after Step 5 the chord $x_3x_4$ is not displayed and the right endpoint of the longer interval for $x_4$ is not $z^*_H$-reusable since edge $x_4x_5$ is not external. Further note that after Step 7 vertex $x_5$ is represented by four intervals.[]{data-label="fig:counterexample"}](counterexample) Step 6: Case IIb, $X = \{x_7,x_8\}$. : Assign $v_1 = x_8$ an interval in the displayed interval for $x = x_3$, and assign $v_2 = x_7$ an interval in the displayed interval for $v_3 = x_6$. As the chord $xy = x_3x_4$ is not displayed, but assigned to an almost $z^*_H$-reusable endpoint $b$ of $y = x_4$, we are in subcase (2c) of [@Sch-83]. Here it is concluded in [@Sch-83] that the vertex $u$ covering $b$ must be $u = v_k = x_6$ due to the externality of edge $ux$. Hence it would suffice to place overlapping displayed intervals for $v_1$ and $v_2$ in an unused portion of the line and add intervals for the inner neighbors of $v_1,v_2$ in the displayed portion for $v_1$, $v_2$, and $v_1v_2$, according to the subset of these vertices that they are adjacent to. However, in our example we have $u = x_5$, which at least appears only twice so far, but which is an inner neighbor of $v_1$ and $v_2$, and thus has to spend its third interval in the displayed portion of $v_1v_2$. We end up with chord $xy$ still assigned to the endpoint $b$ that is covered by $u$, but neither is $ux$ external, nor is $u$ appearing only twice so far. In particular, we are not prepared to insert a future vertex adjacent to $x$ and $y$, as we illustrate in Step 7. Step 7: Case IIa, $X = \{x_9\}$. : The chord $xy = x_3x_4$ is not displayed, but assigned to the endpoint $b$ of $x = x_4$, which is covered by $u = x_5$ with $u$ however being internal and appearing three times already. Thus, if we apply the proposed modification as illustrated in Figure \[fig:reusable-end\], we split one interval of $u$ into two, causing $f(u)$ to consist of four intervals. This concludes our example. Let us remark that the fourth interval for $x_5$ in Step 7 of Figure \[fig:counterexample\] might seem superfluous, but is actually needed to give $u$ a displayed portion. This in turn is necessary as there might be a neighbor $w$ of $x_5$ of degree $1$. For example, with $w$ inside face $x_5x_6x_7$, it would have been inserted between Steps 5 and 6, according to Case I with $z_H = x_5$, spending one interval in the displayed portion of $u$. Also note that in this particular example other ways of assigning endpoints in Step 5 would have allowed the process to continue. However, the first author together with Daniel Gonçalves tried some time to obtain new invariants to fix this induction but did not succeed. Concluding remarks {#sec:conclusion} ================== As mentioned above, Axenovich *et al.* [@Axe-13] used some steps of the construction from Section \[sec:new\] to prove that every graph $G$ whose edges decompose into $k-1$ forests and another forest of maximum degree $2$ admits a $k$-interval representation of depth at most $2$. Note that if $G$ is a $4$-connected triangulation with outer triangle $\Delta_{\rm out}$, the decomposition of $G - E(\Delta_{\rm out})$ given by Lemma \[lem:inner-decomposition\] can be easily extended to a decomposition of all edges in $G$ into two forests and a path, which immediately gives the following. \[thm:4-connected-case\] If $G$ is planar and $4$-connected, then $i_2(G) \leq 3$. Given that the largest interval number among all planar graphs is $3$, while the largest track number is $4$, it remains open to determine whether the largest local track number among all planar graphs is $3$ or $4$, c.f., [@Kna-16 Question 19]. We feel that the gluing of several triangulations along separating triangles is likely to be possible along the lines discussed in [@Kna-16] for planar $3$-trees. However, finding a $3$-local track representation of a $4$-connected planar triangulation, strengthening Theorem \[thm:4-connected-case\], seems to be more difficult. What is the largest local track number among all planar graphs? #### Acknowledgments. We would like to thank Maria Axenovich and Daniel Gonçalves for fruitful discussions. We also thank the anonymous reviewers for their comments and suggestions. Moreover, we thank Ed Scheinerman and Douglas West for their helpful comments on an earlier version of this manuscript and for providing us a copy of Ed’s PhD thesis. The second author was partially supported by ANR projects ANR-16-CE40-0009-01 and ANR-17-CE40-0015 and by the Spanish Ministerio de Economía, Industria y Competitividad, through grant RYC-2017-22701.

--- abstract: 'The normalization of Bethe eigenstates for the totally asymmetric simple exclusion process on a ring of $L$ sites is studied, in the large $L$ limit with finite density of particles, for all the eigenstates responsible for the relaxation to the stationary state on the KPZ time scale $T\sim L^{3/2}$. In this regime, the normalization is found to be essentially equal to the exponential of the action of a scalar free field. The large $L$ asymptotics is obtained using the Euler-Maclaurin formula for summations on segments, rectangles and triangles, with various singularities at the borders of the summation range.' author: - Sylvain Prolhac title: Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process --- (0,0)(10,-65) (0,0)[(0,0)(170,0)(170,20)(0,20)]{} [Introduction]{} \[section introduction\] Understanding the large scale evolution of macroscopic systems from their microscopic dynamics is one of the central aims of statistical physics out of equilibrium. Much progress has been happening toward this goal for systems in the one-dimensional KPZ universality class [@HHZ1995.1; @SS2010.4; @KK2010.1; @C2011.1], which describes the fluctuations in some specific regimes for the height of the interface in growth models, the current of particles in driven diffusive systems and the free energy for directed polymers in random media. The totally asymmetric simple exclusion process (TASEP) [@D1998.1; @GM2006.1] belongs to KPZ universality. On the infinite line, the current fluctuations in the long time limit [@J2000.1] are equal to the ones that have been obtained from other models, in particular polynuclear growth model [@PS2000.2], directed polymer in random media [@BCP2014.1], and from the Kardar-Parisi-Zhang equation [@KPZ1986.1] itself using the replica method [@CLDR2010.1; @D2010.1]. On a finite system, the stationary large deviations of the current for periodic TASEP [@DL1998.1] agree with the ones from the replica method [@BD2000.1] and with the ones for open TASEP at the transition separating the maximal current phase with the high and low density phases [@GLMV2012.1]. Much less is currently known about the crossover between fluctuations on the infinite line and in a finite system, see however [@LK2006.1; @BPPP2006.1; @GMGB2007.1; @B2009.1; @PBE2011.1; @MSS2012.1; @MSS2012.2]. The crossover takes place on the relaxation scale with times $T$ of order $L^{3/2}$ characteristic of KPZ universality in $1+1$ dimension. The aim of the present paper is to compute the large $L$ limit of the normalization of the Bethe eigenstates of TASEP that contribute to the relaxation regime. Our main result is that this limit depends on the eigenstate essentially through the free action of a field $\varphi$ built by summing over elementary excitations corresponding to the eigenstate. This result can be used to derive an exact formula for the current fluctuations in the relaxation regime [@P2014.2]. The paper is organized as follows. In section \[section TASEP\], we briefly recall the master equation generating the time evolution of TASEP and its deformation which allows to count the current of particles. In section \[section Bethe ansatz\], we summarize some known facts about Bethe ansatz for periodic TASEP, and state our main result (\[asymptotics norm\]) about the asymptotics of the norm of Bethe eigenstates. In section \[section Euler-Maclaurin\], we state the Euler-Maclaurin formula for summation on segments, triangles and rectangles with various singularities at the borders of the summation range. The Euler-Maclaurin formula is then used in section \[section asymptotics\] to compute the asymptotics of the normalization of Bethe states. In appendix \[appendix zeta\], some properties of simple and double Hurwitz zeta functions are summarized. [Periodic TASEP]{} \[section TASEP\] We consider TASEP with $N$ hard-core particles on a periodic lattice of $L$ sites. The continuous time dynamics consists of particle hopping from any site $i$ to the next site $i+1$ with rate $1$ if the destination site is empty. Since TASEP is a Markov process, the time evolution of the probability $P_{T}(\mathcal{C})$ to observe the system at time $T$ in the configuration $\mathcal{C}$ is generated by a master equation. A deformation of the master equation can be considered [@DL1998.1] to count the total integrated current of particles $Q_{T}$, defined as the total number of hops of particles up to time $T$. Defining $F_{T}(\mathcal{C})=\sum_{Q=0}^{\infty}{\mathrm{e}}^{\gamma Q}P_{T}(\mathcal{C},Q)$ where $P_{T}(\mathcal{C},Q)$ is the joint probability to have the system in configuration $\mathcal{C}$ with $Q_{T}=Q$, one has $$\label{master eq F} \frac{\partial}{\partial T}\,F_{T}(\mathcal{C})=\sum_{\mathcal{C}'\neq\mathcal{C}}\Big[{\mathrm{e}}^{\gamma}w(\mathcal{C}\leftarrow\mathcal{C}')F_{T}(\mathcal{C}')-w(\mathcal{C}'\leftarrow\mathcal{C})F_{T}(\mathcal{C})\Big]\;.$$ The hopping rate $w(\mathcal{C}'\leftarrow\mathcal{C})$ is equal to $1$ if the configuration $\mathcal{C}'$ can be obtained from $\mathcal{C}$ by moving one particle from a site $i$ to $i+1$, and is equal to $0$ otherwise. The deformed master equation (\[master eq F\]) reduces to the usual master equation for the probabilities $P_{T}(\mathcal{C})$ when the fugacity $\gamma$ is equal to $0$. It can be encoded in a deformed Markov operator $M(\gamma)$ acting on the configuration space of dimension $\Omega={{L \choose N}}$ in the sector with $N$ particles. Gathering the $F_{T}(\mathcal{C})$ in a vector $|F_{T}\rangle$, one can write $$\label{master eq |F>} \partial_{T}|F_{T}\rangle=M(\gamma)F_{T}\;.$$ The deformed master equation (\[master eq F\]), (\[master eq |F&gt;\]) is known [@DL1998.1] to be integrable in the sense of quantum integrability, also called stochastic integrability [@S2012.1] in the context of an evolution generated by a non-Hermitian stochastic operator. At $\gamma=0$, the eigenvalue of the first excited state (gap) has been shown to scale as $L^{-3/2}$ using Bethe ansatz [@GS1992.1; @GM2004.1; @GM2005.1]. The whole spectrum has also been studied [@P2013.1], and in particular the region with eigenvalues scaling as $L^{-3/2}$ [@P2014.1]. In this article, we study the normalization of the corresponding eigenstates, which is needed for the calculation of fluctuations of $Q_{T}$ on the relaxation scale $T\sim L^{3/2}$ [@P2014.2]. There $$\label{GF[M]} \langle{\mathrm{e}}^{\gamma Q_{T}}\rangle=\frac{\langle\mathcal{C}|{\mathrm{e}}^{TM(\gamma)}|\mathcal{C}_{0}\rangle}{\langle\mathcal{C}|{\mathrm{e}}^{TM}|\mathcal{C}_{0}\rangle}\;$$ is evaluated by inserting a decomposition of the identity operator in terms of left and right normalized eigenvectors. Throughout the paper, we consider the thermodynamic limit $L,N\to\infty$ with fixed density of particles $$\label{rho} \rho=\frac{N}{L}\;,$$ and fixed rescaled fugacity $$\label{s} s=\sqrt{\rho(1-\rho)}\,\gamma\,L^{3/2}\;$$ according to the relaxation scale $T\sim L^{3/2}$ in one-dimensional KPZ universality. [Bethe ansatz]{} \[section Bethe ansatz\] In this section, we recall some known facts about Bethe ansatz for periodic TASEP, and state our main result about the asymptotics of the normalization of Bethe eigenstates. [Eigenvalues and eigenvectors]{} Bethe ansatz is one of the main tools that have been used to obtain exact results about dynamical properties of TASEP. It allows to diagonalize the $N$ particle sector of the generator of the evolution $M(\gamma)$ in terms of $N$ (complex) momenta $q_{j}$, $j=1,\ldots,N$. The eigenvectors are then written as sums over all $N!$ permutations assigning momenta to the particles. On the infinite line, the momenta are integrated over on some continuous curve in the complex plane [@TW2008.1]. For a finite system on the other hand, only a discrete set of $N$-tuples of momenta are allowed, as is usual for particles in a box. Writing $y_{j}=1-{\mathrm{e}}^{\gamma}{\mathrm{e}}^{{\mathrm{i}}q_{j}}$, one can show that for the system with periodic boundary conditions, the complex numbers $y_{j}$, $j=1,\ldots,N$ have to satisfy the *Bethe equations* $$\label{BE[y]} {\mathrm{e}}^{L\gamma}(1-y_{j})^{L}=(-1)^{N-1}\prod_{k=1}^{N}\frac{y_{j}}{y_{k}}\;.$$ We use the shorthand $r$ to refer to the sets of $N$ *Bethe roots* $y_{j}$ solving the Bethe equations. The eigenstates are indexed by $r$. The corresponding eigenvalue of $M(\gamma)$ is equal to $$\label{E[y]} E_{r}(\gamma)=\sum_{j=1}^{N}\frac{y_{j}}{1-y_{j}}\;.$$ By translation invariance of the model, each eigenstate of $M(\gamma)$ is also eigenstate of the translation operator. The corresponding eigenvalue is $${\mathrm{e}}^{2{\mathrm{i}}\pi p_{r}/L}={\mathrm{e}}^{N\gamma}\prod_{j=1}^{N}(1-y_{j})\;,$$ with total momentum $p_{r}\in\mathbb{Z}$. The coefficients of the right and left (unnormalized) eigenvectors for a configuration with particles at positions $1\leq x_{1}<\ldots<x_{N}\leq L$ are given by the determinants $$\begin{aligned} \label{psiR[y,x]} && \langle x_{1},\ldots,x_{N}|\psi_{r}(\gamma)\rangle=\det\Big(y_{k}^{-j}(1-y_{k})^{x_{j}}{\mathrm{e}}^{\gamma x_{j}}\Big)_{j,k=1,\ldots,N}\\ \label{psiL[y,x]} && \langle\psi_{r}(\gamma)|x_{1},\ldots,x_{N}\rangle=\det\Big(y_{k}^{j}(1-y_{k})^{-x_{j}}{\mathrm{e}}^{-\gamma x_{j}}\Big)_{j,k=1,\ldots,N}\;.\end{aligned}$$ These determinants are antisymmetric under the exchange of the $y_{j}$’s, and are thus divisible by the Vandermonde determinant of the $y_{j}$’s. In particular, for the configuration $\mathcal{C}_{X}$ with particles at positions $(X,X+1,\ldots,X+N-1)$, they reduce to $$\begin{aligned} \label{psiR[y,X]} && \langle\mathcal{C}_{X}|\psi_{r}(\gamma)\rangle={\mathrm{e}}^{\frac{2{\mathrm{i}}\pi p_{r}X}{L}}{\mathrm{e}}^{\frac{N(N-1)\gamma}{2}}\bigg(\prod_{j=1}^{N}y_{j}^{-N}\bigg)\!\!\prod_{1\leq j<k\leq N}\!\!\!\!(y_{j}-y_{k})\\ \label{psiL[y,X]} && \langle\psi_{r}(\gamma)|\mathcal{C}_{X}\rangle={\mathrm{e}}^{-\frac{2{\mathrm{i}}\pi p_{r}X}{L}}{\mathrm{e}}^{-\frac{N(N-1)\gamma}{2}}\bigg(\prod_{j=1}^{N}\frac{y_{j}}{(1-y_{j})^{N-1}}\bigg)\!\!\prod_{1\leq j<k\leq N}\!\!\!\!(y_{k}-y_{j})\;.\end{aligned}$$ Based on numerical solutions, the expressions above for the eigenvectors and eigenvalues are only valid for generic values of $\gamma$. For specific values of $\gamma$, some eigenstates might be missing. Those can be identified, by adding a small perturbation to $\gamma$, as cases where several $y_{j}$’s coincide, which imply that the determinants in (\[psiR\[y,x\]\]) and (\[psiL\[y,x\]\]) vanish. This is in particular the case for the stationary eigenstate at $\gamma=0$: in the limit $\gamma\to0$, all $y_{j}$’s converge to $0$ as $\gamma^{1/N}$. This will not be a problem here, as one can always add a small perturbation to $\gamma$ when needed, see also [@NW2014.1] for a discussion in XXX and XXZ spin chain. [Normalization of Bethe eigenstates]{} The eigenvectors (\[psiR\[y,x\]\]), (\[psiL\[y,x\]\]) are not normalized. In order to write the decomposition of the identity $${\mathbf{1}}=\sum_{r}\frac{|\psi_{r}(\gamma)\rangle\,\langle\psi_{r}(\gamma)|}{\langle\psi_{r}(\gamma)|\psi_{r}(\gamma)\rangle}\;,$$ one needs to compute the scalar products $\langle\psi_{r}(\gamma)|\psi_{r}(\gamma)\rangle$ between left and right eigenstates corresponding to the same Bethe roots (and hence same eigenvalue). Several results are known, both for on-shell (Bethe roots satisfying the Bethe equations) and off-shell (arbitrary $y_{j}$’s) scalar products. We write explicitly the dependency of the Bethe vectors (\[psiR\[y,x\]\]) and (\[psiL\[y,x\]\]) on the $y_{j}$’s as $\psi(\vec{y})$. For arbitrary complex numbers $y_{j}$, $w_{j}$, $j=1,\ldots,N$, it was shown [@B2009.1; @MS2013.1] that $$\label{norm[y,w] off off} \langle\psi(\vec{w})|\psi(\vec{y})\rangle=\bigg(\prod_{j=1}^{N}\frac{1-y_{j}}{y_{j}}\,\frac{w_{j}^{N}}{(1-w_{j})^{L}}\bigg)\det\Bigg(\frac{\frac{(1-w_{k})^{L}}{w_{k}^{N-1}}-\frac{(1-y_{j})^{L}}{y_{j}^{N-1}}}{y_{j}-w_{k}}\Bigg)_{j,k=1,\ldots,N}\;.$$ For the mixed on-shell / off-shell case, where the $y_{j}$’s verify the Bethe equations while the $w_{j}$’s are arbitrary, one has the Slavnov determinant [@S1989.1] $$\label{norm[y,w] off on} \langle\psi(\vec{w})|\psi(\vec{y})\rangle=(-1)^{N}\bigg(\prod_{j=1}^{N}\frac{(1-y_{j})^{L+1}}{y_{j}^{N}(1-w_{j})^{L}}\bigg)\bigg(\prod_{j=1}^{N}\prod_{k=1}^{N}(y_{j}-w_{k})\bigg) \det\Big(\partial_{y_{i}}\mathcal{E}(w_{j},\vec{y})\Big)_{i,j=1,\ldots,N}\;,$$ where the derivative with respect to $y_{i}$ is taken before setting the $y_{j}$’s equal to solutions of the Bethe equations. The quantity $\mathcal{E}(\lambda,\vec{y})$ is the eigenvalue of the transfer matrix associated to TASEP with spectral parameter $\lambda$ $$\mathcal{E}(\lambda,\vec{y})=\prod_{j=1}^{N}\frac{1}{1-\lambda^{-1}y_{j}}+{\mathrm{e}}^{L\gamma}(1-\lambda)^{L}\prod_{j=1}^{N}\frac{1}{1-\lambda y_{j}^{-1}}\;.$$ Finally, for left and right eigenvectors with the same on-shell Bethe roots $y_{j}$ satisfying Bethe equations, the scalar product is equal to the Gaudin determinant [@GMCW1981.1; @K1982.1] $$\label{norm[y]} \langle\psi_{r}(\gamma)|\psi_{r}(\gamma)\rangle=(-1)^{N}\Big(\prod_{j=1}^{N}(1-y_{j})\Big)\\ \det\bigg(\partial_{y_{i}}\log\Big((1-y_{j})^{L}\prod_{k=1}^{N}\frac{y_{k}}{y_{j}}\Big)\bigg)_{i,j=1,\ldots,N}\;.$$ Very similar determinantal expressions to (\[norm\[y,w\] off on\]) and (\[norm\[y\]\]) also exist for more general integrable models, in particular ASEP with particles hopping in both directions. The determinantal formula (\[norm\[y,w\] off off\]) for the fully off-shell case seems so far only available for the special case of TASEP. In this paper, we only consider the on-shell scalar product (\[norm\[y\]\]), which can be simplified further by computing the derivative with respect to $y_{i}$ and using the identity $$\det(\alpha_{i}+\beta_{i}\delta_{i,j})_{i,j=1,\ldots,N}=\bigg(\prod_{j=1}^{N}\beta_{j}\bigg)\bigg(1+\sum_{j=1}^{N}\frac{\alpha_{j}}{\beta_{j}}\bigg)\;,$$ where $\delta$ is Kronecker’s delta symbol. The scalar product is then equal to $$\langle\psi_{r}(\gamma)|\psi_{r}(\gamma)\rangle=\frac{L}{N}\bigg(\sum_{j=1}^{N}\frac{y_{j}}{N+(L-N)y_{j}}\bigg)\prod_{j=1}^{N}\Big(L-N+\frac{N}{y_{j}}\Big)\;.$$ This normalization is somewhat arbitrary since it depends on the choice of the normalization in the definitions (\[psiR\[y,x\]\]), (\[psiL\[y,x\]\]). We consider then the configuration $\mathcal{C}_{X}$ with particles at positions $(X,X+1,\ldots,X+N-1)$ and define $$\mathcal{N}_{r}(\gamma) =\Omega\,\frac{\langle\mathcal{C}_{X}|\psi_{r}(\gamma)\rangle\langle\psi_{r}(\gamma)|\mathcal{C}_{X}\rangle}{\langle\psi_{r}(\gamma)|\psi_{r}(\gamma)\rangle}\;,$$ with $\Omega={{L \choose N}}$ the total number of configurations. One has $$\label{norm} \mathcal{N}_{r}(\gamma) =(-1)^{\frac{N(N-1)}{2}}{\mathrm{e}}^{-2{\mathrm{i}}\pi p_{r}(\rho-\frac{1}{L})} \frac{{\mathrm{e}}^{N(N-1)\gamma}\Omega}{N^{N}(\prod_{j=1}^{N}y_{j})^{N-2}}\,\frac{\prod_{j=1}^{N}\prod_{k=j+1}^{N}(y_{j}-y_{k})^{2}}{\Big(\frac{1}{N}\sum_{j=1}^{N}\frac{y_{j}}{\rho+(1-\rho)y_{j}}\Big)\prod_{j=1}^{N}\Big(1+\frac{1-\rho}{\rho}\,y_{j}\Big)}\;.$$ Since this formula is based on the rather involved proof of (\[norm\[y\]\]) obtained in [@K1982.1] for the slightly different case of the XXZ spin chain, we checked it numerically starting from (\[psiR\[y,x\]\]), (\[psiL\[y,x\]\]) for all systems with $2\leq L\leq10$, $1\leq N\leq L-1$ and all eigenstates. We used the method described in the next section to solve the Bethe equations. Generic values were chosen for the parameter $\gamma$. Perfect agreement was found with (\[norm\]). In the basis of configurations, all the elements of the left stationary eigenvector at $\gamma=0$ are equal since $M(0)$ is a stochastic matrix. The same is true for the right stationary eigenvector due to a property of pairwise balance verified by periodic TASEP [@SRB1996.1]. Denoting the stationary state by the index $0$, this implies $\mathcal{N}_{0}(0)=1$. The main goal of this article is the calculation of the asymptotics (\[asymptotics norm\]) of (\[norm\]) for large $L$ with fixed density of particles $\rho$ and rescaled fugacity $s$ for the first eigenstates beyond the stationary state. [Solution of the Bethe equations]{} The Bethe equations of TASEP can be solved in a rather simple way using the fact that they almost decouple, since the right hand side of (\[BE\[y\]\]) can be written as $y_{j}^{N}$ times a symmetric function of the $y_{k}$’s independent of $j$. The strategy [@GS1992.1; @DL1998.1] is then to give a name to that function of the $y_{k}$’s and treat it as a parameter independent of the Bethe roots, than is subsequently fixed using its explicit expression in terms of the $y_{k}$’s. This procedure can be conveniently written [@PP2007.1; @P2014.1] by introducing the function $$\label{g} g:y\mapsto\frac{1-y}{y^{\rho}}\;.$$ Indeed, defining the quantity $$\label{b[y]} b=\gamma+\frac{1}{L}\,\sum_{j=1}^{N}\log y_{j}\;$$ and taking the power $1/L$ of the Bethe equations (\[BE\[y\]\]), we observe that there must exist wave numbers $k_{j}$, integers (half-integers) if $N$ is odd (even) such that $$\label{g(y)} g(y_{j})=\exp\Big(\frac{2{\mathrm{i}}\pi k_{j}}{L}-b\Big)\;.$$ Inverting the function $g$ leads to a rather explicit solution of the Bethe equations as $$\label{y[k]} y_{j}=g^{-1}\Big(\exp\Big(\frac{2{\mathrm{i}}\pi k_{j}}{L}-b\Big)\Big)\;.$$ This expression is very convenient for large $L$ asymptotic analysis using the Euler-Maclaurin formula. [First excited states]{} The stationary state corresponds to the choice $k_{j}=k_{j}^{0}$, $j=1,\ldots,N$ with $$\label{k0} k_{j}^{0}=j-\frac{N+1}{2}\;.$$ This choice closely resembles the Fermi sea of a system of spinless fermions. We call *first excited states* the (infinitely many) eigenstates of $M(\gamma)$ having a real part scaling as $L^{-3/2}$ in the thermodynamic limit $L\to\infty$ with fixed density of particles $\rho$ and purely imaginary rescaled fugacity $s$. These eigenstates correspond to sets $\{k_{j},j=1,\ldots,N\}$ close to the stationary choice (\[k0\]). They are built by removing from $\{k_{j}^{0},j=1,\ldots,N\}$ a finite number of $k_{j}$’s located at a finite distance of $\pm N/2$ and adding the same number of $k_{j}$’s at a finite distance of $\pm N/2$ outside of the interval $[-N/2,N/2]$. The first eigenstates are characterized by an equal number of $k_{j}$’s removed and added on each side. In particular, the choice $k_{j}=j-(N-1)/2$ leads to a larger eigenvalue, ${\operatorname{Re}}E(0)\sim L^{-2/3}$ [@P2013.1], and thus does not belong to the first eigenstates. Numerical checks seem to support the fact that no other choices for the $k_{j}$’s lead to eigenvalues with a real part scaling as $L^{-3/2}$, although a proof of this is missing. The first excited states can be described by four finite sets of positive half-integers $A_{0}^{\pm},A^{\pm}\subset\mathbb{N}+\tfrac{1}{2}$: the set of $k_{j}$’s removed from (\[k0\]) are $\{N/2-a,a\in A_{0}^{+}\}$ and $\{-N/2+a,a\in A_{0}^{-}\}$, while the set of $k_{j}$’s added are $\{N/2+a,a\in A^{+}\}$ and $\{-N/2-a,a\in A^{-}\}$, see figure \[fig choice k\]. The cardinals of the sets verify the constraints $$\label{m+-} m_{r}^{+}=|A_{0}^{+}|=|A^{+}| \quad\text{and}\quad m_{r}^{-}=|A_{0}^{-}|=|A^{-}|\;.$$ We call $m_{r}=m_{r}^{+}+m_{r}^{-}$. In the following, we use the notation $r$ as a shorthand for $(A_{0}^{+},A^{+},A_{0}^{-},A^{-})$ to refer to the corresponding excited state. The total momentum of an eigenstate, $p_{r}=\sum_{j=1}^{N}k_{j}$, can be written in terms of the four sets as $p_{r}=\sum_{a\in A_{0}^{+}}a+\sum_{a\in A^{+}}a-\sum_{a\in A_{0}^{-}}a-\sum_{a\in A^{-}}a$. Only the excited states having an eigenvalue with real part scaling as $L^{-3/2}$ contribute to the relaxation for times $T\sim L^{3/2}$: the other eigenstates with larger eigenvalue only give exponentially small corrections when $L\to\infty$. This statement needs however some more justification since, in principle, it could be that the number of higher excited states becomes so large that $TE_{r}(\gamma)$ becomes negligible compared to the “entropy” of the spectrum in the expansion of (\[GF\[M\]\]) over the eigenstates. This entropy was studied in [@P2013.1] at $\gamma=0$ for the bulk of the spectrum with eigenvalues scaling proportionally to $L$. It was shown that the number of eigenvalues with a real part $-Le$ grows as $\exp(Ls(e))$ with $s(e)\sim e^{2/5}$ for small $e$. Assuming that the $2/5$ exponent still holds for eigenvalues scaling as $L^{-\alpha}$ with $-1<\alpha<3/2$, the contribution to (\[GF\[M\]\]) of the entropic part is of order $\exp(L^{(3-2\alpha)/5})$, which is always negligible compared to the contribution of ${\mathrm{e}}^{TE_{r}(\gamma)}\sim\exp(L^{3/2-\alpha})$ except at $\alpha=3/2$. (150,70)(0,-15) (0,40)[(40,0)(110,0)(110,5)(40,5)]{} (0,40)[(1,0)[150]{}]{} (0,45)[(1,0)[150]{}]{} (10,40)(5,0)[13]{}[(0,1)[5]{}]{} (140,40)(-5,0)[13]{}[(0,1)[5]{}]{} (2,42)[$\ldots$]{} (73,42)[$\ldots$]{} (143,42)[$\ldots$]{} (20,10)[(0,0)(5,0)(5,5)(0,5)]{} (35,10)[(0,0)(5,0)(5,5)(0,5)]{} (45,10)[(0,0)(5,0)(5,5)(0,5)]{} (0,10)[(55,0)(90,0)(90,5)(55,5)]{} (100,10)[(0,0)(5,0)(5,5)(0,5)]{} (115,10)[(0,0)(5,0)(5,5)(0,5)]{} (120,10)[(0,0)(5,0)(5,5)(0,5)]{} (130,10)[(0,0)(5,0)(5,5)(0,5)]{} (0,10)[(1,0)[150]{}]{} (0,15)[(1,0)[150]{}]{} (10,10)(5,0)[13]{}[(0,1)[5]{}]{} (140,10)(-5,0)[13]{}[(0,1)[5]{}]{} (2,12)[$\ldots$]{} (73,12)[$\ldots$]{} (143,12)[$\ldots$]{} (42.5,38)[(-0.9,-1)[20]{}]{} (52.5,38)[(-0.65,-1)[14.5]{}]{} (92.5,38)[(1.1,-1)[24]{}]{} (97.5,38)[(1.1,-1)[24]{}]{} (107.5,38)[(1.1,-1)[24]{}]{} (21.4,4.5)[$\frac{7}{2}$]{} (36.4,4.5)[$\frac{1}{2}$]{} (41.4,4.5)[$\frac{1}{2}$]{} (51.4,4.5)[$\frac{5}{2}$]{} (91.4,4.5)[$\frac{7}{2}$]{} (96.4,4.5)[$\frac{5}{2}$]{} (106.4,4.5)[$\frac{1}{2}$]{} (116.4,4.5)[$\frac{3}{2}$]{} (121.4,4.5)[$\frac{5}{2}$]{} (131.4,4.5)[$\frac{9}{2}$]{} (21,2)[$\underbrace{\hspace{18\unitlength}}$]{} (41,2)[$\underbrace{\hspace{13\unitlength}}$]{} (91,2)[$\underbrace{\hspace{18\unitlength}}$]{} (111,2)[$\underbrace{\hspace{23\unitlength}}$]{} (30,-4)[$A^{-}$]{} (47.5,-4)[$A_{0}^{-}$]{} (100,-4)[$A_{0}^{+}$]{} (122.5,-4)[$A^{+}$]{} (75,50)[(1,0)[35]{}]{} (75,50)[(-1,0)[35]{}]{} (72,52)[$\sim L$]{} (40,-10)[(1,0)[15]{}]{} (40,-10)[(-1,0)[20]{}]{} (35,-15)[$\sim L^{0}$]{} (110,-10)[(1,0)[25]{}]{} (110,-10)[(-1,0)[20]{}]{} (107,-15)[$\sim L^{0}$]{} [Field phi]{} The eigenvalue corresponding to the eigenstate $r$ can be nicely written in terms of a function $\eta_{r}$ [@P2014.1]. From the result stated in section \[section expansion norm\] about the normalization of Bethe states, the function $\varphi_{r}(u)=-(2\pi)^{-3/2}\eta_{r}'(\tfrac{u}{2\pi})$ seems in fact the “good” object to describe the first excited states. It is defined by $$\begin{aligned} \label{phi[A,zeta]} && \varphi_{r}(u)=2\sqrt{\pi}\Big({\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta\big(-\frac{1}{2},\frac{1}{2}+\frac{{\mathrm{i}}u}{2\pi}\big)+{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta\big(-\frac{1}{2},\frac{1}{2}-\frac{{\mathrm{i}}u}{2\pi}\big)\Big)\\ &&\hspace{13mm} +{\mathrm{i}}\sqrt{2}\Big(\sum_{a\in A_{0}^{+}}\sqrt{u+2{\mathrm{i}}\pi a}+\sum_{a\in A^{-}}\sqrt{u+2{\mathrm{i}}\pi a} -\sum_{a\in A_{0}^{-}}\sqrt{u-2{\mathrm{i}}\pi a}-\sum_{a\in A^{+}}\sqrt{u-2{\mathrm{i}}\pi a}\Big)\;.\nonumber\end{aligned}$$ The Hurwitz zeta function (\[Hurwitz zeta\]) can be seen as a kind of renormalization of an infinite contribution of the Fermi sea to $\varphi_{r}$. Indeed, using (\[sum\_power\[zeta\]\]), (\[Hurwitz zeta asymptotics\]) and introducing the quantities $\chi_{a}^{\pm}(u)=\pm{\mathrm{i}}\sqrt{2u\pm4{\mathrm{i}}\pi a}$, we observe that $\varphi_{r}$ can be written as a sum over momenta of elementary excitations $k_{j}$ near $\pm N/2$ as $$\varphi_{r}(u)=\lim_{M\to\infty}\Big(-\frac{4\sqrt{2\pi}}{3}M^{3/2}+\frac{\sqrt{2}u}{\sqrt{\pi}}\sqrt{M}+\sum_{a\in B_{M}^{+}}\chi_{a}^{-}(u)+\sum_{a\in B_{M}^{-}}\chi_{a}^{+}(u)\Big)\;,$$ with $B_{M}^{\pm}=\big(\{-M+{\tfrac{1}{2}},-M+\tfrac{3}{2},\ldots,-{\tfrac{1}{2}}\}\backslash(-A_{0}^{\pm})\big)\cup A^{\pm}$. The $\zeta$ functions are responsible for branch points $\pm{\mathrm{i}}\pi$ for the function $\varphi_{r}$. The square roots provide additional branch points in $\pm2{\mathrm{i}}\pi(\mathbb{N}+{\tfrac{1}{2}})$. We define in the following $\varphi_{r}$ with branch cuts $[{\mathrm{i}}\pi,{\mathrm{i}}\infty)$ and $(-{\mathrm{i}}\infty,-{\mathrm{i}}\pi]$. Using the relation between polylogarithms and Hurwitz zeta function, the field for the stationary state can be written as $$\label{phi0[Li]} \varphi_{0}(u)=-\frac{1}{\sqrt{2\pi}}\,{\mathrm{Li}}_{3/2}(-{\mathrm{e}}^{u})\;,$$ which is valid for ${\operatorname{Re}}u<0$, and for ${\operatorname{Re}}u>0$ with $|{\operatorname{Im}}u|<\pi$. Our main result (\[asymptotics norm\]), which expresses the asymptotics of the norm of Bethe eigenstates in terms of the free action of $\varphi_{r}$, seems to indicate that $\varphi_{r}$ should be interpreted as a field, whose physical meaning is unclear at the moment. [Large L expansion of the parameter b]{} \[section expansion b\] For all first excited states, the quantity $b$ converges in the thermodynamic limit to $b_{0}$ [@P2014.1], equal to $$\label{b0} b_{0}=\rho\log\rho+(1-\rho)\log(1-\rho)\;.$$ Writing the correction as $$\label{b[c]} b=b_{0}+\frac{2\pi c}{L}\;,$$ a small generalization of [@P2014.1] to non-zero rescaled fugacity $s$ leads to the large $L$ expansion $$\label{phi(c)} \varphi_{r}(2\pi c)\simeq s-\frac{2{\mathrm{i}}\pi(1-2\rho)p_{r}}{3\sqrt{\rho(1-\rho)}\sqrt{L}}\;.$$ This expansion follows from applying the Euler-Maclaurin formula to (\[b\[y\]\]), with Bethe roots $y_{j}$ given by (\[y\[k\]\]). [Large L expansion of the eigenvalues]{} \[section expansion E\] Another small extension of [@P2014.1] to nonzero rescaled fugacity $s$ gives the expansion up to order $L^{-3/2}$ of the eigenvalue $E_{r}(\gamma)$ as $$\label{asymptotics E} E_{r}(\gamma)\simeq\frac{s\sqrt{\rho(1-\rho)}}{\sqrt{L}}-\frac{2{\mathrm{i}}\pi(1-2\rho)p_{r}}{L} +\frac{\sqrt{\rho(1-\rho)}}{L^{3/2}}\lim_{\Lambda\to\infty}\Big(D^{\Lambda}+\int_{-\Lambda}^{2\pi c}{\mathrm{d}}u\,\varphi_{r}(u)\Big)\;,$$ where $$\label{DLambda} D^{\Lambda}=\frac{4\sqrt{2}m_{r}}{3}\,\Lambda^{3/2}-2\sqrt{2}{\mathrm{i}}\pi\Big(\sum_{a\in A_{0}^{+}}a+\sum_{a\in A^{-}}a-\sum_{a\in A_{0}^{-}}a-\sum_{a\in A^{+}}a\Big)\sqrt{\Lambda}\;$$ cancels the divergent contribution from the integral. This expansion follows from the application of the Euler-Maclaurin formula to (\[E\[y\]\]). [Large L expansion of the norm of Bethe eigenstates]{} \[section expansion norm\] In section \[section asymptotics\], we derive the large $L$ asymptotics (\[asymptotics (Xi3\*Xi4)/(Xi1\*Xi2)\]) for the normalization of Bethe states, with $b$ written as (\[b\[c\]\]) and $c$ arbitrary. Writing the solution of (\[phi(c)\]) at leading order in $L$ as $2\pi c=\varphi_{r}^{-1}(s)$, the asymptotics of the normalization of Bethe states is obtained as $$\begin{aligned} \label{asymptotics norm} && \mathcal{N}_{r}(\gamma)\simeq{\mathrm{e}}^{-2{\mathrm{i}}\pi\rho p_{r}}\,{\mathrm{e}}^{-s\sqrt{\rho(1-\rho)}\sqrt{L}}\nonumber\\ &&\hspace{13mm} \times\frac{(\pi^{2}/4)^{m_{r}^{2}}}{(-4\pi^{2})^{m_{r}}}\,\omega(A_{0}^{+})^{2}\omega(A_{0}^{-})^{2}\omega(A^{+})^{2}\omega(A^{-})^{2}\omega(A_{0}^{+},A_{0}^{-})^{2}\omega(A^{+},A^{-})^{2}\\ &&\hspace{13mm} \times\frac{{\mathrm{e}}^{\varphi_{r}^{-1}(s)}}{\sqrt{2\pi}\,\varphi_{r}'(\varphi_{r}^{-1}(s))}\, \lim_{\Lambda\to\infty}\exp\Big(-2m_{r}^{2}\log\Lambda+\int_{-\Lambda}^{\varphi_{r}^{-1}(s)}{\mathrm{d}}u\,(\varphi_{r}'(u))^{2}\Big)\;,\nonumber\end{aligned}$$ with the combinatorial factors $$\label{omega(A)} \omega(A)=\prod_{\substack{a,a'\in A\\a<a'}}(a-a') \quad\text{and}\quad \omega(A,A')=\prod_{a\in A}\prod_{a'\in A'}(a+a')\;.$$ This is the main technical result of the paper. The field $\varphi_{r}$ is defined by (\[phi\[A,zeta\]\]). One recovers the stationary value $\mathcal{N}_{0}(0)=1$ using the fact that the solution $c$ of $\varphi_{0}(2\pi c)=s$ for the stationary state goes to $-\infty$ when $s$ goes to $0$ and the expression (\[phi0\[Li\]\]) of $\varphi_{0}$ as a polylogarithm. For technical reasons, our derivation of (\[asymptotics norm\]) requires ${\operatorname{Re}}c>0$ for all the other eigenstates. This implies that ${\operatorname{Re}}s$ can not be too small. Numerical resolution of $\varphi_{r}(2\pi c)=s$ for the first eigenstates seem to indicate that the condition ${\operatorname{Re}}s\geq0$ is always sufficient. [Numerical checks of the asymptotic expansion]{} \[section numerics\] Bulirsch-Stoer (BST) algorithm (see *e.g.* [@HS1988.1]) is an extrapolation method for convergence acceleration of algebraically converging sequences $q_{L}$, $L\in\mathbb{N}^{*}$. It assumes that $q_{L}$ behaves for large $L$ as $p_{0}+p_{1}L^{-\omega}+p_{2}L^{-2\omega}+\ldots$ for some exponent $\omega>0$. Given values $q_{j}$, $j=1,\ldots,M$ of the sequence, the algorithm provides an estimation of the limit $p_{0}$, together with an estimation of the error. In the usual case where one does not know the value of the parameter $\omega$, it has to be estimated by trying to minimize the estimation of the error, which requires some educated guesswork. In the case considered in this paper, however, we know that the norm of the eigenstates has an asymptotic expansions in $1/\sqrt{L}$. One can then set from the beginning $\omega=1/2$ when applying BST algorithm. The convergence of the estimation of $p_{0}$ to its exact value is then exponentially fast in the number $M$ of values of the sequence supplied to the algorithm, although the larger $M$ is, the more precision is needed for the $q_{j}$’s due to fast propagation of rounding errors. BST algorithm thus allows to check to very high accuracy the asymptotics obtained. We used BST algorithm in order to check (\[asymptotics norm\]) for all $139$ first eigenstates with $\sum_{a\in A_{0}^{+}}a+\sum_{a\in A^{+}}a+\sum_{a\in A_{0}^{-}}a+\sum_{a\in A^{-}}a\leq6$ (giving $57$ different values for the norms due to degeneracies). We computed numerically the exact formula (\[norm\]) in terms of the Bethe roots, divided by all the factors of the asymptotics except the exponential of the integral, for several values of the system size $L$ and fixed density of particles $\rho$. We then compared the result of BST algorithm for each value of $\rho$ with the numerical value of the exponential of the integral, which was computed by cutting the integral into three pieces: from $-\infty$ to $-1$ with the integrand $\varphi_{r}'(u)^{2}+2m_{r}^{2}/u$, from $-1$ to $0$ with the integrand $\varphi_{r}'(u)^{2}$ and from $0$ to $\varphi_{r}^{-1}(s)$ with the integrand $\varphi_{r}'(u)^{2}$. The first piece absorbs the divergence at $u\to-\infty$, while the last two pieces make sure that the path of integration does not cross the branch cuts of $\varphi_{r}$. All the computations were done with the generic value $s=0.2+{\mathrm{i}}$ for the rescaled fugacity. Solving numerically the equation (\[b\[y\]\]) for $b$, calculating the Bethe roots from (\[y\[k\]\]), inverting the field $\varphi_{r}$, and evaluating numerically the integrals are relatively costly in computer time, especially with a large number of digits. The exact formula was computed with $200$ significant digits, for $\rho=1/2$ with $L=12,14,16,\ldots,200$, for $\rho=1/3$ with $L=18,21,24,\ldots,300$, for $\rho=1/4$ with $L=24,28,32,\ldots,400$, and for $\rho=1/5$ with $L=30,35,40,\ldots,500$. The estimated relative error from BST algorithm was lower than $10^{-50}$ in all cases. Comparing with the numerical evaluation of the integrals, we found a perfect agreement within at least $50$ digits in relative accuracy. [Current fluctuations]{} From the asymptotics (\[asymptotics norm\]) of the norm, we are now in position to write the large $L$, $T$ limit with $$T=\frac{t\,L^{3/2}}{\sqrt{\rho(1-\rho)}}\;$$ of the generating function for the current fluctuations, in the case of an evolution conditioned on the initial condition $\mathcal{C}_{X}$ with particles at position $(X,X+1,\ldots,X+N-1)$ and on the final condition $\mathcal{C}_{Y}$ with particles at position $(Y,Y+1,\ldots,Y+N-1)$. The distance between $X$ and $Y$ is taken as $$\label{Y-X} Y-X=(1-2\rho)T+(x+\rho)L\;.$$ The first term comes from the term of order $L^{-1}$ in the eigenvalue (\[asymptotics E\]) and corresponds to the group velocity, while the second term defines a rescaled distance $x$ on the ring. The current fluctuations are then defined as $$\xi_{t,x}=\frac{Q_{T}-JLT}{\sqrt{\rho(1-\rho)}L^{3/2}}\;,$$ where the mean value of the current per site $J$ has to be set equal to $$\label{J} JLT=\rho(1-\rho)LT-\rho(1-\rho)L^{2}\;.$$ Its leading term $\rho(1-\rho)$ is the stationary value of the current. The negative correction $-\rho(1-\rho)L^{2}$ is caused by the beginning and the end of the evolution, during which the particles can not easily move due to the initial and final conditions chosen. Its value follows [@P2015.1] from Burgers’ equation. Multiplying the norm (\[asymptotics norm\]) by ${\mathrm{e}}^{2{\mathrm{i}}\pi p_{r}(Y-X)/L}$ to change the final state from $\mathcal{C}_{X}$ to $\mathcal{C}_{Y}$, the choices (\[Y-X\]) and (\[J\]) imply for the generating function of the current fluctuations $G_{t,x}(s)=\langle{\mathrm{e}}^{\gamma(Q_{T}-JLT)}\rangle=\langle{\mathrm{e}}^{s\xi_{t,x}}\rangle$ the large $L$, $T$ limit $$G_{t,x}(s)\to\frac{1}{Z_{t}}\sum_{r}\omega_{r}^{2}\,\frac{{\mathrm{e}}^{2{\mathrm{i}}\pi p_{r}x}\,{\mathrm{e}}^{\varphi_{r}^{-1}(s)}}{\varphi_{r}'(\varphi_{r}^{-1}(s))}\,\exp\Bigg[\lim_{\Lambda\to\infty} R^{\Lambda}+\int_{-\Lambda}^{\varphi_{r}^{-1}(s)}{\mathrm{d}}u\,\Big(\varphi_{r}'(u)^{2}+t\,\varphi_{r}(u)\Big)\Bigg]\;.$$ The sum is over the infinitely many first excited states $r$ of $M(\gamma)$, characterized by the four sets of positive half-integers $A_{0}^{+}$, $A_{0}^{-}$, $A^{+}$, $A^{-}$ with the equalities on their number of elements $m_{r}^{+}=|A_{0}^{+}|=|A^{+}|$, $m_{r}^{-}=|A_{0}^{-}|=|A^{-}|$. The function $\varphi_{r}$ is defined in (\[phi\[A,zeta\]\]). The normalization $Z_{t}$ is such that the generating function is equal to $1$ when the rescaled fugacity $s=0$. The regularization of the integral at $u\to-\infty$ is $R^{\Lambda}=tD^{\Lambda}-2m_{r}^{2}\log\Lambda$ with $D^{\Lambda}$ given by (\[DLambda\]). The combinatorial factor $\omega_{r}$ is equal to $$\omega_{r}^{2}=\frac{(-1)^{m_{r}}(\pi^{2}/4)^{m_{r}^{2}}}{(4\pi^{2})^{m_{r}}}\, \omega(A_{0}^{+})^{2}\omega(A_{0}^{-})^{2}\omega(A^{+})^{2}\omega(A^{-})^{2} \omega(A_{0}^{+},A_{0}^{-})^{2}\omega(A^{+},A^{-})^{2}\;,$$ with factors defined in (\[omega(A)\]) and $m_{r}=m_{r}^{+}+m_{r}^{-}$. Taking $s$ purely imaginary inside the generating function, the probability distribution $P_{\xi}$ of the random variable $\xi_{t,x}$ can be extracted by Fourier transform $$P_{\xi}(w)=\int_{-\infty}^{\infty}\frac{{\mathrm{d}}s}{2\pi}\,{\mathrm{e}}^{{\mathrm{i}}sw}G_{t,x}(-{\mathrm{i}}s)\;.$$ We observe that the integral over $s$ can be nicely replaced by an integral over $d=\varphi_{r}^{-1}(s)$ on some curve in the complex plane. The Jacobian of this change of variables precisely cancels the denominator $\varphi_{r}'(\varphi_{r}^{-1}(s))$ in the generating function. [Euler-Maclaurin formula]{} \[section Euler-Maclaurin\] In order to obtain the large $L$ limit for the normalization of the eigenstates, we need to compute the asymptotics of various sums (and products) with a summation range growing as $L$ and a summand involving the summation index $j$ as $j/L$. Such asymptotics can be performed using the Euler-Maclaurin formula [@H1949.1]. It turns out that the sums considered here have various singularities (square root, logarithm and worse) at both ends of the summation range, for which the most naive version of the Euler-Maclaurin formula (\[EM regular\]) does not work. We discuss here some adaptations of the Euler-Maclaurin formula to logarithmic singularities (Stirling’s formula), non-integer powers (Hurwitz zeta function), and logarithm of a difference of square roots ($\sqrt{\text{Stirling}}$ formula). We begin with one-dimensional sums, and consider then sums on some two dimensional domains necessary to treat the Vandermonde determinant of Bethe roots in (\[norm\]). [One-dimensional sums]{} [Functions without singularities]{} The Euler-Maclaurin formula gives an asymptotic expansion for the difference between a Riemann sum and the corresponding integral. Let $M$ and $L$ be positive integers, $\mu=M/L$ their ratio, and $f$ a function with no singularities in a region which contains the segment $[0,\mu]$. Then, for large $M$, $L$ with fixed $\mu$, the Euler-Maclaurin formula states that $$\label{EM regular} \sum_{j=1}^{M}f\Big(\frac{j+d}{L}\Big)\simeq L\Big(\int_{0}^{\mu}{\mathrm{d}}u\,f(u)\Big)+(\mathcal{R}_{L}f)[\mu,d]-(\mathcal{R}_{L}f)[0,d]\;,$$ where the remainder term is expressed in terms of the Bernoulli polynomials $B_{\ell}$ as $$\label{EM remainder term} (\mathcal{R}_{L}f)[\mu,d]=\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d+1)f^{(\ell-1)}(\mu)}{\ell!L^{\ell-1}}\;.$$ A simple derivation of (\[EM regular\]) using Hurwitz zeta function $\zeta$ (\[Hurwitz zeta\]) consists in replacing the function $f$ by its Taylor series at $0$ in the sum. In order to perform the summation over $j$ at each order in the Taylor series, we use $$\label{sum_power[zeta]} \sum_{j=1}^{M}(j+d)^{\nu}=\zeta(-\nu,d+1)-\zeta(-\nu,M+d+1)\;,$$ which follows directly from the definition (\[Hurwitz zeta\]) for $\nu<-1$, and then for all $\nu\neq-1$ by analytic continuation. It gives $$\sum_{j=1}^{M}f\Big(\frac{j+d}{L}\Big)=\sum_{k=0}^{\infty}f^{(k)}(0)\,\frac{\zeta(-k,d+1)-\zeta(-k,M+d+1)}{k!L^{k}}\;.$$ The $\zeta$ function whose argument depends on $M$ can be expanded for large $M=\mu L$ using (\[Hurwitz zeta asymptotics\]). This leads to $$\sum_{j=1}^{M}f\Big(\frac{j+d}{L}\Big) \simeq\sum_{k=0}^{\infty}f^{(k)}(0)\,\frac{\zeta(-k,d+1)}{k!L^{k}} +L\sum_{k=0}^{\infty}f^{(k)}(0)\,\frac{\mu^{k+1}}{(k+1)!} -\sum_{r=0}^{\infty}\frac{\zeta(-r,d+1)}{r!L^{r}}\sum_{k=0}^{\infty}f^{(k+r)}(0)\,\frac{\mu^{k}}{k!}\;.$$ The second term on the right is the Taylor series at $\mu=0$ of the integral from $0$ to $\mu$ of $f$, while in the last term, we recognize the Taylor expansion at $\mu=0$ of $f^{(r)}(\mu)$. Expressing the remaining $\zeta$ function in terms of Bernoulli polynomials using (\[Bernoulli\[zeta\]\]), we arrive at (\[EM regular\]). [Logarithmic singularity: Stirling’s formula]{} For functions having a singularity at the origin, (\[EM regular\]) can no longer be used, since the derivatives at $0$ of the function become infinite. A well known example is Stirling’s formula for the $\Gamma$ function, for which one has a logarithmic singularity. One has the identity $$\label{sum_log[Gamma]} \sum_{j=1}^{M}\log(j+d)=\log\Gamma(M+d+1)-\log\Gamma(d+1)\;,$$ valid for $d+1\not\in\mathbb{R}^{-}$, where the log $\Gamma$ function $\log\Gamma(z)$ is defined as the analytic continuation with $\log\Gamma(1)=0$ of $\log(\Gamma(z))$ to $\mathbb{C}$ minus the branch cut $\mathbb{R}^{-}$. Then, Stirling’s formula can be stated as the asymptotic expansion $$\label{EM Stirling} \sum_{j=1}^{M}\log\Big(\frac{j+d}{L}\Big) \simeq L\Big(\int_{0}^{\mu}{\mathrm{d}}u\,\log u\Big)+(\mathcal{R}_{L}\log)[\mu,d]+(d+{\tfrac{1}{2}})\log L+\log\sqrt{2\pi}-\log\Gamma(d+1)\;.$$ This has the same form as the Euler-Maclaurin formula for a regular function (\[EM regular\]), except for the remainder term at $0$, which involves the non trivial constant $\log\sqrt{2\pi}$ when $d=0$. The constant term is analytic in $d$ except for the branch cut of $\log\Gamma$ if $\log\Gamma$ is interpreted as the log $\Gamma$ function and not the logarithm of the $\Gamma$ function. We use this prescription in the rest of the paper. [Non-integer power singularity: Hurwitz zeta function]{} Another example of singularities is non-integer power functions. Using (\[sum\_power\[zeta\]\]) for $\nu\neq-1$ and $$\label{sum_inverse[Gamma]} \sum_{j=1}^{M}\frac{1}{j+d}=-\frac{\Gamma'(d+1)}{\Gamma(d+1)}+\frac{\Gamma'(M+d+1)}{\Gamma(M+d+1)}\;$$ for $\nu=-1$, the asymptotics of $\zeta$ (\[Hurwitz zeta asymptotics\]) and of $\Gamma$ give $$\label{EM power} \sum_{j=1}^{M}\Big(\frac{j+d}{L}\Big)^{\nu} \simeq L\Big({\approx\!\!\!\!\!\!\!\int}_{0}^{\mu}{\mathrm{d}}u\,u^{\nu}\Big)+(\mathcal{R}_{L}(\cdot)^{\nu})[\mu,d] +\left\{\begin{array}{cll} \frac{\zeta(-\nu,d+1)}{L^{\nu}} & \quad & \nu\neq-1\\ -L\,\frac{\Gamma'(d+1)}{\Gamma(d+1)} & \quad & \nu=-1 \end{array}\right.\;,$$ where $(\cdot)^{\nu}$ denotes the function $x\mapsto x^{\nu}$. The modified integral is equal to $$\label{mint} {\approx\!\!\!\!\!\!\!\int}_{0}^{\mu}{\mathrm{d}}u\,u^{\nu}= \left\{\begin{array}{cll} \frac{\mu^{\nu+1}}{\nu+1} & \quad & \nu\neq-1\\ \log\mu-\log L^{-1} & \quad & \nu=-1 \end{array}\right.\;.$$ It is equal to the usual, convergent, definition of the integral only in the case $\nu>-1$. Both cases in (\[EM power\]) can be unified by replacing $\zeta$ by $\tilde{\zeta}$ defined in (\[Hurwitz zeta tilde\]). [Logarithm of a sum of two square roots: sqrt(Stirling) formula]{} In the calculation of the asymptotics of the normalization of Bethe states, more complicated singularities appear, with functions that depend themselves on $L$. We define $$\label{alpha+-} \alpha_{\pm}(u,q)=\log(\sqrt{u}\pm\sqrt{q})\;.$$ One has the asymptotic expansion $$\begin{aligned} \label{EM sqrt(Stirling)} && \sum_{j=1}^{M}\alpha_{\pm}\Big(\frac{j+d}{L},\frac{q}{L}\Big) \simeq L\Big(\int_{0}^{\mu}{\mathrm{d}}u\,\alpha_{\pm}(u,q/L)\Big) +(\mathcal{R}_{L}\alpha_{\pm}(\cdot,q/L))[\mu,d] +\frac{q}{2}-\frac{q\log q}{2}\\ &&\hspace{32mm} +\frac{{\mathrm{i}}\pi(1\mp1)q}{2}\,{\operatorname{sgn}}(\arg q) +\frac{(d+\frac{1}{2})\log L}{2}+\frac{\log(2\pi)}{4}-\frac{\log\Gamma(d-q+1)}{2}\nonumber\\ &&\hspace{32mm} \pm\int_{0}^{q}{\mathrm{d}}u\,\frac{\zeta(\frac{1}{2},d+u-q+1)}{2\sqrt{u}}\;,\nonumber\end{aligned}$$ where the first integral is equal to $$\int_{0}^{\mu}{\mathrm{d}}u\,\alpha_{\pm}(u,q)=-\frac{\mu}{2}\pm\sqrt{\mu}\sqrt{q}+\frac{q\log q}{2}+\frac{{\mathrm{i}}\pi(-1\pm1)q}{2}\,{\operatorname{sgn}}(\arg q)+(\mu-q)\log(\sqrt{\mu}\pm\sqrt{q})\;.$$ The argument of $q$ is taken in the interval $(-\pi,\pi)$. The path of integration for the second integral in (\[EM sqrt(Stirling)\]) is required to avoid the branch cuts of the integrand coming from the square root and the $\zeta$ function. The asymptotic expansion (\[EM sqrt(Stirling)\]) is a kind of square root version of Stirling’s formula since $\alpha_{+}(u,q)+\alpha^{-}(u,q)=\log(u-q)$. Indeed, adding (\[EM sqrt(Stirling)\]) for $\alpha_{+}$ and $\alpha_{-}$ gives (\[EM Stirling\]) after using the property $$\label{EM remainder add} \Big(\mathcal{R}_{L}f\Big(\cdot-\frac{q}{L}\Big)\Big)\big[\mu,d\big]=(\mathcal{R}_{L}f)\big[\mu-\frac{q}{L},d\big]=(\mathcal{R}_{L}f)[\mu,d-q]+L \int_{\mu-\frac{q}{L}}^{\mu}\!\!\!{\mathrm{d}}u\,f(u)\;,$$ which follows from the relation (\[Bernoulli sum\]) satisfied by the Bernoulli polynomials. The expansion (\[EM sqrt(Stirling)\]) is a bit more complicated to show than (\[EM Stirling\]) or (\[EM power\]). It can be derived by using the summation formula $$\begin{aligned} \label{sum_log_sqrt[Gamma,zeta]} && \sum_{j=1}^{M}\log(\sqrt{j+d}\pm\sqrt{q})= \frac{\log\Gamma(M+d-q+1)}{2}-\frac{\log\Gamma(d-q+1)}{2}\\ &&\hspace{40mm} \pm\int_{0}^{q}{\mathrm{d}}u\,\frac{\zeta(\frac{1}{2},d-q+u+1)-\zeta(\frac{1}{2},M+d-q+u+1)}{2\sqrt{u}}\;,\nonumber\end{aligned}$$ which can be proved starting from the identity $$\label{d log(sqrt-sqrt)} \partial_{\lambda}\log(\sqrt{j+d+\lambda}\pm\sqrt{q+\lambda})=\pm\frac{1}{2\sqrt{j+d+\lambda}\,\sqrt{q+\lambda}}\;.$$ Indeed, summing (\[d log(sqrt-sqrt)\]) over $j$ using (\[sum\_power\[zeta\]\]) and integrating over $\lambda$, there exist a quantity $K_{M}(d,q)$, independent of $\lambda$, such that $$\sum_{j=1}^{M}\log(\sqrt{j+d+\lambda}\pm\sqrt{q+\lambda}) =K_{M}(d,q)\pm\int_{0}^{\lambda}{\mathrm{d}}u\,\frac{\zeta(\frac{1}{2},d+u+1)-\zeta(\frac{1}{2},M+d+u+1)}{2\sqrt{q+u}}\;.$$ The constant of integration can be fixed from the special case $\lambda=-q$, using (\[sum\_log\[Gamma\]\]). Taking $\lambda=0$ in the previous equation finally gives (\[sum\_log\_sqrt\[Gamma,zeta\]\]). The asymptotic expansion (\[EM sqrt(Stirling)\]) is a consequence of the summation formula (\[sum\_log\_sqrt\[Gamma,zeta\]\]). Using (\[sum\_power\[zeta\]\]) and (\[sum\_log\[Gamma\]\]) we rewrite (\[sum\_log\_sqrt\[Gamma,zeta\]\]) as $$\sum_{j=1}^{M}\log(\sqrt{j+d}\pm\sqrt{q})= \frac{1}{2}\,\sum_{j=1}^{M}\log(j+d-q) \pm\int_{0}^{q}\frac{{\mathrm{d}}u}{2\sqrt{u}}\,\sum_{j=1}^{M}\frac{1}{\sqrt{j+d-q+u}}\;,$$ where the integration is on a contour that avoids the branch cuts of the square roots. The asymptotic expansions (\[EM Stirling\]) and (\[EM power\]) give $$\begin{aligned} && \sum_{j=1}^{M}\log\Big(\sqrt{\frac{j+d}{L}}\pm\sqrt{\frac{q}{L}}\Big)\simeq \frac{L}{2}\,\Big(\int_{0}^{\mu}{\mathrm{d}}u\,\log u\Big) +{\tfrac{1}{2}}(d-q+{\tfrac{1}{2}})\log L+\frac{\log(2\pi)}{4}-\frac{\log\Gamma(d-q+1)}{2}\nonumber\\ &&\hspace{42mm} \pm\int_{0}^{q}{\mathrm{d}}u\,\frac{\zeta(\tfrac{1}{2},d-q+u+1)}{2\sqrt{u}} \pm\sqrt{L}\Big(\int_{0}^{q}\frac{{\mathrm{d}}u}{2\sqrt{u}}\Big)\Big(\int_{0}^{\mu}\frac{{\mathrm{d}}v}{\sqrt{v}}\Big)\\ &&\hspace{42mm} \pm\frac{1}{2\sqrt{L}}\int_{0}^{q}\frac{{\mathrm{d}}u}{\sqrt{u}}\Big(\mathcal{R}_{L}\frac{1}{\sqrt{\cdot}}\Big)[\mu,d-q+u]+\frac{(\mathcal{R}_{L}\log)[\mu,d-q]}{2}\;.\nonumber\end{aligned}$$ The operator $\mathcal{R}_{L}$, defined in (\[EM remainder term\]), is linear. From (\[EM remainder add\]), it verifies $$\int_{0}^{q}{\mathrm{d}}u\,h(u)\,(\mathcal{R}_{L}f)[\mu,d-q+u]=\Big(\mathcal{R}_{L}\Big(\int_{0}^{q}{\mathrm{d}}u\,h(u)f(\cdot+\tfrac{u}{L})\Big)\Big)[\mu,d-q]+L\int_{0}^{q}{\mathrm{d}}u\,h(u)\int_{\mu}^{\mu+\frac{u}{L}}{\mathrm{d}}v\,f(v)\;.$$ Applying this property to $h(u)=f(u)=u^{-1/2}$ and using (\[d log(sqrt-sqrt)\]) to integrate inside the operator $\mathcal{R}_{L}$, one has $$\begin{aligned} && \pm\frac{1}{2\sqrt{L}}\int_{0}^{q}\frac{{\mathrm{d}}u}{\sqrt{u}}\Big(\mathcal{R}_{L}\frac{1}{\sqrt{\cdot}}\Big)[\mu,d-q+u] +\frac{(\mathcal{R}_{L}\log)[\mu,d-q]}{2}\\ && =\Big(\mathcal{R}_{L}\log\big(\sqrt{\cdot}\pm\sqrt{q/L}\big)\Big)[\mu,d] -L\int_{0}^{q/L}{\mathrm{d}}u\,\log(\sqrt{\mu}\pm\sqrt{u})\;.\nonumber\end{aligned}$$ After some simplifications, we arrive at (\[EM sqrt(Stirling)\]). [Singularities at both ends]{} In all the cases described so far in this section, we observe that the asymptotic expansion can always be written as $$\label{EM regular singular} \sum_{j=1}^{M}f\Big(\frac{j+d}{L}\Big) \simeq L\Big(\int_{0}^{\mu}{\mathrm{d}}u\,f(u)\Big) +(\mathcal{R}_{L}f)[\mu,d] +(\mathcal{S}_{L}f)[d]\;,$$ with some regularization needed when the integral does not converge. This is true in general since one can always decompose the sum from $1$ to $M$ as a sum from $1$ to $\varepsilon L$ plus a sum from $\varepsilon L+1$ to $M$. For all $\varepsilon>0$ such that $\varepsilon L$ is an integer, the asymptotics of the second sum is given by (\[EM regular\]) and the limit $\varepsilon\to0$ can be written as (\[EM regular singular\]) with a singular part $(\mathcal{S}_{L}f)[d]$ independent of $\mu$. Let us now consider the case of a function $f$, with singularities at both $0$ and $\rho=N/L$. The singularities are specified by functions $\mathcal{S}_{0}=\mathcal{S}_{L}f$ and $\mathcal{S}_{\rho}=\mathcal{S}_{L}f(\rho-\cdot)$ in (\[EM regular singular\]). Splitting the sum into two parts at $M=\mu L$ leads to $$\sum_{j=1}^{N}f\Big(\frac{j+d}{L}\Big)=\sum_{j=1}^{M}f\Big(\frac{j+d}{L}\Big)+\sum_{j=1}^{N-M}f\Big(\rho-\frac{j-d-1}{L}\Big)\;.$$ One can use (\[EM regular singular\]) on both parts. From (\[Bernoulli symmetry\]), the regular remainder terms at $\mu$ cancels: $(\mathcal{R}_{L}f)[\mu,d] +(\mathcal{R}_{L}f(\rho-\cdot))[\rho-\mu,-d-1]=0$, leaving only the integral and the singular terms: $$\sum_{j=1}^{N}f\Big(\frac{j+d}{L}\Big) \simeq L\Big(\int_{0}^{\rho}{\mathrm{d}}u\,f(u)\Big)+\mathcal{S}_{0}[d]+\mathcal{S}_{\rho}[-d-1]\;.$$ In particular, for $$\label{f[fk]} f(x)=\alpha\log x+\sum_{k=-1}^{\infty}f_{k}x^{k/2}=\overline{\alpha}\log(\rho-x)+\sum_{k=-1}^{\infty}\overline{f}_{k}(\rho-x)^{k/2}\;,$$ one has the asymptotic expansion $$\begin{aligned} \label{EM sqrt sqrt} && \sum_{j=1}^{N}f\Big(\frac{j+d}{L}\Big)\simeq L\Big(\int_{0}^{\rho}{\mathrm{d}}u\,f(u)\Big) +(\alpha-\overline{\alpha})(d+{\tfrac{1}{2}})\log L +(\alpha+\overline{\alpha})\log\sqrt{2\pi} -\alpha\log\Gamma(d+1)\nonumber\\ &&\hspace{25mm} -\overline{\alpha}\log\Gamma(-d)+\sum_{k=-1}^{\infty}f_{k}\,\frac{\zeta(-k/2,d+1)}{L^{k/2}}+\sum_{k=-1}^{\infty}\overline{f}_{k}\,\frac{\zeta(-k/2,-d)}{L^{k/2}}\;.\end{aligned}$$ Similarly let us consider a function $f$ with square root singularities and singularities as a sum of two square roots, both at $0$ and $\rho$: $$f(x)=\Big(\sqrt{x}+\sigma_{0}\sqrt{\frac{q_{0}}{L}}\Big)h_{0}(x)=\Big(\sqrt{\rho-x}+\sigma_{1}\sqrt{\frac{q_{1}}{L}}\Big)h_{1}(\rho-x)\;.$$ The parameters $q_{0}$ and $q_{1}$ are complex numbers, $\sigma_{0}$ and $\sigma_{1}$ are equal to $1$ or $-1$. The functions $h_{0}$ and $h_{1}$ have only square root singularities at $0$: $$h_{0}(x)=\exp\Big(\sum_{r=0}^{\infty}h_{0,r}x^{r/2}\Big) \quad\text{and}\quad h_{1}(x)=\exp\Big(\sum_{r=0}^{\infty}h_{1,r}x^{r/2}\Big)\;,$$ with coefficients $h_{0,r}$ and $h_{1,r}$ which may depend on $L$. Using (\[EM sqrt(Stirling)\]) and (\[EM power\]), one finds after some simplifications the asymptotic expansion $$\begin{aligned} \label{EM sqrt(Stirling) sqrt(Stirling)} && \sum_{j=m_{0}}^{N-m_{1}}\log f\Big(\frac{j+d}{L}\Big) \simeq L\Big(\int_{0}^{\rho}{\mathrm{d}}u\,\log f(u)\Big) +\frac{m_{0}+m_{1}-1}{2}\,\log L+\log\sqrt{2\pi}\nonumber\\ &&\hspace{1mm} +\frac{q_{0}}{2}-\frac{q_{0}\log q_{0}}{2}+\frac{{\mathrm{i}}\pi(1-\sigma_{0})q_{0}}{2}\,{\operatorname{sgn}}(\arg q_{0})-\frac{\log\Gamma(m_{0}+d-q_{0})}{2}+\sigma_{0}\int_{0}^{q_{0}}{\mathrm{d}}u\,\frac{\zeta(\frac{1}{2},m_{0}+d-q_{0}+u)}{2\sqrt{u}}\nonumber\\ &&\hspace{1mm} +\frac{q_{1}}{2}-\frac{q_{1}\log q_{1}}{2}+\frac{{\mathrm{i}}\pi(1-\sigma_{1})q_{1}}{2}\,{\operatorname{sgn}}(\arg q_{1})-\frac{\log\Gamma(m_{1}-d-q_{1})}{2}+\sigma_{1}\int_{0}^{q_{1}}{\mathrm{d}}u\,\frac{\zeta(\frac{1}{2},m_{1}-d-q_{1}+u)}{2\sqrt{u}}\nonumber\\ &&\hspace{1mm} +\sum_{r=0}^{\infty}\frac{h_{0,r}\,\zeta(-r/2,m_{0}+d)}{L^{r/2}} +\sum_{r=0}^{\infty}\frac{h_{1,r}\,\zeta(-r/2,m_{1}-d)}{L^{r/2}}\;.\end{aligned}$$ The integers $m_{0}$ and $m_{1}$ were added in order to treat singularities that may appear for $j$ close to $1$ and $N$ such that $f((j+d)/L)=0$. Their contribution to (\[EM sqrt(Stirling) sqrt(Stirling)\]) come from (\[sum\_power\[zeta\]\]) and (\[sum\_log\_sqrt\[Gamma,zeta\]\]). We assumed that for large $L$, the coefficients $h_{0,r}$ and $h_{1,r}$ do not grow too fast when $r$ increases. In this paper, we only use (\[EM sqrt(Stirling) sqrt(Stirling)\]) with coefficients $h_{0,r}$ and $h_{1,r}$ that have a finite limit when $L$ goes to $\infty$, with an expansion in powers of $1/\sqrt{L}$. Also, we only need the expansion up to order $L^{0}$. One has $$\begin{aligned} && \sum_{r=0}^{\infty}\frac{h_{0,r}\,\zeta(-r/2,m_{0}+d)}{L^{r/2}} +\sum_{r=0}^{\infty}\frac{h_{1,r}\,\zeta(-r/2,m_{1}-d)}{L^{r/2}}\\ && =\frac{1-m_{0}-m_{1}}{2}\,\log L +({\tfrac{1}{2}}-m_{0}-d)\log\frac{\sigma_{0}f(0)}{\sqrt{q_{0}}} +({\tfrac{1}{2}}-m_{1}+d)\log\frac{\sigma_{1}f(\rho)}{\sqrt{q_{1}}} +{\operatorname{\mathcal{O}}}\Big(\frac{1}{\sqrt{L}}\Big)\;.\nonumber\end{aligned}$$ [Two-dimensional sums]{} Generalizations of the Euler-Maclaurin formula can also be used in the case of summations over two indices. Things are however more complicated than in the one-dimensional case because the way to handle those sums depends on the two-dimensional domain of summation, and because of the new kinds of singularities that can happen at singular points of the boundary. We consider here only the case of rectangles $\{(j,j'),1\leq j\leq M,1\leq j'\leq M'\}$ and triangles $\{(j,j'),1\leq j<j'\leq M\}$ that are needed for the asymptotic expansion of the normalization. [Rectangle with square root singularities at a corner]{} We consider a function of two variables $f$ with square root singularities at the point $(0,0)$ $$\label{f sqrt sqrt} f(u,v)=\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}f_{k,k'}u^{k/2}v^{k'/2}\;,$$ and define two auxiliary functions on the edges of the square $$g_{k}(v)=\sum_{k'=0}^{\infty}f_{k,k'}v^{k'/2} \quad\text{and}\quad h_{k'}(u)=\sum_{k=0}^{\infty}f_{k,k'}u^{k/2}\;.$$ Then, taking $M=\mu L$ and $N=\rho L$, (\[sum\_power\[zeta\]\]) gives the large $L$ asymptotic expansion $$\begin{aligned} \label{EM rectangle sqrt} && \sum_{j=1}^{M}\sum_{j'=1}^{N}f\Big(\frac{j+d}{L},\frac{j'+d'}{L}\Big) \simeq L^{2}\int_{0}^{\mu}{\mathrm{d}}u\int_{0}^{\rho}{\mathrm{d}}v\,f(u,v)\nonumber\\ && +\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d+1)}{\ell!L^{\ell-2}}\int_{0}^{\rho}{\mathrm{d}}v\,f^{(\ell-1,0)}(\mu,v) +\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d'+1)}{\ell!L^{\ell-2}}\int_{0}^{\mu}{\mathrm{d}}u\,f^{(0,\ell-1)}(u,\rho)\nonumber\\ && +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d+1)}{L^{\frac{k}{2}-1}}\int_{0}^{\rho}{\mathrm{d}}v\,g_{k}(v) +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d'+1)}{L^{\frac{k}{2}-1}}\int_{0}^{\mu}{\mathrm{d}}u\,h_{k}(u)\\ && +\sum_{\ell,\ell'=1}^{\infty}\frac{B_{\ell}(d+1)}{\ell!L^{\ell-1}}\,\frac{B_{\ell'}(d'+1)}{\ell'!L^{\ell'-1}}\,f^{(\ell-1,\ell'-1)}(\mu,\rho) +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d+1)}{L^{\frac{k}{2}}}\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d'+1)}{\ell!L^{\ell-1}}\,g_{k}^{(\ell-1)}(\rho)\nonumber\\ && +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d'+1)}{L^{\frac{k}{2}}}\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d+1)}{\ell!L^{\ell-1}}\,h_{k}^{(\ell-1)}(\mu) +\sum_{k,k'=0}^{\infty}\frac{\zeta(-k/2,d+1)\zeta(-k/2,d'+1)}{L^{\frac{k+k'}{2}}}\,f_{k,k'}\;.\nonumber\end{aligned}$$ The first term with the double integral is related to the full square, the next four terms with a single integral to the four edges of the square, and the four last terms to the four corners of the square. [Triangle with square root singularities at a corner]{} We consider again a function of two variables $f$ with square root singularities at $(0,0)$ as in (\[f sqrt sqrt\]), and define $$f_{k}(v)=\sum_{k'=0}^{\infty}f_{k,k'}v^{k'/2}\;.$$ One has the asymptotic expansion $$\begin{aligned} \label{EM triangle sqrt} && \sum_{j=1}^{M}\sum_{j'=j+1}^{M}f\Big(\frac{j+d}{L},\frac{j'+d'}{L}\Big) \simeq L^{2}\int_{0}^{\mu}{\mathrm{d}}u\int_{u}^{\mu}{\mathrm{d}}v\,f(u,v)\\ && +\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d'+1)}{\ell!L^{\ell-2}}\,\partial_{\mu}^{\ell-1}\int_{0}^{\mu}{\mathrm{d}}u\,f(u,\mu) +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d+1)}{L^{\frac{k}{2}-1}}\int_{0}^{\mu}{\mathrm{d}}v\,f_{k}(v)\nonumber\\ &&\hspace{55mm} +\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d-d')}{\ell!L^{\ell-2}}\,{\approx\!\!\!\!\!\!\!\int}_{0}^{\mu}{\mathrm{d}}u\,f^{(\ell-1,0)}(u,u)\nonumber\\ && +\sum_{\ell,\ell'=1}^{\infty}\frac{B_{\ell}(d-d')}{\ell!L^{\ell-1}}\,\frac{B_{\ell'}(d'+1)}{\ell'!L^{\ell'-1}}\,\partial_{\mu}^{\ell'-1}f^{(\ell-1,0)}(\mu,\mu) +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d+1)}{L^{\frac{k}{2}}}\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d'+1)}{\ell!L^{\ell-1}}\,f_{k}^{(\ell-1)}(\mu)\nonumber\\ &&\hspace{75mm} +\sum_{k,k'=0}^{\infty}\frac{f_{k,k'}}{L^{\frac{k+k'}{2}}}\,\tilde{\zeta}_{0}(-k/2,-k'/2;d+1,d'+1)\;.\nonumber\end{aligned}$$ The modified integral is defined as in (\[mint\]), after expanding near $u=0$. The modified double Hurwitz zeta function $\tilde{\zeta}_{0}$ is defined in appendix \[appendix zeta\]. The first term in (\[EM triangle sqrt\]) corresponds to the whole triangle, the next three terms to the three edges, and the last three terms to the three corners. As usual, (\[EM triangle sqrt\]) can be shown by expanding $f$ near the point $(0,0)$. At each order in the expansion, the summation over $j,j'$ can be performed in terms of double Hurwitz zeta functions using $$\begin{aligned} \label{sum_power[zeta double]} && \sum_{j=1}^{M}\sum_{j'=j+1}^{M}(j+d)^{\nu}(j'+d')^{\nu'}=\zeta(-\nu,-\nu';d+1,d'+1)\\ &&\hspace{5mm} +\zeta(-\nu',-\nu;M+d'+1,M+d)-\zeta(-\nu,d+1)\zeta(-\nu',M+d'+1)\;.\nonumber\end{aligned}$$ The summation formula (\[sum\_power\[zeta double\]\]) can be shown by using the decomposition $$\begin{picture}(115,25) \put(0,0){\color{lightgray}\polygon*(0,0)(10,10)(0,10)} \put(0,0){\polygon(0,0)(10,10)(0,10)} \put(30,0){\color{lightgray}\polygon*(0,0)(25,25)(0,25)} \put(30,0){\polygon(0,0)(10,10)(0,10)} \put(55,0){\color{lightgray}\polygon*(10,10)(25,25)(25,10)} \put(55,0){\polygon(0,0)(10,10)(0,10)} \put(90,0){\color{lightgray}\polygon*(0,10)(25,10)(25,25)(0,25)} \put(90,0){\polygon(0,0)(10,10)(0,10)} \put(20,3){$=$} \put(45,3){$+$} \put(80,3){$-$} \end{picture}\;,$$ writing $$\begin{aligned} && \sum_{j=1}^{M}\sum_{j'=j+1}^{M}(j+d)^{\nu}(j'+d')^{\nu'} =\sum_{j=1}^{\infty}\sum_{j'=j+1}^{\infty}(j+d)^{\nu}(j'+d')^{\nu'}\\ && +\sum_{j'=M+1}^{\infty}\sum_{j=j'}^{\infty}(j+d)^{\nu}(j'+d')^{\nu'} -\sum_{j=1}^{\infty}\sum_{j'=M+1}^{\infty}(j+d)^{\nu}(j'+d')^{\nu'}\;,\nonumber\end{aligned}$$ provided that $\nu'<-1$ and $\nu+\nu'<-2$ to ensure the convergence of the infinite sums. From the definition (\[Hurwitz zeta double\]) of double Hurwitz zeta functions, this leads to (\[sum\_power\[zeta double\]\]). By analytic continuation, (\[sum\_power\[zeta double\]\]) is valid for all $\nu$, $\nu'$ different from the poles of simple and double $\zeta$. It is also valid when replacing the double $\zeta$ by their modified values $\tilde{\zeta}_{\alpha}$ and $\tilde{\zeta}_{1-\alpha}$ when $2-s-s'\in\mathbb{N}$. Indeed, using (\[Bernoulli symmetry\]), we observe that the quantity $\tilde{\zeta}_{\alpha}(s,s';z,z')+\tilde{\zeta}_{1-\alpha}(s',s;M+z',M+z-1)$ is not divergent on the line $2-s-s'\in\mathbb{N}$, and is independent of $\alpha$. One has $$\lim_{s+s'\to2-n}(\zeta(s,s';z,z')-\zeta(s',s;M+z',M+z-1)) =\tilde{\zeta}_{\alpha}(s,s';z,z')-\tilde{\zeta}_{1-\alpha}(s',s;M+z',M+z-1)\;$$ for arbitrary direction in the convergence of $s+s'$ to $2-n$ and for arbitrary $\alpha$. Expanding for large $M=\mu L$ using (\[Hurwitz zeta asymptotics\]) and (\[Hurwitz zeta double asymptotics\]), a tedious calculation finally leads to the large $L$ asymptotic expansion (\[EM triangle sqrt\]). [Triangle with square root singularities at all corners]{} We consider a function $f(u,v)$ of two variables, analytic in the interior of the domain $\{(u,v),0<u<v<\rho\}$ and with square root singularities at the points $(0,0)$, $(\rho,\rho)$, $(0,\rho)$. We define $g(u,v)=f(\rho-v,\rho-u)$ and $h(u,v)=f(u,\rho-v)$. The expansions near the singularities are given by $$f(u,v)=\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}f_{k,k'}u^{k/2}v^{k'/2}\;,\quad g(u,v)=\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}g_{k,k'}u^{k/2}v^{k'/2}\;,\quad h(u,v)=\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}h_{k,k'}u^{k/2}v^{k'/2}\;.$$ We also define $$f_{\nu}(v)=\sum_{k'=0}^{\infty}f_{\nu,k'}v^{k'/2}\;,\qquad g_{\nu}(v)=\sum_{k'=0}^{\infty}g_{\nu,k'}v^{k'/2}\;.$$ In order to handle the singularities at the corners, we decompose the triangle as $$\begin{aligned} && \sum_{j=1}^{N}\sum_{j'=j+1}^{N}f\Big(\frac{j+d}{L},\frac{j'+d'}{L}\Big) =\sum_{j=1}^{M}\sum_{j'=j+1}^{M}f\Big(\frac{j+d}{L},\frac{j'+d'}{L}\Big)\\ && +\sum_{j'=1}^{N-M}\sum_{j=j'+1}^{N-M}g\Big(\frac{j'-d'-1}{L},\frac{j-d-1}{L}\Big) +\sum_{j=1}^{M}\sum_{j'=1}^{N-M}h\Big(\frac{j+d}{L},\frac{j'-d'-1}{L}\Big)\;.\nonumber\end{aligned}$$ Then, for large $N=\rho L$ and $M=\mu L$, using (\[EM rectangle sqrt\]) and (\[EM triangle sqrt\]) and combining all the terms leads to the large $L$ asymptotic expansion $$\begin{aligned} \label{EM triangle sqrt sqrt sqrt} && \sum_{j=1}^{N}\sum_{j'=j+1}^{N}f\Big(\frac{j+d}{L},\frac{j'+d'}{L}\Big)\simeq L^{2}\int_{0}^{\rho}{\mathrm{d}}u\int_{u}^{\rho}{\mathrm{d}}v\,f(u,v)\\ && +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,d+1)}{L^{k/2-1}}\int_{0}^{\rho}{\mathrm{d}}v\,f_{k}(v) +\sum_{k=0}^{\infty}\frac{\zeta(-k/2,-d')}{L^{k/2-1}}\int_{0}^{\rho}{\mathrm{d}}v\,g_{k}(v)\nonumber\\ && +\sum_{\ell=1}^{\infty}\frac{B_{\ell}(d-d')}{\ell!L^{\ell-2}}\Big({\approx\!\!\!\!\!\!\!\int}_{0}^{\mu}{\mathrm{d}}v\,f^{(l-1,0)}(v,v)+\sum_{m=1}^{\ell-1}(-1)^{\ell-m}f^{(m-1,l-m-1)}(\mu,\mu)\nonumber\\ &&\hspace{70mm} +(-1)^{\ell-1}{\approx\!\!\!\!\!\!\!\int}_{\mu}^{\rho}{\mathrm{d}}v\,f^{(0,l-1)}(v,v)\Big)\nonumber\\ && +\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}\frac{f_{k,k'}}{L^{\frac{k+k'}{2}}}\,\tilde{\zeta}_{0}(-k/2,-k'/2,d+1,d'+1) +\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}\frac{g_{k',k}}{L^{\frac{k+k'}{2}}}\,\tilde{\zeta}_{0}(-k'/2,-k/2,-d',-d)\nonumber\\ &&\hspace{72mm} +\sum_{k=0}^{\infty}\sum_{k'=0}^{\infty}\frac{h_{k,k'}}{L^{\frac{k+k'}{2}}}\,\zeta(-k/2,d+1)\zeta(-k'/2,-d')\;.\nonumber\end{aligned}$$ The modified integral is defined as in (\[mint\]), after expanding near $v=0$ and $v=\rho$. The expansion is independent of the arbitrary parameter $\mu$, $0<\mu<\rho$ that splits the modified integral. [Triangle with logarithmic singularity on an edge: Barnes function]{} A two dimensional generalization of Stirling’s formula for the $\Gamma$ function is $$\begin{aligned} \label{EM triangle log} && \sum_{j=1}^{N}\sum_{j'=j+1}^{N}\log(j'-j+d)\simeq\frac{N^{2}\log N}{2}-\frac{3N^{2}}{4}+d\,N\log N\\ && +N\big(\log\sqrt{2\pi}-d-\log\Gamma(d+1)\big)+\Big(\frac{d^{2}}{2}-\frac{1}{12}\Big)\log N+\zeta'(-1)+d\log\sqrt{2\pi}-\log G(d+1)\;,\nonumber\end{aligned}$$ where $\zeta$ is Riemann’s zeta function and $\log G$ is the log Barnes function, equal to the analytic continuation of the logarithm of the Barnes function $G$. The constant term is usually written in terms of the Glaisher-Kinkelin constant $A=\exp(\tfrac{1}{12}-\zeta'(-1))$. The expansion (\[EM triangle log\]) follows from the identity $G(z+1)=\Gamma(z)G(z)$ and the asymptotics of Barnes function for large argument $$\label{Barnes G asymptotics} \log G(N+1)\simeq\frac{N^{2}\log N}{2}-\frac{3N^{2}}{4}+\frac{\log(2\pi)}{2}\,N-\frac{\log N}{12}+\zeta'(-1)\;.$$ [Square with logarithm of a sum of square roots]{} We consider two dimensional generalizations of the $\sqrt{\text{Stirling}}$ formula (\[EM sqrt(Stirling)\]). One has the asymptotics $$\begin{aligned} \label{EM rectangle log(sqrt+sqrt)} && \sum_{j=1}^{N}\sum_{j'=1}^{N}\log(\sqrt{j+d}+\sqrt{j'+d})\simeq \frac{N^{2}\log N}{2}+\frac{N^{2}}{4}+(2d+1)N\\ &&\hspace{7mm} +4\sqrt{N}\zeta(-{\tfrac{1}{2}},d+1)-\frac{\log N}{24} +(d+{\tfrac{1}{2}})^{2}+\int_{0}^{d+{\tfrac{1}{2}}}{\mathrm{d}}u\,\frac{\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+u)^{2}}{2}+\kappa_{0}\;,\nonumber\end{aligned}$$ with $\kappa_{0}\approx-0.128121307412384$. In order to show this, we introduce $M=\mu N$, $0<\mu<1$ and decompose the sum as $$\sum_{j=1}^{N}\sum_{j'=1}^{N} \;=\;\sum_{j=1}^{M}\sum_{j'=1}^{M} \;+\sum_{j=M+1}^{N}\sum_{j'=1}^{M} \;+\;\sum_{j=1}^{M}\sum_{j'=M+1}^{N} +\sum_{j=M+1}^{N}\sum_{j'=M+1}^{N}\;.$$ The last three terms in the right hand side can be evaluated using Euler-Maclaurin formula in a rectangle with only square root singularities, using (\[EM rectangle sqrt\]) and $$\begin{aligned} && \log(\sqrt{u+\alpha}+\sqrt{v+\beta})\\ && =\partial_{u}\Big(-\frac{u}{2}+\sqrt{u+\alpha}\sqrt{v+\beta}+(u+\alpha-v-\beta)\log(\sqrt{u+\alpha}+\sqrt{v+\beta})\Big)\nonumber\\ && =\partial_{u}\partial_{v}\Big(-\frac{3uv}{4}+\frac{(u+\alpha)^{3/2}\sqrt{v+\beta}}{2}+\frac{\sqrt{u+\alpha}(v+\beta)^{3/2}}{2}-\frac{(u+\alpha-v-\beta)^{2}}{2}\log(\sqrt{u+\alpha}+\sqrt{v+\beta})\Big)\;\nonumber\end{aligned}$$ to compute the integrals. One finds up to order $0$ in $N$ $$\begin{aligned} && \sum_{j=1}^{N}\sum_{j'=1}^{N}\log(\sqrt{j+d}+\sqrt{j'+d'}) -\sum_{j=1}^{M}\sum_{j'=1}^{M}\log(\sqrt{j+d}+\sqrt{j'+d'})\\ && \simeq\Big(\frac{1}{4}-\frac{\mu^{2}}{4}-\frac{\mu^{2}\log\mu}{2}\Big)N^{2} +(2d+1)(1-\mu)N+4(1-\sqrt{\mu})\zeta(-{\tfrac{1}{2}},d+1)\sqrt{N}+\frac{\log\mu}{24}\;.\nonumber\end{aligned}$$ The limit $\mu\to0$ leads to the divergent terms in (\[EM rectangle log(sqrt+sqrt)\]). The term at order $N^{0}$ follows from the summation formula (\[sum\_power\[zeta\]\]) applied to the identity $$\partial_{\lambda}\log\big(\sqrt{j+d+\lambda}+\sqrt{j'+d+\lambda}\big)=\frac{1}{2\sqrt{j+d+\lambda}\sqrt{j'+d+\lambda}}\;.$$ The remaining constant of integration $\kappa_{0}$ can be evaluated numerically with high precision using BST algorithm, as described in section \[section numerics\]. [Square with logarithm of a sum of square roots (2)]{} One has the asymptotics $$\begin{aligned} \label{EM rectangle log(sqrt+sqrt) i} && \sum_{j=1}^{N}\sum_{j'=1}^{N}\log\big(\sqrt{-{\mathrm{i}}(j+d)}+\sqrt{{\mathrm{i}}(j'+d')}\big) \simeq\frac{N^{2}\log N}{2}+\Big(-\frac{3}{4}+\log2\Big)N^{2}\\ && +\big((d+d'+1)\log2-{\mathrm{i}}(d-d')\big)N-2{\mathrm{i}}\sqrt{N}\big(\zeta(-{\tfrac{1}{2}},d+1)-\zeta(-{\tfrac{1}{2}},d'+1)\big)+\Big(\frac{1}{24}-\frac{(d+d'+1)^{2}}{4}\Big)\log N\nonumber\\ && -\frac{{\mathrm{i}}(d-d')(d+d'+1)}{2}+\int_{d'+{\tfrac{1}{2}}}^{0}{\mathrm{d}}u\,\frac{\zeta({\tfrac{1}{2}},d+1+u)\zeta({\tfrac{1}{2}},d'+1-u)}{2{\mathrm{i}}}+\kappa_{1}(d+d')\;.\nonumber\end{aligned}$$ When $d+d'=-1$, the constant of integration is $\kappa_{1}(-1)\approx0.05382943932689441$. The derivation is essentially identical to the one of (\[EM rectangle log(sqrt+sqrt)\]). [Large L asymptotics]{} \[section asymptotics\] In this section, we compute the large $L$ asymptotics of the quantities $$\begin{aligned} \label{Xi1} && \Xi_{1}=\frac{1}{N}\sum_{j=1}^{N}\frac{y_{j}}{\rho+(1-\rho)y_{j}}\;,\\ \label{Xi2} && \Xi_{2}=\prod_{j=1}^{N}\Big(1+\frac{1-\rho}{\rho}\,y_{j}\Big)\;,\\ \label{Xi3} && \Xi_{3}=\prod_{j=1}^{N}\prod_{k=j+1}^{N}\frac{y_{j}-y_{k}}{y_{j}^{0}-y_{k}^{0}}\;,\\ \label{Xi4} && \Xi_{4}=\prod_{j=1}^{N}\prod_{k=j+1}^{N}\big(y_{j}^{0}-y_{k}^{0}\big)\;,\end{aligned}$$ for $y_{j}$’s given by (\[y\[k\]\]), $k_{j}$’s constructed in terms of sets $A_{0}^{\pm}$, $A^{\pm}$, and with the correction (\[b\[c\]\]) to $b$. The parameter $c$ will be in this section an arbitrary complex number that is **not** required to verify (\[phi(c)\]). [Function Phi]{} We introduce the function $\Phi$ defined from $g$ (\[g\]) by $$\label{Phi} \Phi(u)=g^{-1}\big({\mathrm{e}}^{-b_{0}+2{\mathrm{i}}\pi u}\big)\;.$$ Writing the parameter $b$ as in (\[b\[c\]\]), the Bethe roots can be expressed as $$y_{j}=\Phi\Big(\frac{k_{j}+{\mathrm{i}}c}{L}\Big)\;.$$ In particular, for the stationary eigenstate, one has $$y_{j}^{0}=\Phi\Big(-\frac{\rho}{2}+\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\Big)\;.$$ For the first eigenstates, the $k_{j}$’s added ($\pm(N/2+a)$, $a\in A^{\pm}$) correspond to $y_{j}=\Phi(\pm\tfrac{\rho}{2}+\frac{{\mathrm{i}}(c\mp{\mathrm{i}}a)}{L})$, while the $k_{j}$’s removed ($\pm(N/2-a)$, $a\in A_{0}^{\pm}$) correspond to $y_{j}=\Phi(\pm\tfrac{\rho}{2}+\frac{{\mathrm{i}}(c\pm{\mathrm{i}}a)}{L})$. These expressions are suited for the use of the Euler-Maclaurin formula to compute the asymptotics of sums of the form $\sum_{j=1}^{N}f(y_{j})$ and $\sum_{j=1}^{N}\sum_{j'=j+1}^{N}f(y_{j},y_{j'})$. The function $\Phi$ verifies $\Phi(\pm\rho/2)=-\tfrac{\rho}{1-\rho}$. The points $\pm\rho/2$ are branch points of the function $\Phi$. The expansion of $\Phi$ around them is given by $$\begin{aligned} \label{Phi expansion} && \Phi\big(\pm(\tfrac{\rho}{2}-u)\big)\simeq -\frac{\rho}{1-\rho}\bigg( 1-\frac{\sqrt{2}(1\mp{\mathrm{i}})\sqrt{\pi}\sqrt{u}}{\sqrt{\rho(1-\rho)}} \mp\frac{4{\mathrm{i}}\pi(1+\rho)u}{3\rho(1-\rho)}\\ &&\hspace{41mm} +\frac{\sqrt{2}(1\pm{\mathrm{i}})\pi^{3/2}(1+11\rho+\rho^{2})u^{3/2}}{9(\rho(1-\rho))^{3/2}} +\frac{8\pi^{2}(1+\rho)(1-25\rho+\rho^{2})u^{2}}{135\rho^{2}(1-\rho)^{2}} \bigg)\;.\nonumber\end{aligned}$$ A useful property of the function $\Phi$ is that its derivative can be expressed in terms of $\Phi$ alone. One has $$\label{Phi'} \Phi'(u)=-2{\mathrm{i}}\pi\,\frac{\Phi(u)(1-\Phi(u))}{\rho+(1-\rho)\Phi(u)}\;.$$ [Asymptotics of Xi1]{} The Bethe roots $y_{j}$ can be replaced by $y_{j}^{0}$ in (\[Xi1\]), up to corrections obtained by summing over the sets $A_{0}^{\pm}$, $A^{\pm}$. Writing $b$ as (\[b\[c\]\]), the summand is equal to $y_{j}/(\rho+(1-\rho)y_{j})=f(\rho/2+(k_{j}+{\mathrm{i}}c)/L)$ with $f(u)=\Phi(u-\tfrac{\rho}{2})/(\rho+(1-\rho)\Phi(u-\tfrac{\rho}{2}))$. One has $$\begin{aligned} \label{Xi1[A]} && \sum_{j=1}^{N}\frac{y_{j}}{\rho+(1-\rho)y_{j}} =\sum_{j=1}^{N}f\Big(\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\Big) +\sum_{a\in A^{-}}f\Big(\frac{{\mathrm{i}}(c+{\mathrm{i}}a)}{L}\Big) -\sum_{a\in A_{0}^{-}}f\Big(\frac{{\mathrm{i}}(c-{\mathrm{i}}a)}{L}\Big)\\ &&\hspace{56mm} +\sum_{a\in A^{+}}f\Big(\rho+\frac{{\mathrm{i}}(c-{\mathrm{i}}a)}{L}\Big) -\sum_{a\in A_{0}^{+}}f\Big(\rho+\frac{{\mathrm{i}}(c+{\mathrm{i}}a)}{L}\Big)\;.\nonumber\end{aligned}$$ The function $f$ verifies (\[f\[fk\]\]) with first coefficients equal to $\alpha=\overline{\alpha}=0$ and $$f_{-1}=-\frac{(1-{\mathrm{i}})\sqrt{\rho}}{2^{3/2}\sqrt{\pi}\sqrt{1-\rho}} \quad\text{and}\quad \overline{f}_{-1}=-\frac{(1+{\mathrm{i}})\sqrt{\rho}}{2^{3/2}\sqrt{\pi}\sqrt{1-\rho}}\;.$$ From (\[EM sqrt sqrt\]), the expansion up to order $\sqrt{L}$ of the sum over $j$ in the right hand side of (\[Xi1\[A\]\]) is $$\sum_{j=1}^{N}f\Big(\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\Big) \simeq L\Big(\int_{0}^{\rho}{\mathrm{d}}u\,f(u)\Big) +\sqrt{L}\big(f_{-1}\zeta(\tfrac{1}{2},\tfrac{1}{2}+{\mathrm{i}}c)+\overline{f}_{-1}\zeta(\tfrac{1}{2},\tfrac{1}{2}-{\mathrm{i}}c)\big)\;.$$ The integral can be computing by making the change of variables $z=\Phi(u-\tfrac{\rho}{2})$, as explained in appendix \[appendix integrals\]. The residue calculation gives $\int_{0}^{\rho}{\mathrm{d}}u\,f(u)=0$. At leading order in $L$, using (\[Phi expansion\]) to treat the sums over $a$, and (\[zeta’\]), we can express $\Xi_{1}$ at leading order in terms of the derivative of the function $\varphi_{r}$ (\[phi\[A,zeta\]\]). We find $$\label{Asymptotics 1} \frac{1}{N}\sum_{j=1}^{N}\frac{y_{j}}{\rho+(1-\rho)y_{j}}\simeq\frac{\varphi_{r}'(2\pi c)}{\sqrt{\rho(1-\rho)}\,\sqrt{L}}\;.$$ [Asymptotics of Xi2]{} We consider the logarithm of $\Xi_{2}$, defined in (\[Xi2\]). The calculation of the large $L$ asymptotics follows closely the one for $\Xi_{1}$, except one needs to push the expansion of the sum up to the constant term in $L$ in order to get the prefactor of the exponential in $\Xi_{2}$. The summand is equal to $\log(1+y_{j}(1-\rho)/\rho)=f(\rho/2+(k_{j}+{\mathrm{i}}c)/L)$ with $f(u)=\log(1+\Phi(u-\tfrac{\rho}{2})(1-\rho)/\rho)$. One has again $$\begin{aligned} \label{Xi2[A]} && \sum_{j=1}^{N}\log\Big(1+\frac{1-\rho}{\rho}\,y_{j}\Big) =\sum_{j=1}^{N}f\Big(\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\Big) +\sum_{a\in A^{-}}f\Big(\frac{{\mathrm{i}}(c+{\mathrm{i}}a)}{L}\Big) -\sum_{a\in A_{0}^{-}}f\Big(\frac{{\mathrm{i}}(c-{\mathrm{i}}a)}{L}\Big)\\ &&\hspace{62mm} +\sum_{a\in A^{+}}f\Big(\rho+\frac{{\mathrm{i}}(c-{\mathrm{i}}a)}{L}\Big) -\sum_{a\in A_{0}^{+}}f\Big(\rho+\frac{{\mathrm{i}}(c+{\mathrm{i}}a)}{L}\Big)\;.\nonumber\end{aligned}$$ The function $f$ verifies (\[f\[fk\]\]) with first coefficients equal to $\alpha=\overline{\alpha}=1/2$, $f_{-1}=\overline{f}_{-1}=0$ and $$f_{0}=\log\frac{(1+{\mathrm{i}})\sqrt{2\pi}}{\sqrt{\rho(1-\rho)}} \quad\text{and}\quad \overline{f}_{0}=\log\frac{(1-{\mathrm{i}})\sqrt{2\pi}}{\sqrt{\rho(1-\rho)}}\;.$$ From (\[EM sqrt sqrt\]), the expansion up to order $L^{0}$ of the sum over $j$ in the right hand side of (\[Xi2\[A\]\]) is $$\begin{aligned} \label{tmp Xi2} && \sum_{j=1}^{N}f\Big(\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\Big) \simeq L\Big(\int_{0}^{\rho}{\mathrm{d}}u\,f(u)\Big) +\log(2\pi)-\frac{\log\Gamma(\frac{1}{2}+{\mathrm{i}}c)}{2}-\frac{\log\Gamma(\frac{1}{2}-{\mathrm{i}}c)}{2}\\ &&\hspace{75mm} +f_{0}\,\zeta(0,\tfrac{1}{2}+{\mathrm{i}}c) +\overline{f}_{0}\,\zeta(0,\tfrac{1}{2}-{\mathrm{i}}c)\;.\nonumber\end{aligned}$$ Again, the integral can be computing as explained in appendix \[appendix integrals\]. One finds $\int_{0}^{\rho}{\mathrm{d}}u\,f(u)=0$. At leading order in $L$, using (\[Phi expansion\]) and the constraint (\[m+-\]) to treat the sums over $a$, and $\zeta(0,z)=\tfrac{1}{2}-z$ (\[Bernoulli\[zeta\]\]), one can express the expansion of $\log \Xi_{2}$ to order $L^{0}$. After taking the exponential and using Euler’s reflection formula $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$ to eliminate the $\Gamma$ functions, we obtain $$\label{asymptotics Xi2} \prod_{j=1}^{N}\Big(1+\frac{1-\rho}{\rho}\,y_{j}\Big)\simeq\sqrt{1+{\mathrm{e}}^{2\pi c}}\,\frac{\big(\prod_{a\in A^{+}}\sqrt{c-{\mathrm{i}}a}\big)\big(\prod_{a\in A^{-}}\sqrt{c+{\mathrm{i}}a}\big)}{\big(\prod_{a\in A_{0}^{+}}\sqrt{c+{\mathrm{i}}a}\big)\big(\prod_{a\in A_{0}^{-}}\sqrt{c-{\mathrm{i}}a}\big)}\;.$$ The factor $\sqrt{1+{\mathrm{e}}^{2\pi c}}$ has infinitely many branch cuts ${\mathrm{i}}(n+{\tfrac{1}{2}})+\mathbb{R}^{+}$, $n\in\mathbb{Z}$. Since $\log\Gamma$ is interpreted as the log $\Gamma$ function and not the logarithm of the $\Gamma$ function in (\[tmp Xi2\]), $\sqrt{1+{\mathrm{e}}^{2\pi c}}$ has to be understood for ${\operatorname{Re}}c>0$ as the analytic continuation in $c$ from the real axis $(-1)^{\big\lfloor\frac{{\operatorname{Im}}(4\pi c+2{\mathrm{i}}\pi)}{4\pi}\big\rfloor}\sqrt{1+{\mathrm{e}}^{2\pi c}}$ with $\lfloor x\rfloor$ the largest integer smaller or equal to $x$. This corresponds to choosing instead the branch cuts $(-{\mathrm{i}}\infty,-{\mathrm{i}}/2]\cup[{\mathrm{i}}/2,\infty)$ for $\sqrt{1+{\mathrm{e}}^{2\pi c}}$. [Asymptotics of Xi3]{} We consider the quantity $\Xi_{3}$, defined in (\[Xi3\]). For the stationary state, one has $\Xi_{3}=1$. We focus on the other first eigenstates, and take the parameter $c$ with ${\operatorname{Re}}c>0$: this constraint is verified for the solution of (\[phi(c)\]) as long as the real part of $s$ is not too negative. In particular, it seems to to be valid for all $s$ with ${\operatorname{Re}}s\geq0$. This is not the case for the stationary state, for which the solution of (\[phi(c)\]) at leading order in $L$ is $c\to-\infty$ when $s\to0$. Replacing $y_{j}$ and $y_{k}$ by $y_{j}^{0}$ and $y_{k}^{0}$ in the definition (\[Xi3\]) of $\Xi_{3}$, with corrections coming from the sets $A_{0}^{\pm}$, $A^{\pm}$, one has $$\begin{aligned} \label{V/V0[Phi]} && \prod_{j=1}^{N}\prod_{k=j+1}^{N}\frac{y_{j}-y_{k}}{y_{j}^{0}-y_{k}^{0}}=\text{(finite products)}\\ &&\hspace{32mm} \times\frac {\prod\limits_{a\in A^{+}}\prod\limits_{j=1}^{N}\Big(\Phi\big(-\frac{\rho}{2}+\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\big)-\Phi\big(\frac{\rho}{2}+\frac{a+{\mathrm{i}}c}{L}\big)\Big)} {\prod\limits_{a\in A_{0}^{+}}\prod\limits_{\substack{j=1\\n-j\neq a-{\tfrac{1}{2}}}}^{N}\Big(\Phi\big(-\frac{\rho}{2}+\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\big)-\Phi\big(\frac{\rho}{2}-\frac{a-{\mathrm{i}}c}{L}\big)\Big)}\nonumber\\ &&\hspace{32mm} \times \frac {\prod\limits_{a\in A^{-}}\prod\limits_{j=1}^{N}\Big(\Phi\big(-\frac{\rho}{2}+\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\big)-\Phi\big(-\frac{\rho}{2}-\frac{a-{\mathrm{i}}c}{L}\big)\Big)} {\prod\limits_{a\in A_{0}^{-}}\prod\limits_{\substack{j=1\\j\neq a+{\tfrac{1}{2}}}}^{N}\Big(\Phi\big(-\frac{\rho}{2}+\frac{j-\frac{1}{2}+{\mathrm{i}}c}{L}\big)-\Phi\big(-\frac{\rho}{2}+\frac{a+{\mathrm{i}}c}{L}\big)\Big)}\;.\nonumber\end{aligned}$$ The $(\text{finite products})$ factor contains all the factors with only contributions from the sets $A_{0}^{\pm}$, $A^{\pm}$ and no product over $j$ between $1$ and $N$. Their asymptotics can be computed from (\[Phi expansion\]) and (\[m+-\]). At leading order in $L$, one finds $$\begin{aligned} && (\text{finite products})\simeq\Big(\frac{(1-\rho)^{3/2}}{2{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}}\,\sqrt{L}\Big)^{m_{r}} \Big(\prod_{a\in A_{0}^{+}}(-1)^{a-\frac{1}{2}}\Big)\Big(\prod_{a\in A_{0}^{-}}(-1)^{a+\frac{1}{2}}\Big)\\ &&\hspace{25mm} \times\prod_{\sigma\in\{-,+\}}\!\!\! \frac {\prod\limits_{\substack{a,a'\in A^{\sigma}\\\sigma a<\sigma a'}}\!\!\Big(\sqrt{c-\sigma{\mathrm{i}}a}-\sqrt{c-\sigma{\mathrm{i}}a'}\Big)\prod\limits_{\substack{a,a'\in A_{0}^{\sigma}\\\sigma a>\sigma a'}}\!\!\Big(\sqrt{c+\sigma{\mathrm{i}}a}-\sqrt{c+\sigma{\mathrm{i}}a'}\Big)} {\prod\limits_{a\in A^{\sigma}}\prod\limits_{a'\in A_{0}^{\sigma}}\!\!\Big(\sqrt{c-\sigma{\mathrm{i}}a}-\sqrt{c+\sigma{\mathrm{i}}a'}\Big)} \nonumber\\ &&\hspace{25mm} \times \frac {\prod\limits_{a\in A^{+}}\prod\limits_{a'\in A^{-}}\Big(\sqrt{c-{\mathrm{i}}a}+\sqrt{c+{\mathrm{i}}a'}\Big)\prod\limits_{a\in A_{0}^{+}}\prod\limits_{a'\in A_{0}^{-}}\Big(\sqrt{c+{\mathrm{i}}a}+\sqrt{c-{\mathrm{i}}a'}\Big)} {\prod\limits_{a\in A^{+}}\prod\limits_{a'\in A_{0}^{-}}\Big(\sqrt{c-{\mathrm{i}}a}+\sqrt{c-{\mathrm{i}}a'}\Big)\prod\limits_{a\in A_{0}^{+}}\prod\limits_{a'\in A^{-}}\Big(\sqrt{c+{\mathrm{i}}a}+\sqrt{c+{\mathrm{i}}a'}\Big)} \;.\nonumber\end{aligned}$$ After taking the logarithm, the factors that contain a product over $j$ in (\[V/V0\[Phi\]\]) take the form of a sum for $j$ between $1$ and $N$ of $\log f((j-\frac{1}{2}+{\mathrm{i}}c)/L)$ where $$f(x)=\Phi\big(-\frac{\rho}{2}+x\big)-\Phi\big(\frac{\sigma\rho}{2}+\frac{{\mathrm{i}}c+\sigma'a}{L}\big)\;,$$ with parameters, $\sigma,\sigma'\in\{+1,-1\}$, $a\in\mathbb{N}+\tfrac{1}{2}$. The function $f$ has a singularity when either $x$ or $\rho-x$ is of order $1/L$: using (\[Phi expansion\]) and the assumption ${\operatorname{Re}}c>0$, one has for large $L$ $$\begin{aligned} f(x/L)&&\simeq\frac{\sqrt{2}(1+{\mathrm{i}})\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}\,\Big(\sqrt{\frac{x}{L}}+\sigma\sqrt{\frac{{\mathrm{i}}c+\sigma'a}{L}}\Big)\\ f(\rho-x/L)&&\simeq\frac{\sqrt{2}(1-{\mathrm{i}})\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}\,\Big(\sqrt{\frac{x}{L}}-\sigma\sqrt{-\frac{{\mathrm{i}}c+\sigma'a}{L}}\Big)\;.\nonumber\end{aligned}$$ This is precisely the type of singularities that can be treated by (\[EM sqrt(Stirling)\]). Since both ends of the summation range exhibit this singularity, one can directly use (\[EM sqrt(Stirling) sqrt(Stirling)\]), with coefficients $\sigma_{0}=\sigma$, $\sigma_{1}=-\sigma$, $q_{0}={\mathrm{i}}c+\sigma'a$, $q_{1}=-{\mathrm{i}}c-\sigma'a$. Introducing nonnegative integers $m$ and $m'$ to avoid terms in the sum for which $f((j-\frac{1}{2}+{\mathrm{i}}c)/L)=0$, we find the large $L$ asymptotics $$\begin{aligned} && \sum_{j=m}^{N-m'}\log f\Big(\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}\Big) \simeq\rho\log\!\Big(\frac{\rho}{1-\rho}\Big)L+\sigma\frac{2{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}\sqrt{c-\sigma'{\mathrm{i}}a}}{\sqrt{1-\rho}}\,\sqrt{L}+\frac{m+m'-1}{2}\,\log L\\ &&\hspace{35mm} +\log\sqrt{2\pi}+\frac{2\pi(1-2\rho)c}{3(1-\rho)}-\frac{\pi(1-5\rho)\sigma'{\mathrm{i}}a}{6(1-\rho)} +\frac{{\mathrm{i}}\pi(1-m+m')}{4}\nonumber\\ &&\hspace{35mm} -(m+m'-1)\log\frac{2\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}-\frac{\log\Gamma(m-{\tfrac{1}{2}}-\sigma'a)}{2}-\frac{\log\Gamma(m'+{\tfrac{1}{2}}+\sigma'a)}{2}\nonumber\\ &&\hspace{35mm} +\sigma\int_{\sigma'{\mathrm{i}}a}^{c}{\mathrm{d}}u\,\frac{{\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},m-{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},m'+{\tfrac{1}{2}}-{\mathrm{i}}u)}{2\sqrt{u-\sigma'{\mathrm{i}}a}}\;.\nonumber\end{aligned}$$ There, the integral giving the leading order in $L$ was computed as explained at the beginning of appendix \[appendix integrals\] by making the change of variable $z=\Phi(u-\frac{\rho}{2})$. One has $$\begin{aligned} && \int_{0}^{\rho}{\mathrm{d}}u\,\log f(u)=\rho\log\Big(-\Phi\Big(\frac{\sigma\rho}{2}+\frac{{\mathrm{i}}c+\sigma'a}{L}\Big)\Big)\\ &&\hspace{22mm} \simeq\rho\log\frac{\rho}{1-\rho}+\sigma\frac{2{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}\sqrt{c-\sigma'{\mathrm{i}}a}}{\sqrt{1-\rho}\sqrt{L}}+\frac{2\pi(1-2\rho)(c-\sigma'{\mathrm{i}}a)}{3(1-\rho)L}\;.\nonumber\end{aligned}$$ We write $\sum'$ for the full sum between $1$ and $N$ minus any divergent term as in (\[V/V0\[Phi\]\]). Its expansion up to order $0$ in $L$ is $$\begin{aligned} \label{tmp Xi3} && \sum_{j=1}^{N}\!'\,\log f\Big(\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}\Big) \simeq\rho\log\!\Big(\frac{\rho}{1-\rho}\Big)L+\sigma\frac{2{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}\sqrt{c-\sigma'{\mathrm{i}}a}}{\sqrt{1-\rho}}\,\sqrt{L}+\frac{1-\sigma\sigma'}{4}\,\log L\\ &&\hspace{30mm} +\log\sqrt{2\pi}+\frac{2\pi(1-2\rho)c}{3(1-\rho)}-\frac{\pi(1-5\rho)\sigma'{\mathrm{i}}a}{6(1-\rho)}+\frac{{\mathrm{i}}\pi(\sigma-\sigma')}{8}\nonumber\\ &&\hspace{30mm} -\frac{1-\sigma\sigma'}{2}\,\log\frac{2\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}-\frac{\log\Gamma(m-{\tfrac{1}{2}}-\sigma'a)}{2}-\frac{\log\Gamma(m'+{\tfrac{1}{2}}+\sigma'a)}{2}\nonumber\\ &&\hspace{30mm} +\sigma\int_{\sigma'{\mathrm{i}}a}^{c}{\mathrm{d}}u\,\frac{{\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},m-{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},m'+{\tfrac{1}{2}}-{\mathrm{i}}u)}{2\sqrt{u-\sigma'{\mathrm{i}}a}}\nonumber\\ &&\hspace{30mm} +\sum_{j=1}^{m-1}\!\!'\,\log\big(\sqrt{j-{\tfrac{1}{2}}+{\mathrm{i}}c}+\sigma\sqrt{{\mathrm{i}}c+\sigma'a}\big)+\sum_{j=1}^{m'}\!'\,\log\big(\sqrt{j-{\tfrac{1}{2}}-{\mathrm{i}}c}-\sigma\sqrt{-{\mathrm{i}}c-\sigma'a}\big)\;.\nonumber\end{aligned}$$ The integers $m$ and $m'$ must be taken large enough so that the arguments of the $\Gamma$ functions do not belong to $-\mathbb{N}$. They are also needed to ensure the convergence of the integral at $u=\sigma'{\mathrm{i}}a$, since $\zeta({\tfrac{1}{2}},-n+\varepsilon)\sim\varepsilon^{-1/2}$ for $n\in\mathbb{N}$. The branch cuts of the integrand as a function of $u$ are chosen equal to $({\mathrm{i}}\infty,{\mathrm{i}}(m-{\tfrac{1}{2}})]$, $(-{\mathrm{i}}\infty,-{\mathrm{i}}(m'+{\tfrac{1}{2}})]$ and $(-\infty,\sigma'{\mathrm{i}}a]$. Since the left hand side of (\[tmp Xi3\]) is independent of $m$, $m'$, one would like to eliminate them also in the right hand side. This can be done using the relation $$\begin{aligned} \label{tmp identity Xi3 shift m} &&\int_{-\Lambda}^{q}{\mathrm{d}}u\,\frac{{\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},m-{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},m'+{\tfrac{1}{2}}-{\mathrm{i}}u)}{2\sqrt{u-\sigma'{\mathrm{i}}a}}\\ && =\int_{-\Lambda}^{q}{\mathrm{d}}u\,\frac{{\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)}{2\sqrt{u-\sigma'{\mathrm{i}}a}} -\frac{{\mathrm{i}}\pi(m+m'-1)}{4}\nonumber\\ && -\sum_{j=1}^{m-1}\Big(\log\big(\sqrt{j-{\tfrac{1}{2}}+{\mathrm{i}}q}+\sqrt{\sigma'a+{\mathrm{i}}q}\big)-\log\big(\sqrt{\Lambda+{\mathrm{i}}(j-{\tfrac{1}{2}})}+\sigma'\sqrt{\Lambda+\sigma'{\mathrm{i}}a}\big)\Big)\nonumber\\ && +\sum_{j=1}^{m'}\Big(\log\big(\sqrt{j-{\tfrac{1}{2}}-{\mathrm{i}}q}+\sqrt{-\sigma'a-{\mathrm{i}}q}\big)-\log\big(\sqrt{\Lambda-{\mathrm{i}}(j-{\tfrac{1}{2}})}-\sigma'\sqrt{\Lambda+\sigma'{\mathrm{i}}a}\big)\Big)\;,\nonumber\end{aligned}$$ which follows from (\[sum\_log\_sqrt\[Gamma,zeta\]\]) and is valid when ${\operatorname{Re}}q>0$ and ${\operatorname{Re}}\Lambda>0$ with $|{\operatorname{Im}}\Lambda|<a$ if the path of integration is required to avoid all branch cuts. We decompose the integral from $\sigma'{\mathrm{i}}a$ to $c$ in (\[tmp Xi3\]) as an integral from $-\Lambda$ to $c$ minus the limit $\varepsilon\to0$, $\varepsilon>0$ of an integral from $-\Lambda$ to $\sigma'{\mathrm{i}}a+\varepsilon$. After using (\[tmp identity Xi3 shift m\]) for $q=c$ and $q=\sigma'{\mathrm{i}}a+\varepsilon$ the limit $\Lambda\to\infty$ of the integrals become convergent. The limit $\varepsilon\to0$ of the integral between $-\infty$ and $\sigma'{\mathrm{i}}a+\varepsilon$ (with a contour of integration that avoids the branch cut of $\zeta$) is given by the identity $$\int_{-\infty}^{\sigma'{\mathrm{i}}a+\varepsilon}{\mathrm{d}}u\,\frac{{\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)}{2\sqrt{u-\sigma'{\mathrm{i}}a}} \underset{\varepsilon\to0}{\simeq}\sigma'\log\sqrt{\varepsilon}+\sigma'\log\sqrt{8\pi}+{\mathrm{i}}\pi a\;,$$ that was obtained numerically. The divergent contribution $\log\sqrt{\varepsilon}$ cancels with terms in the sums of (\[tmp identity Xi3 shift m\]) at $j=a+{\tfrac{1}{2}}$. After several simplifications, (\[tmp Xi3\]) becomes $$\begin{aligned} && \sum_{j=1}^{N}\!'\,\log f\Big(\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}\Big) \simeq\rho\log\!\Big(\frac{\rho}{1-\rho}\Big)L+\sigma\frac{2{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}\sqrt{c-\sigma'{\mathrm{i}}a}}{\sqrt{1-\rho}}\,\sqrt{L}+\frac{1-\sigma\sigma'}{4}\,\log L\\ &&\hspace{34mm} -\frac{1-\sigma\sigma'}{2}\,\log\Big(\frac{\sqrt{\rho}}{2\sqrt{\pi}(1-\rho)^{3/2}\sqrt{c-\sigma'{\mathrm{i}}a}}\Big)+\frac{2\pi(1-2\rho)(c-\sigma'{\mathrm{i}}a)}{3(1-\rho)}\nonumber\\ &&\hspace{34mm} -(\sigma-\sigma'){\mathrm{i}}\pi(a-\tfrac{1}{4})+\sigma\int_{-\infty}^{c}{\mathrm{d}}u\,\frac{{\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)}{2\sqrt{u-\sigma'{\mathrm{i}}a}}\;.\nonumber\end{aligned}$$ Putting everything together, we obtain for the product of the four factors of $\Xi_{3}$ containing products over $j$ in (\[V/V0\[Phi\]\]) $$\begin{aligned} && {\mathrm{e}}^{-\frac{2{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}}{\sqrt{1-\rho}}\Big(\sum_{a\in A_{0}^{+}}\sqrt{c+{\mathrm{i}}a}+\sum_{a\in A^{-}}\sqrt{c+{\mathrm{i}}a}-\sum_{a\in A_{0}^{-}}\sqrt{c-{\mathrm{i}}a}-\sum_{a\in A^{+}}\sqrt{c-{\mathrm{i}}a}\Big)\,\sqrt{L}}\\ && \times\Big(\frac{\sqrt{\rho}}{2\sqrt{\pi}(1-\rho)^{3/2}\sqrt{L}}\Big)^{m_{r}} {\mathrm{i}}^{m_{r}^{+}-m_{r}^{-}} {\mathrm{e}}^{-\frac{2{\mathrm{i}}\pi(1-2\rho)p_{r}}{3(1-\rho)}} \Big(\prod_{a\in A_{0}^{+}}\frac{1}{\sqrt{c+{\mathrm{i}}a}}\Big) \Big(\prod_{a\in A_{0}^{-}}\frac{1}{\sqrt{c-{\mathrm{i}}a}}\Big)\nonumber\\ && \times\exp\Bigg[-\frac{1}{2}\int_{-\infty}^{c}{\mathrm{d}}u\,\Big({\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)\Big)\nonumber\\ &&\hspace{25mm} \times\bigg(\sum_{a\in A_{0}^{+}}\frac{1}{\sqrt{u+{\mathrm{i}}a}}+\sum_{a\in A^{-}}\frac{1}{\sqrt{u+{\mathrm{i}}a}}-\sum_{a\in A_{0}^{-}}\frac{1}{\sqrt{u-{\mathrm{i}}a}}-\sum_{a\in A^{+}}\frac{1}{\sqrt{u-{\mathrm{i}}a}}\bigg)\Bigg]\;.\nonumber\end{aligned}$$ [Asymptotics of Xi4]{} We write $$\label{Xi4[sum]} \prod_{j=1}^{N}\prod_{j'=j+1}^{N}\big(y_{j}^{0}-y_{j'}^{0}\big)={\mathrm{i}}^{\frac{N(N-1)}{2}}\exp\bigg(\sum_{j=1}^{N}\sum_{j'=j+1}^{N}\log(-{\mathrm{i}}(y_{j}^{0}-y_{j'}^{0}))\bigg)\;,$$ where $\log$ is the usual determination of the logarithm with branch cut $\mathbb{R}^{-}$. The (clockwise) contour on which the $y_{j}^{0}$’s condense in the complex plane is represented in figure \[fig contour Phi\]. The factor $-{\mathrm{i}}$ in the logarithm ensures that the branch cut of the logarithm is not crossed. Since $y_{j}^{0}\to-\rho/(1-\rho)$ when $j\ll L$ and when $N-j\ll L$, the quantity $y_{j}^{0}-y_{j'}^{0}$ goes to $0$ at the three corners of the triangle $\{(j,j'),1\leq j<j'\leq N\}$. To avoid extra singularities and allow us to use (\[EM triangle sqrt sqrt sqrt\]), we consider first the regular part $$S_{\text{reg}}=\sum_{j=1}^{N}\sum_{j'=j+1}^{N}f\Big(\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L},\frac{j'-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}\Big)\;$$ with $f$ defined by $$f(u,v)=\log\bigg(\!-{\mathrm{i}}\Big(\Phi\big(-\frac{\rho}{2}+u\big)-\Phi\big(-\frac{\rho}{2}+v\big)\Big)\,\frac{(\sqrt{u}+\sqrt{v})(\sqrt{\rho-u}+\sqrt{\rho-v})}{(v-u)(\sqrt{-{\mathrm{i}}u}+\sqrt{{\mathrm{i}}(\rho-v)})}\bigg).$$ Using the same notations as in (\[EM triangle sqrt sqrt sqrt\]), one has $$\begin{aligned} && f_{0,0}=\log\frac{4{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}\;,\qquad g_{0,0}=\log\Big(-\frac{4{\mathrm{i}}\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}\Big)\;,\qquad h_{0,0}=\log\Big(\frac{2\sqrt{\pi}\sqrt{\rho}}{(1-\rho)^{3/2}}\Big)\;,\\ && f_{1}(v)=\frac{1}{\sqrt{v}}+\frac{{\mathrm{i}}}{\sqrt{\rho-v}}-\frac{2\sqrt{{\mathrm{i}}}\sqrt{\pi}\sqrt{\rho}}{\sqrt{1-\rho}(\rho+(1-\rho)\Phi(v-\rho/2))}\;,\nonumber\\ && g_{1}(v)=\frac{1}{\sqrt{v}}-\frac{{\mathrm{i}}}{\sqrt{\rho-v}}-\frac{2\sqrt{-{\mathrm{i}}}\sqrt{\pi}\sqrt{\rho}}{\sqrt{1-\rho}(\rho+(1-\rho)\Phi(\rho/2-v))}\;,\nonumber\\ && f_{2}(v)=-\frac{1}{2(\rho-v)}+\frac{\sqrt{\rho-v}}{2\sqrt{\rho}\,v}-\frac{2{\mathrm{i}}\pi\rho}{(1-\rho)(\rho+(1-\rho)\Phi(v-\rho/2))^{2}}+\frac{4{\mathrm{i}}\pi(1+\rho)}{3(1-\rho)(\rho+(1-\rho)\Phi(v-\rho/2))}\;,\nonumber\\ && g_{2}(v)=-\frac{1}{2(\rho-v)}+\frac{\sqrt{\rho-v}}{2\sqrt{\rho}\,v}+\frac{2{\mathrm{i}}\pi\rho}{(1-\rho)(\rho+(1-\rho)\Phi(\rho/2-v))^{2}}-\frac{4{\mathrm{i}}\pi(1+\rho)}{3(1-\rho)(\rho+(1-\rho)\Phi(\rho/2-v))}\;.\nonumber\end{aligned}$$ When the two arguments of $f$ are equal, one finds the limits $$\begin{aligned} && f(v,v)=\log\Big(\frac{8\pi\sqrt{v}\sqrt{\rho-v}\,\Phi(v-\rho/2)(1-\Phi(v-\rho/2))}{(\sqrt{-{\mathrm{i}}v}+\sqrt{{\mathrm{i}}(\rho-v)})(\rho+(1-\rho)\Phi(v-\rho/2))}\Big)\;,\\ && f^{(1,0)}(v,v)=-\frac{1}{2\rho}+\frac{{\mathrm{i}}\pi}{1-\rho}+\frac{1}{4v}-\frac{1}{4(\rho-v)}+\frac{{\mathrm{i}}\sqrt{\rho-v}}{2\rho\sqrt{v}}-\frac{{\mathrm{i}}\pi\rho}{(1-\rho)(\rho+(1-\rho)\Phi(v-\rho/2))^{2}}\;,\nonumber\\ && f^{(0,1)}(v,v)=\frac{1}{2\rho}+\frac{{\mathrm{i}}\pi}{1-\rho}+\frac{1}{4v}-\frac{1}{4(\rho-v)}+\frac{{\mathrm{i}}\sqrt{v}}{2\rho\sqrt{\rho-v}}-\frac{{\mathrm{i}}\pi\rho}{(1-\rho)(\rho+(1-\rho)\Phi(v-\rho/2))^{2}}\;.\nonumber\end{aligned}$$ One can use (\[EM triangle sqrt sqrt sqrt\]) for the asymptotic expansion. After some simplifications, which involve in particular the calculation of several simple and double integrals explained in appendix \[appendix integrals\], and using the explicit values (\[Bernoulli\[zeta\]\]) and (\[Hurwitz zeta double 00\]) for $\zeta(0,z)$, $\zeta(-1,z)$ and $\tilde{\zeta}_{0}(0,0,z,z')$, one finds $$\begin{aligned} && S_{\text{reg}}\simeq L^{2}\Big(\frac{9\rho^{2}}{8}+\frac{\rho^{2}\log\rho}{4}+\frac{\rho(1-\rho)\log(1-\rho)}{2}\Big)\\ &&\hspace{7mm} +L\Big(-\frac{\rho\log\rho}{4}-\frac{(1-\rho)\log(1-\rho)}{2}+\frac{\rho}{2}-\frac{\rho\log(8\pi)}{2}+\frac{\pi\rho c}{2}\Big)\nonumber\\ &&\hspace{7mm} +\sqrt{L}\Big(2\sqrt{\rho}-\frac{\sqrt{2\pi}\sqrt{\rho}}{\sqrt{1-\rho}}\Big)\Big((1+{\mathrm{i}})\zeta(-{\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}c)+(1-{\mathrm{i}})\zeta(-{\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}c)\Big) +\Big(-\frac{1}{8}+\frac{\log2}{12}-\frac{\pi c}{2}-c^{2}\Big)\;.\nonumber\end{aligned}$$ The singular part $S_{\text{sng}}$ of the sum in (\[Xi4\[sum\]\]) is equal to $$\begin{aligned} \label{tmp Xi4 singular part} && \sum_{j=1}^{N}\sum_{j'=j+1}^{N}\log\frac{\Big(\frac{j'-j}{L}\Big)\Big(\sqrt{-{\mathrm{i}}\,\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}}+\sqrt{{\mathrm{i}}\,\frac{j'-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}}\Big)}{\Big(\sqrt{\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}}+\sqrt{\frac{j'-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}}\Big)\Big(\sqrt{\rho-\frac{j-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}}+\sqrt{\rho-\frac{j'-{\tfrac{1}{2}}+{\mathrm{i}}c}{L}}\Big)}\\ && =N\log2-\frac{N(N-1)}{4}\log L+\log(G(N+1))+\frac{1}{4}\log\frac{\Gamma(N+{\tfrac{1}{2}}+{\mathrm{i}}c)\Gamma(N+{\tfrac{1}{2}}-{\mathrm{i}}c)}{\Gamma({\tfrac{1}{2}}+{\mathrm{i}}c)\Gamma({\tfrac{1}{2}}-{\mathrm{i}}c)}\nonumber\\ && +\sum_{j=1}^{N}\sum_{j'=1}^{N}\log\big(\sqrt{-{\mathrm{i}}(j-{\tfrac{1}{2}}+{\mathrm{i}}c)}+\sqrt{{\mathrm{i}}(j'-{\tfrac{1}{2}}-{\mathrm{i}}c)}\big)\nonumber\\ && -\sum_{j=1}^{N}\sum_{j'=1}^{N}1_{\{j+j'>N\}}\log\big(\sqrt{-{\mathrm{i}}(j-{\tfrac{1}{2}}+{\mathrm{i}}c)}+\sqrt{{\mathrm{i}}(j'-{\tfrac{1}{2}}-{\mathrm{i}}c)}\big)\nonumber\\ && -\frac{1}{2}\sum_{j=1}^{N}\sum_{j'=1}^{N}\log(\sqrt{j-{\tfrac{1}{2}}+{\mathrm{i}}c}+\sqrt{j'-{\tfrac{1}{2}}+{\mathrm{i}}c}) -\frac{1}{2}\sum_{j=1}^{N}\sum_{j'=1}^{N}\log(\sqrt{j-{\tfrac{1}{2}}-{\mathrm{i}}c}+\sqrt{j'-{\tfrac{1}{2}}-{\mathrm{i}}c})\;,\nonumber\end{aligned}$$ where $G$ is Barnes function $G(N+1)=\prod_{j=1}^{N-1}j!$, whose large $N$ expansion is given by (\[Barnes G asymptotics\]). The expansions of the remaining sums are obtained from (\[EM rectangle log(sqrt+sqrt) i\]), (\[EM rectangle log(sqrt+sqrt)\]) and $$\begin{aligned} && \sum_{j=1}^{N}\sum_{j'=1}^{N}1_{\{j+j'>N\}}\log\Big(\sqrt{-{\mathrm{i}}(j-{\tfrac{1}{2}}+{\mathrm{i}}c)}+\sqrt{{\mathrm{i}}(j'-{\tfrac{1}{2}}-{\mathrm{i}}c)}\Big) \simeq\frac{N^{2}\log N}{4}+\Big(\log2-\frac{5}{8}\Big)N^{2}\\ &&\hspace{85mm} +\frac{N\log N}{4}-\frac{(4-\pi)c}{2}\,N-\Big(\frac{1+\log2}{24}-\frac{\pi c}{4}\Big)\;,\nonumber\end{aligned}$$ which can be derived by cutting the triangle into two triangles and a rectangle and using (\[EM rectangle sqrt\]) and (\[EM triangle sqrt\]). The expansions (\[EM rectangle log(sqrt+sqrt) i\]) and (\[EM rectangle log(sqrt+sqrt)\]) contribute the constants $\kappa_{1}(-1)$ and $\kappa_{0}$ as $\kappa_{1}(-1)-\kappa(0)\approx$0.1819507467 3927841. These constants can be evaluated numerically with very large precision using BST algorithm as described in section \[section numerics\]. From numerical computations, we conjecture the identity $$\kappa_{1}(-1)-\kappa(0)=\frac{1}{12}-\zeta'(-1)-\frac{\log2}{8}-\int_{-\infty}^{0}{\mathrm{d}}u\,\Big({\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)\Big)^{2}\;,$$ that was checked within $100$ significant digits. Our derivation of the asymptotics (\[asymptotics norm\]) of the norm $\mathcal{N}_{r}(\gamma)$ does not rely heavily on this numerical conjecture since the value of $\kappa_{1}(-1)-\kappa(0)$ can in fact also be inferred from the stationary value $\mathcal{N}_{0}(0)=1$. Gathering the various contributions to the singular term, one finds $$\begin{aligned} && S_{\text{sng}}\simeq\Big(\frac{\log\rho}{4}-\frac{9}{8}\Big)\rho^{2}L^{2}+\frac{\rho L\log L}{2}+\Big(\frac{\log\rho}{4}-\frac{1}{2}+\frac{\log(8\pi)}{2}+\frac{\pi c}{2}\Big)\rho L\nonumber\\ &&\hspace{10mm} -2\sqrt{\rho}\sqrt{L}\Big((1+{\mathrm{i}})\zeta(-{\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}c)+(1-{\mathrm{i}})\zeta(-{\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}c)\Big)\nonumber\\ &&\hspace{10mm} +\frac{1}{8}-\frac{\log2}{12}+\frac{\log(2\pi)}{4}-\frac{\pi c}{4}+c^{2}-\frac{\log\Gamma({\tfrac{1}{2}}+{\mathrm{i}}c)}{4}-\frac{\log\Gamma({\tfrac{1}{2}}-{\mathrm{i}}c)}{4}\nonumber\\ &&\hspace{10mm} -\frac{1}{4}\int_{-\infty}^{c}{\mathrm{d}}u\,\Big({\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)\Big)^{2}\;.\end{aligned}$$ Putting the regular and the singular terms together, taking the exponential, and using Euler’s reflection formula $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$ to eliminate the $\Gamma$ functions, we finally obtain $$\begin{aligned} && \prod_{j=1}^{N}\prod_{j'=j+1}^{N}(y_{j}^{0}-y_{j'}^{0}) \simeq{\mathrm{i}}^{\frac{N(N-1)}{2}}{\mathrm{e}}^{\frac{\rho bL^{2}}{2}}{\mathrm{e}}^{\frac{\rho L\log L}{2}}{\mathrm{e}}^{-\frac{(1-\rho)\log(1-\rho)L}{2}}\\ &&\hspace{30mm} \times\exp\Big(-\frac{2\sqrt{\pi}\sqrt{\rho}}{\sqrt{1-\rho}}\big({\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta(-{\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}c)+{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta(-{\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}c)\big)\sqrt{L}\Big)\nonumber\\ &&\hspace{30mm} \times{\mathrm{e}}^{-\pi c}(1+{\mathrm{e}}^{2\pi c})^{1/4} \,\exp\Bigg[-\frac{1}{4}\int_{-\infty}^{c}{\mathrm{d}}u\,\Big({\mathrm{e}}^{{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}+{\mathrm{i}}u)-{\mathrm{e}}^{-{\mathrm{i}}\pi/4}\zeta({\tfrac{1}{2}},{\tfrac{1}{2}}-{\mathrm{i}}u)\Big)^{2}\Bigg]\;,\nonumber\end{aligned}$$ where $(1+{\mathrm{e}}^{2\pi c})^{1/4}$ has to be interpreted once again as the analytic continuation in $c$ from the real axis as explained after (\[asymptotics Xi2\]). [Asymptotics of (Xi3\^2\*Xi4\^2)/(Xi1\*Xi2)]{} We observe that many simplifications occur when multiplying $\Xi_{3}$ and $\Xi_{4}$ if we replace the $\zeta$ function by the eigenstate-dependent function $\varphi_{r}$ defined in (\[phi\[A,zeta\]\]). One has $$\begin{aligned} && \prod_{j=1}^{N}\prod_{k=j+1}^{N}(y_{j}-y_{k}) \simeq{\mathrm{i}}^{\frac{N(N-1)}{2}} {\mathrm{e}}^{\frac{\rho b L^{2}}{2}} {\mathrm{e}}^{\frac{\rho L \log L}{2}} {\mathrm{e}}^{-\frac{(1-\rho)\log(1-\rho)L}{2}} {\mathrm{e}}^{-\frac{\sqrt{\rho}\varphi_{r}(2\pi c)\sqrt{L}}{\sqrt{1-\rho}}} {\mathrm{e}}^{-\frac{2{\mathrm{i}}\pi(1-2\rho)p_{r}}{3(1-\rho)}}\\ &&\hspace{5mm} \frac{(\pi/2)^{m_{r}^{2}}}{(2\pi)^{m_{r}}} \,{\mathrm{e}}^{{\mathrm{i}}\pi\big(\sum_{a\in A_{0}^{+}}a-\sum_{a\in A_{0}^{-}}a\big)} \omega(A_{0}^{+})\omega(A_{0}^{-})\omega(A^{+})\omega(A^{-})\omega(A_{0}^{+},A_{0}^{-})\omega(A^{+},A^{-})\nonumber\\ &&\hspace{5mm} \frac{(1+{\mathrm{e}}^{2\pi c})^{1/4}}{{\mathrm{e}}^{\pi c}} \frac{\big(\prod_{a\in A^{+}}(c-{\mathrm{i}}a)^{1/4}\big)\big(\prod_{a\in A^{-}}(c+{\mathrm{i}}a)^{1/4}\big)}{\big(\prod_{a\in A_{0}^{+}}(c+{\mathrm{i}}a)^{1/4}\big)\big(\prod_{a\in A_{0}^{-}}(c-{\mathrm{i}}a)^{1/4}\big)} \,\exp\!\Big(\lim_{\Lambda\to\infty}-m_{r}^{2}\log\Lambda+\int_{-\Lambda}^{2\pi c}{\mathrm{d}}u\,\frac{(\varphi_{r}'(u))^{2}}{2}\Big) \;,\nonumber\end{aligned}$$ where the combinatorial factors $\omega$ are defined in (\[omega(A)\]). The limit $\Lambda\to\infty$ is needed to define the integral, which is divergent when $u=-\infty$ (except for the stationary state $m_{r}^{+}=m_{r}^{-}=0$) since $\varphi_{r}'(u)\sim u^{-1/2}$ when $u\to-\infty$. Even more simplifications occur after dividing $\Xi_{3}^{2}\Xi_{4}^{2}$ by $\Xi_{1}\Xi_{2}$. One finally finds $$\begin{aligned} \label{asymptotics (Xi3*Xi4)/(Xi1*Xi2)} && \frac{\prod_{j=1}^{N}\prod_{k=j+1}^{N}(y_{j}-y_{k})^{2}} {\Big(\frac{1}{N}\sum_{j=1}^{N}\frac{y_{j}}{\rho+(1-\rho)y_{j}}\Big)\Big(\prod_{j=1}^{N}\Big(1+\frac{1-\rho}{\rho}\,y_{j}\Big)\Big)} \simeq {\mathrm{e}}^{\rho\,bL^{2}}{\mathrm{e}}^{\rho\,L\log L}{\mathrm{e}}^{-(1-\rho)\log(1-\rho)L}\,\exp\Big(\!-\frac{2\sqrt{\rho}\,\varphi_{r}(2\pi c)}{\sqrt{1-\rho}}\,\sqrt{L}\Big)\nonumber\\ &&\hspace{4mm} \times\frac{(-1)^{\frac{N(N-1)}{2}+m_{r}}}{(\pi^{2}/4)^{-m_{r}^{2}}(4\pi^{2})^{m_{r}}}\,\exp\Big(-\frac{4{\mathrm{i}}\pi(1-2\rho)p_{r}}{3(1-\rho)}\Big)\,\omega(A_{0}^{+})^{2}\omega(A_{0}^{-})^{2}\omega(A^{+})^{2}\omega(A^{-})^{2}\omega(A_{0}^{+},A_{0}^{-})^{2}\omega(A^{+},A^{-})^{2}\nonumber\\ &&\hspace{4mm} \times\frac{\sqrt{\rho(1-\rho)}\sqrt{L}}{{\mathrm{e}}^{2\pi c}\,\varphi_{r}'(2\pi c)}\,\exp\!\Big(\lim_{\Lambda\to\infty}-2m_{r}^{2}\log\Lambda+\int_{-\Lambda}^{2\pi c}{\mathrm{d}}u\,(\varphi_{r}'(u))^{2}\Big)\;.\end{aligned}$$ We observe that several factors depending on $c$ have cancelled: the only dependency on $c$ left are $\varphi_{r}(2\pi c)$, ${\mathrm{e}}^{2\pi c}\varphi_{r}'(2\pi c)$, and the upper limit of the integral. [Conclusions]{} It was shown in [@P2014.1] that the first eigenvalues of TASEP are naturally expressed in terms of a function $\eta$ constructed from the elementary excitations characterizing the eigenstate. We extend that result here by showing that $\varphi=\eta'$ can be identified as a scalar field: indeed, we observe that the normalization of the corresponding Bethe eigenstates can be expressed in terms of the exponential of the free action of $\varphi$. This might hint at a field theoretic description of current fluctuations on the relaxation scale $T\sim L^{3/2}$. The field $\varphi$ is equal to a sum of square roots corresponding to elementary excitations over a Fermi sea, plus Hurwitz zeta functions corresponding to a kind of renormalization of the contribution of the Fermi sea. In our calculations, the contributions leading to these two parts of the field need to be treated separately. It is only at the end of the calculation that everything combines perfectly at several places to give exactly the same field $\varphi$ everywhere. It would be very nice to find a simpler derivation that makes it clearer why the field $\varphi$ should appears in the end, and to explain all the other unexpected cancellations that happen between the asymptotic expansions of seemingly very different factors. [Hurwitz zeta function and double Hurwitz zeta function]{} \[appendix zeta\] In this appendix, we summarize some properties of Hurwitz zeta function and double Hurwitz zeta function. [Hurwitz zeta function]{} [Definitions]{} Hurwitz zeta function is defined for ${\operatorname{Re}}s>1$ by $$\label{Hurwitz zeta} \zeta(s,z)=\sum_{j=0}^{\infty}(j+z)^{-s}\;.$$ For $z\not\in\mathbb{R}^{-}$, it can be analytically continued to a meromorphic function of $s$ with a simple pole at $s=1$ with residue equal to $1$, independent of $z$. It is convenient for using the Euler-Maclaurin formula to define a modification $\tilde{\zeta}$ of $\zeta$ such that $\tilde{\zeta}(s,z)=\zeta(s,z)$ when $s\neq1$ and $$\label{Hurwitz zeta tilde} \tilde{\zeta}(1,z)=\lim_{s\to1}\zeta(s,z)-\frac{1}{s-1}=-\frac{\Gamma'(z)}{\Gamma(z)}\;.$$ The modified function $\tilde{\zeta}$ is not continuous at $s=1$. It obeys however the property $$\lim_{s\to1}\big(\zeta(s,z)-\zeta(s,z')\big)=\tilde{\zeta}(s,z)-\tilde{\zeta}(s,z')\;.$$ [Derivative]{} Hurwitz zeta function verifies $$\label{zeta'} \partial_{z}\zeta(s,z)=-s\zeta(s+1,z)\;.$$ [Bernoulli polynomials]{} Hurwitz zeta function is related to Bernoulli polynomials. For $r\in\mathbb{N}^{*}$, one has $$\label{Bernoulli[zeta]} B_{r}(z)=-r\zeta(1-r,z)\;,$$ which is a polynomial in $z$ of degree $r$. The Bernoulli polynomials form an Appell sequence, *i.e.* $B'_{r}(x)=rB_{r-1}(x)$, or equivalently $$\label{Bernoulli sum} B_{r}(x+y)=\sum_{m=0}^{r}{{r \choose m}}B_{m}(x)y^{r-m}\;.$$ They also verify the symmetry relation $$\label{Bernoulli symmetry} B_{r}(x+1)=(-1)^{r}B_{r}(-x)\;.$$ [Asymptotic expansions]{} When its second argument becomes large, Hurwitz zeta has the asymptotic expansion $$\label{Hurwitz zeta asymptotics} \zeta(s,M+x) \simeq-\frac{1}{1-s}\sum_{\ell=0}^{\infty}{{1-s \choose \ell}}\frac{B_{\ell}(x)}{M^{\ell+s-1}}\;,$$ while for $\tilde{\zeta}$, Stirling’s formula leads to $$\label{Hurwitz zeta tilde asymptotics} \tilde{\zeta}(1,M+x)\simeq-\log M+\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell}}{\ell}\,\frac{B_{\ell}(x)}{M^{\ell}}\;.$$ [Double Hurwitz zeta function]{} A two-dimensional generalization, the double Hurwitz zeta function, can be defined as $$\label{Hurwitz zeta double} \zeta(s,s';z,z')=\sum_{j=0}^{\infty}\sum_{j'=j+1}^{\infty}(j+z)^{-s}(j'+z')^{-s'}\;.$$ The sum converges for $z$ and $z'$ outside $\mathbb{R}^{-}$ when ${\operatorname{Re}}s'>1$ and ${\operatorname{Re}}(s+s')>2$. Unlike the usual Hurwitz zeta function, there does not seem to be a standard accepted notation here, partly due to the fact that several natural two dimensional generalizations can be considered. [Analytic continuation]{} The analytic continuation to arbitrary $s$, $s'$ can be made [@M2002.1] using the Mellin-Barnes integral formula, which can be stated for $-{\operatorname{Re}}s<p<0$, $\lambda\not\in\mathbb{R}^{-}$ as $$\label{Mellin-Barnes} (1+\lambda)^{-s}=\int_{p-{\mathrm{i}}\infty}^{p+{\mathrm{i}}\infty}\frac{{\mathrm{d}}w}{2{\mathrm{i}}\pi}\,\frac{\Gamma(s+w)\Gamma(-w)}{\Gamma(s)}\,\lambda^{w}\;,$$ and which follows from closing the contour of integration on the right and calculating the residues on the positive real axis when $|\lambda|<1$. Rewriting (\[Hurwitz zeta double\]) as $$\zeta(s,s';z,z') =\sum_{j=0}^{\infty}\sum_{j'=0}^{\infty}(j+z)^{-s}(j'+z'-z+1)^{-s'}\Big(1+\frac{j+z}{j'+z'-z+1}\Big)^{-s'}\;,$$ applying (\[Mellin-Barnes\]) to the factor $(1+\tfrac{j+z}{j'+z'-z+1})^{-s'}$, shifting $w$ by $-s'$, and using (\[sum\_power\[zeta\]\]) to compute the sums over $j$ and $j'$ (provided $1<{\operatorname{Re}}w<{\operatorname{Re}}(s+s'-1)$) leads to $$\zeta(s,s';z,z') =\int_{p-{\mathrm{i}}\infty}^{p+{\mathrm{i}}\infty}\frac{{\mathrm{d}}w}{2{\mathrm{i}}\pi}\,\frac{\Gamma(w)\Gamma(s'-w)\zeta(s+s'-w,z)\zeta(w,z'-z+1)}{\Gamma(s')}\;$$ with $1<p<{\operatorname{Re}}s'$. The contour of integration can be shifted to $0<p<1$ by taking the residue coming from the simple pole with residue $1$ of $\zeta(w,z'-z+1)$ at $w=1$. Shifting again the contour of integration to the left we pick the residues at $w=-k$, $k\in\mathbb{N}$ coming from $\Gamma(w)$ (residue $(-1)^{k}/k!$ at $w=-k$). Shifting $k$ by $1$, one finally finds in terms of Bernoulli polynomials (\[Bernoulli\[zeta\]\]) $$\begin{aligned} \label{Hurwitz zeta double sum integral} && \zeta(s,s';z,z')=\frac{1}{s'-1}\sum_{\ell=0}^{m+1}{{1-s' \choose \ell}}\zeta(s+s'+\ell-1,z)B_{\ell}(z'-z+1)\nonumber\\ &&\hspace{20mm} +\int_{-m-{\tfrac{1}{2}}-{\mathrm{i}}\infty}^{-m-{\tfrac{1}{2}}+{\mathrm{i}}\infty}\frac{{\mathrm{d}}w}{2{\mathrm{i}}\pi}\,\frac{\Gamma(w)\Gamma(s'-w)\zeta(s+s'-w,z)\zeta(w,z'-z+1)}{\Gamma(s')}\;.\end{aligned}$$ The remaining integral is analytic in the domain $\{(s,s'),{\operatorname{Re}}s'>-m-{\tfrac{1}{2}},{\operatorname{Re}}(s+s')>-m+{\tfrac{1}{2}}\}$. It implies that $\zeta(s,s';z,z')$ is a meromorphic function of $s$, $s'$ with (possible) poles at $s'=1$ and $s+s'=2-n$, $n\in\mathbb{N}$. When approaching the pole at $s+s'=2-n$, one has (when $s,s'\not\in1-\mathbb{N}$) $$\begin{aligned} && \zeta(s+\alpha\varepsilon,s'+(1-\alpha)\varepsilon;z,z') \underset{\varepsilon\to0}{\simeq}\frac{1}{s'-1}{{1-s' \choose n}}B_{n}(z'-z+1)\\ &&\hspace{50mm} \times\Big(\frac{1}{\varepsilon}-\frac{\Gamma'(z)}{\Gamma(z)}-(1-\alpha)\frac{\Gamma'(1-s')}{\Gamma(1-s')}+(1-\alpha)\frac{\Gamma'(s)}{\Gamma(s)}\Big)\;.\nonumber\end{aligned}$$ The arbitrary parameter $\alpha$ characterizes the direction in which $(s,s')$ approaches the line $s+s'=2-n$. We define a modified version $\tilde{\zeta}_{\alpha}$ of double Hurwitz zeta, equal to $\zeta$ when $2-s-s'\not\in\mathbb{N}$, and to $$\tilde{\zeta}_{\alpha}(s,s';z,z')=\lim_{\varepsilon\to0}\Big(\zeta(s+\alpha\varepsilon,s'+(1-\alpha)\varepsilon;z,z') -\frac{1}{s'-1}{{1-s' \choose n}}\frac{B_{n}(z'-z+1)}{\varepsilon}\Big)\;$$ when $s+s'=2-n$, $n\in\mathbb{N}$. More explicitly $$\begin{aligned} \label{Hurwitz zeta double tilde sum integral} && \tilde{\zeta}_{\alpha}(s,s';z,z')=\frac{1}{s'-1}\sum_{\substack{\ell=0\\(\ell\neq n)}}^{m+1}{{1-s' \choose \ell}}\zeta(\ell-n+1,z)B_{\ell}(z'-z+1)\\ &&\hspace{20mm} -\frac{1}{s'-1}{{1-s' \choose n}}B_{n}(z'-z+1)\bigg(\frac{\Gamma'(z)}{\Gamma(z)}+(1-\alpha)\Big(\frac{\Gamma'(1-s')}{\Gamma(1-s')}-\frac{\Gamma'(s)}{\Gamma(s)}\Big)\bigg)\nonumber\\ &&\hspace{20mm} +\int_{-m-{\tfrac{1}{2}}-{\mathrm{i}}\infty}^{-m-{\tfrac{1}{2}}+{\mathrm{i}}\infty}\frac{{\mathrm{d}}w}{2{\mathrm{i}}\pi}\,\frac{\Gamma(w)\Gamma(s'-w)\zeta(s+s'-w,z)\zeta(w,z'-z+1)}{\Gamma(s')}\;,\nonumber\end{aligned}$$ with $m\geq n-1$, $m\geq-{\operatorname{Re}}s'-{\tfrac{1}{2}}$ and when $s,s'\not\in1-\mathbb{N}$. For $\alpha=1$, this corresponds to replacing the simple Hurwitz zeta function in the summation in (\[Hurwitz zeta double sum integral\]) by its modified value (\[Hurwitz zeta tilde\]). [Double Bernoulli polynomials]{} The modified double zeta functions can be extended to $s\in-\mathbb{N}$, $s'\in-\mathbb{N}$. There, the integral vanishes because of the $\Gamma(s')$ in the denominator and $\tilde{\zeta}_{\alpha}(s,s';z,z')$ become polynomials in $z$, $z'$. Using (\[Bernoulli\[zeta\]\]), we obtain ($s+s'=2-n$) $$\label{Hurwitz zeta double integers} \tilde{\zeta}_{\alpha}(s,s';z,z')=\frac{1}{s'-1}\sum_{\ell=0}^{n-1}{{1-s' \choose \ell}}\frac{B_{\ell}(z'-z+1)B_{n-\ell}(z)}{\ell-n} +B_{n}(z'-z+1)\frac{(-1)^{s}(1-\alpha)}{(1-s)(1-s'){{2-s-s' \choose 1-s}}}\;.$$ In particular, at $s=s'=0$, one has $$\label{Hurwitz zeta double 00} \tilde{\zeta}_{\alpha}(0,0;z,z')=-\frac{1+\alpha}{12}+\frac{\alpha(z-z')}{2}-\frac{\alpha z^{2}}{2}+\frac{(1-\alpha)(z')^{2}}{2}+\alpha zz'\;.$$ Unlike the one-dimensional case, there is no unique natural way to define double Bernoulli numbers and polynomials because of the arbitrary parameter $\alpha$. [Asymptotic expansion]{} The expressions (\[Hurwitz zeta double sum integral\]) gives the large $M$ asymptotic expansion of $\zeta(s,s';z+M,z'+M)$ using the one (\[Hurwitz zeta asymptotics\]) for simple Hurwitz zeta since $\zeta(s+s'-w,M+z)\sim M^{-m+{\tfrac{1}{2}}-s-s'}$ in the integral can be made arbitrarily small by taking $m$ large enough. This gives the asymptotic expansion $$\label{Hurwitz zeta double asymptotics} \zeta(s,s';M+z,M+z')\simeq\sum_{\ell=0}^{\infty}\sum_{m=0}^{\infty}\frac{{{1-s' \choose \ell}}}{1-s'}\,\frac{{{2-s-s'-\ell \choose m}}}{2-s-s'-\ell}\,\frac{B_{m}(z)B_{\ell}(z'-z+1)}{M^{\ell+m+s+s'-2}}\;,$$ valid when $s'\neq1$ and $s+s'\not\in2-\mathbb{N}$, and similarly when $s+s'\in2-\mathbb{N}$ from (\[Hurwitz zeta double tilde sum integral\]) and (\[Hurwitz zeta double integers\]). [Calculation of various integrals]{} \[appendix integrals\] The various asymptotic expansions obtained in this paper using the Euler-Maclaurin formula involve integrals. Most of them have an integrand that depends on the variable of integration $u\in[0,\rho]$ only through $\Phi(u-\tfrac{\rho}{2})$, with $\Phi$ defined in (\[Phi\]). Such integrals can be computed by making the change of variables $z=\Phi(u-\tfrac{\rho}{2})$. From (\[Phi’\]), the Jacobian is given by $${\mathrm{d}}u=-\frac{{\mathrm{d}}z}{2{\mathrm{i}}\pi}\Big(\frac{\rho}{z}+\frac{1}{1-z}\Big)\;.$$ The variable $z$ lives on the clockwise contour $\overline{\mathcal{C}_{0}}$, which starts and ends at $z=-\rho/(1-\rho)$ for $u=0$ and $u=\rho$. The contour encloses $0$ but not $1$, see figure \[fig contour Phi\]. Hence, for the simplest integrands that do not involve branch cuts, the calculation of the integral reduces to a simple residue calculation. In the rest of this appendix, we treat some slightly more complicated integrals on two dimensional domains, with integrands having branch cuts that cross the contour of integration. We use the notation $\mathcal{C}_{0}=\{\Phi(\tfrac{\rho}{2}-u),0\leq u\leq\rho\}$ for the counter clockwise contour corresponding to $\overline{\mathcal{C}_{0}}$. ![Contour on which the Bethe roots $y_{j}$ accumulate in the complex plane for the first eigenstates in the thermodynamic limit with density of particles $\rho=1/3$. It crosses the negative real axis at $-\rho/(1-\rho)$. For any value of $\rho$, the contour encloses $0$ but not $1$.[]{data-label="fig contour Phi"}](ContourPhi.eps){width="100mm"} [A double integral]{} We consider the double integral $$I_{1}=\int_{0}^{\rho}{\mathrm{d}}u\,\int_{u}^{\rho}{\mathrm{d}}v\,\log\Big(-{\mathrm{i}}\big(\Phi(u-\tfrac{\rho}{2})-\Phi(v-\tfrac{\rho}{2})\big)\Big)\;.$$ Making the changes of variables $z=\Phi(u-\tfrac{\rho}{2})$ and $w=\Phi(v-\tfrac{\rho}{2})$ leads to $$\label{I double tmp 1} I_{1}=\int_{\overline{\mathcal{C}_{0}}}\frac{{\mathrm{d}}z}{2{\mathrm{i}}\pi}\,\Big(\frac{\rho}{z}+\frac{1}{1-z}\Big)\int_{z}^{-\tfrac{\rho}{1-\rho}}\frac{{\mathrm{d}}w}{2{\mathrm{i}}\pi}\,\Big(\frac{\rho}{w}+\frac{1}{1-w}\Big)\log(-{\mathrm{i}}(z-w))\;.$$ The inner integral can be computed in terms of the dilogarithm function ${\mathrm{Li}}_{2}$. Indeed, using ${\mathrm{Li}}_{2}'(w)=-w^{-1}\log(1-w)$, we observe that the function $F_{z}$ defined by $$F_{z}(w)=\rho\Big({\mathrm{Li}}_{2}\big(\frac{z}{w}\big)+\frac{(\log w)^{2}}{2}+\frac{{\mathrm{i}}\pi}{2}\,\log w\Big) -\Big({\mathrm{Li}}_{2}\big(\frac{1-z}{1-w}\big)+\frac{(\log(1-w))^{2}}{2}-\frac{{\mathrm{i}}\pi}{2}\,\log(1-w)\Big)\;$$ verifies $$F_{z}'(w)=\Big(\frac{\rho}{w}+\frac{1}{1-w}\Big)\log(-{\mathrm{i}}(z-w))\;.$$ The contour of integration for $w$ in (\[I double tmp 1\]) does not cross the branch cuts coming from the dilogarithm. The integration over $z$ can be done by taking the residue at $0$ for all the terms such that the contour does not cross a branch cut. Using ${\mathrm{Li}}_{2}(0)=0$, one finds $$\begin{aligned} && I_{1}=\frac{\rho}{2}\int_{\mathcal{C}_{0}}\frac{{\mathrm{d}}z}{(2{\mathrm{i}}\pi)^{2}}\,\Big(\frac{\rho}{z}+\frac{1}{1-z}\Big)\big((\log z)^{2}+{\mathrm{i}}\pi\log z\big)\\ &&\hspace{4mm} +\frac{\rho}{2{\mathrm{i}}\pi}\Big(\frac{{\mathrm{i}}\pi b_{0}}{2}-\frac{(1-\rho)\pi^{2}}{6}+\frac{(1-\rho)(\log(1-\rho))^{2}}{2}-\frac{\rho(\log\rho)^{2}}{2}+\rho\log\rho\log(1-\rho)+{\mathrm{Li}}_{2}(1-\rho)\Big)\;.\nonumber\end{aligned}$$ The last integral can be computed using the fact that the contour $\mathcal{C}_{0}$ intersects the negative real axis at $z=-\rho/(1-\rho)$ and the identities $(\log z)/z=\partial_{z}(\log z)^{2}/2$, $(\log z)^{2}/z=\partial_{z}(\log z)^{3}/3$, $(\log z)/(z-1)=\partial_{z}({\mathrm{Li}}_{2}(z)+\log z\log(1-z))$ and $(\log z)^{2}/(z-1)=\partial_{z}(-2{\mathrm{Li}}_{3}(z)+2\log z\,{\mathrm{Li}}_{2}(z)+(\log z)^{2}\log(1-z))$. After some simplifications, one finds $$I_{1}=\frac{\rho b_{0}}{2}\;,$$ with $b_{0}$ defined in (\[b0\]). [Another double integral]{} We consider the double integral $$I_{2}=\int_{0}^{\rho}{\mathrm{d}}u\,\int_{u}^{\rho}{\mathrm{d}}v\,\log\frac{(\sqrt{u}+\sqrt{v})(\sqrt{\rho-u}+\sqrt{\rho-v})}{(v-u)(\sqrt{-{\mathrm{i}}u}+\sqrt{{\mathrm{i}}(\rho-v)})}\;.$$ It can be rewritten as $$I_{2}=\int_{0}^{\rho}{\mathrm{d}}u\,\int_{u}^{\rho}{\mathrm{d}}v\,\log(\sqrt{u}+\sqrt{v}) -\int_{0}^{\rho}{\mathrm{d}}u\,\int_{u}^{\rho}{\mathrm{d}}v\,\log(v-u) -\int_{0}^{\rho}{\mathrm{d}}u\,\int_{u}^{\rho}{\mathrm{d}}v\,\log(\sqrt{-{\mathrm{i}}u}+\sqrt{{\mathrm{i}}(\rho-v)})\;.$$ Using $\log(\sqrt{u}+\sqrt{v})=\partial_{v}((v-u)\log(\sqrt{u}+\sqrt{v})+\sqrt{u v}-v/2)$ and $\log(\sqrt{-{\mathrm{i}}u}+\sqrt{{\mathrm{i}}(\rho-v)})=\partial_{v}(-(u+\rho-v)\log(\sqrt{-{\mathrm{i}}u}+\sqrt{{\mathrm{i}}(\rho-v)})+{\mathrm{i}}\sqrt{u}\sqrt{\rho-v}-v/2)$, one finds $$I_{2}=\frac{9\rho^{2}}{8}-\frac{\rho^{2}\log\rho}{4}\;.$$ [10]{} T. Halpin-Healy and Y.-C. Zhang. 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--- abstract: | Programming and software engineering courses in computer science curricula typically focus on both providing theoretical knowledge of programming languages and best-practices, and developing practical development skills. In a massive course – several hundred students – the teachers are not able to adequately attend to the practical part, therefore process automation and incentives to students must be used to drive the students in the right direction. Our goals was to design an automated programming assignment infrastructure capable of supporting massive courses. The infrastructure should encourage students to apply the key software engineering (SE) practices – automated testing, configuration management, and Integrated Development Environment (IDE) – and acquire the basic skills for using the corresponding tools. We selected a few widely adopted development tools used to support the key software engineering practices and mapped them to the basic activities in our exam assignment management process. This experience report describes the results from the past academic year. The infrastructure we built has been used for a full academic year and supported four exam sessions for a total of over a thousand students. The satisfaction level reported by the students is generally high. author: - Marco Torchiano - Giorgio Bruno bibliography: - 'refs.bib' title: 'Integrating Software Engineering Key Practices into an OOP Massive In-Classroom Course: an Experience Report' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010405.10010489&lt;/concept\_id&gt; &lt;concept\_desc&gt;Applied computing Education&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10011007.10011006.10011071&lt;/concept\_id&gt; &lt;concept\_desc&gt;Software and its engineering Software configuration management and version control systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10011007.10011006.10011066.10011069&lt;/concept\_id&gt; &lt;concept\_desc&gt;Software and its engineering Integrated and visual development environments&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10011007.10011074.10011092.10011093&lt;/concept\_id&gt; &lt;concept\_desc&gt;Software and its engineering Object oriented development&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10011007.10011074.10011099.10011102.10011103&lt;/concept\_id&gt; &lt;concept\_desc&gt;Software and its engineering Software testing and debugging&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; Introduction ============ Software development eventually consists in delivering working code [@beck2001manifesto]. Computer science and the software-related part of computer engineering should teach programming and on top of that provide software engineering skills. At Politecnico di Torino, Italy, the first course introducing a “*modern*” programming language is the Object Oriented Programming (OOP) course where the language of choice is Java. For historical reasons the BSc degree in computer engineering does not include a Software Engineering course, therefore we decided to provide the basic Software Engineering knowledge in the OOP course. The students attending the course are Millennials: they were born before 1996. While previous programming courses in the curriculum adopted paper-based exams, we opted for a computer-based exam to leverage the technology familiarity of “digital natives”. The course, in addition to the Java language (version 8), provides an introduction to UML [@Rumbaugh2004], design patterns [@GoF] and basic software engineering practices. It basically follows the indications provided in [@CS2013]. The three key practices that we integrated in the course are: - automated testing: represents a clear step from informally trying the program to a formalized and repeatable verification activity, - configuration management: introduces a standard way of versioning code and keeping a common shared repository, - integrated development environment: provides basic features supporting coding, e.g. code completion, language specific presentation, automatic incremental build, error highlighting, and automatic code refactoring [@fowler1999refactoring]. Such practices are meant to develop software testing and software configuration skills as recommended in SWECOM 1.0 [@SWECOM1_0]. Moreover, automated testing appears particularly suited to responde to the Millennials’ need for frequent feedback [@Myers2010]. A particularly tough challenge in the introduction of such practices is represented by the size of the course: the largest of the three parallel instances counts hundreds of newly enrolled students (330 for a.y. 2017/18). The course is taught in presence and offers practical sessions in the lab facilities of the university. We can define it as a Massive In-Classroom Course (MICC). A MICC, as opposed to a MOOC, exhibits the following characteristics: - the numbers are smaller than on-line courses, but still large for regular university courses; - while there are videolectures, they just record the lectures held in classroom therefore they are neither primarily designed nor optimized for autonomous fruition; - the practical organization must enable anybody to attend in person all the educational activities (both lectures and labs); - it is not necessarily open, although all materials for the OOP course are freely available online. This paper reports the experience in integrating a few key software engineering practices in the OOP course by means of specific technologies. In particular we show how the devised solution was able to address both organizational and learning objectives. On one side, such technologies support the management of assignments in the course, on the other side they are required to perform essential tasks thus stimulating the students to acquire the related basic skills. First of all, section \[sec:context\] presents the context of the course and the detailed motivation that lead us to this course implementation. Then section \[sec:technology\] provides the details of how such key SE practices have been realized in the course. After that, section \[sec:discussion\] discusses the educational implications, the main issues encountered, and the lessons learned. Context and Motivation {#sec:context} ====================== The Object-Oriented Programming course is located in the second year of the Bachelor degree in Computer Engineering[^1] at Politecnico di Torino. The first year encompasses fundamental topics for all engineering disciplines (maths, computer science, physics), the second year introduces general ICT and computer engineering topics: circuit theory, algorithms and data structures, object-oriented programming, and databases. The third year focuses on more advanced topics, e.g. operating systems, computer networks, communications, electronics. The OOP course introduces Object-Oriented programming using the Java programming language and provides basic knowledge of software engineering. The main contents are: - Basic OO features (1 ECTS[^2]) including the OO paradigm, Java, classes and attributes, visibility, basic types, and practical skills concerning the Eclipse IDE. - Inheritance, interfaces, and advanced features (2 ECTS), including functional interfaces, lambda expressions, exceptions, and generic types. - Standard libraries (3 ECTS) including the Collections framework, streams, files, dates, threads, and GUIs. - Software Engineering principles (2 ECTS), including the Software life cycle, UML, Design Patterns, Configuration management, Testing. The latter two subtopics provide basic skills in Subversion[^3] and JUnit[^4]. The course consists of over 70 hours in the classroom, including both lectures introducing the topics and live coding sessions presenting and discussing programming assignment solutions, and 20 hours in the lab dedicated to the development of programming assignments. While one could argue that the hours in the lab should be much more, this is not practically possible since there are limited lab facilities that are shared with several other courses, e.g. there are 20 replicas of the basic Computer Science course with about 200 students each. The OOP course is run in three replicas, two in Italian and one in English, with 330, 270, and 115 enrolled students respectively. The exam process is sketched by the activity diagram shown in Figure 1 and encompasses a few steps: 1. the teacher prepares an initial project and uploads it; 2. during the exam, the students develop a small program[^5] in two hours, while sitting in the lab; 3. at the end of the exam the students must submit their program; 4. meanwhile, the teacher has prepared an acceptance test suite; 5. after the exam, the teacher assesses the functionality of the program versus the test suite; 6. the students have to fix or complete the program in order to make it pass all the tests in the suite; 7. the teacher grades the work done by the students. The assignment consists of: - a requirements document, usually made up of four or five sections that are designed to be implemented incrementally, because the features required in a section make use of the ones defined in previous sections. The requirements describe a set of classes and their methods; - an initial project, containing skeletal classes, i.e. classes with the methods called by the tests but with minimal bodies returning fixed values (e.g. `null`); the project can be opened with the reference IDE and is syntactically correct; - an example class, containing a `main()` method that exercises the most relevant methods described in the requirements. It is intended to clarify the requirements and to provide the students with a basic testing tool. {width="0.75\linewidth"} The evaluation is computed on the basis of the functional compliance – both in terms of correctness and completeness – of the program. Such an approch has been inspired by the agile manifesto principle “*Working software is the primary measure of progress*” [@beck2001manifesto]. More in detail, the tests are packaged into a `.jar` file containing both class files and source files. This is done to avoid both unintended and malicious modifications to the test suite. In practice the grade is computed on the basis of two indicators: - Percentage of acceptance tests passed by the lab version ($S$), - Code churn ($M$) applied to make the program pass all the tests. Code churn [@Khoshgoftaar1996] is the amount of added and modified lines of code; it is a very simple measure of the quantity of code modification. The former indicator provides a coarse grained assessment of the functional compliance from an end-user point of view, the latter represents a fine grained evaluation and is a proxy measure of the rework needed to fix defects (correctness) and to complete unimplemented features (completeness). The basic formula to compute the grade is: $$Grade = c_0 + c_1 \cdot \left( S + (1-S) \frac{c_2}{c_2+M} \right)$$ Where the constants $c_0$, $c_1$ and $c_2$ are adjusted case by case based on the difficulty of the exam. Given the above formula: - when a large amount of modifications is applied the grade is essentially defined by the percentage of tests passed - as the percentage of passed tests get lower the component inversely proportional to the modifications gets a higher weight. An important aspect of this evaluation approach is that the completeness and correctness of the program delivered in the lab is evaluated by comparison. The reference program is the fully working version submitted from home, after the exam; it is a natural evolution carried out by the same student who wrote it initially in the lab. A possible alternative would be to use a predefined solution developed by the teacher as a reference, but its adequacy could be low for the following reasons: - since there is no single solution for any given problem, the comparison with a predefined solution could penalize different – possibly even better – solutions; - the amount of work needed to complete the program and to fix defects can reasonably be estimated only by comparing the original one with the evolved version. The approach also encourages the students to understand the requirements, identify a design and then work on the requirements, one by one, developing fully working code (possibly just for a subset of the requirements) rather than write a complete solution in a single *big-bang*, which typically does not work. The rationale behind such an approach is that, in real-world terms, it is better to have a program that performs correctly on a subset of requirements that a program that is almost complete but crashes at the beginning and eventually does nothing. The above assessment method is fully automated and can be applied to large numbers of delivered projects producing objective and unbiased grades. The current approach presented in this paper is an evolution of the one developed originally in 2003 and described in [@Torchiano2009]. The approach was updated in response to several challenges: - the number of students enrolled raised significantly in the latest years from around 200 per year to roughly 700 in the current academic year, making this course a real Massive In-Classroom Course; therefore the solution must be scalable and robust; - the teaching staff is very limited: three teachers lecturing three parallel tracks (to fit lecture halls hosting 250 students at most), plus three teaching assistants supporting the students in the labs; - the lectures are video recorded and this encourages the students to attend the course remotely. In particular the students should be able to work autonomously on their assignments – e.g. at home – due both to personal reasons and to crowded labs; therefore the assignment management framework must be based on tools that can be easily installed on their PCs; - the instruments and tools should enable the students to acquire skills directly usable in a real-world setting; therefore the tools should be widely adopted in practitioner communities; - the course content was extended to include basic SE practices, so as to encourage the students to adopt or at least become acquainted with basic software engineering practices, i.e.: - automated testing, - configuration management, - integrated development environment (IDE). To better characterize the learning outcomes, we can refer to taxonomies that describe curricula objectives in terms of topics and levels of understanding. In particular, Bloom’s taxonomy [@bloom1956taxonomy; @anderson2001taxonomy] classifies learning achievements into six different cognitive levels: knowledge, comprehension, application, analysis, synthesis and evaluation. While the educational goals in the programming part of the course clearly address all the six levels of the taxonomy, the software engineering part only addresses the lower levels of the taxonomy. Technological platform {#sec:technology} ====================== The technological solution we developed is based on a few technologies that both implement software engineering best-practices and cover a key role in the exam and assignment management process described above. The SE areas we decided to cover with technologies and assignment related activities are: - Automated Testing using JUnit - Configuration Management using Subversion - IDE as Eclipse In addition we had to set for a robust method for authentication and authorization to be used during exams. We decided to use SVN authentication as the basic technology. Automated Testing {#testing} ----------------- Testing is a key technique for the Verification and Validation phase in any software development process [@runeson2006survey], in particular automated unit testing has gained much attention in recent years. JUnit is the de-facto standard for writing automated tests in Java [@beck1998test]. While its original purpose was to write unit tests, it is also widely used as the basis for UI testing and end-to-end tests. In our approach, JUnit is used to evaluate the functional compliance of assignments. The basic measure is the proportion of passed test cases. The JUnit execution report is the standard feedback the students receive both when they complete their lab assignments during the course and right after the exam. In terms of test automation, the main challenge is that we need to test a huge number of programs. The peculiarity is that while in a regular industrial setting we have large test suites to be executed on a single program (or parts thereof), in our course we have a single (small) test suite to be run against several hundred similar programs that provide different implementations of the same classes. A technical obstacle is that – at least in theory – we ought to start a new Java Virtual Machine (JVM) for every project, load the tests and the classes making up the program, and run the tests. Unfortunately the VM startup and class loading are very heavy tasks. The solution we devised to achieve a reasonable scalability is to use a hierarchy of Java class loaders as shown in Figure 2. {width="1\linewidth"} Class loaders are classes responsible for finding and loading classes whenever the VM needs them. A bootstrap class loader is always present and it searches the classes in the predefined *classpath*. Class loaders are generally organized in a delegation hierarchy, so that if a specific class loader is not able to find a class, recursively delegates the search to its parent. In our approach a dedicated class loader class has been developed to load the test classes from a given `.jar` file. In addition we developed a project class loader that loads classes from the project path; an instance is created for each project. The test execution starts from this more specific class loader; when a test class is needed it delegates the test class loader to get it. Since test classes are common to all the assignments, in most cases the test class loader finds them in the cache. Such a solution has several advantages: - the test classes are loaded only once by the test class loader, - any project specific class loader is isolated from the others so that classes with the same name can be loaded in the projects without any interference, - overall a single VM can be used for testing several projects. Owing to this approach, testing projects can proceed at a rate of two projects per second on a standard Ubuntu VM with 4 cores and 8GB RAM. Each project typically counts five to eight classes while the test suite includes 20 test cases at least. In terms of acquired skills and knowledge, the students are not required to write tests – the main reason being the short time available for the exam, i.e. 2 hours – though they must be able to: (i) import a test suite, (ii) run the tests, (iii) understand the tests results, and (iv) identify the cause of the test failures. In particular, for the latter ability, the students have to know what the assert statements mean, how an expected exception is tested, and in general they must be able to read a failure or error message, as well as to understand a stack trace in order to locate the origin of a failure, and also to interpret the test code to figure out the conditions that led to the failure. The implementation of the infrastructure includes 67 Java classes for a total of 6700 LOCs. Configuration Management ------------------------ Subversion (Svn) is a widespread centralized version control system [@collins2004version]. Although its adoption has recently decreased in favor of more modern distributed systems, such as *git* [@swicegood2008pragmatic], Svn is still widely used in industry and as far as our course is concerned it is easier to use, thus less error prone, and simpler to manage. In our approach Subversion is used to give the assignments to the students as well as to collect their implementations, and this takes place both during the course and at the exam. While Svn can support concurrent development with a Copy-Modify-Merge approach, and can manage different threads of execution using branches, the course makes use only of the basic versioning features. In practice the assignment life cycle is supported by Svn as follows: 1. the teacher commits an initial version of a Java project together with an acceptance test suite to a *master* repository, 2. the initial project is committed to all student repositories by the teacher using a simple script, 3. the students check-out the initial project and start working on it, 4. the students commit the results of their work to their own repositories, 5. the teacher checks out the latest version of the projects available in the repositories and runs the tests on them. The latter step is performed using the multiple classloaders approach described in the previous sub-section. The main challenges faced in customizing Svn for the purpose of the course were as follows: - the students must have isolated personal repositories, so that no interference can occur by mistake; - during the exam, the students must not be able to access other students’ repositories, to avoid plagiarism; - during the exam, the students must be able to access their repositories as soon as the exam begins; therefore the repositories must be created in advance; - the students must not be able to keep working after the exam deadline has elapsed. The isolation can be obtained by means of a single repository containing one subfolder per student and adequate permissions. Alternatively, one repository per student can be created with the student having access to her own repository only. While the former is more efficient, it is less isolated: every time a student performs a commit, the revision number is incremented for every other students too. For this reason, even if it is more expensive we opted for the one repository per student solution. During the course, a student sharing his credential with a colleague is generally not a problem and can foster collaboration. But, such behavior must be prevented during the exam. The solution is to create a new repository for each student who signed up for the exam. Then each repository is populated with a copy of the initial project and the credentials for the repositories are handed to the students at the beginning of the exam. The creation of an Svn repository, on our server, typically takes 4-5 seconds, therefore the repositories must be created in advance, at the beginning of the course and before each exam session. During the exam, students are allowed to commit their projects as many times as they wish. At the end of the exam, the teacher annotates the actual end time; only commits performed before the end time are taken into account. In addition Svn was used to make the tests available to the students on a dedicated test repository. Sometimes errors can be found in tests after the reports have been sent to the students. By using Subversion we can update the test `.jar` inside the repository and notify the students via email. In terms of acquired skills and knowledge, the students must learn a few basic tasks: - performing the check-out of a project from a repository, - performing the commit of a project to a repository. In addition, during the course the students are encouraged to perform frequent commits when developing a project. During the exam, they are invited to commit after implementing each requirement and explicitly instructed that is *safer* to commit 10 minutes before the deadline. The goal of the course is to make the students familiar with the elementary configuration management operations that are at the basis of any workflow they will adopt in the future. The management of the repositories has been implemented using 16 scripts in bash and python, for a total of 910 and 827 LOCs respectively. Java IDE -------- The usage of an IDE is often an implicit assumption when writing code. In our course we opted for Eclipse[^6] because historically it was one of the most widespread IDEs and due to the fact that it is an open-source product. The Eclipse Java IDE is the reference IDE that is taught during the course. The configuration management and testing tasks are performed by the students using the plug-ins for this IDE. Eclipse is installed in all labs and the students are encouraged to install it on their machines. We observe that while Eclipse comes with a built-in JUnit plug-in, – oddly enough – it has no default built-in plug-in for Subversion. Therefore an additional plug-in (Subversive) has to be installed on top of the default Java IDE. [@llll@]{} Level & Testing & Configuration Management & IDE Remember & **JUnit framework elements** & **Svn operation** & **Eclipse features** Understand & **Semantics of test methods and assert statements** & **Semantics of commands** & **Main tasks (e.g. compile, run, etc.)** Apply & **Execute test suite** & **Perform *check-out* and *commit*** & **Develop and run** Analyze & **Understand test results** & *Understand outcome of operations* & **Understand error messages** Evaluate & **Identify failure causes** & Identify conflict causes & **Identify defects or problems** Create & Write tests & Merge conflicts & Set-up a project Discussion {#sec:discussion} ========== Learning objectives ------------------- Table \[tab:levels\] reports the six taxonomy levels and the corresponding capabilities addressed with respect to the three key SE areas included in our course. In the table, the capabilities addressed by the course are shown in **bold**, those partially addressed in *italic*, and the others, not addressed, in a regular font. The cognitive levels addressed are first needed in the lab assignments the students have to perform during the course and then they are required in the exam. Therefore we are confident that the students passing the exam achieved those levels to a good degree of completeness. The *Analyze* level for configuration management is only partly addressed because no concurrent development is used in the course, therefore no conflict will take place: this is an activity students learn in lectures but never experience in practice. For this reason, the *Evaluate* level is not addressed either. The *Create* level is not addressed for any of the three key areas. As to testing, writing tests is a time consuming activity that cannot fit in the tight schedule (2 hours) allowed for the exam. As far as configuration management is concerned, the lack of concurrent development makes it impossible to apply merge operations. Regarding the Java IDE, all assignments start with students importing pre-defined Eclipse projects from Svn, therefore the project set-up phase is not put into practice. Concerning the basic skills we observed one important point: even though most students are able to perform correct Svn operations, they tend to apply a minimalistic workflow. Students are encouraged to perform a commit after completing each requirement section, nevertheless most of them tend to perform fewer commits, just the barely minimum to abide by the exam rules: a commit at the end of the exam session and a commit after the session. Given an assignment whose requirements contain $r$ sections, the recommended process entails at least $r+1$ commits: one for each requirement section plus one from home after the exam. This is a very simple, though approximate, criterion to identify compliant students. We analyzed the number of commits performed by the students on their exam repositories. Out of 1008 repositories – corresponding to the bookings – we found that 25% of them contained only the initially project and no student commit. These are untouched projects: students that either booked the exam but did not show up or decided to quit during the exam. Excluding the untouched projects, the distribution of the number of commits for the students who actually attended the exams is shown in Figure \[fig:commits\]. {width="95.00000%"} There is a small percentage of students (7.7%) who performed just 1 commit, i.e. they committed a version in the lab during the exam but did not completed their programs at home; they are the exam dropouts. A larger share of the students (92.3%) performed at least two commits, i.e. one in the lab and one from home. Table \[tab:compliant\] reports, for each exam session, the number of touched repositories and the proportion of dropouts and compliant students. We observe that overall 39% of students complied with the recommended process. The first two exam sessions – closely following the end of the course – exhibit a higher compliance, 44% and 41% respectively. **Session** **Students** **Dropout** **Compliant** ------------- -------------- ------------- --------------- June 2017 334 7.5% 44.3% July 2017 258 6.6% 41.5% Sept 2017 101 15.8% 22.8% Jan 2018 63 0.0% 23.8% *All* 756 7.7% 38.8% : Process compliant students[]{data-label="tab:compliant"} Issues ------ The first instance of the course using the infrastructure described above was given in a.y. 2016/17. The set-up was used both during the course (from March to June 2017) and for the exam sessions. We managed four exam sessions (June, July and September 2017, and January 2018) for a total of 629 exams graded. Given the huge number of students we encountered several problems. We summarize here the main issues that emerged during and after the exam sessions: - Several students after checking-out realized that Eclipse did not provide editing support (e.g. code completion). This is typically due to the fact they did check-out the whole repository and not just the folder containing the Eclipse project; as a consequence, Eclipse is not able to recognize the folder as a Java project and thus cannot provide all Java-related supporting features. - After the exam some students got a test report showing many failures they could not find in their projects. The cause for this lies in a late commit, i.e. a commit performed after the exam deadline. - After the exam, some students got no test reports because the projects submitted contained errors that prevented a successful compilation. Despite the invitation issued 10 minutes before the end of the exam, several students continued to work on the code rather than checking the code for errors. - Sometimes students get compilation errors they were not able to see in the lab within their IDE. In our experience this is due to a few causes: - the Eclipse uses its own (incremental) compiler that in a few cases – e.g. type inference for generic types – behaves differently from the Oracle JDK compiler we use to compile the project before testing; - the Eclipse IDE, when suggesting imports in case of undefined classes or interfaces, usually provides a list of all compatible elements, e.g. for `Collections` it includes of course the `java.util.Collections` class as well as, e.g. `com.sun.xml.internal.ws.policy. privateutil.PolicyUtils.Collections`. The latter class is usually not present in a clean JDK installation and the corresponding `import` is marked as an error during compilation. - During the first exam session, the students in a lab were not able to connect to the Svn repository. This problem was caused by a misconfiguration in the web proxy and firewall in just one lab. As a side note, we also encountered some weird issues unrelated to the topics covered in the course. A few students in every exam session typically call for help because suddenly the editor in Eclipse is overwriting their code instead of inserting new characters: this is due to the fact that the students inadvertently pushed the *Ins* button on the keyboard thus switching from insert to overwrite mode. We speculate that such *magic Ins key* problem is due to some students being used to small factor laptop keyboards that do not have a dedicated *Ins* key. Another issue that emerged while discussing with colleages is the suitability of the Eclipse IDE. In the last year the Eclipse market share[^7] (40%) appears to be shrinking in favor of IntelliJ IDEA (46%), which according to colleagues provide a more modern and usable environment. Lessons learned --------------- We collected a number of critical issues that we intend to overcome in the next version of the course. **Students are not able to use the basic tools**: this is particularly true for Subversion (as reported above) but sometimes it happens they are not familiar with Eclipse or even with the PCs available in the university lab. The lesson we learned is that the countermeasure is to force or provide incentives for the students to get familiar with the tools *before* sustaining the exam. Currently the assignments proposed to the students during the course are not mandatory. A possible mitigation to this problem may be to give additional points in the final grade if the students complete a specific assignment that requires basic skills (e.g. Subversion). **The development environment might differ in part from the testing environment**: this is typically due to the compiler (Eclipse own compiler vs. JDK javac), the *classpath* (Eclipse Java project vs. clean JDK), or the operating systems (Windows in the lab vs. Ubuntu for the test server). The consequences of this issue can be significantly reduced by implementing a simplified *Continuous Integration* [@duvall2007continuous] infrastructure. Every commit goes through compilation and testing and the results are reported back to the students. Such a feedback would enable the students to understand what the problem is in the testing environment. **Students assume they can work in a new environment just because they used a similar one**: several students – because of the crowded lecture rooms and labs, the availability of video recorded lectures, and the possibility of performing assignments on their own PCs – tend not to attend all lectures and labs. As a consequence, the day of the exam turns out to be the first time they use the lab equipment. The (presumed) tech savyness and confidence of Millennials apparently bring them to overestimate their knowledge. However, forcing the students to work on their assignments in the lab, would restrict their freedom, and possibly overload both the facility and the teaching assistants. It is important to make sure the environment the students re-create on their machines is as close as possible to the lab environment. This can be achieved by defining very well the reference environment – IDE and JDK version – as well as ensuring the latest version – the one that the students will download most likely – is installed in the lab too. **Scalable and reliable automation requires a lot of effort for the infrastructure**: even if the three cornerstone technologies are quite sound and mature, their usage in the course is peculiar and requires dedicated workflows to be designed as well as a suitable infrastructure to be developed. For this course, during several years, over 10KLOC of code were written mostly in Java but also in Python, Bash shell and Html. The recommendation is to use existing tools as far as possible but also to be prepared for a large effort in infrastructure development. **Whenever you rely on a server, never underestimate the network**: we performed tests in two (out of six) labs used for the exam, but not in the one that turned out to have the issue. The recommendation is of course that extensive testing must be performed in the field. Conclusions =========== This paper presented a report on the experience in integrating three key Software Engineering practices – automated testing, configuration management, and integrated development environment – into a large OOP course. The key practices play a twofold role: first, they are instrumental to achieve a set of educational goals, second, they are the cornerstones of the infrastructure supporting assignment management both during the course and at the exams. The resources required to run the course consist in a linux server hosting the Subversion repositories, the scripts, and runnning the test correction procedure. In addition labs large enough are required with PCs hosting the Eclipse IDE. All the required software is open-source and the additional custom software can be provided upon request. The course, as reported, has been run once in a.y. 2016/17, although it builds on almost 15 years of experience. The anonymous student satisfaction questionnaires resulted in 90% students being overall satisfied for the educational part. The global satisfaction level – also including the logistics – is at 83%, mainly due to the crowded classes and labs. For the next edition of the course we plan to put into practice the lessons learned, the most important being the introduction of a light-weight continuous integration feature. Moreover for future editions we will have to consider a possible evolution of the adopted technologies (e.g. IDE and configuration mangement), taking into account both the ease of use and the popularity. [^1]: Computer Engineering BSc. syllabus: [https://didattica.polito.it/ pls/portal30/gap.a\_mds.espandi2?p\_a\_acc=2018&p\_sdu=37& p\_cds=10&p\_lang=EN](https://goo.gl/UMEu4y) [^2]: ECTS stands for European Credit Transfer and Accumulation System and is a standard means for comparing the “volume of learning based on the defined learning outcomes and their associated workload” for higher education across the European Union [^3]: https://subversion.apache.org [^4]: http://junit.org/junit4/ [^5]: A few examples of the required program are available\ at: http://softeng.polito.it/courses/02JEY/exams/ [^6]: http://www.eclipse.org [^7]: http://www.baeldung.com/java-in-2017